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Corollary 6.8 Suppose \( F : {}_{R}\mathbf{M} \rightarrow {}_{S}\mathbf{M} \) is a functor.\n\na) If \( F \) is covariant, then \( {\mathcal{L}}_{0}F \approx F \) if and only if \( F \) is right exact.\n\nb) If \( F \) is covariant, then \( {\mathcal{R}}_{0}F \approx F \) if and only if \( F \) is left exact.\n\nc) If \( F \) is contravariant, then \( {\mathcal{L}}^{0}F \approx F \) if and only if \( F \) is left exact.\n\nd) If \( F \) is contravariant, then \( {\mathcal{R}}^{0}F \approx F \) if and only if \( F \) is right exact.
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Proof: The \
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No
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Proposition 6.9 Suppose given the commutative diagram\n\n\n\nwith exact columns and diagonals in \( {}_{R}\mathbf{M} \), and with \( P \) and \( {P}^{\prime \prime } \) projective.\n\nThen there exist fillers\n\n\n\nforming a commutative diagram.
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Proof: Choose any fillers for\n\n\n\nand define\n\n\[ \n{\varphi }^{\prime }\left( {x,{x}^{\prime \prime }}\right) = {\iota \varphi }\left( x\right) + \psi \left( {x}^{\prime \prime }\right) \n\] \n\nand\n\n\[ \n{\varphi }^{\prime \prime } = {\pi \psi } \n\] \n\nCommutativity of the three-dimensional diagram follows from commutativity of the five faces containing purported fillers (three triangles and two \
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Yes
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Corollary 6.10 Suppose the diagram\n\n\n\n(with entries in \( {}_{R}\mathbf{M} \) ) is commutative, with exact rows. Then, given simultaneous projective resolutions of \( B,{B}^{\prime },{B}^{\prime \prime } \) and \( C,{C}^{\prime },{C}^{\prime \prime } \), there exist fillers forming a commutative diagram with exact rows and columns:\n\n
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Proof: In \( {}_{R} \) Sh, let a boldface letter (e.g., \( \mathbf{C} \) ) denote a short exact sequence of left \( R \) -modules denoted with the plainface letter, with primes attached (e.g., \( 0 \rightarrow C \rightarrow {C}^{\prime } \rightarrow {C}^{\prime \prime } \rightarrow 0 \) ). Also, FOR THIS PROOF ONLY, let \
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Yes
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Proposition 6.12 Suppose \( 0 \rightarrow B\overset{\iota }{ \rightarrow }{B}^{\prime }\overset{\pi }{ \rightarrow }{B}^{\prime \prime } \rightarrow 0 \) is short exact in \( {}_{R}\mathbf{M} \) , and suppose \( A \in {\mathbf{M}}_{R} \) . Let \( {\delta }_{n} : {\operatorname{Tor}}_{n}\left( {A,{B}^{\prime \prime }}\right) \rightarrow {\operatorname{Tor}}_{n - 1}\left( {A, B}\right) \) be the connecting homomorphism of Corollary 3.17, obtained from a flat resolution of \( A \) . Let \( {\widetilde{\delta }}_{n} : {\operatorname{Tor}}_{n}\left( {A,{B}^{\prime \prime }}\right) \rightarrow {\operatorname{Tor}}_{n - 1}\left( {A, B}\right) \) be the connecting homomorphism of Theorem 6.6(a), obtained from a simultaneous resolution of \( 0 \rightarrow B \rightarrow {B}^{\prime } \rightarrow {B}^{\prime \prime } \rightarrow 0 \), via \( {\operatorname{Tor}}_{n}\left( {A, \bullet }\right) = {\mathcal{L}}_{n}\left( {A \otimes }\right) \) . Then \( {\widetilde{\delta }}_{n} = {\left( -1\right) }^{n}{\delta }_{n} \) .
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Proof: In this proof, a capital script letter, for example, \( \mathcal{C} \), will denote a chain complex. If \( \mathcal{C} \) is a chain complex \( \left\langle {{C}_{i},{d}_{i}}\right\rangle ,{d}_{i} : {C}_{i} \rightarrow {C}_{i - 1} \), denote by \( \widehat{\mathcal{C}} \) the chain complex \( \left\langle {{C}_{i},{\left( -1\right) }^{i + 1}{d}_{i}}\right\rangle \) . Let \[ \cdots \rightarrow {F}_{n} \rightarrow {F}_{n - 1} \rightarrow \cdots \rightarrow {F}_{1} \rightarrow {F}_{0} \rightarrow A \rightarrow 0 \] be a flat resolution of \( A \) . Let \( \mathcal{A} \) denote the complex \[ \cdots \rightarrow {F}_{n} \otimes B \rightarrow {F}_{n - 1} \otimes B \rightarrow \cdots \rightarrow {F}_{1} \otimes B \rightarrow {F}_{0} \otimes B \rightarrow 0 \rightarrow 0, \] with indices shifted, so that the \( i \) th group is \( {F}_{i - 1} \otimes B \) (and the 0 th group is 0). Similarly, let \( {\mathcal{A}}^{\prime } \) (respectively, \( {\mathcal{A}}^{\prime \prime } \) ) be the complex with \( {B}^{\prime } \) (respectively, \( \left. {B}^{\prime \prime }\right) \) replacing \( B \) . (We need this dimension shift to match up with the notation in Proposition 3.9.) Let \[ \cdots \rightarrow {P}_{n} \rightarrow {P}_{n - 1} \rightarrow \cdots \rightarrow {P}_{1} \rightarrow {P}_{0} \rightarrow 0 \] denote a simultaneous resolution of \( 0 \rightarrow B \rightarrow {B}^{\prime } \rightarrow {B}^{\prime \prime } \rightarrow 0 \), constructed via Proposition 6.5. Let \( \mathcal{D} \) denote the complex \[ \cdots \rightarrow A \otimes {P}_{n} \rightarrow A \otimes {P}_{n - 1} \rightarrow \cdots \rightarrow A \otimes {P}_{1} \rightarrow A \otimes {P}_{0} \rightarrow 0 \rightarrow 0 \] (again with the dimension shift), and similarly let \( {\mathcal{D}}^{\prime } \) (respectively, \( {\mathcal{D}}^{\prime \prime } \) ) denote the (deleted) complex obtained by tensoring the resolution of \( {B}^{\prime } \) (respectively, \( {B}^{\prime \prime } \) ) with \( A \) . Finally, let \( \mathcal{C} \) denote the complex \( \left\langle {{C}_{n},{D}_{n}}\right\rangle \), where we use the form described after the proof of Corollary 3.10: \[ {C}_{i, j} = {F}_{i - 1} \otimes {P}_{j - 1} \] \[ {C}_{n} = {\bigoplus }_{i = 1}^{n}{C}_{i, n - i + 1},, n \geq 1 \] \[ {D}_{n} = {d}_{n} + {\left( -1\right) }^{n + 1}{\partial }_{n} : {C}_{n} \rightarrow {C}_{n - 1}, n \geq 2 \] with \( {C}_{0} = 0,{D}_{1} = 0 \), and \( {d}_{n} \) and \( {\partial }_{n} \) as in Proposition 3.9. Note that we have chain maps \( \mathcal{C} \rightarrow \mathcal{D} \) (as well as \( {\mathcal{C}}^{\prime } \rightarrow {\mathcal{D}}^{\prime } \) and \( {\mathcal{C}}^{\prime \prime } \rightarrow {\mathcal{D}}^{\prime \prime } \) ) thanks to incorporating the required dimension shifts into \( \mathcal{D},{\mathcal{D}}^{\prime } \), and \( {\mathcal{D}}^{\prime \prime } \) . (Consult the array preceeding Proposition 3.9.) Also, since \( {\left( -1\right) }^{n + 1}{D}_{n} =
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Yes
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Proposition 6.13 Suppose ZFC is consistent. Then strong conglomerate theory is consistent if and only if the existence of strongly inaccessible cardinals is undecidable in ZFC.
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Proof: For the if part, suppose the existence of strongly inaccessible cardinals is consistent with ZFC. There then exists a model \( M \) for ZFC which contains a strongly inaccessible cardinal \( \kappa \) by the consistency theorem of mathematical logic. For a model of strong conglomerate theory, let conglomerates be the universe of sets in the model \( M \), while the collection of classes is \( \mathcal{P}\left( {V}_{\kappa }\right) \), and the collection of sets is \( \left\{ {A \subset {V}_{\kappa } : \left| A\right| < \kappa }\right\} = {V}_{\kappa } \). On the other hand, if strong conglomerate theory is consistent, let \( U \) be a universe of conglomerates in some model. \( U \) exists again by the consistency theorem. Let \( C \) be the conglomerate of classes, and \( S \) the class of sets in this model. Then \( U \) is a model of ZFC, so \( U \) has its own \
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No
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Proposition 6.14 (Yoneda Lemma) If \( F \) is a covariant functor from \( {}_{R}\mathbf{M} \) to \( \mathbf{{Ab}} \), then for all \( B \in {}_{R}\mathbf{M}, F\left( B\right) \approx \operatorname{Nat}\left( {\operatorname{Hom}\left( {B, \bullet }\right), F}\right) \) . The isomorphism sends \( \tau \in \operatorname{Nat}\left( {\operatorname{Hom}\left( {B, \bullet }\right), F}\right) \) to \( {\tau }_{B}\left( {i}_{B}\right) \) . (Note: \( {\tau }_{C} : \operatorname{Hom}\left( {B, C}\right) \rightarrow \) \( F\left( C\right) \) .)
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Proof: First, note that \( \tau \mapsto {\tau }_{B}\left( {i}_{B}\right) \) is a homomorphism, which is immediate from the definition of the sum of two natural transformations. Furthermore, for any \( C \in {}_{R}\mathbf{M} \) and \( f \in {\operatorname{Hom}}_{R}\left( {B, C}\right) ,{\operatorname{Hom}}_{R}\left( {B, f}\right) \) is post-composition by \( f \) (abbreviated \( {f}_{ * } \) ) in \( {\operatorname{Hom}}_{\mathbb{Z}}\left( {{\operatorname{Hom}}_{R}\left( {B, B}\right) ,{\operatorname{Hom}}_{R}\left( {B, C}\right) }\right) \) . Hence,\n\n\n\ncommutes. But now suppose \( {\tau }_{B}\left( {i}_{B}\right) = 0 \) and look where \( f \) goes:\n\n\[ 0 = F\left( f\right) \left( 0\right) = F\left( f\right) {\tau }_{B}\left( {i}_{B}\right) = {\tau }_{C}\left( {{f}_{ * }{i}_{B}}\right) = {\tau }_{C}\left( f\right) . \]\n\nSince \( C \) and \( f \) are arbitrary, \( \tau \equiv 0 \) . Hence the kernel of the homomorphism is trivial, so it is one-to-one. (The kernel is a conglomerate subgroup. Elementary theorems from group theory remain theorems for conglomerate groups.)\n\nFinally, we must show that the homomorphism is onto. Given \( x \in F\left( B\right) \) , and \( C \in {}_{R}\mathbf{M} \) ; define \( {\tau }_{C}^{x}\left( f\right) = F\left( f\right) \left( x\right) \) when \( f \in \operatorname{Hom}\left( {B, C}\right) \) . (This has the right values. \( F\left( f\right) \in \operatorname{Hom}\left( {F\left( B\right), F\left( C\right) }\right) \), so that \( {\tau }_{C}^{x}\left( f\right) \in F\left( C\right) \) . That is, \( {\tau }_{C}^{x} \) maps \( \operatorname{Hom}\left( {B, C}\right) \) to \( F\left( C\right) \) .) Clearly \( {\tau }_{B}^{x}\left( {i}_{B}\right) = x \) ; we must check that \( {\tau }^{x} \) is natural. If \( C, D \in {}_{R}\mathbf{M} \) and \( g \in \operatorname{Hom}\left( {C, D}\right) \), we must check commutativity of\n\n\n\nIf \( f \in \operatorname{Hom}\left( {B, C}\right) \), then\n\n\[ F\left( g\right) {\tau }_{C}^{x}\left( f\right) = F\left( g\right) F\left( f\right) \left( x\right) = F\left( {gf}\right) \left( x\right) = {\tau }_{D}^{x}\left( {gf}\right) = {\tau }_{D}^{x} \circ {g}_{ * }\left( f\right) . \]\n\nThe preceding, surprisingly, does not even use additivity of \( F \) . The next result, the final one of this chapter, requires even more than additivity.
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Yes
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Proposition 7.1 Suppose A is an additive category, and suppose \( \left( {A;{\varphi }_{1},{\varphi }_{2},{\pi }_{1},{\pi }_{2}}\right) \) is a biproduct of \( {A}_{1} \) and \( {A}_{2} \) . Then \( {\pi }_{1}{\varphi }_{2} = 0 \), and \( {\pi }_{2}{\varphi }_{1} = 0 \). Also,
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Proof: Identical, word for word, with the proof of Proposition 2.1 (except that \( \mathbf{A} \) replaces \( {}_{R}\mathbf{M} \) and the letter \( A \) replaces \( B \) ).
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No
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Proposition 7.1 Suppose A is an additive category, and suppose \( \left( {A;{\varphi }_{1},{\varphi }_{2},{\pi }_{1},{\pi }_{2}}\right) \) is a biproduct of \( {A}_{1} \) and \( {A}_{2} \) . Then \( {\pi }_{1}{\varphi }_{2} = 0 \), and \( {\pi }_{2}{\varphi }_{1} = 0 \). Also,\n\n\[ \n{A}_{1}\overset{{\varphi }_{1}}{ \rightarrow }A\overset{{\varphi }_{2}}{ \leftarrow }{A}_{2} \n\]\n\ndefines a direct sum, and\n\n\[ \n{A}_{1}\overset{{\pi }_{1}}{ \leftarrow }A\overset{{\pi }_{2}}{ \rightarrow }{A}_{2} \n\]\n\ndefines a (direct) product.
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Proof: Identical, word for word, with the proof of Proposition 2.1 (except that \( \mathbf{A} \) replaces \( {}_{R}\mathbf{M} \) and the letter \( A \) replaces \( B \) ).
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No
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Proposition 7.2 Suppose \( \mathbf{A} \) is an additive category, and suppose\n\n\[ \n{A}_{1}\overset{{\varphi }_{1}}{ \rightarrow }A\overset{{\varphi }_{2}}{ \leftarrow }{A}_{2}\n\]\n\ndefines a direct sum in \( \mathbf{A} \) . Then there exist unique \( {\pi }_{j} \in \operatorname{Hom}\left( {A,{A}_{j}}\right), j = \) \( 1,2 \), such that \( \left( {A;{\varphi }_{1},{\varphi }_{2},{\pi }_{1},{\pi }_{2}}\right) \) is a biproduct of \( {A}_{1} \) and \( {A}_{2} \) .
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Proof: From Proposition 7.1 above, \( {\pi }_{1} \) and \( {\pi }_{2} \) are fillers for\n\n\n\n\n\nand these fillers exist uniquely by the definition of a direct sum (coproduct). All we must now check is that \( {\varphi }_{1}{\pi }_{1} + {\varphi }_{2}{\pi }_{2} = {i}_{A} \) . Set \( \psi = {\varphi }_{1}{\pi }_{1} + {\varphi }_{2}{\pi }_{2} \) . Note that\n\n\[ \n\psi {\varphi }_{1} = \left( {{\varphi }_{1}{\pi }_{1} + {\varphi }_{2}{\pi }_{2}}\right) {\varphi }_{1}\n\]\n\n\[ \n= {\varphi }_{1}{\pi }_{1}{\varphi }_{1} + {\varphi }_{2}{\pi }_{2}{\varphi }_{1}\n\]\n\n\[ \n= {\varphi }_{1}{i}_{{A}_{1}} + {\varphi }_{2}0\n\]\n\n\[ \n= {\varphi }_{1}\n\]\n\nSimilarly, \( \psi {\varphi }_{2} = {\varphi }_{2} \) . That is, \( \psi \) is a filler for\n\n\n\nBut this filler is unique, and \( {i}_{A} \) works, so \( \psi = {i}_{A} \) .
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Yes
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Corollary 7.3 Suppose \( \\mathbf{A} \) is an additive category, and suppose\n\n\[ \n{A}_{1}\\overset{{\\pi }_{1}}{ \\leftarrow }A\\overset{{\\pi }_{2}}{ \\rightarrow }{A}_{2}\n\]\n\ndefines a product in \( \\mathbf{A} \) . Then there exist unique \( {\\varphi }_{j} \\in \\operatorname{Hom}\\left( {{A}_{j}, A}\\right), j = 1,2 \) , such that \( \\left( {A;{\\varphi }_{1},{\\varphi }_{2},{\\pi }_{1},{\\pi }_{2}}\\right) \) is a biproduct of \( {A}_{1} \) and \( {A}_{2} \) .
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Proof: \( A,{\\pi }_{1} \), and \( {\\pi }_{2} \) define a coproduct in the additive category \( {\\mathbf{A}}^{\\text{op }} \) ; use Proposition 7.2.
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No
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Proposition 7.4 Suppose \( \\mathbf{A} \) and \( {\\mathbf{A}}^{\\prime } \) are two additive categories, and suppose \( \\mathbf{A} \) contains a biproduct of any two objects. Suppose \( F : \\mathbf{A} \\rightarrow {\\mathbf{A}}^{\\prime } \) is a covariant functor. Then the following are equivalent.\n\ni) \( F \) is additive, that is, \( F\\left( {f + g}\\right) = F\\left( f\\right) + F\\left( g\\right) \) for any \( f, g \\in \\operatorname{Hom}(A \) , \( B);A, B \\in \\mathbf{A} \) .\n\nii) \( F\\left( {{A}_{1} \\oplus {A}_{2}}\\right) \\approx F\\left( {A}_{1}\\right) \\oplus F\\left( {A}_{2}\\right) \) for all \( {A}_{1},{A}_{2} \\in \\mathbf{A} \) .\n\niii) \( F\\left( {A \\oplus A}\\right) \\approx F\\left( A\\right) \\oplus F\\left( A\\right) \) for all \( A \\in \\mathbf{A} \) .
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Proof: (i) \\(\\Rightarrow\\) (ii) \\(\\Rightarrow\\) (iii) \\(\\Rightarrow\\) (i) works the same way here as it did in Proposition 6.1. The technical point-that \( F\\left( {\\pi }_{1}\\right) \) and \( F\\left( {\\pi }_{2}\\right) \) are the \( {\\pi }_{1}^{\\prime } \) and \( {\\pi }_{2}^{\\prime } \) for which \( \\left( {F\\left( {A \\oplus A}\\right) ;F\\left( {\\varphi }_{1}\\right), F\\left( {\\varphi }_{2}\\right) ,{\\pi }_{1}^{\\prime },{\\pi }_{2}^{\\prime }}\\right) \) is a biproduct in \( {\\mathbf{A}}^{\\prime } \\) -is even the same. To see this, we must establish that \( F\\left( {\\pi }_{1}\\right) \) and \( F\\left( {\\pi }_{2}\\right) \) are fillers for the appropriate diagrams in the proof of Proposition 7.2. That is, we must check that \( {i}_{F\\left( A\\right) } = F\\left( {\\pi }_{1}\\right) F\\left( {\\varphi }_{1}\\right) = F\\left( {\\pi }_{2}\\right) F\\left( {\\varphi }_{2}\\right) \), while \( 0 = \\) \( F\\left( {\\pi }_{1}\\right) F\\left( {\\varphi }_{2}\\right) = F\\left( {\\pi }_{2}\\right) F\\left( {\\varphi }_{1}\\right) \\) . But \( {i}_{F\\left( A\\right) } = F\\left( {i}_{A}\\right) = F\\left( {{\\pi }_{1}{\\varphi }_{1}}\\right) = F\\left( {\\pi }_{1}\\right) F\\left( {\\varphi }_{1}\\right) \\) ; similarly, \( {i}_{F\\left( A\\right) } = F\\left( {\\pi }_{2}\\right) F\\left( {\\varphi }_{2}\\right) \\) . Also, \( F\\left( {\\pi }_{1}\\right) F\\left( {\\varphi }_{2}\\right) = F\\left( {{\\pi }_{1}{\\varphi }_{2}}\\right) = F\\left( 0\\right) \\), and similarly \( F\\left( {\\pi }_{2}\\right) F\\left( {\\varphi }_{1}\\right) = F\\left( 0\\right) \\), so it suffices to show that \( F\\left( 0\\right) = 0 \\), that is, \( F \) (zero morphism) \( = \\) zero morphism. Since the zero morphism is precisely the morphism which factors through \
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Yes
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Proposition 7.5 Suppose \( \mathbf{A} \) is a pre-Abelian category; \( A, B \in \mathbf{A};f \in \) \( \operatorname{Hom}\left( {A, B}\right) \) . Let \( j : K \rightarrow A \) be a kernel of \( f \), and \( p : B \rightarrow D \) a cokernel of \( f \) .\n\na) \( K \) and \( D \) are unique up to isomorphism.\n\nb) \( j \) is monic and \( p \) is epic.\n\nc) If \( \varphi \in \operatorname{Hom}\left( {B, C}\right) \), and \( \varphi \) is monic, then \( j : K \rightarrow A \) is a kernel for \( {\varphi f} \) . If \( \psi \in \operatorname{Hom}\left( {C, A}\right) \), and \( \psi \) is epic, then \( p : B \rightarrow D \) is a cokernel for \( {f\psi } \) .\n\nd) \( f \) is monic \( \Leftrightarrow K = 0 \) ; \( f \) is epic \( \Leftrightarrow D = 0 \) .
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Proof: (a) and (b) were done earlier. For (c), note that \( {\varphi fj} = 0 \), and if \( g \in \operatorname{Hom}\left( {E, A}\right) \), then \( {\varphi fg} = 0 \Leftrightarrow {fg} = 0 \), so exactly the same morphisms are asked to factor through \( K \) as through \( \ker \left( {\varphi f}\right) \) . Again, \( D = \operatorname{coker}{f\psi } \) works the same way in \( {\mathbf{A}}^{\text{op }} \) .\n\nFinally, for (d),\n\n\( f \) is monic \( \Leftrightarrow \forall C,\forall g \in \operatorname{Hom}\left( {C, A}\right) : \left( {{fg} = 0 \Leftrightarrow g = 0}\right) \)\n\n\( \Leftrightarrow \forall C,\forall g \in \operatorname{Hom}\left( {C, A}\right) : \left( {{fg} = 0 \Leftrightarrow g\text{ factors through }0}\right) \)\n\n\( \Leftrightarrow 0 \) is a kernel of \( f \) .\n\nBy (a), \( K \) is a zero object. The result for \( D \) is the same in \( {\mathbf{A}}^{\text{op }} \) .
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Yes
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Proposition 7.8 Suppose A is a pre-Abelian category with enough projectives. Then any object in \( \mathbf{A} \) has a projective resolution which can be chosen with a choice function. If \( B,{B}^{\prime } \in \mathbf{A} \) and \( \varphi \in \operatorname{Hom}\left( {B,{B}^{\prime }}\right) \), and if \( \left\langle {{P}_{n},{d}_{n}}\right\rangle \) is a projective resolution of \( B \) and \( \left\langle {{P}_{n}^{\prime },{d}_{n}^{\prime }}\right\rangle \) is a projective resolution of \( {B}^{\prime } \) , then there exist fillers \( {\varphi }_{n} \in \operatorname{Hom}\left( {{P}_{n},{P}_{n}^{\prime }}\right) \) making\n\n\n\ncommutative. Further, if \( {\varphi }_{n}^{\prime } \in \operatorname{Hom}\left( {{P}_{n},{P}_{n}^{\prime }}\right) \) also serve as fillers, then \( {\varphi }_{n} \) and \( {\varphi }_{n}^{\prime } \) are homotopic, that is, there exist \( {D}_{n} \in \operatorname{Hom}\left( {{P}_{n},,{P}_{n + 1}^{\prime }}\right) \) (with \( \left. {{D}_{-1} = 0}\right) \) such that \( {\varphi }_{n} - {\varphi }_{n}^{\prime } = {d}_{n + 1}^{\prime }{D}_{n} + {D}_{n - 1}{d}_{n} \) .
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Proof: This goes much like Proposition 3.1, without the images (or elements). It turns out that all this does is make things more inductive.\n\nTo begin with, projective resolutions do exist. Choose a projective \( {P}_{0} \) and an epimorphism \( \pi : {P}_{0} \rightarrow B \) . Let \( {j}_{1} : {K}_{1} \rightarrow {P}_{0} \) be a kernel for \( \pi \) . Choose a projective \( {P}_{1} \) and an epimorphism \( {p}_{1} : {P}_{1} \rightarrow {K}_{1} \) ; set \( {d}_{1} = {j}_{1}{p}_{1} \) . Let \( {j}_{2} : {K}_{2} \rightarrow {P}_{1} \) be a kernel for \( {d}_{1} \) . (Note: Since \( {j}_{1} \) is monic, \( {j}_{2} : {K}_{2} \rightarrow {P}_{1} \) is a kernel for \( {p}_{1} \) as well, by Proposition 7.5(c).) Choose a projective \( {P}_{2} \) and an epimorphism \( {p}_{2} : {P}_{2} \rightarrow {K}_{2} \) ; set \( {d}_{2} = {j}_{2}{p}_{2} \) . Et cetera.\n\nTrue, an infinite number of choices from (possibly) proper classes are made, something that cannot be accomplished directly using a choice function. Nevertheless, one can bypass this using the Zermello hierarchy (Section 6.6). If \( B \in \mathbf{A} \), let \( \sigma \left( B\right) \) be the smallest ordinal \( \sigma \) such that \( {V}_{\sigma } \cap \mathbf{A} \) contains a projective \( P \) for which \( \operatorname{Hom}\left( {P, B}\right) \) contains an epimorphism. Using the axiom of choice, choose \( P\left( B\right) \in {V}_{\sigma \left( B\right) } \cap \mathbf{A} \) (a set) and \( {\pi }^{B} \) an epimorphism in \( \operatorname{Hom}\left( {P\left( B\right), B}\right) \) . Similarly, choose a kernel for \( {\pi }^{B} \) from a smallest \( {V}_{\sigma } \cap \mathbf{A} \) . Note that, by recursion, the above constructions actually choose (using a function) a specific projective resolution for any \( B \in \mathbf{A} \) .\n\nNext, fillers. To begin with, recall that \
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Yes
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Corollary 7.9 Suppose \( \mathbf{A} \) is a pre-Abelian category with enough injectives. Then any object in \( \mathbf{A} \) has an injective resolution which can be chosen with a choice function. If \( B,{B}^{\prime } \in \mathbf{A} \) and \( \varphi \in \operatorname{Hom}\left( {B,{B}^{\prime }}\right) \), and if \( \left\langle {{E}_{n},{\partial }_{n}}\right\rangle \) is an injective resolution of \( B \) and \( \left\langle {{E}_{n}^{\prime },{\partial }_{n}^{\prime }}\right\rangle \) is an injective resolution of \( {B}^{\prime } \), then there exist fillers \( {\varphi }_{n} \in \operatorname{Hom}\left( {{E}_{n},{E}_{n}^{\prime }}\right) \) making\n\n\n\ncommutative. Further, any two collections of fillers are homotopic.
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## Proof: Quote Proposition 7.8 for \( {\mathbf{A}}^{\text{op }} \) .
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No
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Proposition 7.11 Let A be a pre-Abelian category that satisfies either \( {Ab} \) -monic or \( {Ab} \) -epic. Then\na) \( \mathbf{A} \) is balanced.\nb) If \( A, B \in \mathbf{A} \), and \( f \in \operatorname{Hom}\left( {A, B}\right) \), then there exist \( C \in \mathbf{A}, p \in \) \( \operatorname{Hom}\left( {A, C}\right) \), and \( j \in \operatorname{Hom}\left( {C, B}\right) \) such that \( f = {jp} \), where \( j \) is monic and \( p \) is epic.
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Proof: Assume Ab-monic; for Ab-epic, work in \( {\mathbf{A}}^{\text{op }} \)\n\nFor (a), suppose \( f \in \operatorname{Hom}\left( {A, B}\right) \) is a bimorphism. Since \( f \) is monic, there exists a \( g \in \operatorname{Hom}\left( {B, C}\right) \) for which \( f : A \rightarrow B \) is a kernel of \( g \) . Since \( {gf} = 0 \) and \( f \) is epic, \( g = 0 \) . That is, \( f : A \rightarrow B \) is a kernel of the zero map. Since \( {i}_{B} : B \rightarrow B \) is also a kernel, \( f \) is an isomorphism by Proposition 7.5(a).\n\nNow (b). Assume \( f : A \rightarrow B \) is given. Let \( q : B \rightarrow D \) be a cokernel of \( f \) , and let \( j : C \rightarrow B \) denote a kernel of \( q \) . We have a diagram\n\n\n\nwith a (unique) filler \( p \), since \( {qf} = 0\left( {q = \operatorname{coker}f}\right) \Rightarrow p \) exists \( \left( {j = \ker q}\right) \) . By Proposition \( {7.5}\left( \mathrm{\;b}\right), j \) is monic; we must show that \( p \) is epic. Suppose \( g \in \operatorname{Hom}\left( {C, E}\right) \), and \( {gp} = 0 \) :\n\n\n\nLet \( e : K \rightarrow C \) denote a kernel for \( g : C \rightarrow E \) . We now have a commutative diagram\n\n\n\nwith a filler \( d \) since \( {gp} = 0 \) . Now \( e \) and \( j \) are monic, so \( {je} \) is monic, hence is a kernel of some \( h \in \operatorname{Hom}\left( {B, F}\right) \) :\n\n\n\nBut now watch this:\n\n\( {hje} = 0 \Rightarrow {hjed} = 0 \Rightarrow {hf} = 0 \Rightarrow h \) factors through \( D \)\n\n\n\nIn this diagram, \( {hj} = {kqj} = 0 \) since \( {qj} = 0 \) . \( \left( {j : C \rightarrow B\text{is a kernel for}q\text{.}}\right) \) It follows that \( j \) factors through \( {je} : K \rightarrow B \) since \( {je} \) is a kernel of \( h : B \rightarrow F \) :\n\n\n\nNow we have that \( j = {je\varphi } \Rightarrow {e\varphi } = {i}_{C} \) since \( j \) is monic. Also, \( {e\varphi e} = {i}_{C}e = \) \( e{i}_{K} \), so \( {\varphi e} = {i}_{K} \) since \( e \) is monic. That is, \( e \) is an isomorphism with inverse \( \varphi \) . But \( e : K \rightarrow C \) was a kernel for \( g : C \rightarrow E \), so \( g = 0 \) . Since \( g \) was arbitrary with \( {gp} = 0, p \) is epic.
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Yes
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Proposition 7.12 Suppose A is a pre-Abelian category. Then the following are equivalent:\n\ni) A satisfies \( {Ab} \) -monic.\n\nii) \( \mathbf{A} \) is balanced, and cokernel-exact sequences are kernel-exact.\n\niii) If \( 0 \rightarrow A \rightarrow B \rightarrow C \) is cokernel-exact, then \( A \rightarrow B \) is a kernel for \( B \rightarrow C \) .\n\niv) If \( A \rightarrow B \) is monic, with cokernel \( B \rightarrow D \), then \( B \rightarrow D \) has kernel \( A \rightarrow B \) .
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Proof: (i) \( \Rightarrow \) (ii). If \( \mathbf{A} \) satisfies \( \mathrm{{Ab}} \) -monic, then bimorphisms are isomorphisms by Proposition 7.11(a). To show that cokernel-exact sequences are kernel-exact, we appeal to the construction in the proof of Proposition 7.11(b). Suppose\n\n\[ A\xrightarrow[]{f}B\xrightarrow[]{g}C \]\n\nis cokernel-exact. Filling in \( K = \) a kernel of \( B \rightarrow C \) and \( D = \) a cokernel of \( A \rightarrow B \) yields\n\n\n\nSince \( l \) is monic, \( q \) and \( {lq} = g \) have the same kernel, namely \( K \) . By the proof of Proposition 7.11(b), \( p \) is epic, so \( A \rightarrow B \rightarrow C \) is kernel-exact.\n\n(ii) \( \Rightarrow \) (iii). If \( 0 \rightarrow A \rightarrow B \rightarrow C \) is cokernel-exact, then in the presence of (ii) it is kernel-exact, so that \( A \rightarrow B \) is a kernel for \( B \rightarrow C \) (since also bimorphisms are isomorphisms).\n\n(iii) \( \Rightarrow \) (iv). If \( A \rightarrow B \) is monic with cokernel \( B \rightarrow D \), then by definition \( A \rightarrow B \rightarrow D \) is cokernel-exact. \( 0 \rightarrow A \rightarrow B \) is also cokernel-exact, so \( 0 \rightarrow A \rightarrow B \rightarrow D \) is cokernel-exact; assuming (iii), \( A \rightarrow B \) is a kernel for \( B \rightarrow D \) .\n\nFinally,(iv) \( \Rightarrow \) (i) is trivial.\n\nCondition (iv) is sometimes written \
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Yes
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Corollary 7.13 Suppose A is a pre-Abelian category. Then the following are equivalent:\n\ni) A satisfies \( {Ab} \) -epic.\n\nii) \( \mathbf{A} \) is balanced, and kernel-exact sequences are cokernel-exact.\n\niii) If \( A \rightarrow B \rightarrow C \rightarrow 0 \) is kernel-exact, then \( B \rightarrow C \) is a cokernel for \( A \rightarrow B \) .\n\niv) If \( A \rightarrow B \) is epic with kernel \( K \rightarrow A \), then \( K \rightarrow A \) has cokernel \( A \rightarrow B \) .
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Proof: Apply Proposition 7.12 to \( {\mathbf{A}}^{\text{op }} \) .
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No
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Proposition 7.14 Suppose A is a pre-Abelian category with a separating class of projectives.\n\na) Given \( f : A \rightarrow B \) in \( \mathbf{A} : f \) is monic if and only if for all projectives \( P \) and all \( \varphi : P \rightarrow A,{f\varphi } = 0 \Rightarrow \varphi = 0 \) .
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(a) If \( f \) is monic, then \( {f\varphi } = 0 \Rightarrow \varphi = 0 \) by definition. Suppose \( f \) is not monic. Let \( j : K \rightarrow A \) be a kernel for \( f \) . There exists a projective \( P \) and \( \psi : P \rightarrow K \) for which \( {j\psi } \neq {0\psi } = 0 \), since \( j \neq 0 \) . Set \( \varphi = {j\psi } \) . Then \( {f\varphi } = {fj\psi } = {0\psi } = 0 \), but \( \varphi \neq 0 \) .
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Yes
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Proposition 7.15 Suppose A is pre-Abelian with a separating class of pro-jectives, and suppose \( f : A \rightarrow B \) and \( g : B \rightarrow C \) in \( \mathbf{A} \), with \( A \rightarrow B \rightarrow C \) kernel-exact. Then \( A \rightarrow B \rightarrow C \) is cokernel-exact.
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Proof: Let \( \pi : B \rightarrow D \) denote a cokernel for \( f \), and \( h : D \rightarrow C \) satisfy \( {h\pi } = g \) . Let \( P \) be projective, and suppose \( \psi : P \rightarrow D \) satisfies \( {h\psi } = 0 \) . A filler \( \varphi \) exists for\n\n\n\nsince \( \pi \) is epic, and a filler \( \eta \) exists for\n\n\n\nsince \( A \rightarrow B \rightarrow C \) is kernel-exact (Proposition 7.14(c)). But now \( \psi = {\pi \varphi } = \) \( {\pi f\eta } = {0\eta } = 0 \) . Since \( P \) and \( \psi \) are arbitrary, \( h \) is monic by Proposition \( {7.14}\left( \mathrm{a}\right) \) .
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Yes
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Corollary 7.16 Suppose A is pre-Abelian with a separating class of pro-jectives. Suppose \( f : A \rightarrow B \) in \( \mathbf{A} \) . Let \( j : K \rightarrow A \) denote a kernel for \( f \) , and \( \pi : A \rightarrow D \) a cokernel for \( j \) . Let \( g : D \rightarrow B \) satisfy \( {g\pi } = f \) (possible since \( {fj} = 0 \) ). Then \( g \) is monic. Hence, \( f = {g\pi } \) is a composite of an epimorphism followed by a monomorphism.
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Proof: \( K \rightarrow A \rightarrow B \) is kernel-exact by definition, so it is cokernel-exact. This just means that \( g \) is monic.
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Yes
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Proposition 7.17 Suppose A is a balanced pre-Abelian category with a separating class of projectives. Then \( \\mathbf{A} \) satisfies \( {Ab} \) -epic.
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Proof: A satisfies condition (ii) in Corollary 7.13.
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No
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Corollary 7.18 Suppose A is a balanced pre-Abelian category with a separating class of projectives and a coseparating class of injectives. Then \( \\mathbf{A} \) is Abelian.
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Proof: Both A and A \( {}^{\\text{op }} \) satisfy Ab-epic by Proposition 7.17, since a cosep-arating class of injectives in \( \\mathbf{A} \) becomes a separating class of projectives in \( {\\mathbf{A}}^{\\text{op }} \) . But if \( {\\mathbf{A}}^{\\text{op }} \) satisfies Ab-epic, then \( \\mathbf{A} \) satisfies Ab-monic, so \( \\mathbf{A} \) satisfies both Ab-monic and Ab-epic.
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Yes
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Proposition 7.21 Suppose \( 0 \rightarrow B \rightarrow {B}^{\prime } \rightarrow {B}^{\prime \prime } \rightarrow 0 \) is kernel-exact in a pre-Abelian category \( \mathbf{A} \) with enough projectives. Given projective resolutions of \( B \) and \( {B}^{\prime \prime } \) : there exist morphisms \( {\pi }^{\prime } : {P}_{0} \oplus {P}_{0}^{\prime \prime } \rightarrow {B}^{\prime } \) and \( {d}_{n}^{\prime } : {P}_{n} \oplus {P}_{n}^{\prime \prime } \rightarrow {P}_{n - 1} \oplus {P}_{n - 1}^{\prime \prime } \) such that is commutative with kernel-exact rows and columns. (The vertical morphisms are the obvious ones.)
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Proof: First, find a filler \( f \) for using the fact that \( {P}_{0}^{\prime \prime } \) is projective. Now suppose \( \left( {{P}_{0}^{\prime };\varphi ,{\varphi }^{\prime \prime },\rho ,{\rho }^{\prime \prime }}\right) \) is a biproduct of \( {P}_{0} \) with \( {P}_{0}^{\prime \prime } \) (so that \( {P}_{0}^{\prime } \) can serve as \( {P}_{0} \oplus {P}_{0}^{\prime \prime } \) ). The first two and these columns are exact. (Exactness of the \( {P}_{0} \) column is easily checked; we shall return to this in Section 7.8.) We now set \( {\pi }^{\prime } = f{\rho }^{\prime \prime } + {j\pi \rho } \) . Note that \( {\pi }^{\prime }\varphi = f{\rho }^{\prime \prime }\varphi + {j\pi \rho \varphi } = 0 + {j\pi } = {j\pi } \), while \( p{\pi }^{\prime } = {pf}{\rho }^{\prime \prime } + {pj\pi \rho } = \) \( {\pi }^{\prime \prime }{\rho }^{\prime \prime } + 0 = {\pi }^{\prime \prime }{\rho }^{\prime \prime }; \) thus, is commutative. Observe that \( {\pi }^{\prime } \) is epic by the \
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No
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Proposition 7.23 Suppose \( \mathbf{A} \) is an additive category, and \( \left( {\varphi ,{\varphi }^{\prime }}\right) \in \) \( \operatorname{Hom}\left( {f, g}\right) \) in \( \mathbf{A}\left( \rightarrow \right) \) . Suppose \( \iota : K \rightarrow A \) is a kernel for \( \varphi \), and \( {\iota }^{\prime } : {K}^{\prime } \rightarrow {A}^{\prime } \) is a kernel for \( {\varphi }^{\prime } \) . Let \( \bar{f} : K \rightarrow {K}^{\prime } \) denote the induced morphism\n\n\n\nThen \( \left( {\iota ,{\iota }^{\prime }}\right) : \left( {K,\bar{f},{K}^{\prime }}\right) \rightarrow \left( {A, f,{A}^{\prime }}\right) \) is a kernel for \( \left( {\varphi ,{\varphi }^{\prime }}\right) \) .
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Proof: Suppose \( \left( {\psi ,{\psi }^{\prime }}\right) \) maps \( \left( {C, h,{C}^{\prime }}\right) \) to \( \left( {A, f,{A}^{\prime }}\right) \) in \( \mathbf{A}\left( \rightarrow \right) \), with \( \left( {\varphi ,{\varphi }^{\prime }}\right) \) \( \left( {\psi ,{\psi }^{\prime }}\right) = \left( {0,0}\right) \) . Then \( \psi \) factors through \( K \) and \( {\psi }^{\prime } \) through \( {K}^{\prime } \), yielding the diagram\n\n\n\n\[ \psi = \iota \bar{\psi } \]\n\n\[ {\psi }^{\prime } = {\iota }^{\prime }\overline{{\psi }^{\prime }} \]\n\nThis diagram is commutative, since \( {\iota }^{\prime }\bar{f}\bar{\psi } = {f\iota }\bar{\psi } = {f\psi } = {\psi }^{\prime }h = {\iota }^{\prime }{\bar{\psi }}^{\prime }h \), so that \( \bar{f}\bar{\psi } = {\bar{\psi }}^{\prime }h \) since \( {\iota }^{\prime } \) is monic. (The lefthand square is the only one at issue.) Finally, \( \bar{\psi } \) and \( {\bar{\psi }}^{\prime } \) are each unique, so that the pair \( \left( {\bar{\psi },{\bar{\psi }}^{\prime }}\right) \) is unique.
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Yes
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Corollary 7.24 Suppose \( \mathbf{A} \) is an additive category, and \( \left( {\varphi ,{\varphi }^{\prime }}\right) \in \operatorname{Hom}\left( {f, g}\right) \) in \( \mathbf{A}\left( \rightarrow \right) \) . Suppose \( \pi : B \rightarrow D \) is a cokernel for \( \varphi \), and \( {\pi }^{\prime } : {B}^{\prime } \rightarrow {D}^{\prime } \) is a cokernel for \( {\varphi }^{\prime } \) . Let \( \widetilde{g} : D \rightarrow {D}^{\prime } \) denote the induced morphism\n\n\n\nThen \( \left( {\pi ,{\pi }^{\prime }}\right) : \left( {B, g,{B}^{\prime }}\right) \rightarrow \left( {D,\widetilde{g},{D}^{\prime }}\right) \) is a cokernel for \( \left( {\varphi ,{\varphi }^{\prime }}\right) \) .
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Proof: Essentially the same as Proposition 7.23, with arrows reversed. (Look in \( \mathbf{A}{\left( \rightarrow \right) }^{\text{op }} \approx {\mathbf{A}}^{\text{op }}\left( \rightarrow \right) \) .)
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No
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Corollary 7.25 Suppose A is a pre-Abelian category, and suppose \( \left( {\varphi ,{\varphi }^{\prime }}\right) \in \) \( \operatorname{Hom}\left( {f, g}\right) \) in \( \mathbf{A}\left( \rightarrow \right) \) . Then \( \left( {\varphi ,{\varphi }^{\prime }}\right) \) is monic in \( \mathbf{A}\left( \rightarrow \right) \) if and only if \( \varphi \) and \( {\varphi }^{\prime } \) are each monic in \( \mathbf{A} \) .
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Proof: \( \left( {\varphi ,{\varphi }^{\prime }}\right) \) is monic if and only if its kernel \( \left( {\ker \varphi ,\bar{f},\ker {\varphi }^{\prime }}\right) \) is zero (using earlier notation); this happens if and only if \( \varphi \) and \( {\varphi }^{\prime } \) are each monic.
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Yes
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Corollary 7.26 Suppose \( \mathbf{A} \) is a pre-Abelian category, and suppose \( \left( {\varphi ,{\varphi }^{\prime }}\right) \in \) \( \operatorname{Hom}\left( {f, g}\right) \) . Then \( \left( {\varphi ,{\varphi }^{\prime }}\right) \) is epic in \( \mathbf{A}\left( \rightarrow \right) \) if and only if \( \varphi \) and \( {\varphi }^{\prime } \) are each epic in \( \mathbf{A} \) .
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Proof: \( \left( {\varphi ,{\varphi }^{\prime }}\right) \) is epic if and only if its cokernel \( \left( {\operatorname{coker}\varphi ,\widetilde{g},\operatorname{coker}{\varphi }^{\prime }}\right) \) is zero (again using earlier notation); this happens if and only if \( \varphi \) and \( {\varphi }^{\prime } \) are each epic.
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Yes
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Proposition 7.28 Suppose \( \mathbf{A} \) is a pre-Abelian category. Then \( \mathbf{A}\left( \rightarrow \right) \) is pre-Abelian. Furthermore, if \( \mathbf{A} \) is Abelian, then so is \( \mathbf{A}\left( \rightarrow \right) \) .
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Proof: If \( \mathbf{A} \) is pre-Abelian, then \( \mathbf{A}\left( \rightarrow \right) \) has biproducts (Proposition 7.27), kernels (Proposition 7.23), and cokernels (Corollary 7.24). Hence, \( \mathbf{A}\left( \rightarrow \right) \) is pre-Abelian.\n\nSuppose \( \mathbf{A} \) is Abelian. If \( \left( {\varphi ,\psi }\right) \) is monic, then \( \varphi \) and \( \psi \) are each monic (Corollary 7.25). Let \( \pi \) denote a cokernel for \( \varphi \), and \( \rho \) a cokernel for \( \psi \) . Then \( \left( {\pi ,\rho }\right) \) is a cokernel for \( \left( {\varphi ,\psi }\right) \) (Corollary 7.24). Furthermore, \( \varphi \) is a kernel for \( \pi \), and \( \psi \) is a kernel for \( \rho \) (Proposition 7.12(iv)) since \( \varphi \) and \( \psi \) are monic. Hence, \( \left( {\varphi ,\psi }\right) \) is a kernel for \( \left( {\pi ,\rho }\right) \) (Proposition 7.23). All put together, \( \mathbf{A}\left( \rightarrow \right) \) satisfies Ab-monic; Ab-epic is similar.
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Yes
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Proposition 7.29 Suppose A is pre-Abelian, and suppose \( P \) and \( {P}^{\prime } \) are projective in \( \mathbf{A} \) . Then \( P \rightarrow P \oplus {P}^{\prime } \) is projective in \( \mathbf{A}\left( \rightarrow \right) \) .
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Proof: This is done in two stages. First, \( 0 \rightarrow {P}^{\prime } \) is shown to be projective, then \( P \rightarrow P \) is shown to be projective. Their coproduct \( P \rightarrow P \oplus {P}^{\prime } \) is then projective (see Exercise 21).\n\nSuppose \( \left( {\rho ,{\rho }^{\prime }}\right) \) is epic in \( \mathbf{A}\left( \rightarrow \right) \) :\n\n\n\nGiven \( {\psi }^{\prime } : {P}^{\prime } \rightarrow {A}^{\prime } \), we need fillers for\n\n\n\nproducing a commutative diagram. Any \( {\mu }^{\prime } \) serving as a filler for\n\n\n\nwill do. ( \( {P}^{\prime } \) is projective, and this triangle is the only nontrivial part.) This takes care of \( 0 \rightarrow {P}^{\prime } \) .\n\nAgain, suppose \( \left( {\rho ,{\rho }^{\prime }}\right) \) is epic in \( \mathbf{A}\left( \rightarrow \right) \) . Given \( \left( {\psi ,{\psi }^{\prime }}\right) : {i}_{P} \rightarrow f \), we need fillers \( \mu \) and \( {\mu }^{\prime } \) such that\n\n\n\ngives a commutative diagram. To do this, find a filler \( \mu \) for\n\n\n\nThe far triangle is commutative. Setting \( {\mu }^{\prime } = {g\mu } \) makes the top rectangle\n\n\n\ncommutative. Finally, for the near triangle\n\n\n\nnote that \( {\rho }^{\prime }{\mu }^{\prime } = {\rho }^{\prime }{g\mu } = {f\rho \mu } = {f\psi } = {\psi }^{\prime } \), so this triangle is also commutative.
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No
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Corollary 7.30 (to proof of Proposition 7.29) Suppose \( \mathbf{A} \) is a pre-Abelian category in which \( P,{P}^{\prime } \), and \( {P}^{\prime \prime } \) are projective. Suppose the diagram\n\n\n\nis given in \( \mathbf{A} \) with \( \rho ,{\rho }^{\prime } \), and \( {\rho }^{\prime \prime } \) all epic. Finally, suppose the commutative diagram (with fillers \( \mu ,{\mu }^{\prime } \) chosen)\n\n\n\nis given. Then there exists a filler \( {\mu }^{\prime \prime } \) for\n\n\n\ngiving a commutative diagram.
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Proof: From the point of view of the far wedge, this consists of lifting two things. First, given \( \mu \), the morphism \( {g\mu } \) makes\n\n\n\ncommutative (as noted in the earlier proof), so \( {g\mu } \) is the filler for this wedge. Furthermore, from the proof of Proposition 7.29, any filler \( {\mu }^{\prime \prime } \) for\n\n\n\nyields a commutative diagram. The filler to use is \( {g}^{\prime }{\mu }^{\prime } \) from\n\n\n\ncombined with any filler for\n\n\n\nThis will make the far wedge (and, by symmetry, the near one as well) commute.
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Yes
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Proposition 7.31 Suppose \( \mathbf{A} \) is a pre-Abelian category.\n\na) If \( \mathbf{A} \) has enough projectives, then so does \( \mathbf{A}\left( \rightarrow \right) \) .
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Proof: For starters, suppose \( P\overset{h}{ \rightarrow }P \oplus {P}^{\prime }\overset{l}{ \leftarrow }{P}^{\prime } \) is a coproduct, and \( {\pi }^{\prime } \) : \( {P}^{\prime } \rightarrow {A}^{\prime } \) and \( \varphi : {A}^{\prime } \rightarrow D \) satisfy \( \varphi {\pi }^{\prime } \neq 0 \) . Suppose \( \rho : P \rightarrow {A}^{\prime } \) is any morphism, and form a filler \( \theta \) using the coproduct construction:\n\n\n\nIn this, \( 0 \neq \varphi {\pi }^{\prime } = {\varphi \theta l} \), so \( {\varphi \theta } \neq 0 \), too. In particular, if \( {\pi }^{\prime } \) is epic, then (letting \( \varphi \) float among all nonzero morphisms) \( \theta \) is also epic.\n\nTo use this in proving (a), let \( f : A \rightarrow {A}^{\prime } \) be given in \( \mathbf{A}\left( \rightarrow \right) \), and suppose \( \pi : P \rightarrow A \) is epic, while \( {\pi }^{\prime } : {P}^{\prime } \rightarrow {A}^{\prime } \) is epic, with \( P \) and \( {P}^{\prime } \) projective. Then (setting \( \rho = {f\pi } \) )\n\n\n\nis (horizontally) epic in \( \mathbf{A}\left( \rightarrow \right) \) .
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Yes
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Corollary 7.32 Suppose \( \mathbf{A} \) is a pre-Abelian category.\na) If \( \mathbf{A} \) has enough injectives, then so does \( \mathbf{A}\left( \rightarrow \right) \).\nb) If \( \mathbf{A} \) has a coseparating class of injectives, then so does \( \mathbf{A}\left( \rightarrow \right) \).\nc) If \( \mathbf{A} \) has a coseparating set of injectives, then so does \( \mathbf{A}\left( \rightarrow \right) \).\nd) If \( \mathbf{A} \) has an injective coseparator, then so does \( \mathbf{A}\left( \rightarrow \right) \).
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Proof: Quote Proposition 7.31 in \( \mathbf{A}{\left( \rightarrow \right) }^{\text{op }} \approx \left( {\mathbf{A}}^{\text{op }}\right) \left( \rightarrow \right) \) . (Proposition 7.22).
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Yes
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Lemma 7.33 Suppose A is pre-Abelian, and suppose\n\n\n\nis a commutative diagram in \( \\mathbf{A} \) with\n\ni) \( j : K \\rightarrow B \) a kernel for \( g : B \\rightarrow C \) ,\n\nii) \( {f}^{\\prime } \) monic, and\n\niii) \( \\varphi \) and \( \\eta \) monic.\n\nThen \( \\psi \) is monic.
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Proof: Let \( l : L \\rightarrow B \) denote a kernel for \( \\psi : B \\rightarrow {B}^{\\prime } \) ; we shall show that \( l = 0 \), which will imply that \( L = 0 \) and \( \\psi \) is monic. The relevant diagram is\n\n\n\nNow \( {\\psi l} = 0 \\Rightarrow 0 = {g}^{\\prime }{\\psi l} = {\\eta gl} \\Rightarrow {gl} = 0 \) since \( \\eta \) is monic. Hence, \( l \) factors through \( K \), since \( j : K \\rightarrow B \) is a kernel for \( g : B \\rightarrow C \) :\n\n\n\nBut now \( l = {j\\tau } \\Rightarrow 0 = {\\psi l} = {\\psi j\\tau } = {f}^{\\prime }{\\varphi \\tau } \\Rightarrow {\\varphi \\tau } = 0 \), since \( {f}^{\\prime } \) is monic \( \\Rightarrow \\tau = 0 \), since \( \\varphi \) is monic. Hence, \( l = {j\\tau } = 0 \) .
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Yes
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Proposition 7.34 Suppose \( \mathbf{A} \) is an Abelian category, and suppose\n\n\n\ncommutes and has exact rows in \( \mathbf{A} \) . Then\n\na) If \( \varphi \) and \( \eta \) are monic, then so is \( \psi \) .\n\nb) If \( \varphi \) and \( \eta \) are epic, then so is \( \psi \) .\n\nc) If \( \varphi \) and \( \eta \) are isomorphisms, then so is \( \psi \) .
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Proof: For (a), note that Lemma 7.33 applies. (Proposition 7.12(iii) guarantees that \( A \rightarrow B \) is the kernel of \( B \rightarrow C \) .) For (b), apply (a) to \( {\mathbf{A}}^{\text{op }} \) . For (c), use (a) and (b) together with the fact that \( \mathbf{A} \) is balanced (Proposition 7.11(a)).
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Yes
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Lemma 7.35 Suppose \( \mathbf{A} \) is pre-Abelian, \( j : K \rightarrow A \) and \( f : A \rightarrow B \) are monic, and \( \varphi \in \operatorname{Hom}\left( {B, C}\right) \), producing\n\n\n\nSuppose \( {fj} \) is a kernel for \( \varphi \) . Then \( j \) is a kernel for \( {\varphi f} \) .
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Proof: We show \( j \) has the required universal property. Suppose \( g : D \rightarrow A \) has the property that \( \left( {\varphi f}\right) g = 0 \) . Then \( \varphi \left( {fg}\right) = 0 \), so \( {fg} \) factors through \( K \)\n\n\n\nvia some morphism \( \psi \) . Thus \( {fj\psi } = {fg} \Rightarrow {j\psi } = g \) since \( f \) is monic, and\n\n\n\ncommutes. Finally, \( \psi \) is unique since \( j \) is monic.
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Yes
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Proposition 7.36 Suppose A is an Abelian category, \( \pi : A \rightarrow B \) is epic, and \( \varphi \in \operatorname{Hom}\left( {B, C}\right) \) . If \( \pi \) and \( {\varphi \pi } \) have the same kernel(s), then \( \varphi \) is monic.
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Proof: Suppose first that \( \varphi \) is epic. Let \( j : K \rightarrow A \) denote a kernel for both \( \pi \) and \( {\varphi \pi } \) . Then \( \pi \) and \( {\varphi \pi } \) are both cokernels for \( j \) by part (iv) of Corollary 7.13. Hence, there is an isomorphism \( \psi : B \rightarrow C \) such that\n\n\n\ncommutes, by uniqueness of cokernels. But \( {\varphi \pi } = {\psi \pi } \Rightarrow \varphi = \psi \), since \( \pi \) is epic, so \( \varphi \) is an isomorphism.\n\nFor general \( \varphi \), write \( \varphi = {fp} \) according to Proposition 7.11(b), where \( f \) is monic and \( p \) is epic. Then kernels of \( {\varphi \pi } = {fp\pi } \) coincide with kernels of \( {p\pi } \) since \( f \) is monic, so the first part of the proof applies (with \( p \) replacing \( \varphi ) \) . Thus, \( p \) is an isomorphism, and \( \varphi = {fp} \) is monic.
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No
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Proposition 7.37 Suppose \( \mathbf{A} \) is an Abelian category, and suppose\n\n\n\ncommutes and has exact rows in \( \mathbf{A} \) . Then \( \psi \) monic \( \Rightarrow \eta \) monic.
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Proof: Assume \( \psi \) is monic. Note that \( j : A \rightarrow B \) is a kernel for \( \pi : B \rightarrow C \) by Proposition 7.12(iii), and likewise \( {\psi j} = {j}^{\prime } \) is a kernel for \( {\pi }^{\prime } \) . By Lemma \( {7.35}, j \) is a kernel for \( {\pi }^{\prime }\psi = {\eta \pi } \) . Thus, \( j : A \rightarrow B \) is a kernel for both \( {\eta \pi } \) and \( \pi \), so \( \eta \) is monic by Proposition 7.36.
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Yes
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Corollary 7.38 Suppose \( \mathbf{A} \) is an Abelian category, and suppose\n\n\n\ncommutes and has exact rows in \( \mathbf{A} \) . Then \( \psi \) epic \( \Rightarrow \varphi \) epic.
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Proof: Apply Proposition 7.37 to \( {\mathbf{A}}^{\text{op }} \) .
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Yes
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Lemma 7.39 Suppose \( \mathbf{A} \) is Abelian, and\n\n\n\nis a commutative square in \( \mathbf{A} \) with \( \varphi \) epic and \( \psi \) monic. Then \( f \) and \( {f}^{\prime } \) have isomorphic images.
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## Proof: Consider\n\n\n\nIn the righthand square, \( j,{j}^{\prime } \), and \( \psi \) are all monic, so \( \theta \) is monic by the discussion following Proposition 7.35. Similarly, consideration of the left-hand square shows that \( \theta \) is epic. Hence, \( \theta \) is an isomorphism since \( \mathbf{A} \) is balanced (Proposition 7.11(a)).
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Yes
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Lemma 7.42 (Monic 4-Lemma) Suppose \( \mathbf{A} \) is an Abelian category, and suppose\n\n\n\nis commutative in \( \mathbf{A} \) with exact rows. Assume \( {\varphi }_{1} \) is epic, while \( {\varphi }_{2} \) and \( {\varphi }_{4} \) are monic. Then \( {\varphi }_{3} \) is monic.
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Proof: Let \( {K}_{j} = \) kernel of \( {f}_{j + 1} \approx \) image of \( {f}_{j} \) (Lemma 7.40), and \( {K}_{j}^{\prime } = \) kernel of \( {g}_{j + 1} \) . Note that by Lemma 7.39, \( {K}_{1} \approx {K}_{1}^{\prime } \), so we have a diagram\n\n\n\nwith short exact rows (since \( {K}_{2} \approx \) image \( {f}_{2} \), and \( {K}_{2}^{\prime } \approx \) image of \( {g}_{2} \) ). Since \( {\varphi }_{2} \) is monic, \( \psi \) is also monic by Proposition 7.37. Finally, we now have\n\n\n\nwith exact rows, in which the hypotheses of Lemma 7.33 are satisfied. Hence, \( {\varphi }_{3} \) is monic.
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Yes
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Corollary 7.43 (Epic 4-Lemma) Suppose \( \mathbf{A} \) is an Abelian category, and suppose\n\n\n\n is commutative in \( \mathbf{A} \) with exact rows. Assume \( {\varphi }_{4} \) is monic, while \( {\varphi }_{1} \) and \( {\varphi }_{3} \) are epic. Then \( {\varphi }_{2} \) is epic.
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Proof: Apply Lemma 7.42 to \( {\mathbf{A}}^{\text{op }} \) .
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No
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Proposition 7.44 (5-Lemma for Abelian categories) Suppose A is an Abelian category, suppose\n\n\n\nis commutative in \( \\mathbf{A} \) with exact rows, and suppose\n\ni) \( {\\varphi }_{2} \) and \( {\\varphi }_{4} \) are isomorphisms,\n\nii) \( {\\varphi }_{1} \) is epic, and\n\niii) \( {\\varphi }_{5} \) is monic.\n\nThen \( {\\varphi }_{3} \) is an isomorphism.
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Proof: \( {\\varphi }_{3} \) is monic by Lemma 7.42, and epic by Corollary 7.43, so \( {\\varphi }_{3} \) is an isomorphism, since \( \\mathbf{A} \) is balanced (Proposition 7.11(a)).
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Yes
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Theorem 7.45 Suppose \( \mathbf{A} \) is an Abelian category in which \( f : A \rightarrow B \) and \( g : B \rightarrow C \) satisfy \( {gf} = 0 \) . Let \( j : K \rightarrow B \) denote a kernel for \( g \), and \( \pi : B \rightarrow D \) a cokernel for \( f \) . Let\n\n\n\n\n\ndenote the resulting factorization, with \( p : K \rightarrow H \) a cokernel for \( \bar{f} \) and \( \kappa : \bar{H} \rightarrow D \) a kernel for \( \bar{g} \) . Then the induced \( \tau \) is an isomorphism.
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Proof: A construction similar to the one preceeding the theorem (actually the same in \( {\mathbf{A}}^{\text{op }} \) ) yields a diagram\n\n\n\n\n\nwhere \( p : K \rightarrow H \) is a cokernel for \( {\varphi \rho } \) and for \( \varphi \), and \( \kappa : \bar{H} \rightarrow D \) is a kernel for \( {k\psi } \) and for \( \psi \) . Using this, we may delete \( f \) and \( g \) to get\n\n\n\nThis diagram contains enough information to produce the isomorphism \( \tau \) of \( H \) with \( \bar{H} \) .\n\nTo start in this direction, note that we have the following:\n\n\( \pi = \operatorname{coker}f = \operatorname{coker}\left( {j\varphi \rho }\right) = \operatorname{coker}\left( {j\varphi }\right) \) so that \( {j\varphi } = \ker \pi \),\n\n\( j = \ker g = \ker \left( {k\psi \pi }\right) = \ker \left( {\psi \pi }\right) \) so that \( {\psi \pi } = \operatorname{coker}\left( j\right) \),\n\n\[ p = \operatorname{coker}\varphi \text{, and} \]\n\n\[ \kappa = \ker \psi . \]\n\nNow \( {\pi j\varphi } = 0 \Rightarrow {\pi j} \) factors through coker \( \varphi \), that is, through \( H \) :\n\n\n\nClaim: In the preceeding diagram, in an Abelian category, \( \sigma \) is monic.\n\nProof of claim: Twist the diagram to\n\n\n\nFrom earlier remarks, the rows are short exact, so that \( \sigma \) is monic by Proposition 7.37. End of proof of claim.\n\nContinuing, note that \( 0 = {\psi \pi j} = {\psi \sigma p} \), so that \( {\psi \sigma } = 0 \), since \( p \) is epic. Hence, \( \sigma \) factors through \( \bar{H} = \ker \psi \) :\n\n\n\n\n\nNow \( \sigma = {\kappa \tau } \) is monic, so \( \tau \) is monic. Similarly, looking in \( {\mathbf{A}}^{\text{op }},{\tau p} \) (which corresponds to \( \sigma \) ) is epic, so \( \tau \) is epic. That is, \( \tau \) is a bimorphism, hence is an isomorphism (Abelian categories are balanced).
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Yes
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Proposition 7.47 (Ker-Coker Exact Sequence) Suppose A is an Abelian category in which\n\n\n\nis commutative and has exact rows. Extend to include kernels and cokernels of the vertical arrows:\n\n\n\nThen this diagram has exact rows. Furthermore, there is a naturally defined \( \delta : {K}_{3} \rightarrow {D}_{1} \) such that\n\n\[ \n{K}_{1}\overset{\bar{\varphi }}{ \rightarrow }{K}_{2}\overset{\bar{\pi }}{ \rightarrow }{K}_{3}\overset{\delta }{ \rightarrow }{D}_{1}\overset{\bar{\psi }}{ \rightarrow }{D}_{2}\overset{\bar{\rho }}{ \rightarrow }{D}_{3} \n\]\n\nis exact. Finally, \( \bar{\varphi } \) is monic if \( \varphi \) is monic, and \( \bar{\rho } \) is epic if \( \rho \) is epic.
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Proof: First of all, let \( {\varphi }^{\prime } : {A}_{1}^{\prime } \rightarrow {A}_{2} \) denote a kernel for \( \pi \) ; one has induced fillers \( {f}_{1}^{\prime } \) and \( {\varphi }^{ * } \):\n\n\n\nwhere \( {f}_{1}^{\prime } \) occurs since \( \psi : {B}_{1} \rightarrow {B}_{2} \) is a kernel for \( \rho \) ,(and \( \rho {f}_{2}{\varphi }^{\prime } = {f}_{3}\pi {\varphi }^{\prime } = 0 \) , so that \( {f}_{2}{\varphi }^{\prime } \) factors through \( {B}_{1} \) ) and \( {\varphi }^{ * } \) is induced because \( {\pi \varphi } = 0 \) . This is really the induced diagram for images discussed in Section 7.6 and is commutative. Letting \( {j}_{1}^{\prime } : {K}_{1}^{\prime } \rightarrow {A}_{1}^{\prime } \) denote a kernel for \( {f}_{1}^{\prime } \), we also get induced kernel maps \( {\bar{\varphi }}^{ * } \) and \( {\bar{\varphi }}^{\prime } \):\n\n\n\nSince \( {\bar{\varphi }}^{\prime }{\bar{\varphi }}^{ * } : {K}_{1} \rightarrow {K}_{2} \) fills in where \( \bar{\varphi } \) did, and \( \bar{\varphi } \) is unique, we get that \( \bar{\varphi } = {\bar{\varphi }}^{\prime }{\bar{\varphi }}^{ * } \) . Now \( {\bar{\varphi }}^{ * } \) is epic by Proposition 7.46(b), since \( {\varphi }^{ * } \) is epic (kernel-exactness of \( {A}_{1} \rightarrow {A}_{2} \rightarrow {A}_{3} \) ), \( {j}_{1}^{\prime } \) is a kernel for \( {f}_{1}^{\prime } \), and \( {j}_{1} \) is a kernel for \( {f}_{1} = {f}_{1}^{\prime }{\varphi }^{ * } \) . Thus, if we show that \( {\bar{\varphi }}^{\prime } \) is a kernel for \( \bar{\pi } \), we obtain kernel-exactness of \( {K}_{1} \rightarrow {K}_{2} \rightarrow {K}_{3} \) by definition. Furthermore, if \( \varphi \) is monic, we can take \( {A}_{1}^{\prime } = {A}_{1} \), and \( {K}_{1}^{\prime } = {K}_{1} \), giving \( \bar{\varphi } = {\bar{\varphi }}^{\prime } \) monic.\n\n\( \bar{\pi }{\bar{\varphi }}^{\prime } = 0 \) since \( {j}_{3}\bar{\pi }{\bar{\varphi }}^{\prime } = \pi {\varphi }^{\prime }{j}_{1}^{\prime } = 0 \), and \( {j}_{3} \) is monic. To show that \( {\bar{\varphi }}^{\prime } \) is a kernel for \( \bar{\pi } \), let \( g : C \rightarrow {K}_{2} \) be such that \( \bar{\pi }g = 0 \) . It suffices to show that \( g \) factors uniquely through \( {K}_{1}^{\prime } \) . Now \( 0 = {j}_{3}\bar{\pi }g = \pi {j}_{2}g \), so \( {j}_{2}g \) factors through \( {A}_{1}^{\prime } \), since \( {\varphi }^{\prime } \) is a kernel for \( \pi \) :\n\n\n\nhas rows that are short exact and columns that are underexact. Let \( {H}_{n} \) (respectively, \( {H}_{n}^{\prime },{H}_{n}^{\prime \prime } \) ) denote the homology of the vertical arrows at \( {B}_{n} \) (respectively, \( {B}_{n}^{\prime },{B}_{n}^{\prime \prime } \) ). Then there are naturally defined morphisms \( {\delta }_{n} \) : \( {H}_{n}^{\prime \prime } \rightarrow {H}_{n - 1} \) such that \n\nis exact.
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Proof: Consider the portion \n\nof the diagram. Let \( {D}_{n + 1} \) (respectively, \( {D}_{n + 1}^{\prime },{D}_{n + 1}^{\prime \prime } \) ) denote cokernels of \( {d}_{n + 2} \) (respectively, \( {d}_{n + 2}^{\prime },{d}_{n + 2}^{\prime \prime } \) ). Let \( {K}_{n} \) (respectively, \( {K}_{n}^{\prime },{K}_{n}^{\prime \prime } \) ) denote kernels of \( {d}_{n} \) (respectively, \( {d}_{n}^{\prime },{d}_{n}^{\prime \prime } \) ). For example, the first column gives\n\n\n\nin which the filler \( {\bar{d}}_{n + 1} \) exists by first factoring \( {d}_{n + 1} \) through \( {j}_{n} \) (possible since \( \left. {{d}_{n}{d}_{n + 1} = 0}\right) \) producing \( {\alpha }_{n + 1} \), then factoring \( {\alpha }_{n + 1} \) through \( {D}_{n + 1} \) (possible since \( {j}_{n}{\alpha }_{n + 1}{d}_{n + 2} = {d}_{n + 1}{d}_{n + 2} = 0 \Rightarrow {\alpha }_{n + 1}{d}_{n + 2} = 0,{j}_{n} \) being monic). Note that by definition, \( {H}_{n} \) is the cokernel of \( {\alpha }_{n + 1} = {\bar{d}}_{n + 1}{\sigma }_{n + 1} \) , which is the cokernel of \( {\bar{d}}_{n + 1} \), since \( {\sigma }_{n + 1} \) is epic. Similarly, \( {H}_{n + 1} \) is isomorphic (via Theorem 7.45) to the kernel of \( {j}_{n}{\bar{d}}_{n + 1} \), which is the kernel of \( {\bar{d}}_{n + 1} \), since \( {j}_{n} \) is monic. Defining \( {\bar{d}}_{n + 1}^{\prime } \) and \( {\bar{d}}_{n + 1}^{\prime \prime } \) (as well as \( {\beta }_{n + 1}^{\prime },{\beta }_{n + 1}^{\prime \prime } \) , \( \left. {{\gamma }_{n},{\gamma }_{n}^{\prime \prime }}\right) \), we get the array \n\nThe middle two rows are exact by what we already know from the ker-coker exact sequence. Carrying out the construction of \( {\alpha }_{n + 1},{\beta }_{n + 1},{\gamma }_{n} \), and \( {\bar{d}}_{n + 1} \) in \( \mathbf{A}\left( \rightarrow \right) \) shows that the morphisms in the bottom and top rows really are \( {\left( {\varphi }_{ \bullet }\right) }_{ * } \) and \( {\left( {\pi }_{ \bullet }\right) }_{ * } \) . From the ker-coker exact sequence, we will get a naturally defined \( {\delta }_{n + 1} \) and an exact sequence\n\n\[{H}_{n + 1}\overset{{\left( {\varphi }_{n + 1}\right) }_{ * }}{ \rightarrow }{H}_{n + 1}^{\prime }\overset{{\left( {\pi }_{n + 1}\right) }_{ * }}{ \rightarrow }{H}_{n + 1}^{\prime \prime }\overset{{\delta }_{n + 1}}{ \rightarrow }{H}_{n}\overset{{\left( {\varphi }_{n}\right) }_{ * }}{ \rightarrow }{H}_{n}^{\prime }\overset{{\left( {\pi }_{n}\right) }_{ * }}{ \rightarrow }{H}_{n}^{\prime \prime }\]\n\nwhich (letting \( n \) vary now) will complete the proof once we know \
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Yes
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Theorem 7.49 Suppose \( \mathbf{A} \) is a pre-Abelian category, \( {\mathbf{A}}^{\prime } \) is an Abelian category, and \( F : \mathbf{A} \rightarrow {\mathbf{A}}^{\prime } \) is an additive functor.\n\na) If \( F \) is covariant, \( \mathbf{A} \) has enough projectives, and \( 0 \rightarrow B \rightarrow {B}^{\prime } \rightarrow \) \( {B}^{\prime \prime } \rightarrow 0 \) is kernel-exact, then there is a naturally defined long exact sequence for the left derived functors of \( F \) :
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Proof: We do (a); (c) follows by replacing \( {\mathbf{A}}^{\prime } \) by \( {\left( {\mathbf{A}}^{\prime }\right) }^{\text{op }} \) ,(d) follows by replacing \( \mathbf{A} \) with \( {\mathbf{A}}^{\mathrm{{op}}} \), and (b) follows by making both replacements.\n\nThe crucial point, once again, is a simultaneous projective resolution of \( 0 \rightarrow B \rightarrow {B}^{\prime } \rightarrow {B}^{\prime \prime } \rightarrow 0 \) using Proposition 7.21. Since \( F \) is additive, each\ncolumn (obtained by applying \( F \) to the projectives of Proposition 7.21)\nis split exact by Proposition 7.4. This gives everything except naturality and independence of the simultaneous resolution used. Naturality follows directly from the usual \( \mathbf{A}\left( \rightarrow \right) \) business once independence of the simultaneous resolution is known, but (alas) independence of the simultaneous resolution does not follow directly from working in \( \mathbf{A}\left( \rightarrow \right) \), since a typical \( {P}_{n} \rightarrow {P}_{n}^{\prime } \), as it appears in Proposition 7.8, is not projective in \( \mathbf{A}\left( \rightarrow \right) \) . (See Exercise 20.) This is where that obscure Corollary 7.30 comes in.\n\nSuppose we have two projective resolutions of \( B \), say,\n\n\[ \n\cdots \rightarrow {P}_{2} \rightarrow {P}_{1} \rightarrow {P}_{0} \rightarrow B \rightarrow 0 \n\]\n\nand\n\n\[ \n\cdots \rightarrow {\bar{P}}_{2} \rightarrow {\bar{P}}_{1} \rightarrow {\bar{P}}_{0} \rightarrow B \rightarrow 0. \n\]\n\nOne can choose \( {Q}_{n} \) projective so that\n\n\[ \n\cdots \rightarrow {P}_{2} \oplus {\bar{P}}_{2} \oplus {Q}_{2} \rightarrow {P}_{1} \oplus {\bar{P}}_{1} \oplus {Q}_{1} \rightarrow {P}_{0} \oplus {\bar{P}}_{0} \oplus {Q}_{0} \rightarrow B \rightarrow 0 \n\]\n\nis a projective resolution, and (in fact)
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No
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Proposition 7.50 Suppose A is a pre-Abelian category in which \( \varphi : A \rightarrow \) \( B \) and \( \psi : B \rightarrow D \) are given. Then\n\n\[ 0 \rightarrow \operatorname{Hom}\left( {D, C}\right) \overset{{\psi }^{ * }}{ \rightarrow }\operatorname{Hom}\left( {B, C}\right) \overset{{\varphi }^{ * }}{ \rightarrow }\operatorname{Hom}\left( {A, C}\right) \]\n\nis exact for all \( C \) if and only if \( \psi \) is a cokernel for \( \varphi \) .
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Proof: Suppose first that \( \psi \) is a cokernel for \( \varphi \) . Fix \( C \) . Then \( \psi \) is epic, so \( \forall f \in \operatorname{Hom}\left( {D, C}\right) ,{\psi }^{ * }f = 0 \Rightarrow {f\psi } = 0 \Rightarrow f = 0 \) . That is, \( {\psi }^{ * } \) is one-to-one. Next, \( {\varphi }^{ * }{\psi }^{ * } = {\left( \psi \varphi \right) }^{ * } = {0}^{ * } = 0 \) . Finally, if \( {\varphi }^{ * }f = 0 \) for \( f \in \operatorname{Hom}\left( {B, C}\right) \) , then \( {f\varphi } = 0 \Rightarrow f = \bar{f}\psi \) for a (unique) \( \bar{f} \), since \( \psi \) is a cokernel for \( \varphi \) :\n\n\n\nBut now \( f = \bar{f}\psi = {\psi }^{ * }\left( \bar{f}\right) \).\n\nFor the converse, suppose\n\n\[ 0 \rightarrow \operatorname{Hom}\left( {D, C}\right) \overset{{\psi }^{ * }}{ \rightarrow }\operatorname{Hom}\left( {B, C}\right) \overset{{\varphi }^{ * }}{ \rightarrow }\operatorname{Hom}\left( {A, C}\right) \]\n\nis exact for all \( C \) . Setting \( C = D,0 = {\varphi }^{ * }{\psi }^{ * }\left( {i}_{D}\right) = {i}_{D}{\psi \varphi } = {\psi \varphi } \), so \( {\psi \varphi } = 0 \) . But we also have that for any \( C \), and any \( f \in \operatorname{Hom}\left( {B, C}\right) ,0 = {f\varphi } = \) \( {\varphi }^{ * }\left( f\right) \Rightarrow \exists \) a unique \( \bar{f} \in \operatorname{Hom}\left( {D, C}\right) \) such that \( f = {\psi }^{ * }\left( \bar{f}\right) = \bar{f}\psi \) . This is just the definition of a cokernel.
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Yes
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Proposition 7.54 Suppose A is a pre-Abelian category with enough quasi-projectives. Then any object in \( \mathbf{A} \) has a quasiprojective resolution, which can be chosen using a choice function. If \( B,{B}^{\prime } \in \mathbf{A} \) and \( \varphi \in \operatorname{Hom}\left( {B,{B}^{\prime }}\right) \) , and if \( \left\langle {{Q}_{n},{d}_{n}}\right\rangle \) is a quasiprojective resolution of \( B \), and \( \left\langle {{Q}_{n}^{\prime },{d}_{n}^{\prime }}\right\rangle \) is a quasi-projective resolution of \( {B}^{\prime } \), then there exist fillers \( {\varphi }_{n} \in \operatorname{Hom}\left( {{Q}_{n},{Q}_{n}^{\prime }}\right) \) making\n\n\n\ncommutative. Further, if \( {\varphi }_{n}^{\prime } \in \operatorname{Hom}\left( {{Q}_{n},{Q}_{n}^{\prime }}\right) \) also serve as fillers, then \( {\varphi }_{n} \) and \( {\varphi }_{n}^{\prime } \) are homotopic, that is, there exist \( {D}_{n} \in \operatorname{Hom}\left( {{Q}_{n},{Q}_{n + 1}^{\prime }}\right) \) (with \( \left. {{D}_{-1} = 0}\right) \) such that \( {\varphi }_{n} - {\varphi }_{n}^{\prime } = {d}_{n + 1}^{\prime }{D}_{n} + {D}_{n - 1}{d}_{n} \) .
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Proof: Virtually identical to the proof of Proposition 7.8, with the letter \( Q \) replacing \( P \) ,\
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No
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Proposition 7.55 If A is a pre-Abelian category with enough quasipro-jectives, and \( F \) is a left exact contravariant functor from \( \mathbf{A} \) to an Abelian category, then \( {\pi }^{ * } \) is a natural isomorphism of \( F \) with \( Q{\mathcal{L}}^{0}F \) .
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Proof: Recall that we selected \( \pi \) to be a real live cokernel; it is a cokernel for \( {j}_{1} \), and \( {j}_{1} \) is a kernel for \( \pi \), so\n\n\[ 0 \rightarrow F\left( B\right) \overset{F\left( \pi \right) }{ \rightarrow }F\left( {Q}_{0}\right) \overset{F\left( {j}_{1}\right) }{ \rightarrow }F\left( {K}_{1}\right) \]\n\nis exact in the (target) Abelian category. Similarly,\n\n\[ 0 \rightarrow F\left( {K}_{1}\right) \overset{F\left( {p}_{1}\right) }{ \rightarrow }F\left( {Q}_{1}\right) \overset{F\left( {j}_{2}\right) }{ \rightarrow }F\left( {K}_{2}\right) \]\n\nis exact, so \( F\left( {p}_{1}\right) \) is monic. Hence, \( F\left( {j}_{1}\right) \) and \( F\left( {p}_{1}\right) F\left( {j}_{1}\right) = F\left( {{j}_{1}{p}_{1}}\right) = \) \( F\left( {d}_{1}\right) \) have the same kernel, namely \( F\left( B\right) \) . This is the zeroth homology. \( ▱ \)
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Yes
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Corollary 7.56 Suppose \( \mathbf{A} \) is a balanced pre-Abelian category and \( {\mathbf{A}}^{\prime } \) is Abelian. Suppose \( F : \mathbf{A} \rightarrow {\mathbf{A}}^{\prime } \) is a functor.\n\na) If \( F \) is contravariant and left exact, and \( \mathbf{A} \) has enough projectives, then \( {\mathcal{L}}^{0}F \approx F.
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Proof: (b), (c), and (d) follow from (a) by making substitutions of opposite categories for \( \mathbf{A} \) and/or \( {\mathbf{A}}^{\prime } \) .\n\nFor (a), note that \( \mathbf{A} \) satisfies Ab-epic by Proposition 7.17. But this means that projective \( = \) quasiprojective, and epimorphism \( = \) cokernel \( = \) cokernel \( {}^{ \bullet } \) (Proposition 7.14(b)). Hence, \( {\mathcal{L}}^{0}F \approx Q{\mathcal{L}}^{0}F \approx F.
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No
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Corollary 7.58 Suppose A is a pre-Abelian category with enough quasipro-jectives. Suppose \( j : B \rightarrow {B}^{\prime } \) is a kernel for the cokernel \( p : {B}^{\prime } \rightarrow {B}^{\prime \prime } \) . Given quasiprojective resolutions of \( B \) and \( {B}^{\prime \prime } \) : there exist morphisms \( {\pi }^{\prime } : {Q}_{0} \oplus {Q}_{0}^{\prime \prime } \rightarrow {B}^{\prime } \) and \( {d}_{n}^{\prime } : {Q}_{n} \oplus {Q}_{n}^{\prime \prime } \rightarrow {Q}_{n - 1} \oplus {Q}_{n - 1}^{\prime \prime } \) such that is commutative and consists (horizontally) of quasiprojective resolutions. (The vertical morphisms are the obvious ones.)
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Proof: Virtually identical to the proof of Proposition 7.21. The \
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No
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Corollary 7.59 Suppose A is a pre-Abelian category with enough quasi-projectives. Suppose \( j : B \rightarrow {B}^{\prime } \) is a kernel for the cokernel \( p : {B}^{\prime } \rightarrow {B}^{\prime \prime } \) . Then for any additive contravariant functor \( F \) on \( \mathbf{A} \) with values in an Abelian category, there is a long exact sequence
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Proof: Apply Theorem 7.48 to
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No
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Corollary 7.60 Suppose \( F \) is an additive contravariant functor from a pre-Abelian category \( \mathbf{A} \) with enough quasiprojectives to an Abelian category \( {\mathbf{A}}^{\prime } \) . Then \( Q{\mathcal{L}}^{0}F \) is left exact, and \( Q{\mathcal{L}}^{n}F \) is half exact for all \( n \) .
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Proof: Read it off the diagram in Corollary 7.59, with connecting homomorphisms deleted.
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No
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Corollary 7.61 Suppose \( F \) is an additive contravariant functor from a pre-Abelian category \( \mathbf{A} \) with enough quasiprojectives to an Abelian category \( {\mathbf{A}}^{\prime } \). Then \( Q{\mathcal{L}}^{0}F \approx F \) if and only if \( F \) is left exact.
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Proof: Corollary 7.60 plus Proposition 7.55.
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No
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Corollary 7.62 If A is a pre-Abelian category with enough quasiprojec-tives, then every cokernel \( {}^{ \bullet } \) is a cokernel.
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Proof: Suppose A is pre-Abelian with enough quasiprojectives. Suppose \( p : {B}^{\prime } \rightarrow {B}^{\prime \prime } \) is a cokernel \( {}^{ \bullet } \), and suppose \( j : B \rightarrow {B}^{\prime } \) is a kernel for \( p \) . We shall show that \( p \) is a cokernel by appealing to Proposition 7.50.\n\nSuppose \( F = \operatorname{Hom}\left( {\bullet, C}\right), C \) an object in \( \mathbf{A}.F \) is left exact, so \( F \) is naturally isomorphic to \( Q{\mathcal{L}}^{0}F \) by Proposition 7.55. Now \( 0 \rightarrow Q{\mathcal{L}}^{0}F\left( {B}^{\prime \prime }\right) \rightarrow \) \( Q{\mathcal{L}}^{0}F\left( {B}^{\prime }\right) \rightarrow Q{\mathcal{L}}^{0}F\left( B\right) \) is exact by Corollary 7.59, so \( 0 \rightarrow F\left( {B}^{\prime \prime }\right) \rightarrow \) \( F\left( {B}^{\prime }\right) \rightarrow F\left( B\right) \) is exact, thus (letting \( C \) vary), \( {B}^{\prime } \rightarrow {B}^{\prime \prime } \) is a cokernel for \( B \rightarrow {B}^{\prime } \) by Proposition 7.50.
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Yes
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Proposition 8.2 Suppose \( \mathcal{I} \) is directed, and suppose \( \mathcal{J} \) is cofinal in \( \mathcal{I} \) . Then \( \mathcal{J} \) is directed, and for any limiting system \( \left\langle {{A}_{i},{\phi }_{ij}}\right\rangle \) on \( \mathcal{I} \) ,\n\n\[ \mathop{\lim }\limits_{\mathcal{I}}{A}_{i} = \mathop{\lim }\limits_{\mathcal{J}}{A}_{i} \]\n\nthat is, if either exists, then it is a model for the other.
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Proof: First, \( \mathcal{J} \) is directed: If \( i, j \in \mathcal{J} \), then \( \exists k \in \mathcal{I} \) with \( k \geq i, k \geq j \) , since \( \mathcal{I} \) is directed. But \( \exists l \in \mathcal{J} \) with \( l \geq k \), since \( \mathcal{J} \) is cofinal. But now \( l \geq i, l \geq j \) .\n\nNext, any \( \mathcal{J} \) -target extends uniquely to an \( \mathcal{I} \) -target. Suppose we are given \( L \in \operatorname{objC} \), and \( {\psi }_{j} \in \operatorname{Mor}\left( {L,{A}_{j}}\right) \) whenever \( j \in \mathcal{J} \), with \( {\psi }_{j} = {\phi }_{jk}{\psi }_{k} \) when \( j < k \) in \( \mathcal{J} \) ; that is, suppose \( \left\langle {L,{\psi }_{j}}\right\rangle \) is a \( \mathcal{J} \) -target. Suppose \( i \in \mathcal{I} \) . Define \( {\psi }_{i} \) as \( {\phi }_{ij}{\psi }_{j} \) for any \( j \in \mathcal{J}, j \geq i \) . (This is forced.) The first claim is that this definition is independent of the choice of \( j \) . (This is the point where \
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Yes
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Corollary 8.3 Suppose \( \mathcal{I} \) is directed, and suppose \( \mathcal{J} \) is cofinal in \( \mathcal{I} \) . Then \( \mathcal{J} \) is directed, and for any colimiting system \( \left\langle {{A}_{i},{\phi }_{ij}}\right\rangle \) on \( \mathcal{I} \) ,\n\n\[ \n{\operatorname{colim}}_{\mathcal{I}}{A}_{i} = {\operatorname{colim}}_{\mathcal{J}}{A}_{i} \n\]\n\nthat is, if either exists, then it is a model for the other.
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Proof: Proposition 8.2 in \( {\mathbf{C}}^{\text{op }} \) .
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No
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Corollary 8.4 Suppose \( \mathcal{I} \) has a largest element \( {i}_{0} \) . Then \( \mathop{\lim }\limits_{\mathcal{I}}{A}_{i} = {A}_{{i}_{0}} \) .
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Proof: \( \mathcal{I} \) is directed, since if \( i, j \in \mathcal{I} \), then \( {i}_{0} \geq i \) and \( {i}_{0} \geq j \) . Set \( \mathcal{J} = \left\{ {i}_{0}\right\} \) ; this \( \mathcal{J} \) is cofinal.
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Yes
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Proposition 8.6 Suppose \( \mathcal{I} \) is directed, and suppose \( \left\langle {{A}_{i},{\phi }_{ij}}\right\rangle \) is a colimit-ing system on \( \mathcal{I} \) in \( {}_{R}\mathbf{M} \) . Form \( C = {\operatorname{colim}}_{\mathcal{I}}{A}_{i} \) as in the proof of Proposition 8.5, as \( \left( {{\bigoplus }_{\mathcal{I}}{A}_{i}}\right) /B \) . Suppose \( \left\langle {a}_{i}\right\rangle \in {\bigoplus }_{\mathcal{I}}{A}_{i} \) . Set \( S\left( \left\langle {a}_{i}\right\rangle \right) = \left\{ {i : {a}_{i} \neq 0}\right\} \) . Then \( \left\langle {a}_{i}\right\rangle \in B \) if and only if \( \exists k \in \mathcal{I} \) such that \( k \geq i \) for all \( i \in S\left( \left\langle {a}_{i}\right\rangle \right) \) , and \( {\sum }_{i \in S\left( \left\langle {a}_{i}\right\rangle \right) }{\phi }_{ik}\left( {a}_{i}\right) = 0 \) in \( {A}_{k} \) .
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Proof: The proof is most easily completed by making a series of observations. For this purpose, given \( \left\langle {a}_{i}\right\rangle \in {\bigoplus }_{\mathcal{I}}{A}_{i} \), say that \( \left\langle {a}_{i}\right\rangle \) has property \( P \) if \( \exists k \geq i \) for all \( i \in S\left( \left\langle {a}_{i}\right\rangle \right) \) such that \( {\sum }_{i \in S\left( \left\langle {a}_{\imath }\right\rangle \right) }{\phi }_{ik}\left( {a}_{i}\right) = 0 \) in \( {A}_{k} \) .\n\n1. Every \( {\delta }_{jk}\left( x\right) \) has property \( P \) .\n\nThis is because \( S\left( \left\langle {a}_{i}\right\rangle \right) \subset \{ j, k\} \), and one may use \( k \) for the top index: \( {\phi }_{jk}\left( {a}_{j}\right) + {\phi }_{kk}\left( {-{\phi }_{jk}\left( {a}_{j}\right) }\right) = 0. \)\n\n2. The set of \( \left\langle {a}_{i}\right\rangle \) satisfying property \( P \) forms a submodule of \( {\bigoplus }_{\mathcal{I}}{A}_{i} \).\n\nThis is where \
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No
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Proposition 8.7 Suppose \( F \) is a left adjoint functor from a category \( \widehat{\mathbf{C}} \) to a category \( \mathbf{C} \) . Suppose that \( \mathcal{I} \) is partially ordered, and \( \left\langle {{\widehat{B}}_{i},{\widehat{\phi }}_{ij}}\right\rangle \) is a colim-iting system in \( \widehat{\mathbf{C}} \) on \( \mathcal{I} \) which has a colimit \( \widehat{B} = {\operatorname{colim}}_{\mathcal{I}}{\widehat{B}}_{i} \), with morphisms \( {\widehat{\psi }}_{i} \in \operatorname{Mor}\left( {{\widehat{B}}_{i},\widehat{B}}\right) \) . Then \( F\left( \widehat{B}\right) \) (with the morphisms \( F\left( {\widehat{\psi }}_{i}\right) \) ) constitute a colimit for \( \left\langle {F\left( {\widehat{B}}_{i}\right), F\left( {\widehat{\phi }}_{ij}\right) }\right\rangle \), that is,\n\n\[ F\left( {{\operatorname{colim}}_{\mathcal{I}}{\widehat{B}}_{i}}\right) = {\operatorname{colim}}_{\mathcal{I}}F\left( {\widehat{B}}_{i}\right) \]
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Proof: We have commutative diagrams\n\n\n\nso that given commutative diagrams\n\n\n\nall we have to do is find a unique filler \( \Phi \) for all diagrams\n\n\n\nTo do this, note that (including \( \widehat{F} \) and \( \sigma \) in our data) we have from (iii a) a commutative diagram\n\n\[ \begin{array}{l} {\operatorname{Mor}}_{\mathbf{C}}\left( {F\left( {\widehat{B}}_{j}\right), C}\right) \xrightarrow[]{{\sigma }_{{\widehat{B}}_{j}, C}^{-1}}{\operatorname{Mor}}_{\widehat{\mathbf{C}}}\left( {{\widehat{B}}_{j},\widehat{F}\left( C\right) }\right) \\ {\left. \begin{matrix} F{\left( {\widehat{\phi }}_{\imath j}\right) }^{ * }\left| {\;{\sigma }_{{\widehat{B}}_{\imath }j}^{-1}}\right. \end{matrix}\right| }_{0}^{{\sigma }_{{\widehat{B}}_{\imath }, C}} \\ {\operatorname{Mor}}_{\mathbf{C}}\left( {F\left( {\widehat{B}}_{i}\right), C}\right) \xrightarrow[]{{\sigma }_{{\widehat{B}}_{\imath }, C}^{-1}}{\operatorname{Mor}}_{\widehat{\mathbf{C}}}\left( {{\widehat{B}}_{i},\widehat{F}\left( C\right) }\right) \]
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Yes
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Corollary 8.8 Suppose \( R \) is a ring, and \( A \in {\mathbf{M}}_{R} \) . Then for any partially ordered set \( \mathcal{I} \) and colimiting system \( \left\langle {{B}_{i},{\phi }_{ij}}\right\rangle \) in \( {}_{R}\mathbf{M} \) , \( {\operatorname{colim}}_{\mathcal{I}}\left( {A \otimes {B}_{i}}\right) \approx \) \( A \otimes \left( {{\operatorname{colim}}_{\mathcal{I}}{B}_{i}}\right) \)
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Proof: \( A \otimes \) is a left adjoint.
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No
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Proposition 8.9 Suppose \( R \) is a ring, and suppose \( \mathcal{I} \) is a directed set. Let \( \left\langle {{A}_{i},{\phi }_{ij}}\right\rangle ,\left\langle {{A}_{i}^{\prime },{\phi }_{ij}^{\prime }}\right\rangle \), and \( \left\langle {{A}_{i}^{\prime \prime },{\phi }_{ij}^{\prime \prime }}\right\rangle \) be colimiting systems on \( \mathcal{I} \) in \( {}_{R}\mathbf{M} \) , and suppose \( \left\langle {f}_{i}\right\rangle \) is a morphism from \( \left\langle {{A}_{i},{\phi }_{ij}}\right\rangle \) to \( \left\langle {{A}_{i}^{\prime },{\phi }_{ij}^{\prime }}\right\rangle \), and \( \left\langle {g}_{i}\right\rangle \) is a morphism from \( \left\langle {{A}_{i}^{\prime },{\phi }_{ij}^{\prime }}\right\rangle \) to \( \left\langle {{A}_{i}^{\prime \prime };{\phi }_{ij}^{\prime \prime }}\right\rangle \) . Finally, suppose\n\n\[ \n{A}_{i}\overset{{f}_{i}}{ \rightarrow }{A}_{i}^{\prime }\overset{{g}_{i}}{ \rightarrow }{A}_{i}^{\prime \prime }\n\]\n\nis exact for all \( i \) . Then\n\n\[ \n{\operatorname{colim}}_{\mathcal{I}}{A}_{i}\xrightarrow[]{{\operatorname{colim}}_{\mathcal{I}}{f}_{\imath }}{\operatorname{colim}}_{\mathcal{I}}{A}_{i}^{\prime }\xrightarrow[]{{\operatorname{colim}}_{\mathcal{I}}{g}_{\imath }}{\operatorname{colim}}_{\mathcal{I}}{A}_{i}^{\prime \prime }\n\]\n\nis exact.
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Proof: We use the construction appearing in Propositions 8.5 and 8.6. To simplify matters, copy the notation, with primes applied: \( C = {\operatorname{colim}}_{\mathcal{I}}{A}_{i} = \) \( \left( {{\bigoplus }_{\mathcal{I}}{A}_{i}}\right) /B,{C}^{\prime } = {\operatorname{colim}}_{\mathcal{I}}{A}_{i}^{\prime } = \left( {{\bigoplus }_{\mathcal{I}}{A}_{i}^{\prime }}\right) /{B}^{\prime } \), and so on. Note that the previously defined morphism \( {\operatorname{colim}}_{\mathcal{I}}{f}_{i} \), for example, sends\n\n\[ \n\left\langle {a}_{i}\right\rangle + B \in \left( {{\bigoplus }_{\mathcal{I}}{A}_{i}}\right) /B\n\]\n\nto\n\n\[ \n\left\langle {{f}_{i}\left( {a}_{i}\right) }\right\rangle + {B}^{\prime } \in \left( {{\bigoplus }_{\mathcal{I}}{A}_{i}}\right) /{B}^{\prime }\n\]\n(See Exercise 7; this is really the naturality of Proposition 8.5 in the categorical context of \( \mathcal{I} \) -colimiting systems.) It is clear from this that since \( {g}_{i}{f}_{i} \) sends \( \left\langle {a}_{i}\right\rangle + B \) to \( \left\langle {{g}_{i}{f}_{i}\left( {a}_{i}\right) }\right\rangle + {B}^{\prime \prime } = \langle 0\rangle + {B}^{\prime \prime } \), one has \( \left( {{\operatorname{colim}}_{\mathcal{I}}{g}_{i}}\right) \circ \left( {{\operatorname{colim}}_{\mathcal{I}}{f}_{i}}\right) = \) 0.\n\nSuppose \( \langle {a}_{i}^{\prime }\rangle + {B}^{\prime } \in {C}^{\prime }, \) and suppose \( {\operatorname{colim}}_{\mathcal{I}}{g}_{i}\left( {\langle {a}_{i}^{\prime }\rangle + {B}^{\prime }}\right) = \langle 0\rangle + {B}^{\prime \prime }, \) that is, suppose for \( {S}^{\prime \prime }\left( \left\langle {{g}_{i}\left( {a}_{i}^{\prime }\right) }\right\rangle \right) = \left\{ {i : {g}_{i}\left( {a}_{i}^{\prime }\right) \neq 0}\right\} \) we have a \( j \geq i \) for all \( i \in {S}^{\prime \prime }\left( {\langle {g}_{i}\left( {a}_{i}^{\prime }\right) \rangle }\right) \) such that \( \sum {\phi }_{ij}^{\prime \prime }\left( {{g}_{i}\left( {a}_{i}^{\prime }\right) }\right) = 0.\dot{{S}^{\prime }\left( {\langle {a}_{i}^{\prime }\rangle }\right) } \supset {S}^{\prime \prime }\left( {\langle {g}_{i}\left( {a}_{i}^{\prime }\right) \rangle }\right) . \) Choose \( k \geq j, k \geq \) all \( i \in {S}^{\prime }\left( \left\langle {a}_{i}^{\prime }\right\rangle \right) \) (possible by induction on \( \# {S}^{\prime }\left( \left\langle {a}_{i}^{\prime }\right\rangle \right) \) , since \( \mathcal{I} \) is directed). We have that\n\n\[ \n0 = {\phi }_{jk}^{\prime \prime }\left( {\mathop{\sum }\limits_{{i \in {S}^{\prime \prime }\left( {\langle {g}_{\imath }\left( {a}_{\imath }^{\prime }\right) \rangle }\right) }}{\phi }_{ij}^{\prime \prime }({g}_{i}\left( {a}_{i}^{\prime }\right) }\)
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Yes
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Theorem 8.10 Suppose \( R \) is a ring, and suppose \( A \in {\mathbf{M}}_{R} \) . Let \( \mathcal{I} \) be a directed set, and suppose \( \left\langle {{B}_{i},{\phi }_{ij}}\right\rangle \) is a colimiting system on \( \mathcal{I} \) in \( {}_{R}\mathbf{M} \) . Then for all \( n \) ,
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Proof: Induction on \( n;n = 0 \) is Corollary 8.8. As usual, the \( n = 1 \) case requires special discussion, relevant to the induction step. Suppose\n\n\[ 0 \rightarrow K\overset{\theta }{ \rightarrow }F\overset{\pi }{ \rightarrow }A \rightarrow 0 \]\n\nis exact in \( {\mathbf{M}}_{R} \), with \( F \) flat. For each \( i \), we have exact sequences\n\n\[ 0 \rightarrow {\mathrm{{Tor}}}_{1}\left( {A,{B}_{i}}\right) \overset{{\delta }_{\imath }}{ \rightarrow }K \otimes {B}_{i}\overset{\theta \otimes {B}_{\imath }}{ \rightarrow }F \otimes {B}_{i}\overset{\pi \otimes {B}_{\imath }}{ \rightarrow }A \otimes {B}_{i} \rightarrow 0 \]\n\nfrom which we have a diagram\n\n\[ 0 \succ {\operatorname{colim}}_{\mathcal{I}}{\operatorname{Tor}}_{1}\left( {A,{B}_{i}}\right) \xrightarrow[]{{\operatorname{colim}}_{\mathcal{I}}{\delta }_{\imath }}{\operatorname{colim}}_{\mathcal{I}}\left( {K \otimes {B}_{i}}\right) \xrightarrow[]{{\operatorname{colim}}_{\mathcal{I}}\theta \otimes {B}_{\imath }}{\operatorname{colim}}_{\mathcal{I}}\left( {F \otimes {B}_{i}}\right) \]\n\n\[ 0 > {\operatorname{Tor}}_{1}\left( {A,{\operatorname{colim}}_{\mathcal{I}}{B}_{i}}\right) \overset{\delta }{ \rightarrow }K \otimes \left( {{\operatorname{colim}}_{\mathcal{I}}{B}_{i}}\right) \overset{\theta \otimes {\operatorname{colim}}_{\mathcal{I}}{B}_{i}}{ \rightarrow }F \otimes \left( {{\operatorname{colim}}_{\mathcal{I}}{B}_{i}}\right) \]\n\nwith exact rows by Proposition 8.9. Hence\n\n\[ {\operatorname{colim}}_{\mathcal{I}}{\operatorname{Tor}}_{1}\left( {A,{B}_{i}}\right) \approx \ker \left( {{\operatorname{colim}}_{\mathcal{I}}\left( {\theta \otimes {B}_{i}}\right) }\right) \]\n\n\[ \approx \ker \left( {\theta \otimes {\operatorname{colim}}_{\mathcal{I}}{B}_{i}}\right) \]\n\n\[ \approx {\operatorname{Tor}}_{1}\left( {A,{\operatorname{colim}}_{\mathcal{I}}{B}_{i}}\right) \]\n\nThe induction step is easy; for \( n \geq 1 \) ,\n\n\[ {\operatorname{colim}}_{\mathcal{I}}{\mathrm{{Tor}}}_{n + 1}\left( {A,{B}_{i}}\right) \approx {\operatorname{colim}}_{\mathcal{I}}{\mathrm{{Tor}}}_{n}\left( {K,{B}_{i}}\right) \]\n\n\[ \approx {\operatorname{Tor}}_{n}\left( {K,{\operatorname{colim}}_{\mathcal{I}}{B}_{i}}\right) \]\n\n\[ \approx {\operatorname{Tor}}_{n + 1}\left( {A,{\operatorname{colim}}_{\mathcal{I}}{B}_{i}}\right) . \]
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Yes
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Corollary 8.11 Suppose \( R \) is any ring, and suppose \( {B}_{i} \in {}_{R}\mathbf{M} \) satisfy \( F - \dim {B}_{i} \leq n \) for \( i \in \mathcal{I},\mathcal{I} \) a directed set. Suppose \( \left\langle {{B}_{i},{\phi }_{ij}}\right\rangle \) is a colimiting system on \( \mathcal{I} \) . Then \( \mathrm{F} \) -dim \( {\operatorname{colim}}_{\mathcal{I}}{B}_{i} \leq n \) .
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Proof: Apply \( {\operatorname{Tor}}_{n + 1};{\operatorname{Tor}}_{n + 1} \) of each entry in the colimit is zero.
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No
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Lemma 8.13 Suppose \( A \in {}_{R}\mathbf{M} \), with \( A = {A}_{1} \oplus {A}_{2} \), an internal direct sum. Suppose \( {B}_{1} \) is a submodule of \( {A}_{1} \), and \( {B}_{2} \) is a submodule of \( {A}_{2} \) . Suppose \( \theta : {A}_{1}/{B}_{1} \rightarrow {A}_{2}/{B}_{2} \) is a homomorphism, and suppose \( {A}_{1} \) is generated by \( \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \) . For each \( k = 1,\ldots, n \), choose \( {y}_{k} \in {A}_{2} \) such that \( \theta \left( {{x}_{k} + }\right. \) \( \left. {B}_{1}\right) = {y}_{k} + {B}_{2} \) . Let \( B \) be the submodule of \( A \) generated by \( {B}_{1},{B}_{2} \), and \( \left\{ {{x}_{1} - {y}_{1},\ldots ,{x}_{n} - {y}_{n}}\right\} \) . Let \( {\psi }_{i} : {A}_{i}/{B}_{i} \rightarrow A/B \) be the natural maps \( {\psi }_{i}(x + \) \( \left. {B}_{i}\right) = x + B \) . Then \( {\psi }_{2} \) is an isomorphism, and the triangle\n\n\n\nis commutative.
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Proof: Note that \( {\psi }_{2}\theta \left( {{x}_{j} + {B}_{1}}\right) = {\psi }_{2}\left( {{y}_{j} + {B}_{2}}\right) = {y}_{j} + B = {x}_{j} + B = \) \( {\psi }_{1}\left( {{x}_{j} + {B}_{1}}\right) \), since all \( {x}_{j} - {y}_{j} \in B \) . Since \( {\psi }_{2}\theta \) and \( {\psi }_{1} \) agree on a set of generators of \( {A}_{1}/{B}_{1},{\psi }_{2}\theta = {\psi }_{1} \) .\n\nIf \( y \in {A}_{2} \), then \( y + B \) is in the image of \( {\psi }_{2} \) . If \( x \in {A}_{1} \), then \( x + B \in \) \( \mathrm{{im}}{\psi }_{1} = \mathrm{{im}}{\psi }_{2}\theta \subset \mathrm{{im}}{\psi }_{2} \) . Combining, all of \( A/B = \left( {{A}_{1} \oplus {A}_{2}}\right) /B \) is in the image of \( {\psi }_{2} \), so \( {\psi }_{2} \) is onto.\n\nFinally, \( {\psi }_{2}\left( {y + {B}_{2}}\right) = y + B \) for \( y \in {A}_{2} \), so \( {\psi }_{2}\left( {y + {B}_{2}}\right) = 0 + B \Leftrightarrow y \in \)\n\n\( {A}_{2} \cap B \) . If \( y \in {A}_{2} \cap B \), then \( y = {b}_{1} + {b}_{2} + \mathop{\sum }\limits_{{j = 1}}^{n}{r}_{j}\left( {{x}_{j} - {y}_{j}}\right) \), with \( {b}_{1} \in {B}_{1} \) and\n\n\( {b}_{2} \in {B}_{2} \) . But then \( y = \left( {{b}_{1} + \sum {r}_{j}{x}_{j}}\right) + \left( {{b}_{2} - \sum {r}_{j}{y}_{j}}\right) \), so that since \( y \in {A}_{2} \) and \( {A}_{1} \cap {A}_{2} = 0,{b}_{1} + \sum {r}_{j}{x}_{j} = 0 \) . This means that \( \sum {r}_{j}{x}_{j} = - {b}_{1} \in {B}_{1} \), so that \( \sum {r}_{j}{y}_{j} + {B}_{2} = \sum {r}_{j}\theta \left( {{x}_{j} + {B}_{1}}\right) = \theta \left( {\sum {r}_{j}{x}_{j} + {B}_{1}}\right) = \theta \left( 0\right) = 0 \), so \( \sum {r}_{j}{y}_{j} \in {B}_{2} \) as well. Hence \( y = {b}_{2} - \sum {r}_{j}{y}_{j} \in {B}_{2} \), and \( y + {B}_{2} = 0 + {B}_{2} \) . This says that \( {\psi }_{2} \) is one-to-one.
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Yes
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Proposition 8.14 (Lazard) Suppose \( E \in {}_{R}\mathbf{M} \) . There exists a \( \left( {C, D}\right) \) - subquotient system with the following properties (index set \( \mathcal{I} \) and all \( {A}_{i},{B}_{i} \) , \( {\psi }_{i},{\phi }_{ij} \) as in Proposition 8.12):\n\na) \( C/D \approx E \) .\n\nb) \( {A}_{i}/{B}_{i} \) is finitely presented for all \( i \in \mathcal{I} \) .\n\nc) If \( F \) is finitely presented, and if \( \eta : F \rightarrow C/D \) is a homomorphism, then \( \exists i \in \mathcal{I} \) and an isomorphism \( {\eta }^{\prime } : F \rightarrow {A}_{i}/{B}_{i} \) such that the triangle\n\n\n\ncommutes.\n\nd) If \( i \in \mathcal{I} \), and if \( {\psi }_{i} = {\rho \sigma } \), where \( \sigma : {A}_{i}/{B}_{i} \rightarrow F \) and \( \rho : F \rightarrow C/D \) with \( F \) finitely presented, then \( \exists j \in \mathcal{I}, j \geq i \), and an isomorphism \( \tau : F \rightarrow {A}_{j}/{B}_{j} \) such that the diagram\n\n\n\ncommutes.\n\nRemark: (d) is the crucial point. This colimiting system is universal for finitely presented modules and their homomorphisms.
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Proof: Let \( C \) be the free module on the set \( E \times \mathbb{N},\mathbb{N} = \) natural numbers. Define a map \( \pi \) from \( C \) onto \( E \) by sending each \( \left( {e, n}\right) \in E \times \mathbb{N} \) to \( e \) . Let \( D \) be the kernel of this map, and let \( \mathcal{A} \) be the family of submodules of \( C \) generated by finite subsets of \( E \times \mathbb{N} \) (hence, free on finite sets) and let \( \mathcal{B} \) be the family of finitely generated submodules of \( D \) . With this setup, we have a \( \left( {C, D}\right) \) -subquotient system, and \( C/D \approx E \), while each \( {A}_{i}/{B}_{i} \) is finitely presented (since each \( {A}_{i} \) is free on a finite set and each \( {B}_{i} \) is finitely generated). There remain the universality properties (c) and (d).\n\nFor (c), note that \( \left( {0,0}\right) \in \mathcal{I} \), so the zero module appears as an \( {A}_{i}/{B}_{i} \) . But then (d) \( \Rightarrow \) (c), by taking (given \( F \) and \( \eta \) ) \( {A}_{i}/{B}_{i} = 0,\sigma = 0,\rho = \eta \) , and setting \( {\eta }^{\prime } = \tau \) . Hence, we focus on (d). Let \( \sigma, F \), and \( \rho \) be as stated in (d).\n\nChoose \( N \geq \) all \( {n}_{k} \), where \( {A}_{i} \) is generated by \( \left\{ {\left( {{e}_{1},{n}_{1}}\right) ,\ldots \left( {{e}_{l},{n}_{l}}\right) }\right\} \) . Since \( F \) is finitely presented, it is generated by \( \left\{ {{x}_{1},\ldots ,{x}_{m}}\right\} \), where the kernel of the homomorphism from the free module on \( \left\{ {{x}_{1},\ldots ,{x}_{m}}\right\} \) to \( F \) is finitely generated. For each \( j = 1,\ldots, m \), choose \( {y}_{j} \in C \) such that \( \rho \left( {x}_{j}\right) = {y}_{j} + D \), and let \( A \) be the member of \( \mathcal{A} \) which is free on \( \left\{ \left( {\pi \left( {y}_{1}\right), N + }\right. \right. \) \( 1),\ldots ,\left( {\pi \left( {y}_{m}\right), N + m}\right) \} \) . Define \( \alpha : A \rightarrow F \) by setting \( \alpha \left( \left( {\pi \left( {y}_{j}\right), N + j}\right) \right) = \) \( {x}_{j};\alpha \) is onto by construction, and its kernel \( B \) will be finitely generated. Note that \( \pi \left( \left( {\pi \left( {y}_{j}\right), N + j}\right) \right) = \pi \left( {y}_{j}\right) \), so that \( \left( {\pi \left( {y}_{j}\right), N + j}\right) + D = {y}_{j} + D \) . But then \( {\rho \alpha }\left( \left( {\pi \left( {y}_{j}\right), N + j}\right) \right) = \rho \left( {x}_{j}\right) = {y}_{j} + D = \left( {\pi \left( {y}_{j}\right), N + j}\right) + D \) . Thus, \( {\rho \alpha }\left( x\right) = x + D \) for all \( x \in A \), since this equation holds for generators of \( A \) . If \( x \in B \)
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Yes
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Theorem 8.16 (Lazard’s Theorem) Suppose \( R \) is a ring, and \( E \in {}_{R}\mathbf{M} \) . The following are equivalent:\n\ni) \( E \) is flat.\n\nii) For all finitely presented \( F,{F}^{ * } \otimes E \rightarrow \operatorname{Hom}\left( {F, E}\right) \) is an isomorphism.\n\niii) For all finitely presented \( F,{F}^{ * } \otimes E \rightarrow \operatorname{Hom}\left( {F, E}\right) \) is onto.\n\niv) \( E \) is a directed colimit of finitely generated free modules.
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Proof: (iv) \( \Rightarrow \) (i) follows from Corollary 8.11, since free modules are flat. (i) \( \Rightarrow \) (ii) is Proposition 4.18. (ii) \( \Rightarrow \) (iii) is trivial. Finally,(iii) \( \Rightarrow \) (iv): Assume (iii). Let \( \mathcal{I},{A}_{i},{B}_{i},{\psi }_{i},{\phi }_{ij} \) be as in Proposition 8.14. Set\n\n\[ \mathcal{J} = \left\{ {j \in \mathcal{I} : {A}_{j}/{B}_{j}\text{ is free }}\right\} \]\n\nSuppose \( i \in \mathcal{I} \) . Then \( {\psi }_{i} \in \operatorname{Hom}\left( {{A}_{i}/{B}_{i}, C/D}\right) \) lies in the image of \( {\left( {A}_{i}/{B}_{i}\right) }^{ * } \otimes \) \( \left( {C/D}\right) \), so \( \exists {\sigma }_{1},\ldots ,{\sigma }_{n} \in \operatorname{Hom}\left( {{A}_{i}/{B}_{i}, R}\right) \), and \( {\bar{y}}_{1},\ldots ,{\bar{y}}_{n} \in C/D \), with \( {\psi }_{i}\left( \bar{x}\right) = \sum {\sigma }_{k}\left( \bar{x}\right) {\bar{y}}_{k} \) . Define \( \sigma = \left( {{\sigma }_{1},\ldots ,{\sigma }_{n}}\right) : {A}_{i}/{B}_{i} \rightarrow {R}^{n} \), and \( \rho : {R}^{n} \rightarrow \) \( C/D \) by \( \rho \left( {{r}_{1},\ldots ,{r}_{n}}\right) = \sum {r}_{k}{\bar{y}}_{k} \) . Then \( {\psi }_{i} = {\rho \sigma } \), so by Proposition 8.14, \( \exists j \in \mathcal{I}, j \geq i \), and an isomorphism \( \tau : {R}^{n} \rightarrow {A}_{j}/{B}_{j} \), for which the diagram\n\n\n\nis commutative. But observe: \( j \in \mathcal{J} \), since \( {A}_{j}/{B}_{j} \approx {R}^{n} \) is free. This just says that \( \mathcal{J} \) is cofinal in \( \mathcal{I} \) . Hence, by Corollary 8.3 (and Proposition 8.12), \( \mathcal{J} \) is directed, and\n\n\[ E \approx C/D \approx {\operatorname{colim}}_{\mathcal{I}}{A}_{i}/{B}_{i} \approx {\operatorname{colim}}_{\mathcal{J}}{A}_{j}/{B}_{j}. \]
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Yes
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Proposition 8.17 Suppose \( R \) and \( S \) are rings, and suppose \( A \in {}_{R}{\mathbf{M}}_{S} \) . Then for all flat \( B \in {}_{S}\mathbf{M},\mathrm{F} - {\dim }_{R}A{ \otimes }_{S}B \leq \mathrm{F} - {\dim }_{R}A \) .
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Proof: Write \( B \approx {\operatorname{colim}}_{\mathcal{I}}{B}_{i} \), where each \( {B}_{i} \) is free and finitely generated. Lazard’s theorem says that we can do this. Then \( A{ \otimes }_{S}B \approx A{ \otimes }_{S} \) \( \left( {{\operatorname{colim}}_{\mathcal{I}}{B}_{i}}\right) \approx {\operatorname{colim}}_{\mathcal{I}}\left( {A{ \otimes }_{S}{B}_{i}}\right) \n\nInterlude: There is a technical point here. \( A{ \otimes }_{S} \) preserves colimits as a functor from \( {}_{S}\mathbf{M} \) to \( {}_{R}\mathbf{M} \) (Note: It is \( {}_{R}\mathbf{M} \) here, not Ab.) by Proposition 8.7, since \( A{ \otimes }_{S} \) is a left adjoint; the right adjoint functor from \( {}_{R}\mathbf{M} \) to \( {}_{S}\mathbf{M} \) is \( C \mapsto {\operatorname{Hom}}_{R}\left( {A, C}\right) \) by Chapter 2, Exercise 15, the generalization to our situation of Theorem 2.4.\n\nNow, each \( A{ \otimes }_{S}{B}_{i} \) is a direct sum of a finite number of copies of \( A \), so F- \( {\dim }_{R}A{ \otimes }_{S}{B}_{i} = \) F- \( {\dim }_{R}A \) (or 0 if \( {B}_{i} = 0 \) ). Hence, F- \( {\dim }_{R}A{ \otimes }_{S}B = \) \( {\mathrm{F\text{-}{dim}}}_{R}\left( {{\mathrm{{colim}}}_{\mathcal{I}}\left( {A{ \otimes }_{S}{B}_{i}}\right) }\right) \leq {\mathrm{F\text{-}{dim}}}_{R}A \) by Corollary 8.11.
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Yes
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Corollary 8.18 Suppose \( R \) and \( S \) are rings, and suppose \( A \in {}_{R}{\mathbf{M}}_{S} \) and \( B \in {}_{S}\mathbf{M} \) . Suppose \( A \) is flat as a left \( R \) -module, and suppose \( B \) is flat as a left \( S \) -module. Then \( A{ \otimes }_{S}B \) is flat as a left \( R \) -module.
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Proof: F-dim \( A = 0 \) in Proposition 8.17.
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No
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Proposition 8.19 Suppose \( R \) and \( S \) are rings, and suppose \( F : {}_{S}\mathbf{M} \rightarrow \) \( {}_{R}\mathbf{M} \) is an exact, strongly additive covariant functor. Then \( \forall B \in {}_{S}\mathbf{M} \) , \[ \mathrm{F} - {\dim }_{R}F\left( B\right) \leq \mathrm{F} - {\dim }_{S}B + \mathrm{F} - {\dim }_{R}F\left( S\right) . \]
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Proof: Replace \( F \) with \( A{ \otimes }_{S} \), where \( A \in {}_{R}{\mathbf{M}}_{S} \) and \( A \) is flat as a right \( S \) - module; this is possible by Watts’ theorem, where \( A = F\left( S\right) \) . If \( \mathrm{F} - {\dim }_{S}B = \) \( \infty \), there is nothing to prove, so assume \( \mathrm{F} - \dim {}_{S}B < \infty \) . There is a flat resolution of \( B \) : \[ 0 \rightarrow {D}_{n} \rightarrow {D}_{n - 1} \rightarrow \cdots \rightarrow {D}_{1} \rightarrow {D}_{0} \rightarrow B \rightarrow 0 \] which, after applying \( A{ \otimes }_{S} \), yields \[ 0 \rightarrow A{ \otimes }_{S}{D}_{n} \rightarrow A{ \otimes }_{S}{D}_{n - 1} \rightarrow \cdots \rightarrow A{ \otimes }_{S}{D}_{1} \rightarrow A{ \otimes }_{S}{D}_{0} \rightarrow A{ \otimes }_{S}B \rightarrow 0 \] since \( A \) is flat as a right \( S \) -module. But \( \mathrm{F} - {\dim }_{R}\left( {A{ \otimes }_{S}{D}_{k}}\right) \leq \mathrm{F} - {\dim }_{R}A = \) \( \mathrm{F} - {\dim }_{R}F\left( S\right) \) by Proposition 8.17, so \( \mathrm{F} - {\dim }_{R}\left( {A{ \otimes }_{S}B}\right) \leq n + \mathrm{F} - {\dim }_{R}F\left( S\right) \) by Corollary 4.3.
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Yes
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Corollary 8.20 Suppose \( R \) is a commutative ring and \( S \) is an admissible multiplicative subset of \( R \) . \n\n\[ \n\\text{a) For all}B \\in {}_{R}\\mathbf{M},\\mathrm{F} - {\\dim }_{S^{-1}R}S^{-1}B \\leq \\mathrm{F} - {\\dim }_{R}B\\text{.}\n\]
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Proof: For (a), let \( F\\left( B\\right) = S^{-1}B \\approx S^{-1}R \\otimes _{R}B \) . Then F-dim \( {}_{S^{-1}R}F\\left( R\\right) = \) \( F - {\\dim }_{S^{-1}R}S^{-1}R = 0 \), so \( F - {\\dim }_{S^{-1}R}S^{-1}B \\leq F - {\\dim }_{R}B \) by Proposition 8.19.
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Yes
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Lemma 9.1 Suppose \( C \) is a left \( R \) -module, and suppose \( B \) is a submodule of \( C \) . Then the set of essential extensions of \( B \) within \( C \) has a maximal element.
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Proof: Partially order the set of essential extensions of \( B \) within \( C \) by set inclusion. This set is nonempty since \( B \) is an essential extension of itself. To complete the proof, we need only verify that the hypotheses of Zorn's lemma are satisfied. Let \( \mathbf{C} \) be a nonempty chain (under set inclusion) of essential extensions of \( B \), and let \( {C}_{0} \) be the union of the members of \( \mathbf{C}.{C}_{0} \) is a submodule of \( C \), since \( \mathbf{C} \) is a chain, and \( {C}_{0} \) is an extension of \( B \) within \( C \) . It remains to show that \( {C}_{0} \) is an essential extension of \( B \) .\n\nSuppose \( A \) is a nonzero submodule of \( {C}_{0} \), and choose any \( a \in A, a \neq 0 \) . Choose a \( D \in \mathbf{C} \) with \( a \in D \) . Then \( {Ra} \subset D \), so \( {Ra} \cap B \neq 0 \), since \( D \) is an essential extension of \( B \) . But now \( 0 \neq {Ra} \cap B \subset A \cap B \) .
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Yes
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Lemma 9.2 Suppose \( C \) is a left \( R \) -module, and suppose \( B \) is a submodule of \( C \) . Then there is a submodule \( D \) of \( C \) which is maximal with respect to the property that \( D \cap B = 0 \) .
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Proof: Partially order the set of submodules of \( C \) having trivial intersection with \( B \) by set inclusion. This set is nonempty since it includes the zero submodule. The union of a nonempty chain (under set inclusion) of submodules of \( C \) having trivial intersection with \( B \), yields a submodule having trivial intersection with \( B \), so every chain has an upper bound. By Zorn’s lemma, there is a maximal submodule \( D \) with respect to the property that \( D \cap B = 0 \) .
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Yes
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Lemma 9.3 Suppose \( C \) is a left \( R \) -module, and suppose \( B \) is a submodule of \( C \) . Let \( D \) be any submodule of \( C \) that is maximal with respect to the property that \( D \cap B = 0 \) . Let \( \pi : C \rightarrow C/D \) denote the canonical surjection. Then \( \pi \) yields an isomorphism of \( B \) with \( \pi \left( B\right) \), and \( C/D \) is an essential extension of \( \pi \left( B\right) \) .
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Proof: The kernel of \( \pi \) is \( D \), so the kernel of \( \pi \mid B \) is \( D \cap B = 0 \) . Hence, \( \pi \) is one-to-one on \( B \) and yields an isomorphism of \( B \) with \( \pi \left( B\right) \) . Suppose \( {D}^{\prime }/D \) is any nonzero submodule of \( C/D \) . Then \( {D}^{\prime } \cap B \neq 0 \) , since \( D \) is maximal with respect to having trivial intersection with \( B \) . But if \( 0 \neq x \in {D}^{\prime } \cap B \), then \( \pi \left( x\right) = x + D \neq 0 \), since \( \pi \) is one-to-one on \( B \) , while \( \pi \left( x\right) = x + D \in \left( {{D}^{\prime }/D}\right) \cap \pi \left( B\right) \) . Hence, \( 0 \neq \pi \left( B\right) \cap \left( {{D}^{\prime }/D}\right) \) . Since all submodules of \( C/D \) have the form \( {D}^{\prime }/D, C/D \) is an essential extension of \( \pi \left( B\right) = \left( {B + D}\right) /D \) .
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Yes
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Proposition 9.4 Suppose \( E \in {}_{R}\mathbf{M} \) . Then \( E \) is injective if and only if \( E \) has no nontrivial essential extensions.
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Proof: The proof is based on the result from Chapter 2 that injectives are absolute direct summands, and vice versa. First, suppose \( E \) is injective, and suppose \( E \) is a submodule of \( C \) . Then \( E \) is a direct summand of \( C \), since injectives are absolute direct summands. If \( C = E \oplus F \), then \( F \cap E = 0 \) . If \( C \) is also an essential extension of \( E \), then this implies that \( F = 0 \), that is, \( C = E \) . That is, \( C \) cannot be an essential extension of \( E \) unless \( C = E \) .\n\nNow suppose \( E \) has no nontrivial essential extensions. This property can be restated as follows. Suppose \( B \) is a submodule of \( C \) ; make a temporary definition that \( B \) is inessential in \( C \) when the only essential extension of \( B \) in \( C \) is \( B \) itself. Then \
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No
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Lemma 9.5 Suppose \( B \in {}_{R}\mathbf{M} \), and suppose \( B \) is a submodule of both \( C \) and \( E \), where otherwise \( C \) and \( E \) are unrelated. Suppose \( C \) is an essential extension of \( B \), and \( E \) is injective. Then \( E \) contains an isomorphic copy of \( C \) .
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Proof: Define \( \sigma : C \rightarrow E \) as any filler:\n\n\n\ndefined via injectivity of \( E \) . Then \( \sigma \left( b\right) = b \) for \( b \in B \), so \( \ker \sigma \cap B = 0 \) . Since \( C \) is an essential extension of \( B \) , \( \ker \sigma = 0 \) . Thus, \( \sigma \left( C\right) \) is an isomorphic copy of \( C \) inside \( E \) .
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Yes
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Proposition 9.7 Suppose \( E \) is an injective left \( R \) -module, and \( B \) is a submodule of \( E \) . Let \( C \) be any maximal essential extension of \( B \) in \( E \) . Then \( C \) is an injective envelope of \( B \) .
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Proof: First, observe that \( C \) has no nontrivial essential extensions \( {C}^{\prime } \) in \( E \), since \( {C}^{\prime } \) would then be an essential extension of \( B \) (contradicting maximality): \( \;0 \neq A \subset {C}^{\prime } \Rightarrow 0 \neq A \cap C \Rightarrow 0 \neq \left( {A \cap C}\right) \cap B = A \cap B. \n\nNow if \( D \) is any essential extension of \( C \) at all, then by Lemma 9.5, \( E \) contains an isomorphic copy \( {D}^{\prime } \) of \( D \), from which \( {D}^{\prime } = C \) and then \( D = C \) . That is, \( C \) has no nontrivial essential extensions, so \( C \) is injective by Proposition 9.4. Since \( C \) is by definition an essential extension of \( B, C \) is an injective envelope of \( B \) .
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Yes
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Corollary 9.8 Any \( B \in {}_{R}\mathbf{M} \) has an injective envelope.
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Proof: In Proposition 9.7, \( E \) and \( C \) exist by the enough injectives theorem and Lemma 9.1.
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No
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Theorem 9.9 Suppose \( B \in {}_{R}\mathbf{M} \) . Then\n\na) \( B \) has an injective envelope, and any two injective envelopes of \( B \) are isomorphic.\n\nb) If \( E\left( B\right) \) (respectively, \( E\left( C\right) \) ) is an injective envelope of \( B \) (respectively, \( C \) ), then any \( \sigma \in \operatorname{Hom}\left( {B, C}\right) \) has an extension \( \tau \in \operatorname{Hom}(E\left( B\right) \) , \( E\left( C\right) ) \) . Furthermore, if \( \sigma \) is one-to-one or bijective, then so are all such \( \tau \) .\n\nc) Any injective envelope \( E\left( B\right) \) of \( B \) is a largest essential extension of \( B \) in that \( E\left( B\right) \) contains an isomorphic copy of any other essential extension of \( B \) .\n\nd) Any injective envelope \( E\left( B\right) \) of \( B \) is a smallest injective extension of \( B \) in that any other injective extension of \( B \) contains an isomorphic \( \operatorname{copy} \) of \( E\left( B\right) \) .
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Proof: First, the quick deductions. (c) follows directly from Lemma 9.5, as remarked following the proof. (d) follows from the uniqueness part of (a) since any injective extension of \( B \) contains, via Lemma 9.1 and Proposition 9.7, an injective envelope. Finally, (a) follows from Lemma 9.5 and the \
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No
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Proposition 9.10 Suppose \( R \) is any ring for which \( \left\langle {{P}_{i},{d}_{i}}\right\rangle \) is a chain complex in \( {}_{R}\mathbf{M} \) . Denote the homology at \( {P}_{n} \) by \( {H}_{n} \) . Suppose \( C \) is an injective left \( R \) -module. Then the homology of \( \left\langle {\operatorname{Hom}\left( {{P}_{i}, C}\right) ,{d}_{i}^{ * }}\right\rangle \) at \( \operatorname{Hom}\left( {{P}_{n}, C}\right) \) is \( \operatorname{Hom}\left( {{H}_{n}, C}\right) \) .
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Proof: Apply \( \operatorname{Hom}\left( {\bullet, C}\right) \) to the unstandard picture for \( {H}_{n} \) :\n\n\n\nin which all rows and columns are exact, since \( \operatorname{Hom}\left( {\bullet, C}\right) \) is an exact functor ( \( C \) being injective). This is just the standard picture for the homology of \( \left\langle {\operatorname{Hom}\left( {{P}_{n}, C}\right) ,{d}_{n}^{ * }}\right\rangle \), except that it’s upside down. That is,\n\n\[ \operatorname{Hom}\left( {{P}_{n}/{Z}_{n}, C}\right) = \text{ image of }{\bar{d}}_{n}^{ * } \]\n\n\[ \approx \text{image of}{d}_{n}^{ * } \]\n\nand\n\n\[ \operatorname{Hom}\left( {{P}_{n}/{B}_{n}, C}\right) \approx \text{ kernel of }{d}_{n + 1}^{ * } \]\n\nso that \( \operatorname{Hom}\left( {{H}_{n}, C}\right) \approx \) homology of \( \left\langle {\operatorname{Hom}\left( {{P}_{n}, C}\right) ,{d}_{n}^{ * }}\right\rangle \) .
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Yes
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Proposition 9.11 Suppose \( \left\langle {{P}_{i},{d}_{i}}\right\rangle \) is a chain complex of left \( R \) -modules, and suppose that for a particular \( n,{Z}_{n} \) is a direct summand of \( {P}_{n} \) . Then \( \rho : {H}^{n}\left( C\right) \rightarrow \operatorname{Hom}\left( {{H}_{n}, C}\right) \) is onto and splits.
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Proof: The point is to define a homomorphism \( \sigma : \operatorname{Hom}\left( {{H}_{n}, C}\right) \rightarrow {H}^{n}\left( C\right) \) , so that \( {\rho \sigma } \) is the identity on \( \operatorname{Hom}\left( {{H}_{n}, C}\right) \) . We will then have \( \operatorname{im}\rho \supset \operatorname{im}{\rho \sigma } = \) \( \operatorname{Hom}\left( {{H}_{n}, C}\right) \), so that \( \rho \) is onto; also \( \sigma \) will provide the splitting.\n\nTo this end, let \( \pi : {P}_{n} \rightarrow {Z}_{n} \) denote the projection associated with \( {Z}_{n} \) as a direct summand of \( {P}_{n} \) . Suppose \( f \in \operatorname{Hom}\left( {{H}_{n}, C}\right) \) . Then since \( {H}_{n} = {Z}_{n}/{B}_{n} \), we get a unique \( \bar{f} \in \operatorname{Hom}\left( {{Z}_{n}, C}\right) \) such that \( \bar{f} \) is zero on \( {B}_{n} \) , and \( \bar{f}\left( x\right) = f\left( {x + {B}_{n}}\right) \) . Set \( g = \bar{f}\pi \in \operatorname{Hom}\left( {{P}_{n}, C}\right) \) . Now \( \pi \) is the identity on \( {Z}_{n} \), so \( g \) and \( \bar{f} \) agree on \( {Z}_{n} \) . In particular, \( g \) is zero on \( {B}_{n} \), and the \( \widehat{g} \) defined in the discussion of \( \rho \) agrees with \( f \) . So: Set\n\n\[ \sigma \left( f\right) = \bar{f}\pi + \operatorname{im}{d}_{n}^{ * } \in {H}^{n}\left( C\right) \]\n\n\( \sigma \) is again a homomorphism virtually by inspection, and \( {\rho \sigma } \) is the identity on \( \operatorname{Hom}\left( {{H}_{n}, C}\right) \) .
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Yes
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Proposition 9.12 Suppose \( \\left\\langle {{P}_{i},{d}_{i}}\\right\\rangle \) is a chain complex of left \( R \) -modules, and \( C \\in {}_{R}\\mathbf{M} \) . Suppose each \( {P}_{i} \) is projective and I-dim \( C \\leq 1 \) . Then there is a naturally defined exact sequence\n\n\[ 0 \\rightarrow {\\operatorname{Ext}}_{R}^{1}\\left( {{H}_{n - 1}, C}\\right) \\rightarrow {H}^{n}\\left( C\\right) \\overset{\\rho }{ \\rightarrow }\\operatorname{Hom}\\left( {{H}_{n}, C}\\right) \\rightarrow 0. \]
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Proof: There is a short exact injective resolution of \( C \)\n\n\[ 0 \\rightarrow C\\overset{\\iota }{ \\rightarrow }{E}_{0}\\overset{\\pi }{ \\rightarrow }{E}_{1} \\rightarrow 0 \]\n\nsince I-dim \( C \\leq 1 \) (Proposition 4.8). Put these in the second factor of \( \\operatorname{Hom}\\left( {\\bullet , \\bullet }\\right) \), and the \( {P}_{i} \) in the first factor, yielding the diagram\n\n\n\nwith exact columns since each \( {P}_{n} \) is projective. The long homology exact sequence is then\n\n\[ {H}^{n}\\left( C\\right) \\rightarrow {H}^{n}\\left( {E}_{0}\\right) \\rightarrow {H}^{n}\\left( {E}_{1}\\right) \]\n\nfrom which (after applying Proposition 9.10) we extract \n\nyielding the short exact sequence\n\n\[ 0 \\rightarrow \\operatorname{Hom}\\left( {{H}_{n - 1},{E}_{1}}\\right) /\\operatorname{im}{\\left( {\\pi }_{n - 1}\\right) }_{ * } \\rightarrow {H}^{n}\\left( C\\right) \\rightarrow \\ker {\\left( {\\pi }_{n}\\right) }_{ * } \\rightarrow 0. \]\n\nHowever, by Corollary 3.12 (computation of Ext from injective resolutions):\n\n\[ \\operatorname{Hom}\\left( {{H}_{n - 1},{E}_{1}}\\right) /\\operatorname{im}{\\left( {\\pi }_{n - 1}\\right) }_{ * }\n\n\[ = \\operatorname{homologyof}\\left( {\\operatorname{Hom}\\left( {{H}_{n - 1},{E}_{0}}\\right) \\overset{{\\left( {\\pi }_{n - 1}\\right) }_{ * }}{ \\rightarrow }\\operatorname{Hom}\\left( {{H}_{n - 1},{E}_{1}}\\right) }\\right) \\text{ at }\\operatorname{Hom}\\left( {{H}_{n - 1},{E}_{1}}\\right) \]\n\n\[ \\approx {\\operatorname{Ext}}_{R}^{1}\\left( {{H}_{n - 1}, C}\\right) \]\n\nwhile\n\n\[ \\ker {\\left( {\\pi }_{n}\\right) }_{ * }\n\n\[ = \\mathrm{{homologyof}}\\left( {\\mathrm{{Hom}}\\left( {{H}_{n},{E}_{0}}\\right) \\overset{{\\left( {\\pi }_{n}\\right) }_{ * }}{ \\rightarrow }\\mathrm{{Hom}}\\left( {{H}_{n},{E}_{1}}\\right) }\\right) \\mathrm{{at}}\\operatorname{Hom}\\left( {{H}_{n},{E}_{0}}\\right) \]\n\n\[ \\approx {\\operatorname{Ext}}_{R}^{0}\\left( {{H}_{n}, C}\\right) \]\n\n\[ \\approx \\operatorname{Hom}\\left( {{H}_{n}, C}\\right) \]\n\nAll that remains is the task of showing that the induced homomorphism \( {H}^{n}\\left( C\\right) \\rightarrow \\ker {\\left( {\\pi }_{n}\\right) }_{ * } \\approx \\operatorname{Hom}\\left( {{H}_{n}, C}\\right) \) is \( \\rho \), which is left as an exercise. (Use the naturality of \( \\rho \).)
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No
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Theorem 9.13 (Universal Coefficient Theorem Involving Ext) Suppose \( R \) is a left hereditary ring, that is, suppose LG-dim \( R \leq 1 \) . Suppose \( \left\langle {{P}_{i},{d}_{i}}\right\rangle \) is a complex of projective left \( R \) -modules with homology \( {H}_{n} \) at \( {P}_{n} \) . Let \( C \in {}_{R}\mathbf{M} \), and suppose the homology of \( \left\langle {\operatorname{Hom}\left( {{P}_{i}, C}\right) ,{d}_{i}^{ * }}\right\rangle \) is \( {H}^{n}\left( C\right) \) at \( \operatorname{Hom}\left( {{P}_{n}, C}\right) \) . Then there is a naturally defined short exact sequence\n\n\[ 0 \rightarrow {\operatorname{Ext}}_{R}^{1}\left( {{H}_{n - 1}, C}\right) \rightarrow {H}^{n}\left( C\right) \rightarrow \operatorname{Hom}\left( {{H}_{n}, C}\right) \rightarrow 0 \]\n\nwhich splits (although the splitting is not asserted to be natural).
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Proof: I-dim \( C \leq \mathrm{{LG}} \) -dim \( R \leq 1 \), so Proposition 9.12 applies, giving the short exact sequence. To see that Proposition 9.11 also applies, observe that \( {B}_{n - 1} \) is a submodule of the projective module \( {P}_{n - 1} \), so \( {B}_{n - 1} \) is projective by the projective dimension theorem \( \left( {{P}_{n - 1} \rightarrow {P}_{n - 1}/{B}_{n - 1} \rightarrow 0}\right. \) extends to a projective resolution of \( {P}_{n - 1}/{B}_{n - 1} \), where first kernel is \( {B}_{n - 1} \), and P-dim \( \left( {{P}_{n - 1}/{B}_{n - 1}}\right) \leq \) LG-dim \( R \leq 1 \) ). Consequently, \( 0 \rightarrow {Z}_{n} \rightarrow {P}_{n} \rightarrow \) \( {B}_{n - 1} \rightarrow 0 \) splits (see Section 2.3), and \( {Z}_{n} \) is a direct summand of \( {P}_{n} \) .
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Yes
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Corollary 9.14 (Universal Coefficient Formula) Suppose \( R \) is a left hereditary ring, \( \left\langle {{P}_{i},{d}_{i}}\right\rangle \) is a complex of projective left \( R \) -modules with homology \( {H}_{n} \) at \( {P}_{n} \), and \( C \in {}_{R}\mathbf{M} \) . Then the homology \( {H}^{n}\left( C\right) \) of \( \left\langle {\operatorname{Hom}\left( {{P}_{i}, C}\right) ,{d}_{i}^{ * }}\right\rangle \) at \( \operatorname{Hom}\left( {{P}_{n}, C}\right) \) is unnaturally isomorphic to
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\[ {\operatorname{Ext}}_{R}^{1}\left( {{H}_{n - 1}, C}\right) \oplus \operatorname{Hom}\left( {{H}_{n}, C}\right) \]
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Yes
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Proposition 9.15 Suppose \( \left\langle {{F}_{i},{d}_{i}}\right\rangle \) and \( \left\langle {{F}_{i}^{\prime },{d}_{i}^{\prime }}\right\rangle \) are chain complexes in \( {\mathbf{M}}_{R} \) and \( {}_{R}\mathbf{M} \), respectively, where \( R \) is any ring. Suppose that for all \( j,{d}_{j}^{\prime } = \) \( 0 \), and \( {F}_{j}^{\prime } \) is flat. Then the Künneth homomorphism is an isomorphism.
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Proof: If \( {d}_{j}^{\prime } = 0 \) for all \( j \), then \( {d}_{i, j} = 0 \) for all \( i \) and \( j \) so that\n\n\[ {\bar{d}}_{n} = {\bigoplus }_{i + j = n}{\left( -1\right) }^{j}{\partial }_{i, j} \]\n\nand consequently\n\n\[ {\bar{Z}}_{n} \approx {\bigoplus }_{i + j = n}\left( {{Z}_{i} \otimes {F}_{j}^{\prime }}\right) \]\n\n\[ {\bar{B}}_{n} \approx {\bigoplus }_{i + j = n}\left( {{B}_{i} \otimes {F}_{j}^{\prime }}\right) ,\text{ and } \]\n\n\[ {\bar{H}}_{n} \approx {\bigoplus }_{i + j = n}\left( {{Z}_{i}/{B}_{i}}\right) \otimes {F}_{j}^{\prime } \]\n\nsince each \( \otimes {F}_{j}^{\prime } \) is an exact functor \( \left( {F}_{j}^{\prime }\right. \) being flat). Since \( {F}_{j}^{\prime } \approx {H}_{j}^{\prime } \), the Künneth homomorphism yields an isomorphism.
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Yes
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Theorem 9.16 (Künneth Exact Sequence) Suppose \( R \) is a ring, with \( \mathrm{W} - \dim R \leq 1 \) . Suppose \( \left\langle {{F}_{i},{d}_{i}}\right\rangle \) is a chain complex of flat right \( R \) -modules, while \( \left\langle {{F}_{i}^{\prime },{d}_{i}^{\prime }}\right\rangle \) is a chain complex of flat left \( R \) -modules. Let \( {H}_{n} \) denote the homology of \( \left\langle {{F}_{i},{d}_{i}}\right\rangle \) at \( {F}_{n} \), and \( {H}_{n}^{\prime } \) the homology of \( \left\langle {{F}_{i}^{\prime },{d}_{i}^{\prime }}\right\rangle \) at \( {F}_{n}^{\prime } \) . Form the tensor complex\n\n\[ \left\langle {{\bar{F}}_{i},{\bar{d}}_{i}}\right\rangle = \left\langle {{F}_{i},{d}_{i}}\right\rangle \otimes \left\langle {{F}_{i}^{\prime },{d}_{i}^{\prime }}\right\rangle \]\n\nThen the homology \( {\bar{H}}_{n} \) of \( \left\langle {{\bar{F}}_{i},{\bar{d}}_{i}}\right\rangle \) at \( {\bar{F}}_{n} \) fits naturally into a short exact sequence\n\n\[ 0 \rightarrow {\bigoplus }_{i + j = n}{H}_{i} \otimes {H}_{j}^{\prime }\overset{\kappa }{ \rightarrow }{\bar{H}}_{n} \rightarrow {\bigoplus }_{i + j = n - 1}{\operatorname{Tor}}_{1}^{R}\left( {{H}_{i},{H}_{j}^{\prime }}\right) \rightarrow 0. \]
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Proof: Adopt the notation in the discussion preceeding Proposition 9.15. Let \( {\bar{H}}_{n}^{\prime } \) denote the homology at \( {\bar{F}}_{n}^{\prime } \) of the complex \( \left\langle {{\bar{F}}_{i}^{\prime },{\bar{d}}_{i}^{\prime }}\right\rangle \), and \( {\bar{H}}_{n}^{ * } \) the homology at \( {\bar{F}}_{n}^{ * } \) of the complex \( \left\langle {{\bar{F}}_{i}^{ * },{\bar{d}}_{i}^{ * }}\right\rangle \) . We get a long exact sequence\n\n\n\nfrom which we get for each \( n \) a short exact sequence\n\n\[ 0 \rightarrow {\bar{H}}_{n}^{\prime }/\operatorname{im}{\delta }_{n + 1} \rightarrow {\bar{H}}_{n} \rightarrow \ker {\delta }_{n} \rightarrow 0. \]\n\nWe know that\n\n\[ {\bar{H}}_{n}^{\prime } \approx {\bigoplus }_{i + j = n}{H}_{i} \otimes {Z}_{j}^{\prime } \]\n\nand\n\n\[ {\bar{H}}_{n}^{ * } \approx {\bigoplus }_{i + j = n}{H}_{i} \otimes {B}_{j}^{ * } = {\bigoplus }_{i + j = n}{H}_{i} \otimes {B}_{j - 1}^{\prime } \]\n\nby Proposition 9.15. We need to analyze\n\n\[ {\delta }_{n + 1} : {\bar{H}}_{n + 1}^{ * } \rightarrow {\bar{H}}_{n}^{\prime } \]\n\nto identify the kernel and image. Now\n\n\[ {\bar{H}}_{n + 1}^{ * } = {\bigoplus }_{i + j = n + 1}{H}_{i} \otimes {B}_{j}^{ * } = {\bigoplus }_{i + j = n}{H}_{i} \otimes {B}_{j}^{\prime }. \]\n\nGiven \( \sum {u}_{i} \otimes {b}_{j}^{\prime } \in {\bar{H}}_{n + 1}^{ * } \), we compute \( {\delta }_{n + 1}\left( {\sum {u}_{i} \otimes {b}_{j}^{\prime }}\right) \) as follows. Find for each \( j \), a \( {v}_{j}^{\prime } \in {F}_{j + 1}^{\prime } \) such that \( {d}_{j + 1}^{\prime }\left( {v}_{j}^{\prime }\right) = {b}_{j}^{\prime } \) ; this pulls \( \sum {u}_{i} \otimes {b}_{j}^{\prime } \) back to \( \sum {u}_{i} \otimes {v}_{j}^{\prime } \in {\bar{F}}_{n + 1} \) with each \( {u}_{i} \otimes {v}_{j}^{\prime } \in {C}_{i, j + 1} \) . Now each \( {u}_{i} \) represents a member of \( {H}_{i} \), so \( {d}_{i}{u}_{i} = 0 \) and \( {\bar{d}}_{n + 1}\left( {\sum {u}_{i} \otimes {v}_{j}^{\prime }}\right) = \sum {u}_{i} \otimes {d}_{j + 1}^{\prime }{v}_{j}^{\prime } = \sum {u}_{i} \otimes {b}_{j}^{\prime } \) ; we’re back where we started. The map \( {\delta }_{n + 1} \) simply looks at this inside \( \oplus {H}_{i} \otimes {Z}_{j}^{\prime } \), since the \( {\bar{F}}_{n}^{\prime } \rightarrow {\bar{F}}_{n} \) maps are obtained from the set inclusions \( {Z}_{j}^{\prime } \hookrightarrow {F}_{j}^{\prime } \) . That is,\n\n\[ {\delta }_{n + 1} : {\bigoplus }_{i + j = n}{H}_{i} \otimes {B}_{j}^{\prime } \rightarrow {\bigoplus }_{i + j = n}{H}_{i} \otimes {Z}_{j}^{\prime } \]\n\n\[ {\delta }_{n + 1} = {\bigoplus }_{i + j = n}{H}_{i} \otimes \left( {{B}_{j}^{\prime } \hookrightarrow {Z}_{j}^{\prime }}\right) \]
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Yes
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Proposition 9.18 Suppose \( R \) is a ring, with \( \mathrm{{RG}} \) -dim \( R \leq 1 \) and LG-dim \( R \leq 1 \) . Suppose \( \left\langle {{F}_{i},{d}_{i}}\right\rangle \) is a chain complex of projective right \( R \) -modules, while \( \left\langle {{F}_{i}^{\prime },{d}_{i}^{\prime }}\right\rangle \) is a chain complex of projective left \( R \) -modules. Then the Künneth exact sequence splits (although the splitting is not asserted to be natural).
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Proof: Again adopt the notation preceeding Proposition 9.15. Then P-dim \( \left( {{F}_{n - 1}/{B}_{n - 1}}\right) \leq \mathrm{{RG}} \) -dim \( R \leq 1 \), so \( {B}_{n - 1} \) is projective by the projective dimension theorem \( \left( {{F}_{n - 1} \rightarrow {F}_{n - 1}/{B}_{n - 1} \rightarrow 0}\right. \) extends to a projective resolution of \( {F}_{n - 1}/{B}_{n - 1} \) whose first kernel is \( {B}_{n - 1} \) ). Consequently, \( 0 \rightarrow {Z}_{n} \rightarrow {F}_{n} \rightarrow {B}_{n - 1} \rightarrow 0 \) is split exact, giving a homomorphism \( {\pi }_{n} : {F}_{n} \rightarrow {H}_{n} \) as the composite of the quotient following the splitting \( {F}_{n} \rightarrow {Z}_{n} \rightarrow {H}_{n} \) . Similarly, there is a homomorphism \( {\pi }_{n}^{\prime } : {F}_{n}^{\prime } \rightarrow {H}_{n}^{\prime } \) . We thus get\n\n\[ \pi = {\bigoplus }_{i + j = n}{\pi }_{i} \otimes {\pi }_{j}^{\prime } : {\bigoplus }_{i + j = n}{F}_{i} \otimes {F}_{j}^{\prime } \rightarrow {\bigoplus }_{i + j = n}{H}_{i} \otimes {H}_{j}^{\prime } \]\n\nor \( \pi : {\bar{F}}_{n} \rightarrow {\bigoplus }_{i + j = n}{H}_{i} \otimes {H}_{j}^{\prime } \) .\n\nAlmost by inspection, \( {\pi \kappa }\left( {{u}_{i} \otimes {v}_{j}^{\prime }}\right) = {u}_{i} \otimes {v}_{j}^{\prime } \) ; we need only show that \( \pi \) is actually well-defined on \( {\bar{H}}_{n} \), that is, that \( \pi \) sends \( {\bar{B}}_{n} \) to zero. But \( {\pi }_{i} \) sends \( {B}_{i} \) to zero and \( {\pi }_{j}^{\prime } \) sends \( {B}_{j}^{\prime } \) to zero. It follows that \( {\pi }_{i} \otimes {\pi }_{j}^{\prime } \) sends \( {F}_{i} \otimes {B}_{j}^{\prime } \) and \( {B}_{i} \otimes {F}_{j}^{\prime } \) to zero. However,\n\n\[ {\bar{d}}_{n + 1}\left( {{\bigoplus }_{i + j = n + 1}\left( {\mathop{\sum }\limits_{k}{u}_{i, j, k} \otimes {v}_{i, j, k}^{\prime }}\right) }\right) = {\bigoplus }_{i + j = n}\left( {\mathop{\sum }\limits_{k}{u}_{i, j + 1, k} \otimes {d}_{j + 1}^{\prime }\left( {v}_{i, j + 1, k}^{\prime }\right) }\right. \]\n\n\[ \left. {+{\left( -1\right) }^{j}{d}_{i + 1}\left( {u}_{i + 1, j, k}\right) \otimes {v}_{i + 1, j, k}^{\prime }}\right) \in {\bigoplus }_{i + j = n}\left( {{F}_{i} \otimes {B}_{j}^{\prime } + {B}_{i} \otimes {F}_{j}^{\prime }}\right) \]\n\nthat is,\n\n\[ {\bar{B}}_{n} \subset {\bigoplus }_{i + j = n}\left( {{F}_{i} \otimes {B}_{j}^{\prime } + {B}_{i} \otimes {F}_{j}^{\prime }}\right) \subset {\bigoplus }_{i + j = n}\ker \left( {{\pi }_{i} \otimes {\pi }_{j}^{\prime }}\right) . \]\n\nRemark: In the above, \( {F}_{i} \otimes {B}_{j}^{\prime } \) is treated as a submodule of \( {F}_{i} \otimes {F}_{j}^{\prime } \) . This is allowed since \( {F}_{i} \) is flat: Apply \( {F}_{i} \otimes \) to \( 0 \rightarrow {B}_{j}^{\prime } \rightarrow {F}_{j}^{\prime } \) . Similarly, \( {\overrightarrow{B}}_{i} \otimes {F}_{j}^{\prime } \) can be viewed as a submodule of \( {F}_{i} \otimes {F}_{j}^{\prime } \) .
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Yes
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Proposition 9.19 Suppose W-dim \( R \leq 1 \), and suppose \( R \) is flat as a \( \mathbb{Z} \) - module. Suppose \( A \in {\mathbf{M}}_{R}, B \in {}_{R}\mathbf{M} \), and \( G \in \mathbf{{Ab}} \) . Then \( {\operatorname{Tor}}_{1}^{\mathbb{Z}}\left( {B, G}\right) \in \) \( {}_{R}\mathbf{M} \) , and\n\n\[{\operatorname{Tor}}_{1}^{R}\left( {A,{\operatorname{Tor}}_{1}^{\mathbb{Z}}\left( {B, G}\right) }\right) \approx {\operatorname{Tor}}_{1}^{\mathbb{Z}}\left( {{\operatorname{Tor}}_{1}^{R}\left( {A, B}\right), G}\right) .\]\n\nFurthermore, if \( R \) is projective as a \( \mathbb{Z} \) -module, then there is a short exact sequence\n\n\[0 \rightarrow A{ \otimes }_{R}{\operatorname{Tor}}_{1}^{\mathbb{Z}}\left( {B, G}\right)\]\n\n\[ \rightarrow \left( {{\mathrm{{Tor}}}_{1}^{R}\left( {A, B}\right) { \otimes }_{\mathbb{Z}}G}\right) \oplus \left( {{\mathrm{{Tor}}}_{1}^{\mathbb{Z}}\left( {A{ \otimes }_{R}B, G}\right) }\right)\]\n\n\[ \rightarrow {\operatorname{Tor}}_{1}^{R}\left( {A, B{ \otimes }_{\mathbb{Z}}G}\right)\]\n\n\[ \rightarrow 0\]\n\nwhich splits if \( \mathrm{{LG}} - \dim R \leq 1 \) and \( \mathrm{{RG}} - \dim R \leq 1 \) .
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Proof: The idea is to use the Künneth theorem twice, once with \( R \) and once with \( \mathbb{Z} \) . The complexes are the ones appearing in Proposition 3.9 and its corollary. To this end, let\n\n\[ \cdots \overset{{d}_{2}}{ \rightarrow }{P}_{1}\overset{{d}_{1}}{ \rightarrow }{P}_{0} \rightarrow A \rightarrow 0 \]\n\ndenote a free resolution of \( A \), and\n\n\[ \cdots \overset{{d}_{2}^{\prime }}{ \rightarrow }{P}_{1}^{\prime }\overset{{d}_{1}^{\prime }}{ \rightarrow }{P}_{0} \rightarrow B \rightarrow 0 \]\n\na free resolution of \( B \) . (Free objects are easiest to handle here.) Furthermore, let\n\n\[ 0 \rightarrow {Q}_{1}\overset{{\partial }_{1}}{ \rightarrow }{Q}_{0} \rightarrow G \rightarrow 0 \]\n\ndenote a projective (i.e., free) resolution of \( G \) as a \( \mathbb{Z} \) -module.\n\nTo start the game, set\n\n\[ {F}_{n} = {\bigoplus }_{i + j = n}{P}_{i}{ \otimes }_{R}{P}_{j}^{\prime } \]\n\nand\n\n\[ {D}_{n} = {\bigoplus }_{i + j = n}\left( {{i}_{{P}_{\imath }} \otimes {d}_{j}^{\prime } + {\left( -1\right) }^{j}{d}_{i} \otimes {i}_{{P}_{\jmath }^{\prime }}}\right) . \]\n\n\( \left\langle {{F}_{n},{D}_{n}}\right\rangle \) is now a complex whose homology is \( {\operatorname{Tor}}_{n}^{R}\left( {A, B}\right) \) at \( {F}_{n} \), by the proof of Proposition 3.9 and its corollary, since \( \left\langle {{F}_{n},{D}_{n}}\right\rangle = \left\langle {{P}_{i},{d}_{i}}\right\rangle \otimes \)\n\n\( \left\langle {{P}_{j}^{\prime },{d}_{j}^{\prime }}\right\rangle \).\n\nNext, set \( {F}_{n}^{\prime } = {Q}_{n} \) . Only \( {H}_{0}^{\prime } \) is nonzero; it is \( G \) . Now apply the Künneth theorem to the tensor product over \( \mathbb{Z} \), yielding the exact sequences\n\n\[ 0 \rightarrow 0 \rightarrow {\bar{H}}_{2} \rightarrow {\operatorname{Tor}}_{1}^{\mathbb{Z}}\left( {{\operatorname{Tor}}_{1}^{R}\left( {A, B}\right), G}\right) \rightarrow 0, \]\n\n\[ 0 \rightarrow {\operatorname{Tor}}_{1}^{R}\left( {A, B}\right) { \otimes }_{\mathbb{Z}}G \rightarrow {\bar{H}}_{1} \rightarrow {\operatorname{Tor}}_{1}^{\mathbb{Z}}\left( {A{ \otimes }_{R}B, G}\right) \rightarrow 0, \]\n\nand\n\n\[ 0 \rightarrow \left( {A{ \otimes }_{R}B}\right) { \otimes }_{\mathbb{Z}}G \rightarrow {\bar{H}}_{0} \rightarrow 0 \rightarrow 0. \]\n\nThis is allowed since each \( {F}_{n} \) is a direct sum of \( {P}_{i}{ \otimes }_{R}{P}_{j}^{\prime } \), which are in turn direct sums of \( R{ \otimes }_{R}R \) since each \( {P}_{i} \) and \( {P}_{j}^{\prime } \) is free. But \( R{ \otimes }_{R}R \approx R \), which is flat as a \( \mathbb{Z} \) -module by assumption, so each \( {F}_{n} \) is flat as a \( \mathbb{Z} \) -module, and Theorem 9.16 applies. Furthermore, if \( R \) is actually
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Yes
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Theorem 9.22 Suppose \( 0 \rightarrow B \rightarrow {B}^{\prime } \rightarrow {B}^{\prime \prime } \rightarrow 0 \) is short exact in \( {}_{R}\mathbf{M} \).\na) Suppose \( 0 \rightarrow A \rightarrow {A}^{\prime } \rightarrow {A}^{\prime \prime } \rightarrow 0 \) is also short exact in \( {\mathbf{M}}_{R} \). If connecting homomorphisms are as defined in Chapter 3, notation-ally, \( {\delta }_{n} : {\operatorname{Tor}}_{n}\left( {\bullet ,{B}^{\prime \prime }}\right) \rightarrow {\operatorname{Tor}}_{n - 1}\left( {\bullet, B}\right) \), and \( {\widehat{\delta }}_{n} : {\operatorname{Tor}}_{n}\left( {{A}^{\prime \prime }, \bullet }\right) \rightarrow \) \( {\operatorname{Tor}}_{n - 1}\left( {A, \bullet }\right) \), then the diagram\n\n\n\nis commutative.
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Proof: In accordance with Proposition 6.5(a), construct a simultaneous projective resolution of \( 0 \rightarrow B \rightarrow {B}^{\prime } \rightarrow {B}^{\prime \prime } \rightarrow 0 \) :\n\n\n\nSince each \( {P}_{n}^{\prime \prime } \) is projective, the vertical sequences split, and \( {P}_{n}^{\prime } \approx {P}_{n} \oplus {P}_{n}^{\prime \prime } \) as in the statement of Proposition 6.5(a). For part (a), tensor this with \( 0 \rightarrow A \rightarrow {A}^{\prime } \rightarrow {A}^{\prime \prime } \rightarrow 0 \) (and prop it upright); the result is a post diagram. \( {\widehat{\delta }}_{n} \) stays the same, but the connecting homomorphisms \( {\delta }_{n} \) are replaced by \( {\widetilde{\delta }}_{n} = {\left( -1\right) }^{n}{\delta }_{n} \) in accordance with Proposition 6.12. We have that \( {\widehat{\delta }}_{n}{\widetilde{\delta }}_{n + 1} = - {\widetilde{\delta }}_{n}{\widehat{\delta }}_{n + 1} \) by Proposition 9.20, so that \( {\dot{\widehat{\delta }}}_{n}{\delta }_{n + 1} = {\left( -1\right) }^{n + 1}{\widehat{\delta }}_{n}{\widetilde{\delta }}_{n + 1} = \) \( - {\left( -1\right) }^{n + 1}{\widetilde{\delta }}_{n}{\widehat{\delta }}_{n + 1} = {\left( -1\right) }^{n}{\widetilde{\delta }}_{n}{\widehat{\delta }}_{n + 1} = {\delta }_{n}{\widehat{\delta }}_{n + 1}.
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Yes
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Lemma 9.23 Suppose \( \sqcup \) is a product satisfying \( {ZR},{ZL} \), and \( A \) . Then \( \sqcup \) satisfies \( {NR},{NL} \), and \( {NC} \) .
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Proof: For NR, note that\n\n\[ \n{f}^{ * }\left( {u \sqcup v}\right) = \left( {u \sqcup v}\right) \sqcup f \n\]\n\n(ZR)\n\n\[ \n= u \sqcup \left( {v \sqcup f}\right) \n\]\n\n(A)\n\n\[ \n= u \sqcup \left( {{f}^{ * }\left( v\right) }\right) \n\]\n\n(ZR)\n\nNL is similar. As for NC,\n\n\[ \n\left( {{f}^{ * }u}\right) \sqcup v = \left( {u \sqcup f}\right) \sqcup v \n\]\n\n(ZR)\n\n\[ \n= u \sqcup \left( {f \sqcup v}\right) \n\]\n\n(A)\n\n\[ \n\bar{u} = u \sqcup \left( {{f}_{ * }v}\right) \n\]\n\n(ZL)
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Yes
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Lemma 9.24 Suppose \( \sqcup \) is a product satisfying \( {ZR},{NR} \), and \( {CR} \) . Then \( \sqcup \) satisfies \( A \) .
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Proof: Induction on \( \ell \) . The \( \ell = 0 \) case is just NR and ZR:\n\n\[ u \sqcup \left( {v \sqcup f}\right) = u \sqcup \left( {{f}^{ * }v}\right) \]\n\n(ZR)\n\n\[ = {f}^{ * }\left( {u \sqcup v}\right) \]\n\n(NR)\n\n\[ = \left( {u \sqcup v}\right) \sqcup f \]\n\n(ZR)\n\nAs for \( \ell - 1 \rightarrow \ell \), we use CR. Choose a projective \( P \) and an epimorphism \( \pi : P \rightarrow B \) with kernel \( K \) . Then \( w = {\delta }_{\ell }\left( {w}^{\prime }\right) \) for some \( {w}^{\prime } \in {\operatorname{Ext}}^{\ell - 1}\left( {K, C}\right) \).\n\nBy the induction hypothesis, \( \left( {u \sqcup v}\right) \sqcup {w}^{\prime } = u \sqcup \left( {v \sqcup {w}^{\prime }}\right) \). We have\n\n\[ \left( {u \sqcup v}\right) \sqcup w = \left( {u \sqcup v}\right) \sqcup {\delta }_{\ell }\left( {w}^{\prime }\right) \]\n\n(def.)\n\n\[ = {\delta }_{\ell + m + n}\left( {\left( {u \sqcup v}\right) \sqcup {w}^{\prime }}\right) \]\n\n(CR)\n\n\[ = {\delta }_{\ell + m + n}\left( {u \sqcup \left( {v \sqcup {w}^{\prime }}\right) }\right) \]\n\n(ind. hyp.)\n\n\[ = u \sqcup {\delta }_{\ell + m}\left( {v \sqcup \left( {w}^{\prime }\right) }\right) \]\n\n(CR)\n\n\[ = u \sqcup \left( {v \sqcup {\delta }_{\ell }\left( {w}^{\prime }\right) }\right) \]\n\n(CR)\n\n\[ = u \sqcup \left( {v \sqcup w}\right) \]\n\n(def.)
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Yes
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Lemma 9.25 Suppose \( \sqcup \) is a product that satisfies \( {ZL} \) and \( {WCL} \) . Then \( \sqcup \) is unique with these properties, and \( \sqcup \) also satisfies \( {LR},{LL},{ZR},{NR} \), and CR. Similarly, any product satisfying \( {ZR} \) and \( {CR} \) necessarily satisfies \( {CL} \) (among others).
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Proof: All are by induction on \( n \) . First, LR, LL, and uniqueness, by induction on \( n \) . If \( n = 0 \), then \( u = f \), and by \( \mathrm{{ZL}}, f \sqcup v = {f}_{ * }v \) . This is forced, and is bilinear. Next, \( n - 1 \rightarrow n \) . If \( u \in {\operatorname{Ext}}^{n}\left( {C, D}\right) \), then \( u = {\bar{\delta }}_{n}\left( {u}^{\prime }\right) \) for some \( {u}^{\prime } \in {\operatorname{Ext}}^{n - 1}\left( {C, E/D}\right) \), and \( u \sqcup v = \left( {{\bar{\delta }}_{n}\left( {u}^{\prime }\right) }\right) \sqcup v = {\bar{\delta }}_{n + m}\left( {{u}^{\prime } \sqcup v}\right) \) is forced (uniqueness) and is bilinear (if for \( {u}_{1} \) we choose \( {u}_{1}^{\prime } \) and for \( {u}_{2} \) we choose \( {u}_{2}^{\prime } \), then for \( {u}_{1} + {u}_{2} \) we can choose \( {u}_{1}^{\prime } + {u}_{2}^{\prime } \) .)
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Yes
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Theorem 9.26 There is a unique product \( \sqcup \) satisfying properties LR, LL, \( {NR},{NL},{NC},{ZR},{ZL},{CR},{CL} \), and \( A \) .
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Proof: For each \( D \in {}_{R}\mathbf{M} \), choose an injective extension \( E \) of \( D \) . Recursively (on \( n \) ) define \( u \sqcup v \) for \( u \in {\operatorname{Ext}}^{n}\left( {C, D}\right) \) and \( v \in {\operatorname{Ext}}^{m}\left( {B, C}\right) \) as follows: If \( n = 0 \), set \( u \sqcup v = {u}_{ * }\left( v\right) \) . ZL is now satisfied. We recursively arrange that WCL is satisfied by setting \( {\bar{\delta }}_{n}\left( u\right) \sqcup v = {\bar{\delta }}_{n + m}\left( {u \sqcup v}\right) \) . Because WCL is so \
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No
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Corollary 9.28 (Nakayama’s Lemma) If \( B \) is a finitely generated left \( R \) -module, \( C \) is a submodule, and \( B = J\left( R\right) B + C \), then \( B = C \) .
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Proof: \( J\left( R\right) B \subset J\left( B\right) \Rightarrow B = J\left( B\right) + C \Rightarrow B = C \) .
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No
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Corollary 9.29 (Nakayama’s Lemma-Alternative Form) If \( B \) is a finitely generated left \( R \) -module for which \( B = J\left( R\right) B \), then \( B = 0 \) .
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Proof: Set \( C = 0 \) in the preceeding corollary.
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No
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Proposition 9.30 Suppose \( R \) is a ring, and \( M \) is a left ideal. Then the following are equivalent:\n\ni) \( R \) is quasilocal with maximal ideal \( M \) .\n\nii) Every \( x \notin M \) has a left inverse.\n\niii) Every \( x \notin M \) has a two-sided inverse.
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Proof: (i) \( \Rightarrow \) (iii): Suppose (i). If \( x \in R \), then \( {Rx} \neq R \Rightarrow {Rx} \subset \) some maximal left ideal, which must be \( M \) . That is, \( {Rx} \neq R \Rightarrow {Rx} \subset M \Rightarrow \) \( x \in M \) . Hence, \( x \notin M \Rightarrow {Rx} = R \Rightarrow x \) has a left inverse, call it \( y \) . Similarly, \( x \notin M \Rightarrow {xR} = R \Rightarrow x \) has a right inverse, call it \( z \) . Final step: \( y = y\left( {xz}\right) = \left( {yx}\right) z = z \) is a two-sided inverse.\n\n(iii) \( \Rightarrow \) (ii) is trivial, so to prove (ii) \( \Rightarrow \) (i), assume (ii). If \( I \) is any left ideal, then \( I ⊄ M \Rightarrow \exists x \in I - M \), for which \( R = {Rx} \subset I \) . That is, \( I ⊄ M \Rightarrow I = R \), or \( I \neq R \Rightarrow I \subset M \) . Hence, \( M \) is a largest proper left ideal, so it is the unique maximal left ideal.
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Yes
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Corollary 9.31 If \( R \) is a ring, then the following are equivalent:\n\ni) \( R \) is quasilocal.\n\nii) Every element of \( R - J\left( R\right) \) is invertible.\n\niii) \( R/J\left( R\right) \) is a division ring.
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Proof: (i) \( \Rightarrow \) (ii) follows from Proposition 9.30(iii). (ii) \( \Rightarrow \) (iii) is direct, since if \( x \notin J\left( R\right) \), then \( {x}^{-1} + J\left( R\right) \) is an inverse to \( x + J\left( R\right) \) in \( R/J\left( R\right) \) . Finally, given (iii), \( R/J\left( R\right) \) has no nontrivial proper left ideals, so if \( M \) is a maximal left ideal, then \( J\left( R\right) \subset M \) by Proposition 9.27, so that \( J\left( R\right) = M \) by the Noether correspondence for ideals containing \( J\left( R\right) \) . Hence, \( J\left( R\right) \) is the unique maximal left ideal, and \( R \) is quasilocal.
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Yes
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Proposition 9.32 Suppose \( R \) is a quasilocal with maximal ideal \( M \), and \( B \) is finitely generated as an \( R \) -module. Then \( \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \) is a minimal set of generators of \( B \) if and only if \( \left\{ {{x}_{1} + {MB},\ldots ,{x}_{n} + {MB}}\right\} \) is a basis of \( B/{MB} \) as a left vector space over the division ring \( R/M \) . In particular, the number \( n \) of generators is unique.
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Proof: This follows directly from a subclaim, that \( \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \) generates \( B \) over \( R \) if and only if \( \left\{ {{x}_{1} + {MB},\ldots ,{x}_{n} + {MB}}\right\} \) generates \( B/{MB} \) over \( R/M \), the reason being that a basis of a left vector space over a division ring is just a minimal set of generators. As for the subclaim, observe that\n\n\[ \left\langle {{x}_{1} + {MB},\ldots ,{x}_{n} + {MB}}\right\rangle = B/{MB} \]\n\n\[ \Leftrightarrow \left\langle {{MB},{x}_{1},\ldots ,{x}_{n}}\right\rangle = B \]\n\n\[ \Leftrightarrow B = {MB} + \left\langle {{x}_{1},\ldots ,{x}_{n}}\right\rangle \]\n\n\[ \Leftrightarrow B = \left\langle {{x}_{1},\ldots ,{x}_{n}}\right\rangle \]\n\nthe last implication via Nakayama's lemma.
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Yes
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Proposition 9.33 Suppose \( R \) is quasilocal with maximal ideal \( M \), and suppose \( \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \) is a minimal set of generators for a finitely generated left \( R \) -module \( B \) . Let \( F \) be free on \( \left\{ {{u}_{1},\ldots ,{u}_{n}}\right\} \), and let \( \pi : F \rightarrow B \) be defined by \( \pi \left( {u}_{j}\right) = {x}_{j} \) . Then \( {MF} \supset \ker \pi \), and \( \pi : F \rightarrow B \) is a projective cover.
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Proof: If \( \sum {r}_{i}{u}_{i} \in \ker \pi \), then all \( {r}_{i} \in M \), since if some \( {r}_{j} \notin M \), then\n\n\[ \n{u}_{j} + \mathop{\sum }\limits_{{i \neq j}}{r}_{j}^{-1}{r}_{i}{u}_{i} \in \ker \pi \Rightarrow {x}_{j} + \mathop{\sum }\limits_{{i \neq j}}{r}_{j}^{-1}{r}_{i}{x}_{i} = 0 \n\]\n\n\[ \n\Rightarrow {x}_{j} \in \left\langle {{x}_{1},\ldots ,{x}_{j - 1},{x}_{j + 1},\ldots {x}_{n}}\right\rangle \n\]\ncontradicting minimality. Finally, if \( f : C \rightarrow F \) is such that \( {\pi f} \) is onto, then \( F = \ker \pi + \operatorname{im}\left( f\right) \subset {MF} + \operatorname{im}\left( f\right) \Rightarrow F = \operatorname{im}\left( f\right) \) by Nakayama’s lemma. \( ▱ \)
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Yes
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Corollary 9.34 If \( R \) is a quasilocal ring, then any finitely generated projective left \( R \) -module is free.
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Proof: Construct \( F \) as above for a finitely generated projective \( P \) . There is an \( f : P \rightarrow F \) such that \( {\pi f} = {i}_{P} \) since \( P \) is projective; this implies that \( f \) is one-to-one, while Proposition 9.33 says that \( f \) is onto.
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No
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