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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 ... | Proof: The \ | No |
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\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{\va... | Yes |
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 \... | 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, le... | Yes |
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 }}\r... | 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}_{... | Yes |
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. | 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 b... | No |
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{N... | 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,... | Yes |
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, | 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 \( B \) ). | No |
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}}{ \... | 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 \( B \) ). | No |
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 = ... | Proof: From Proposition 7.1 above, \( {\pi }_{1} \) and \( {\pi }_{2} \) are fillers for\n\n\n\n\n\nand these fi... | Yes |
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),... | Proof: \( A,{\\pi }_{1} \), and \( {\\pi }_{2} \) define a coproduct in the additive category \( {\\mathbf{A}}^{\\text{op }} \) ; use Proposition 7.2. | No |
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... | 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 \( \... | Yes |
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) \( ... | 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, \( ... | Yes |
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... | 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... | Yes |
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 \( ... | ## Proof: Quote Proposition 7.8 for \( {\mathbf{A}}^{\text{op }} \) . | No |
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}... | 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 \( g \) . Sinc... | Yes |
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 ... | 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[... | Yes |
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 cok... | Proof: Apply Proposition 7.12 to \( {\mathbf{A}}^{\text{op }} \) . | No |
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 \( \varphi : P \rightarrow A,{f\varphi } = 0 \Rightarrow \varphi = 0 \) . | (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 \( \v... | Yes |
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. | 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 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 ... | Proof: \( K \rightarrow A \rightarrow B \) is kernel-exact by definition, so it is cokernel-exact. This just means that \( g \) is monic. | Yes |
Proposition 7.17 Suppose A is a balanced pre-Abelian category with a separating class of projectives. Then \( \\mathbf{A} \) satisfies \( {Ab} \) -epic. | Proof: A satisfies condition (ii) in Corollary 7.13. | No |
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. | 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-... | Yes |
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 } :... | 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}^{\prim... | No |
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}... | 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) ... | Yes |
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 }... | Proof: Essentially the same as Proposition 7.23, with arrows reversed. (Look in \( \mathbf{A}{\left( \rightarrow \right) }^{\text{op }} \approx {\mathbf{A}}^{\text{op }}\left( \rightarrow \right) \) .) | No |
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... | 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. | Yes |
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 \... | 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 ea... | Yes |
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 \( \mathbf{A}\left( \rightarrow \right) \) . | 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 ... | Yes |
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 \( \mathbf{A}\left( \rightarrow \right) \) . | 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 i... | No |
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 \... | 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 th... | Yes |
Proposition 7.31 Suppose \( \mathbf{A} \) is a pre-Abelian category.\n\na) If \( \mathbf{A} \) has enough projectives, then so does \( \mathbf{A}\left( \rightarrow \right) \) . | 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 \(... | Yes |
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 ... | 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). | Yes |
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,... | 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 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\n... | 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)). | Yes |
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 \( {fj} \)... | 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 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. | 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 and has exact rows in \( \mathbf{A} \) . Then \( \psi \) monic \( \Rightarrow \eta \) monic. | 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... | Yes |
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. | Proof: Apply Proposition 7.37 to \( {\mathbf{A}}^{\text{op }} \) . | Yes |
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. | ## 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... | Yes |
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 \... | 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 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 \( {\varphi }_{1} \) ... | Proof: Apply Lemma 7.42 to \( {\mathbf{A}}^{\text{op }} \) . | No |
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} \... | 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)). | Yes |
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\nwhere \( p : K \rightarrow H \) is a cokernel for \( {\varphi \rho }... | Yes |
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\nwhere \( ... | Yes |
Theorem 7.48 Suppose A is an Abelian category in which the array\n\n\n\nhas rows that are short exact and columns that are underexact. Let \( {H}_{n} \) (respectively, \( {H}_{n}^{\prime },{H}_{n}^{\prime \prime } \)... | 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}^{\pr... | Yes |
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... | 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 proje... | No |
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 }^{ * }}{ \righta... | 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 }^{ * } ... | Yes |
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 ... | Proof: Virtually identical to the proof of Proposition 7.8, with the letter \( Q \) replacing \( P \) ,\ | No |
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 \( Q{\mathcal{L}}^{0}F \) . | 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... | Yes |
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 ... | 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 \( = \... | No |
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 \( {... | Proof: Virtually identical to the proof of Proposition 7.21. The \ | No |
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 catego... | Proof: Apply Theorem 7.48 to | No |
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 \( n \) . | Proof: Read it off the diagram in Corollary 7.59, with connecting homomorphisms deleted. | No |
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. | Proof: Corollary 7.60 plus Proposition 7.55. | No |
Corollary 7.62 If A is a pre-Abelian category with enough quasiprojec-tives, then every cokernel \( {}^{ \bullet } \) is a cokernel. | 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\nSup... | Yes |
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{... | 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, a... | Yes |
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} = {\ope... | Proof: Proposition 8.2 in \( {\mathbf{C}}^{\text{op }} \) . | No |
Corollary 8.4 Suppose \( \mathcal{I} \) has a largest element \( {i}_{0} \) . Then \( \mathop{\lim }\limits_{\mathcal{I}}{A}_{i} = {A}_{{i}_{0}} \) . | 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. | Yes |
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 }_{\m... | 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}_{... | No |
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 \( \widehat{\mathbf{C}} \) ... | Proof: We have commutative diagrams\n\n\n\nso that given commutative diagrams\n\n\n\nall we have to do is find a... | Yes |
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 \( {}_{R}\mathbf{M} \) , \( {\operatorname{colim}}_{\mathcal{I}}\left( {A \otimes {B}_{i}}\right) \approx \) \(... | Proof: \( A \otimes \) is a left adjoint. | No |
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 co... | 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... | Yes |
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 \( n \) , | 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... | Yes |
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... | Proof: Apply \( {\operatorname{Tor}}_{n + 1};{\operatorname{Tor}}_{n + 1} \) of each entry in the colimit is zero. | No |
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 \( ... | 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 gene... | Yes |
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}/... | 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 subm... | Yes |
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,{... | 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 8.1... | Yes |
Proposition 8.17 Suppose \( R \) and \( S \) are rings, and suppose \( A \in {}_{R}{\mathbf{M}}_{S} \) . Then for all flat \( B \in {}_{S}\mathbf{M},\mathrm{F} - {\dim }_{R}A{ \otimes }_{S}B \leq \mathrm{F} - {\dim }_{R}A \) . | 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}... | Yes |
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. | Proof: F-dim \( A = 0 \) in Proposition 8.17. | No |
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 ... | 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} - \di... | Yes |
Corollary 8.20 Suppose \( R \) is a commutative ring and \( S \) is an admissible multiplicative subset of \( R \) . \n\n\[ \n\\text{a) For all}B \\in {}_{R}\\mathbf{M},\\mathrm{F} - {\\dim }_{S^{-1}R}S^{-1}B \\leq \\mathrm{F} - {\\dim }_{R}B\\text{.}\n\] | 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. | Yes |
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. | 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 inc... | Yes |
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 \( D \cap B = 0 \) . | 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 t... | Yes |
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 \... | 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 \... | Yes |
Proposition 9.4 Suppose \( E \in {}_{R}\mathbf{M} \) . Then \( E \) is injective if and only if \( E \) has no nontrivial essential extensions. | 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 \( C = E \oplus F \), t... | No |
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 \( C \) . | 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 e... | Yes |
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 \( B \) . | 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... | Yes |
Corollary 9.8 Any \( B \in {}_{R}\mathbf{M} \) has an injective envelope. | Proof: In Proposition 9.7, \( E \) and \( C \) exist by the enough injectives theorem and Lemma 9.1. | No |
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 \operat... | 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 \ | No |
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}\l... | 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 \( \operatorname{Hom}\left( {\bullet, C}\right) \) i... | Yes |
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 s... | 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 \si... | Yes |
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}... | 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 }\... | No |
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}\mat... | 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 t... | Yes |
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 ... | \[ {\operatorname{Ext}}_{R}^{1}\left( {{H}_{n - 1}, C}\right) \oplus \operatorname{Hom}\left( {{H}_{n}, C}\right) \] | Yes |
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 \( j,{d}_{j}^{\prime } = \) \( 0 \)... | 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 ... | Yes |
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 lef... | 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}... | Yes |
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 c... | 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... | Yes |
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\[{\operatorna... | 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... | Yes |
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 conne... | 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... | Yes |
Lemma 9.23 Suppose \( \sqcup \) is a product satisfying \( {ZR},{ZL} \), and \( A \) . Then \( \sqcup \) satisfies \( {NR},{NL} \), and \( {NC} \) . | 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) \sqc... | Yes |
Lemma 9.24 Suppose \( \sqcup \) is a product satisfying \( {ZR},{NR} \), and \( {CR} \) . Then \( \sqcup \) satisfies \( A \) . | 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 \( \ell - 1 \rightarrow \ell \), ... | Yes |
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). | 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 \... | Yes |
Theorem 9.26 There is a unique product \( \sqcup \) satisfying properties LR, LL, \( {NR},{NL},{NC},{ZR},{ZL},{CR},{CL} \), and \( A \) . | 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}... | No |
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 \( B = C \) . | Proof: \( J\left( R\right) B \subset J\left( B\right) \Rightarrow B = J\left( B\right) + C \Rightarrow B = C \) . | No |
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 \( B = 0 \) . | Proof: Set \( C = 0 \) in the preceeding corollary. | No |
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. | 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 inve... | Yes |
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. | 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 n... | Yes |
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 ... | 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 ... | Yes |
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 \righta... | 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\Righ... | Yes |
Corollary 9.34 If \( R \) is a quasilocal ring, then any finitely generated projective left \( R \) -module is free. | 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. | No |
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