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Proposition 9.3. If \( \mathcal{M} : {M}_{1} \supseteq {M}_{2} \supseteq \cdots \) is a filtration on an R-module \( M \) , then \( {\widehat{M}}_{j} = \left\{ {\left( {{x}_{1} + {M}_{1},\ldots ,{x}_{i} + {M}_{i},\ldots }\right) \in {\widehat{M}}_{\mathcal{M}} \mid {x}_{j} \in {M}_{j}}\right\} \) is a submodule of \( {... | Proof. Let \( N = \widehat{M} \) . The alternate description of \( {\widehat{M}}_{j} \) shows that \( {\widehat{M}}_{1} \supseteq \) \( {\widehat{M}}_{2} \supseteq \cdots \) . Also, \( {\widehat{M}}_{j} \) is a submodule of \( \widehat{M} \), since it is the kernel of the homomorphism \( \left( {{x}_{1} + {M}_{1},{x}_{... | Yes |
Proposition 9.5. If \( \varphi : M \rightarrow N \) is surjective, then \( \widehat{\varphi } : {\widehat{M}}_{\mathfrak{a}} \rightarrow {\widehat{N}}_{\mathfrak{a}} \) is surjective. | Proof. Since \( {\mathfrak{a}}^{i}N \) is generated by all \( {ry} \) with \( r \in {\mathfrak{a}}^{i} \) and \( y \in N \) we have \( {\mathfrak{a}}^{i}N = \varphi \left( {{\mathfrak{a}}^{i}M}\right) \) . Let \( \left( {{y}_{1} + \mathfrak{a}N,{y}_{2} + {\mathfrak{a}}^{2}N,\ldots }\right) \in {\widehat{N}}_{\mathfrak{... | Yes |
Lemma 9.7. Let \( \mathcal{M} : {M}_{1} \supseteq {M}_{2} \supseteq \cdots \) and \( \mathcal{N} : {N}_{1} \supseteq {N}_{2} \supseteq \cdots \) be filtrations on \( M \) such that \( {M}_{i} \supseteq {N}_{i} \) and every \( {N}_{i} \) contains some \( {M}_{j} \) . The homomorphisms \( {\rho }_{i} : M/{N}_{i} \rightar... | Proof. The homomorphism \( \widehat{\rho } : {\widehat{M}}_{\mathcal{N}} \rightarrow {\widehat{M}}_{\mathcal{M}} \) induced by \( {\left( {\rho }_{i}\right) }_{i \in I} \) sends \( x = \left( {{x}_{1} + {N}_{1},{x}_{2} + {N}_{2},\ldots }\right) \in {\widehat{M}}_{\mathcal{N}} \) to \( \left( {{x}_{1} + {M}_{1},{x}_{2} ... | Yes |
Proposition 9.8. Let \( R \) be a commutative Noetherian ring, let \( \mathfrak{a} \) be an ideal of \( R \), and let \( M \) be a finitely generated \( R \) -module. There is an isomorphism \( {\widehat{M}}_{\mathfrak{a}} \cong {\widehat{R}}_{\mathfrak{a}}{ \otimes }_{R}M \), which is natural in \( M \) . | Proof. For every \( i > 0 \) there is an isomorphism \( R/{\mathfrak{a}}^{i}{ \otimes }_{R}M \cong M/{\mathfrak{a}}^{i}M \) , which sends \( \left( {r + {\mathfrak{a}}^{i}}\right) \otimes x \) to \( {rx} + {\mathfrak{a}}^{i}M \) and is natural in \( M \) . Hence the projections \( \widehat{R} \rightarrow R/{\mathfrak{a... | Yes |
Corollary 9.9. If \( R \) is Noetherian, then \( {\widehat{R}}_{\mathfrak{a}} \) is a flat \( R \) -module, for every ideal a of \( R \) . | Proof. If \( \mu : A \rightarrow B \) is a monomorphism of \( R \) -modules, and \( A, B \) are finitely generated, then \( {\widehat{R}}_{\mathfrak{a}} \otimes \mu \) is a monomorphism, by 9.6 and 9.8. \( ▱ \) | No |
Proposition 1.1. Every chain transformation \( \varphi : \mathcal{A} \rightarrow \mathcal{B} \) induces a homomorphism \( {H}_{n}\left( \varphi \right) : {H}_{n}\left( \mathcal{A}\right) \rightarrow {H}_{n}\left( \mathcal{B}\right) \), which sends cls \( z \) to cls \( {\varphi }_{n}\left( z\right) \) for all \( z \in ... | Proof. Since \( {\partial }_{n}{\varphi }_{n} = {\varphi }_{n - 1}{\partial }_{n} \) for all \( n \) we have \( {\varphi }_{n}\left( {\operatorname{Im}{\partial }_{n + 1}^{\mathcal{A}}}\right) \subseteq \operatorname{Im}{\partial }_{n + 1}^{\mathcal{B}} \) and \( {\varphi }_{n}\left( {\operatorname{Ker}{\partial }_{n}^... | Yes |
Proposition 1.2. If \( \varphi \) and \( \psi \) are homotopic, then \( {H}_{n}\left( \varphi \right) = {H}_{n}\left( \psi \right) \) for all \( n \) . | Proof. If \( {\varphi }_{n} - {\psi }_{n} = {\partial }_{n + 1}{\sigma }_{n} + {\sigma }_{n - 1}{\partial }_{n} \) for all \( n \), and \( z \in \operatorname{Ker}{\partial }_{n} \), then \( {\varphi }_{n}\left( z\right) - {\psi }_{n}\left( z\right) = \left( {{\partial }_{n + 1}{\sigma }_{n} + {\sigma }_{n - 1}{\partia... | Yes |
Theorem 1.3 (Exact Homology Sequence). Every short exact sequence \( \mathcal{E} \) : \( 0 \rightarrow \mathcal{A} \rightarrow \mathcal{B} \rightarrow \mathcal{C} \rightarrow 0 \) of chain complexes induces an exact sequence\n\n\[ \cdots {H}_{n + 1}\left( \mathcal{C}\right) \rightarrow {H}_{n}\left( \mathcal{A}\right) ... | Proof. Exactness at \( {H}_{n}\left( \mathcal{B}\right) \) is proved by diagram chasing in: \n\nFirst, \( {H}_{n}\left( \psi \right) \left( {{H}_{n}\left( \varphi \right) \left( {\operatorname{cls}a}\right) }\right) ... | Yes |
In Theorem 1.3, a connecting homomorphism \( {H}_{n + 1}\left( \mathcal{C}\right) \overset{{\chi }_{n + 1}}{ \rightarrow } \) \( {H}_{n}\left( \mathcal{A}\right) \) is well defined by \( {\chi }_{n + 1}\operatorname{cls}c = \operatorname{cls}a \) whenever \( c \in \operatorname{Ker}{\partial }_{n + 1}, c = {\psi }_{n +... | Proof. First, \( {\partial }_{n + 1}b = {\varphi }_{n}a \) implies \( {\varphi }_{n - 1}{\partial }_{n}a = {\partial }_{n}{\varphi }_{n}a = {\partial }_{n}{\partial }_{n + 1}b = 0 \) , \( {\partial }_{n}a = 0 \) since \( {\varphi }_{n - 1} \) is injective, and \( a \in \operatorname{Ker}{\partial }_{n} \) . Assume that... | Yes |
Theorem 1.6 (Exact Cohomology Sequence). Let \( G \) be a left \( R \) -module and let \( \mathcal{E} : 0 \rightarrow \mathcal{A} \rightarrow \mathcal{B} \rightarrow \mathcal{C} \rightarrow 0 \) be a short exact sequence of chain complexes of left \( R \) -modules. If every \( {A}_{n} \) is injective, or if every \( {C... | Proof. If every \( {A}_{n} \) is injective, or if every \( {C}_{n} \) is projective, then the sequence \( 0 \rightarrow {A}_{n} \rightarrow {B}_{n} \rightarrow {C}_{n} \rightarrow 0 \) splits; hence the sequence \[ 0 \rightarrow {\operatorname{Hom}}_{R}\left( {{C}_{n}, G}\right) \rightarrow {\operatorname{Hom}}_{R}\lef... | Yes |
Theorem 1.7 (Exact Cohomology Sequence). Let \( \mathcal{A} \) be a chain complex of left \( R \) -modules and let \( \mathcal{E} : 0 \rightarrow G \rightarrow {G}^{\prime } \rightarrow {G}^{\prime \prime } \rightarrow 0 \) be a short exact sequence of left \( R \) -modules. If every \( {A}_{n} \) is projective, then \... | Proof. If every \( {A}_{n} \) is projective, then the sequence \[ 0 \rightarrow {\operatorname{Hom}}_{R}\left( {{A}_{n}, G}\right) \rightarrow {\operatorname{Hom}}_{R}\left( {{A}_{n},{G}^{\prime }}\right) \rightarrow {\operatorname{Hom}}_{R}\left( {{A}_{n},{G}^{\prime \prime }}\right) \rightarrow 0 \] is exact, by XI.2... | Yes |
Lemma 1.8. If \( \mathcal{A}\overset{\varphi }{ \rightarrow }\mathcal{B}\overset{\psi }{ \rightarrow }{\mathcal{C}}^{\prime } \) in Theorem 1.6, then the connecting homomorphism \( {H}^{n}\left( {\mathcal{A}, G}\right) \overset{{\chi }^{n}}{ \rightarrow }{H}^{n + 1}\left( {\mathcal{C}, G}\right) \) is well defined by \... | Proof. The map \( {\chi }^{n} \) is induced by \( {\operatorname{Hom}}_{R}\left( {\mathcal{C}, G}\right) \overset{{\psi }^{ * }}{ \rightarrow }{\operatorname{Hom}}_{R}\left( {\mathcal{B}, G}\right) \overset{{\varphi }^{ * }}{ \rightarrow } \) \( {\operatorname{Hom}}_{R}\left( {\mathcal{A}, G}\right) \), in which \( {\o... | Yes |
Lemma 1.9. If \( G\overset{\varphi }{ \rightarrow }{G}^{\prime }\overset{\psi }{ \rightarrow }{G}^{\prime \prime } \) in Theorem 1.7, then the connecting homomorphism \( {H}^{n}\left( {\mathcal{A},{G}^{\prime \prime }}\right) \overset{{\chi }^{n}}{ \rightarrow }{H}^{n + 1}\left( {\mathcal{A}, G}\right) \) is well defin... | Proof. The map \( {\chi }^{n} \) is induced by \( {\operatorname{Hom}}_{R}\left( {\mathcal{A}, G}\right) \overset{{\varphi }_{ * }}{ \rightarrow }{\operatorname{Hom}}_{R}\left( {\mathcal{A},{G}^{\prime }}\right) \overset{{\psi }_{ * }}{ \rightarrow } \) \( {\operatorname{Hom}}_{R}\left( {\mathcal{A},{G}^{\prime \prime ... | Yes |
Lemma 2.1. Given a diagram with exact rows (solid arrows) in which \( P \) is projective, there exist homomorphisms \( \alpha \) and \( \beta \) (dotted arrows) that make the diagram commutative. | Proof. Since \( P \) is projective, \( {\gamma \sigma } \) factors through the epimorphism \( {\sigma }^{\prime } : {\gamma \sigma } = \) \( {\sigma }^{\prime }\beta \) for some homomorphism \( \beta : P \rightarrow {B}^{\prime } \) . Then \( {\sigma }^{\prime }{\beta \mu } = {\gamma \sigma \mu } = 0 \) ; hence \( {\be... | No |
Theorem 2.2 (Comparison Theorem). Let \( \mathcal{P}\overset{\varepsilon }{ \rightarrow }A \) and \( \mathcal{Q}\overset{\zeta }{ \rightarrow }B \) be projective resolutions, and let \( \varphi : A \rightarrow B \) be a homomorphism. There is a chain transformation \( \bar{\varphi } = {\left( {\varphi }_{n}\right) }_{n... | Proof. Since \( {P}_{0} \) is projective, and \( \zeta : {Q}_{0} \rightarrow B \) is surjective, \( {\varphi \varepsilon } \) factors through \( \zeta \), and \( {\varphi \varepsilon } = \zeta {\varphi }_{0} \) for some \( {\varphi }_{0} : {P}_{0} \rightarrow {Q}_{0} \). From this auspicious start \( {\varphi }_{n} \) ... | Yes |
Lemma 2.3. The diagram below (solid arrows) with exact row and columns, in which \( P \) and \( R \) are projective, can be completed to a commutative \( 3 \times 3 \) diagram (all arrows) with exact rows and columns, in which \( Q \) is projective. | Proof. Since the middle row must split we may as well let \( Q = P \oplus R \), with \( \kappa : p \mapsto \left( {p,0}\right) \) and \( \pi : \left( {p, r}\right) \mapsto r \) . Then \( Q \) is projective and the middle row is exact. Maps \( Q \rightarrow B \) are induced by homomorphisms of \( P \) and \( R \) into \... | Yes |
Proposition 2.4. For every short exact sequence \( 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 \) of modules, and projective resolutions \( \mathcal{P} \rightarrow A,\mathcal{R} \rightarrow C \), there exist a projective resolution \( \mathcal{Q} \rightarrow B \) and a short exact sequence \( 0 \rightarro... | Proof. By 2.3, applied to the given sequence and to \( 0 \rightarrow {K}_{0} = \operatorname{Ker}\varepsilon \rightarrow \) \( {P}_{0}\overset{\varepsilon }{ \rightarrow }A \rightarrow 0,0 \rightarrow {M}_{0} = \operatorname{Ker}\eta \rightarrow {R}_{0}\overset{\eta }{ \rightarrow }C \rightarrow 0 \), there is a commut... | Yes |
Proposition 2.6. For every commutative diagram\n\n\n\nwith short exact rows, projective resolutions \( \mathcal{P} \rightarrow A,\mathcal{R} \rightarrow C,{\mathcal{P}}^{\prime } \rightarrow {A}^{\prime } \) , \( {\m... | Proof. This follows from repeated applications of Lemma 2.5, just as Proposition 2.4 follows from repeated applications of Lemma 2.3. \( ▱ \) | No |
Theorem 2.8 (Comparison Theorem). Let \( A\overset{\eta }{ \rightarrow }\mathcal{J} \) and \( B\overset{\zeta }{ \rightarrow }\mathcal{K} \) be injective resolutions, and let \( \varphi : A \rightarrow B \) be a homomorphism. There is a chain transformation \( {\left( {\varphi }^{n}\right) }_{n \geqq 0} : \mathcal{J} \... |  | No |
Proposition 2.9. For every short exact sequence \( 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 \) of modules, and injective resolutions \( A \rightarrow \mathcal{J}, C \rightarrow \mathcal{L} \), there exist an injective resolution \( B \rightarrow \mathcal{K} \) and a short exact sequence \( 0 \rightarro... |  | No |
Proposition 2.10. For every commutative diagram\n\n\n\nwith short exact rows, injective resolutions \( A \rightarrow \mathcal{J}, C \rightarrow \mathcal{L},{A}^{\prime } \rightarrow {\mathcal{J}}^{\prime } \) , \( {C... | These results are obtained from 2.1, 2.2, 2.4, 2.6 by reversing all arrows. They have largely similar proofs, which may safely be left to readers as exercises. | No |
Proposition 3.1. Let \( F \) be a covariant additive functor. Assign a projective resolution \( {\mathcal{P}}^{A} \rightarrow A \) to every module \( A \) . For every \( n \geqq 0 \), let \( {L}_{n}A = {H}_{n}\left( {F{\mathcal{P}}^{A}}\right) \) ; for every module homomorphism \( \varphi : A \rightarrow B \), let \( {... | Proof. If \( \bar{\varphi } \) and \( \bar{\psi } \) both lift \( \varphi \), then \( \bar{\varphi } \) and \( \bar{\psi } \) are homotopic, by 2.2; since \( F \) is additive, \( F\bar{\varphi } \) and \( F\bar{\psi } \) are homotopic, and \( {H}_{n}\left( {F\bar{\varphi }}\right) = {H}_{n}\left( {F\bar{\psi }}\right) ... | Yes |
Theorem 3.2. Let \( F \) be a covariant additive functor.\n\n(1) If \( P \) is projective, then \( \left( {{L}_{0}F}\right) P \cong {FP} \) and \( \left( {{L}_{n}F}\right) P = 0 \) for all \( n > 0 \) . | Proof. (1). If \( P \) is projective, then \( P \) has a projective resolution\n\n\[ \begin{matrix} \mathcal{P}\overset{\varepsilon }{ \rightarrow }P : \\ \cdots \rightarrow 0 \rightarrow 0 \rightarrow P\overset{\varepsilon }{ \rightarrow }P \rightarrow 0 \end{matrix} \]\n\nin which \( \varepsilon = {1}_{P} \) . Then \... | Yes |
Theorem 3.3. Let \( F \) be a right exact, covariant additive functor, and let \( {G}_{0},{G}_{1},\ldots ,{G}_{n},\ldots \) be a positive connected sequence of covariant functors. For every natural transformation \( {\varphi }_{0} : {G}_{0} \rightarrow F \) there exist unique natural transformations \( {\varphi }_{n} :... | Proof. We construct \( {\varphi }_{n} \) recursively. For every module \( A \) choose a projective presentation \( {\mathcal{E}}^{A} : 0 \rightarrow K\overset{\mu }{ \rightarrow }P \rightarrow A \rightarrow 0 \) (with \( P \) projective). Since \( {\varphi }_{0} \) is natural, and \( F,{L}_{1},\ldots ,{L}_{n},\ldots \)... | Yes |
Theorem 3.4. Let \( {G}_{0},{G}_{1},\ldots ,{G}_{n},\ldots \) be a positive connected sequence of covariant functors. If \( {G}_{n}P = 0 \) whenever \( P \) is projective and \( n > 0 \), then, up to natural isomorphisms, \( {G}_{1},\ldots ,{G}_{n},\ldots \) are the left derived functors of \( {G}_{0} \) . | Proof. First, \( {G}_{0} \) is right exact, by definition. By 3.3, the identity \( {G}_{0} \rightarrow {G}_{0} \) induces natural transformations \( {\varphi }_{n} : {G}_{n} \rightarrow {L}_{n} \) to the derived functors \( {L}_{0} = \) \( {G}_{0},{L}_{1},\ldots ,{L}_{n},\ldots \) of \( {G}_{0} \), which form a commuta... | Yes |
Proposition 4.3. For every projective resolution \( \mathcal{P} \rightarrow A \) and injective resolution \( B \rightarrow \mathcal{J} \) there are natural isomorphisms\n\n\[ \n{\operatorname{Ext}}_{R}^{n}\left( {A, B}\right) \cong {H}^{n}\left( {{\operatorname{Hom}}_{R}\left( {\mathcal{P}, B}\right) }\right) \cong {H}... | This follows from the definition of Ext and the definitions of derived functors. | No |
Proposition 4.9. \( {\operatorname{Ext}}_{\mathbb{Z}}^{n}\left( {A, B}\right) = 0 \) for all \( n \geqq 2 \) and abelian groups \( A \) and \( B \) . | Proof. Let \( R \) be a PID. Every submodule of a free \( R \) -module is free. Hence every \( R \) -module \( A \) has a free resolution \( \mathcal{F} : 0 \rightarrow {F}_{1} \rightarrow {F}_{0} \rightarrow A \) in which \( {F}_{n} = 0 \) for all \( n \geqq 2 \) . Then \( {\operatorname{Hom}}_{R}\left( {{F}_{n}, B}\r... | Yes |
Theorem 5.1. For every right \( R \) -module \( A \) and left \( R \) -module \( B \) and every \( n > 0 \) there is an isomorphism \( {\operatorname{LTor}}_{n}^{R}\left( {A, B}\right) \cong {\operatorname{RTor}}_{n}^{R}\left( {A, B}\right) \), which is natural in \( A \) and \( B \) . | The proof of Theorem 5.1 is similar to that of Theorem 4.1, and may be entrusted to readers. | No |
Proposition 5.3. For every projective resolution \( \mathcal{P} \rightarrow A \) and \( \mathcal{Q} \rightarrow B \) there are natural isomorphisms\n\n\[ \n{\operatorname{Tor}}_{n}^{R}\left( {A, B}\right) \cong {H}_{n}\left( {\mathcal{P}{ \otimes }_{R}B}\right) \cong {H}_{n}\left( {A{ \otimes }_{R}\mathcal{Q}}\right) .... | This follows from the definition of Tor and the definition of left derived functors. | No |
For every right \( R \) -module \( A \) and left \( R \) -module \( B \) there is an isomorphism \( {\operatorname{Tor}}_{n}^{R}\left( {A, B}\right) \cong {\operatorname{Tor}}_{n}^{{R}^{\mathrm{{op}}}}\left( {B, A}\right) \), which is natural in \( A \) and \( B \) . | There is an isomorphism \( A{ \otimes }_{R}B \cong B{ \otimes }_{{R}^{\text{op }}}A, a \otimes b \mapsto b \otimes a \) , which is natural in \( A \) and \( B \) . If \( \mathcal{P} \rightarrow A \) is a projective resolution of \( A \) as a right \( R \) -module, then \( \mathcal{P} \rightarrow A \) is a projective re... | Yes |
Proposition 5.8. \( {\operatorname{Tor}}_{n}^{\mathbb{Z}}\left( {A, B}\right) = 0 \) for all \( n \geqq 2 \) and abelian groups \( A \) and \( B \) . | Proof. More generally, let \( R \) be a PID. Every submodule of a free \( R \) -module is free. Hence every \( R \) -module \( A \) has a free resolution \( \mathcal{F} : 0 \rightarrow {F}_{1} \rightarrow {F}_{0} \rightarrow A \) in which \( {F}_{n} = 0 \) for all \( n \geqq 2 \) . Then \( {F}_{n}{ \otimes }_{R}B = 0 \... | Yes |
Proposition 5.10. For a right \( R \) -module \( A \) the following properties are equivalent: (1) \( A \) is flat; (2) \( {\operatorname{Tor}}_{1}^{R}\left( {A, B}\right) = 0 \) for every left \( R \) -module \( B \) ; (3) \( {\operatorname{Tor}}_{n}^{R}\left( {A, B}\right) = 0 \) for every left \( R \) -module \( B \... | Proof. Let \( \mathcal{Q} \rightarrow B \) be a projective resolution. If \( A \) is flat, then \( A{ \otimes }_{R} - \) is exact, the sequence \( \cdots \rightarrow A{ \otimes }_{R}{Q}_{1} \rightarrow A{ \otimes }_{R}{Q}_{0} \rightarrow A{ \otimes }_{R}B \rightarrow 0 \) is exact, and \( {\operatorname{Tor}}_{n}^{R}\l... | Yes |
Proposition 5.11. A right R-module \( A \) is flat if and only if \( A{ \otimes }_{R}L \rightarrow A{ \otimes }_{R}{}_{R}R \) is injective for every left ideal \( L \) of \( R \) . | Proof. Assume that \( A{ \otimes }_{R}L \rightarrow A{ \otimes }_{R}R \) is injective for every left ideal \( L \) of \( R \) . Then \( 0 \rightarrow L \rightarrow {}_{R}R \rightarrow R/L \rightarrow 0 \) induces an exact sequence\n\n\[ \n{\operatorname{Tor}}_{1}^{R}\left( {A,{}_{R}R}\right) \rightarrow {\operatorname{... | Yes |
Theorem 6.1 (Universal Coefficient Theorem for Cohomology). Let \( R \) be a left hereditary ring; let \( \mathcal{C} \) be a complex of projective left \( R \) -modules, and let \( M \) be any left \( R \) -module. For every \( n \in \mathbb{Z} \) there is an exact sequence\n\n\[ 0 \rightarrow {\operatorname{Ext}}_{R}... | Proof. Every \( {\partial }_{n} : {C}_{n} \rightarrow {C}_{n - 1} \) induces a commutative square \n\nwhere \( {Z}_{n - 1} = \operatorname{Ker}{\partial }_{n - 1},{\iota }_{n - 1} \) and \( {\kappa }_{n - 1} \) are i... | Yes |
Theorem 6.3 (Universal Coefficient Theorem for Homology). Let \( R \) be a right hereditary ring; let \( \mathcal{C} \) be a complex of projective right \( R \) -modules, and let \( M \) be any left \( R \) -module. For every \( n \in \mathbb{Z} \) there is an exact sequence\n\n\[ 0 \rightarrow {H}_{n}\left( \mathcal{C... | Proof. The commutative square and exact sequences \n\n\[ 0 \rightarrow {Z}_{n}\overset{{\iota }_{n}}{ \rightarrow }{C}_{n}\overset{{\pi }_{n}}{ \rightarrow }{B}_{n - 1} \rightarrow 0,\;0 \rightarrow {B}_{n}\overset{{... | Yes |
Proposition 7.2. If \( G \) is finite and \( n \geqq 1 \), then \( {H}^{n}\left( {G, A}\right) \) is torsion, and the order of every element of \( {H}^{n}\left( {G, A}\right) \) divides the order of \( G \) ; if \( A \) is divisible and torsion-free, then \( {H}^{n}\left( {G, A}\right) = 0 \) . | Proof. Let \( u \) be an \( n \) -cochain. Define an \( \left( {n - 1}\right) \) -cochain \( {v}_{{x}_{1},\ldots ,{x}_{n - 1}} = \) \( \mathop{\sum }\limits_{{x \in G}}{u}_{{x}_{1},\ldots ,{x}_{n - 1}, x} \) . Then\n\n\[ \mathop{\sum }\limits_{{x \in G}}{\left( {\delta }^{n}u\right) }_{{x}_{1},\ldots ,{x}_{n}, x} = {x}... | Yes |
Proposition 7.3. For every abelian group \( A \), there is a one-to-one correspondence between group actions of \( G \) on \( A \) by automorphisms, and [unital] \( \mathbb{Z}\left\lbrack G\right\rbrack \) -module structures on \( A \) . | This is similar to Proposition IX.7.3. A group action \( \left( {g, a}\right) \mapsto {ga} \) of \( G \) extends to a module action \( \left( {\mathop{\sum }\limits_{{x \in G}}{k}_{x}x}\right) a = \mathop{\sum }\limits_{{x \in G}}{k}_{x}{xa} \) of \( \mathbb{Z}\left\lbrack G\right\rbrack \) ; readers will verify that t... | No |
Proposition 7.5. For every \( G \) -module \( A \), there is an isomorphism \( {C}^{n}\left( {G, A}\right) \) \( \cong {\operatorname{Hom}}_{\mathbb{Z}\left\lbrack G\right\rbrack }\left( {{B}_{n}, A}\right) \), which is natural in \( A \), such that the square\n\n\[ \n{C}^{n}\left( {G, A}\right) \overset{ \cong }{ \rig... | Proof. Since \( {B}_{n} \) is free on all \( \left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) such that \( {x}_{i} \neq 1 \) for all \( i \), every \( n \) -cochain \( u \) induces a unique homomorphism \( {\theta }^{n}u : {B}_{n} \rightarrow A \) such that\n\n\[ \n\left( {{\theta }^{n}u}\right) \left\lbrack {{x}... | Yes |
Lemma 7.9. If \( G \) is the free group on \( {\left( {x}_{i}\right) }_{i \in I} \), then \( I\left( G\right) \) is a free \( G \) -module, with basis \( {\left( {x}_{i} - 1\right) }_{i \in I} \) . | Proof. Let \( M \) be the submodule of \( I\left( G\right) \) generated by all \( {x}_{i} - 1 \) . We show by induction on the length of the reduced word \( x \in G \) that \( x - 1 \in M \) for all \( x \in G \) : indeed, if \( x \neq 1 \), then either \( x = {x}_{i}y \) or \( x = {x}_{i}^{-1}y \), where \( y \) is sh... | Yes |
Proposition 7.10. \( {H}^{n}\left( {G, A}\right) = 0 \) for all \( n \geqq 2 \) when \( G \) is a free group. | Proof. By Lemma 7.9, \( \cdots \rightarrow 0 \rightarrow 0 \rightarrow I\left( G\right) \rightarrow \mathbb{Z}\left\lbrack G\right\rbrack \rightarrow \mathbb{Z} \) is a projective resolution of \( \mathbb{Z} \) ; hence \( {\operatorname{Ext}}_{\mathbb{Z}\left\lbrack G\right\rbrack }^{n}\left( {\mathbb{Z}, A}\right) = 0... | Yes |
Proposition 8.1. For every module \( B \) and \( n \geqq 1 \) there are isomorphisms\n\n\[{\operatorname{Ext}}^{n + 1}\left( {A, B}\right) \cong {\operatorname{Ext}}^{n}\left( {{K}_{0}, B}\right) \cong {\operatorname{Ext}}^{n - 1}\left( {{K}_{1}, B}\right) \cong \cdots \cong {\operatorname{Ext}}^{1}\left( {{K}_{n - 1},... | Proof. Since all \( {P}_{m} \) are projective, Theorem 4.4 and the exact sequences \( 0 \rightarrow {K}_{0} \rightarrow {P}_{0} \rightarrow A \rightarrow 0,0 \rightarrow {K}_{m} \rightarrow {P}_{m} \rightarrow {K}_{m - 1} \rightarrow 0 \) yield exact sequences that are natural in \( B \), for every \( k, m \geqq 1 \) :... | Yes |
Proposition 8.2. If \( {K}_{0},{K}_{1},\ldots \) and \( {L}_{0},{L}_{1},\ldots \) are the syzygies of a module \( A \) in two projective resolutions \( \mathcal{P} \rightarrow A,\mathcal{Q} \rightarrow A \) of \( A \), then \( {K}_{n} \) and \( {L}_{n} \) are projectively equivalent for all \( n \geqq 0 \) . | Proof. First we prove Schanuel's lemma: in the diagram with exact rows (solid arrows, next page), if \( P \) and \( Q \) are projective and \( \theta \) is an isomorphism, then \( P \oplus L \cong Q \oplus K : \n\n\n... | Yes |
Proposition 8.3. For a module \( A \) the following conditions are equivalent:\n\n(1) A has a projective resolution \( 0 \rightarrow {P}_{n} \rightarrow \cdots \rightarrow {P}_{0} \rightarrow A \rightarrow 0 \) ;\n\n(2) \( {\operatorname{Ext}}^{m}\left( {A, B}\right) = 0 \) for all \( m \geqq n + 1 \) and all modules \... | Proof. (3) implies (4): by 8.1, \( {\operatorname{Ext}}^{1}\left( {{K}_{n - 1}, B}\right) = {\operatorname{Ext}}^{n + 1}\left( {A, B}\right) = 0 \) for all \( B \) ; hence \( {K}_{n - 1} \) is projective, by 4.5. Clearly (1) implies (2),(2) implies (3),(4) implies (5), and (5) implies (1). \( ▱ \) | Yes |
Proposition 8.4. If \( 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 \) is exact, then:\n\n(1) \( \operatorname{pd}B \leqq \max \left( {\operatorname{pd}A,\operatorname{pd}C}\right) \) ;\n\n(2) \( \operatorname{pd}A \leqq \max \left( {\operatorname{pd}B,\operatorname{pd}C - 1}\right) \) ;\n\n(3) \( \operato... | Proof. (1). If \( \operatorname{pd}A \) , \( \operatorname{pd}C \leqq n \), then \( 0 = {\operatorname{Ext}}^{n + 1}\left( {C, M}\right) \rightarrow {\operatorname{Ext}}^{n + 1}\left( {B, M}\right) \) \( \rightarrow {\operatorname{Ext}}^{n + 1}\left( {A, M}\right) = 0 \) is exact for every module \( M \) ; hence \( {\o... | Yes |
Proposition 8.5. \( \operatorname{pd}\left( {{\bigoplus }_{i \in I}{A}_{i}}\right) = \) l.u.b. \( {}_{i \in I}\operatorname{pd}{A}_{i} \) . | The proof is an exercise. | No |
Theorem 9.2. For any ring \( R \), lgld \( R\left\lbrack X\right\rbrack = \lg \operatorname{ld}R + 1 \) . | The proof of Theorem 9.2 uses change of ring constructions. Since \( R \) is a subring of \( R\left\lbrack X\right\rbrack \), every left \( R\left\lbrack X\right\rbrack \) -module is, in particular, a left \( R \) -module. Conversely, every left \( R \) -module \( A \) has a universal left \( R\left\lbrack X\right\rbra... | No |
Lemma 9.3. Let \( A \) be a left \( R \) -module. Every element of \( \bar{A} = R\left\lbrack X\right\rbrack { \otimes }_{R}A \) can be written uniquely as a sum \( \mathop{\sum }\limits_{k}{X}^{k} \otimes {a}_{k} \), where \( {a}_{k} \in A \) for all \( k \) and \( {a}_{k} = 0 \) for almost all \( k \) . So \( \bar{A}... | Proof. As a right \( R \) -module, \( R\left\lbrack X\right\rbrack \) is free with basis \( 1, X,{X}^{2},\ldots \) ; hence 9.3 follows from XI.5.9. \( ▱ \) | No |
Lemma 9.4. If \( P \) is a projective \( R \) -module, then \( \bar{P} \) is a projective \( R\left\lbrack X\right\rbrack \) -module. Conversely, a projective \( R\left\lbrack X\right\rbrack \) -module is also projective as an \( R \) -module. | Proof. If \( F \cong {\bigoplus }_{i \in I}{}_{R}R \) is a free left \( R \) -module, then\n\n\[ \bar{F} \cong {\bigoplus }_{i \in I}R\left\lbrack X\right\rbrack { \otimes }_{RR}R \cong {\bigoplus }_{i \in I}R\left\lbrack X\right\rbrack \]\n\nis a free left \( R\left\lbrack X\right\rbrack \) -module. If now \( P \) is ... | Yes |
Lemma 9.5. \( {\operatorname{pd}}_{R\left\lbrack X\right\rbrack }\bar{A} = {\operatorname{pd}}_{R}A \) for every left \( R \) -module \( A \) . | Proof. If \( {\operatorname{pd}}_{R}A \leqq n \), then \( A \) has a projective resolution \( 0 \rightarrow {P}_{n} \rightarrow \) \( \cdots \rightarrow {P}_{0} \rightarrow A \rightarrow 0 \) (over \( R \) ); since \( R\left\lbrack X\right\rbrack \) is flat as a right \( R \) -module, \( 0 \rightarrow {\bar{P}}_{n} \ri... | Yes |
Lemma 9.6. For every \( R\left\lbrack X\right\rbrack \) -module \( A \) there is an exact sequence \( 0 \rightarrow \bar{A} \rightarrow \bar{A} \rightarrow A \rightarrow 0. \) | Proof. By 9.3, every element of \( \bar{A} \) is a sum \( \mathop{\sum }\limits_{{0 \leqq k \leqq m}}{X}^{k} \otimes {a}_{k} \), where \( {a}_{k} \in A \) . Hence there is an \( R\left\lbrack X\right\rbrack \) -homomorphism \( \sigma : \bar{A} \rightarrow A \) such that \( \sigma \left( {{X}^{k} \otimes a}\right) = {X}... | Yes |
Corollary 9.7. lgld \( R\left\lbrack X\right\rbrack \leqq \operatorname{lgld}R + 1 \) . | Proof. If lgld \( R = \infty \), then, for every \( n < \infty \), some \( R \) -module \( A \) has \( {\operatorname{pd}}_{R}A \geqq \) \( n \), whence \( {\operatorname{pd}}_{R\left\lbrack X\right\rbrack }\bar{A} \geqq n \), by 9.5; thus lgld \( R\left\lbrack X\right\rbrack = \infty \) . If lgld \( R = n < \infty \),... | Yes |
For every left \( R \) -module \( A \) there is a natural isomorphism \( {\operatorname{Hom}}_{R}\left( {A, R}\right) \cong {\operatorname{Ext}}_{R\left\lbrack X\right\rbrack }^{1}\left( {A, R\left\lbrack X\right\rbrack }\right) . | Let \( \mu : R\left\lbrack X\right\rbrack \rightarrow R\left\lbrack X\right\rbrack \) be multiplication by \( X \) . The exact sequence\n\n\[ 0 \rightarrow R\left\lbrack X\right\rbrack \overset{\mu }{ \rightarrow }R\left\lbrack X\right\rbrack \rightarrow R \rightarrow 0 \]\n\nof \( R\left\lbrack X\right\rbrack \) -modu... | Yes |
Lemma 9.9. \( {\operatorname{Ext}}_{R}^{n}\left( {A, R}\right) \cong {\operatorname{Ext}}_{R\left\lbrack X\right\rbrack }^{n + 1}\left( {A, R\left\lbrack X\right\rbrack }\right) \), for every \( R \) -module \( A \) . | Proof. First, we show that \( {\operatorname{Ext}}_{R\left\lbrack X\right\rbrack }^{n}\left( {P, R\left\lbrack X\right\rbrack }\right) = 0 \) for every projective \( R \) -module \( P \) and \( n > 0 \) . By 9.4,9.3, \( \bar{P} \) is a projective \( R\left\lbrack X\right\rbrack \) -module and \( \bar{P}/X\bar{P} \cong ... | Yes |
Theorem 1.2 (Homomorphism Theorem). If \( \varphi : A \rightarrow B \) is a homomorphism of \( R \) -algebras, then \( \operatorname{Im}\varphi \) is a subalgebra of \( B \), Ker \( \varphi \) is an ideal of \( A \), and \( A/\operatorname{Ker}\varphi \cong \operatorname{Im}\varphi \) ; in fact, there is a unique algeb... | Proof. The homomorphism theorems for rings and modules both yield the diagram above, with the same unique isomorphism \( \theta \) ; hence \( \theta \) is an algebra isomorphism. \( ▱ \) | No |
Theorem 1.3 (Factorization Theorem). Let \( I \) be a two-sided ideal of an R-algebra A. Every algebra homomorphism whose kernel contains I factors uniquely through the projection \( A \rightarrow A/I \) : | Proof. The factorization theorems for rings and modules both yield the diagram above, with the same unique homomorphism \( \psi \) ; hence \( \psi \) is an algebra homomorphism. \( ▱ \) | No |
Theorem 1.4 (Homomorphism Theorem). If \( \varphi : A \rightarrow B \) is a homomorphism of graded \( R \) -algebras, then \( \operatorname{Im}\varphi \) is a graded subalgebra of \( B \), Ker \( \varphi \) is a graded ideal of \( A \), and \( A/\operatorname{Ker}\varphi \cong \operatorname{Im}\varphi \) ; in fact, the... |  | No |
Proposition 2.4. Every module homomorphism of an R-module \( M \) into an \( R \) -algebra \( A \) extends uniquely to an algebra homomorphism of \( T\left( M\right) \) into \( A \) . | Proof. Let \( \varphi : M \rightarrow A \) be a module homomorphism. For every \( n \geqq 2 \) , multiplication in \( A \) yields an \( n \) -linear mapping \( \left( {{a}_{1},\ldots ,{a}_{n}}\right) \mapsto \varphi \left( {a}_{1}\right) \cdots \varphi \left( {a}_{n}\right) \) of \( {M}^{n} \) into \( A \), which induc... | Yes |
Corollary 2.5. Every R-algebra that is generated by a submodule \( M \) is isomorphic to a quotient algebra of \( T\left( M\right) \) . | Proof. If \( A = \langle \varphi \left( M\right) \rangle \) in the proof of 2.4, then \( \bar{\varphi } \) is surjective, by 2.1. \( ▱ \) | No |
If \( M \) is the free \( R \) -module on a set \( X \), then \( T\left( M\right) \) is the free \( R \) -algebra on the set \( X \) : every mapping of \( X \) into an \( R \) -algebra \( A \) extends uniquely to an algebra homomorphism of \( T\left( M\right) \) into \( A \) . | This follows from Proposition 2.4, since every mapping of \( X \) into an \( R \) -algebra \( A \) extends uniquely to a module homomorphism of \( M \) into \( A \) . \( ▱ \) | Yes |
Lemma 3.1. \( I \subseteq {\bigoplus }_{n \geqq 2}{T}^{n}\left( M\right) \) . | Proof. Let \( t \in {T}^{m}\left( M\right) \) and \( u \in {T}^{n}\left( M\right) \) . If \( m + n \geqq 2 \), then \( t \otimes u - u \otimes t \in \) \( {\bigoplus }_{n \geq 2}{T}^{n}\left( M\right) \) . If \( m + n < 2 \), then \( m = 0 \) or \( n = 0 \), and \( t \otimes u - u \otimes t = 0 \) since \( r \otimes t ... | Yes |
Proposition 3.3. Every module homomorphism of an \( R \) -module \( M \) into a commutative \( R \) -algebra \( A \) extends uniquely to an algebra homomorphism of \( S\left( M\right) \) into \( A \) . | Proof. Let \( \pi : T\left( M\right) \rightarrow S\left( M\right) \) be the projection and let \( \varphi : M \rightarrow A \) be a module homomorphism. By 2.4, \( \varphi \) extends to an algebra homomorphism \( \psi : T\left( M\right) \rightarrow A \) . Since \( A \) is commutative we have \( \psi \left( {t \otimes u... | Yes |
Corollary 3.5. If \( M \) is the free \( R \) -module on a totally ordered set \( X \), then \( S\left( M\right) \) is a free \( R \) -module, with a basis that consists of all \( {x}_{1}\cdots {x}_{n} \) with \( n \geqq 0 \) , \( {x}_{1},\ldots ,{x}_{n} \in X \), and \( {x}_{1} \leqq \cdots \leqq {x}_{n} \) . | Proof. The monomials of \( R\left\lbrack {\left( {X}_{i}\right) }_{i \in I}\right\rbrack \) constitute a basis of \( R\left\lbrack {\left( {X}_{i}\right) }_{i \in I}\right\rbrack \) as an \( R \) -module. Hence \( S\left( M\right) \) is a free \( R \) -module when \( M \) is a free \( R \) -module, by 3.4. If \( X = {\... | Yes |
Lemma 3.6. Let \( {S}^{n}\left( M\right) = {T}^{n}\left( M\right) /{I}_{n} \), where \( n \geqq 2 \) and \( {I}_{n} \) is the submodule of \( {T}^{n}\left( M\right) = M \otimes \cdots \otimes M \) generated by all \( {a}_{▟} \otimes \cdots \otimes {a}_{\sigma n} - {a}_{1} \otimes \cdots \otimes {a}_{n} \) , where \( {a... | Proof. Let \( \pi : {T}^{n}\left( M\right) \rightarrow {S}^{n}\left( M\right) \) be the projection. Then \( {\mu }_{n}\left( {{a}_{1},\ldots ,{a}_{n}}\right) = \) \( \pi \left( {{a}_{1} \otimes \cdots \otimes {a}_{n}}\right) \) is symmetric, by the choice of \( {I}_{n} \), and \( n \) -linear.\n\nLet \( v : {M}^{n} \ri... | Yes |
Proposition 3.7. For every \( R \) -module \( M, I \) is a graded ideal of \( T\left( M\right) \) , \( I = {\bigoplus }_{n \geqq 0}{I}_{n} \), and \( S\left( M\right) = T\left( M\right) /I = {\bigoplus }_{n \geqq 0}{S}^{n}\left( M\right) \) is a graded \( R \) -algebra. | Proof. Let \( {a}_{1},\ldots ,{a}_{n} \in M \), where \( n \geqq 2 \) . Since \( S\left( M\right) \) is commutative, \( {a}_{▟}\cdots {a}_{\sigma n} = {a}_{1}\cdots {a}_{n} \) for every permutation \( \sigma \), and \( {a}_{▟} \otimes \cdots \otimes {a}_{\sigma n} - {a}_{1} \otimes \) \( \cdots \otimes {a}_{n} \in I \)... | Yes |
Every module homomorphism \( \varphi \) of an \( R \) -module \( M \) into an \( R \) -algebra \( A \) in which \( \varphi \left( M\right) \) is anticommutative extends uniquely to an algebra homomorphism of \( \Lambda \left( M\right) \) into \( A \). | Proof. Let \( \pi : T\left( M\right) \rightarrow \Lambda \left( M\right) \) be the projection. By 2.4, \( \varphi \) extends to an algebra homomorphism \( \psi : T\left( M\right) \rightarrow A \) . Since \( \varphi \left( M\right) \) is anticommutative in \( A \) we have \( \psi \left( {a \otimes a}\right) = \varphi {\... | Yes |
Lemma 4.3. Let \( {\Lambda }^{n}\left( M\right) = {T}^{n}\left( M\right) /{J}_{n} \), where \( n \geqq 2 \) and \( {J}_{n} \) is the submodule of \( {T}^{n}\left( M\right) = M \otimes \cdots \otimes M \) generated by all \( {a}_{1} \otimes \cdots \otimes {a}_{n} \), where \( {a}_{1},\ldots ,{a}_{n} \in M \) and \( {a}_... | Proof. Let \( \pi : {T}^{n}\left( M\right) \rightarrow {\Lambda }^{n}\left( M\right) \) be the projection. Then \( {\mu }_{n}\left( {{a}_{1},\ldots ,{a}_{n}}\right) \) \( = \pi \left( {{a}_{1} \otimes \cdots \otimes {a}_{n}}\right) \) is alternating, by the choice of \( {J}_{n} \), and \( n \) -linear.\n\nLet \( v : {M... | Yes |
Proposition 4.4. For every \( R \) -module \( M, J \) is a graded ideal of \( T\left( M\right) \) , \( J = {\bigoplus }_{n \geqq 0}{J}_{n} \), and \( \Lambda \left( M\right) = T\left( M\right) /J = {\bigoplus }_{n \geqq 0}{\Lambda }^{n}\left( M\right) \) is a graded R-algebra. | Proof. Let \( {a}_{1},\ldots ,{a}_{m} \in M \), where \( m \geqq 2 \) . Since \( M \) is anticommutative in \( \Lambda \left( M\right) ,{a}_{i} = {a}_{j} \) for some \( i \neq j \) implies \( {a}_{1} \land \cdots \land {a}_{m} = \pm {a}_{i} \land {a}_{j} \land {a}_{1} \land \cdots \land {a}_{m} = \) 0 and \( {a}_{1} \o... | Yes |
Proposition 5.1. If \( A \) and \( B \) are \( R \) -algebras, then \( A \otimes B \) is an \( R \) -algebra, in which \( \left( {a \otimes b}\right) \left( {{a}^{\prime } \otimes {b}^{\prime }}\right) = a{a}^{\prime } \otimes b{b}^{\prime } \) for all \( a,{a}^{\prime } \in A \) and \( b,{b}^{\prime } \in B \) . | Proof. The mapping \( \left( {a, b,{a}^{\prime },{b}^{\prime }}\right) \mapsto a{a}^{\prime } \otimes b{b}^{\prime } \) of \( A \times B \times A \times B \) into \( A \otimes B \) is multilinear, since \( \otimes \) and the multiplications on \( A \) and \( B \) are bilinear. Hence there is a unique module homomorphis... | Yes |
Proposition 5.2. If \( \varphi : A \rightarrow {A}^{\prime } \) and \( \psi : B \rightarrow {B}^{\prime } \) are homomorphisms of \( R \) -algebras, then so is \( \varphi \otimes \psi : A \otimes B \rightarrow {A}^{\prime } \otimes {B}^{\prime } \) . | Proof. For all \( a,{a}^{\prime } \in A \) and \( b,{b}^{\prime } \in B \) ,\n\n\[ \left( {\varphi \otimes \psi }\right) \left( {\left( {a \otimes b}\right) \left( {{a}^{\prime } \otimes {b}^{\prime }}\right) }\right) = \varphi \left( {a{a}^{\prime }}\right) \otimes \psi \left( {b{b}^{\prime }}\right) \]\n\n\[ = \varph... | Yes |
Proposition 5.3. If \( A \) and \( B \) are commutative \( R \) -algebras, then \( A \otimes B \) is a commutative R-algebra, and is also an A-algebra and a B-algebra. | Proof. The algebra \( A \otimes B \) is commutative since its generators \( a \otimes b \) com-mmute with each other. Its multiplication is bilinear over \( A : \left( {at}\right) {t}^{\prime } = a\left( {t{t}^{\prime }}\right) = \) \( t\left( {a{t}^{\prime }}\right) \) for all \( a \in A \) and \( t,{t}^{\prime } \in ... | Yes |
Proposition 5.4. If \( A \) is free as an \( R \) -module, with basis \( {\left( {e}_{i}\right) }_{i \in I} \), then \( A \otimes B \) is free as a right \( B \) -module, with basis \( {\left( {e}_{i} \otimes 1\right) }_{i \in I} \) . | Proof. As \( R \) -modules, \( A \cong {\bigoplus }_{i \in I}{}_{R}R \) and \( A \otimes B \cong \left( {{\bigoplus }_{i \in I}{}_{R}R}\right) \otimes B \cong \) \( {\bigoplus }_{i \in I}B \) ; when \( a = \mathop{\sum }\limits_{{i \in I}}{r}_{i}{e}_{i} \), these isomorphisms send \( a \) to \( {\left( {r}_{i}\right) }... | Yes |
Proposition 5.6. Let \( A \) and \( B \) be \( R \) -algebras and let \( \iota : A \rightarrow A \otimes B \) , \( \kappa : B \rightarrow A \otimes B \) be the canonical homomorphisms. For every \( R \) -algebra \( C \) and algebra homomorphisms \( \varphi : A \rightarrow C,\psi : B \rightarrow C \) such that \( \varph... | Proof. If \( \chi \) is an algebra homomorphism and \( \chi \circ \iota = \varphi \) and \( \chi \circ \kappa = \psi \), then \( \chi \left( {a \otimes b}\right) = \chi \left( {\iota \left( a\right) \kappa \left( b\right) }\right) = \varphi \left( a\right) \psi \left( b\right) \) for all \( a, b \) ; therefore \( \chi ... | Yes |
Proposition 5.7 (Noether [1929]). Let \( A \) and \( B \) be \( R \) -algebras. For every abelian group \( M \) there is a one-to-one correspondence between the left \( A \) -, right \( B \) -bimodule structures on \( M \) (with the same actions of \( R \) ) and the left \( A \otimes {B}^{\mathrm{{op}}} \) - module str... | Proof. Let \( M \) be an left \( A \) -, right \( B \) -bimodule, with the same actions of \( R \) on \( M \), so that \( M \) is, in particular, an \( R \) -module. A left \( A \) -, right \( B \) -bimodule structure on \( M \) consists of ring homomorphisms \( \alpha : A \rightarrow {\operatorname{End}}_{\mathbb{Z}}\... | Yes |
Proposition 6.1. Let \( K \subseteq E \) be any field extension. Let \( \alpha \) be algebraic over \( K \) and let \( q = \operatorname{Irr}\left( {\alpha : K}\right) = {q}_{1}^{{m}_{1}}\cdots {q}_{r}^{{m}_{r}} \), where \( {q}_{1},\ldots ,{q}_{r} \in E\left\lbrack X\right\rbrack \) are distinct monic irreducible poly... | Proof. Readers will verify that there is an isomorphism \( E \otimes K\left\lbrack X\right\rbrack \cong E\left\lbrack X\right\rbrack \) that sends \( \gamma \otimes \left( {\mathop{\sum }\limits_{{n \geqq 0}}{a}_{n}{X}^{n}}\right) \) to \( \mathop{\sum }\limits_{{n \geqq 0}}\gamma {a}_{n}{X}^{n} \) . Hence the inclusio... | No |
Proposition 6.2. For every field extension \( K \subseteq E, E{ \otimes }_{K}K\left( {\left( {X}_{i}\right) }_{i \in I}\right) \) is a domain, whose field of fractions is isomorphic to \( E\left( {\left( {X}_{i}\right) }_{i \in I}\right) \) . | Proof. Just this once, let \( X = {\left( {X}_{i}\right) }_{i \in I} \) . As before, \( {a}_{m}{X}^{m} \) denotes the monomial \( {a}_{m}\mathop{\prod }\limits_{{i \in I}}{X}_{i}^{{m}_{i}} \) of \( K\left\lbrack X\right\rbrack \), and similarly in \( E\left\lbrack X\right\rbrack \) . Readers will verify that there is a... | No |
Proposition 6.3. If \( K \subseteq E \subseteq L \) and \( K \subseteq F \subseteq L \) are fields, then \( E \) and \( F \) are linearly disjoint over \( K \) if and only if the homomorphism \( E \otimes F \rightarrow {EF} \subseteq L \) is injective. | Proof. Let \( {\left( {\alpha }_{i}\right) }_{i \in I} \) be a basis of \( E \) over \( K \) . By 5.4, \( {\left( {\alpha }_{i} \otimes 1\right) }_{i \in I} \) is a basis of \( E \otimes F \) over \( F \), which \( \mu \) sends back to \( {\left( {\alpha }_{i}\right) }_{i \in I} \) . If \( \mu \) is injective, then \( ... | Yes |
Corollary 6.4. Let \( K \subseteq E \) and \( L \) be fields. If \( E \) and \( F \) are linearly disjoint over \( K \) and \( E{ \otimes }_{K}L \) is reduced, then \( {EF}{ \otimes }_{F}L \) is reduced. | Proof. By 6.3, \( E{ \otimes }_{K}F \rightarrow {EF} \) is injective, so that \( E{ \otimes }_{K}F \) is a domain; since \( E{ \otimes }_{K}F \) contains both \( E \) and \( F \), its field of fractions is \( {EF} \) . Now, every \( t \in {EF}{ \otimes }_{F}L \) is a finite sum \( t = \mathop{\sum }\limits_{{1 \leqq i ... | Yes |
Theorem 7.1 (Wedderburn). Let \( K \) be a field. A \( K \) -algebra \( A \) is a simple left Artinian \( K \) -algebra if and only if it is isomorphic to \( {M}_{n}\left( D\right) \) for some \( n > 0 \) and division \( K \) -algebra \( D \cong {\operatorname{End}}_{A}^{\mathrm{{op}}}\left( S\right) \), where \( S \) ... | Proof. Let \( A \) be a simple Artinian \( K \) -algebra. By IX.3.8, \( A \cong {M}_{n}\left( D\right) \) for some \( n > 0 \) and some division ring \( D \cong {\operatorname{End}}_{A}^{\mathrm{{op}}}\left( S\right) \), where \( S \) is a simple left \( A \) -module. Since \( K \) is central in \( A, S \) is, in parti... | Yes |
Theorem 7.3 (Noether [1929]). Over any field, the tensor product of two central simple algebras is a central simple algebra. | Proof. Let \( A \) and \( B \) be \( K \) -algebras. Buoyed by 5.5 we identify \( a \) and \( a \otimes 1 \) , \( b \) and \( 1 \otimes b \), for all \( a \in A \) and \( b \in B \), so that \( A \) and \( B \) become subalgebras of \( A{ \otimes }_{K}B \) . Theorem 7.3 then follows from a more detailed result: | No |
Lemma 7.4 (Noether [1929]). Let \( B \) be central simple. If \( A \) is simple, then \( A{ \otimes }_{K}B \) is simple. If \( A \) is central, then \( A{ \otimes }_{K}B \) is central. In fact:\n\n(1) if \( J \neq 0 \) is an ideal of \( A{ \otimes }_{K}B \), then \( J \cap A \neq 0 \) ;\n\n(2) every ideal of \( A{ \oti... | Proof. (1). Choose \( t = {a}_{1} \otimes {b}_{1} + \cdots + {a}_{m} \otimes {b}_{m} \in J \smallsetminus 0 \) (where \( {a}_{i} \in A \) , \( {b}_{i} \in B \) ) so that \( m \) is the least possible. Then \( {a}_{1},\ldots ,{a}_{m} \) are linearly independent over \( K \), and so are \( {b}_{1},\ldots ,{b}_{m} \) . In... | Yes |
Theorem 7.5 (Skolem-Noether [1929]). Let \( A \) be a simple \( K \) -algebra and let B be a central simple \( K \) -algebra, both of finite dimension over \( K \) . Any two homomorphisms \( \varphi ,\psi : A \rightarrow B \) are conjugate (there exists a unit \( u \) of \( B \) such that \( \psi \left( a\right) = {u\v... | Proof. Since \( {\dim }_{K}B \) is finite, then \( B \) is left Artinian; by 7.1, \( B \cong {M}_{n}\left( D\right) \) \( \cong {\operatorname{End}}_{D}\left( S\right) \) for some division \( K \) -algebra \( D \cong {\operatorname{End}}_{B}^{\text{op }}\left( S\right) \), where \( S \) is a simple left \( B \) -module... | Yes |
Theorem 7.6. Let \( A \) be a simple \( K \) -algebra of finite dimension over \( K \) and let \( B \) be a simple subalgebra of \( A \) . The centralizer \( C \) of \( B \) is a simple subalgebra of \( A \) . Moreover, \( B \) is the centralizer of \( C \) and \( {\dim }_{K}A = \left( {{\dim }_{K}B}\right) \left( {{\d... | Proof. As in the proof of 7.5, \( A \cong {M}_{n}\left( D\right) \cong {\operatorname{End}}_{D}\left( S\right) \) for some division \( K \) -algebra \( D \cong {\operatorname{End}}_{A}^{\mathrm{{op}}}\left( S\right) \), where \( S \) is a simple left \( A \) -module. Then \( S \) is a left \( A \) -, right \( D \) -bim... | Yes |
Theorem 7.8 (Frobenius [1877]). A division \( \mathbb{R} \) -algebra that has finite dimension over \( \mathbb{R} \) is isomorphic to \( \mathbb{R},\mathbb{C} \), or the quaternion algebra \( \mathbb{H} \) . | Proof. Let \( D \) be a division \( \mathbb{R} \) -algebra with center \( K \) and let \( F \supseteq K \) be a maximal subfield of \( D \), so that \( \mathbb{R} \subseteq K \subseteq F \subseteq D \) . By 7.7, \( {\dim }_{K}D = \) \( {\left( {\dim }_{K}F\right) }^{2} \) . Now, \( \left\lbrack {F : \mathbb{R}}\right\r... | Yes |
Proposition 1.1. The binary operation \( \land \) on a lower semilattice \( \left( {S, \leqq }\right) \) is idempotent \( \left( {x \land x = x\text{for all}x \in S}\right) \), commutative, associative, and order preserving \( \left( {x \leqq y\text{implies}x \land z \leqq y \land z\text{for all}z}\right) \) . Moreover... | The proof is an exercise. | No |
Proposition 1.4. If \( X \) is a set, and \( L \) is a set of subsets of \( X \) that is closed under intersections and contains \( X \), then \( L \), partially ordered by inclusion, is a lattice. | Proof. Let \( A, B \in L \) . Then \( A \cap B \in L \) is the g.l.b. of \( A \) and \( B \) . The l.u.b. of \( A \) and \( B \) is the intersection of all \( C \in L \) that contain \( A \cup B \) (including \( X \) ), which belongs to \( L \) by the hypothesis. \( ▱ \) | Yes |
Proposition 2.3. Relative to a closure map on a complete lattice \( L \), the set of all closed elements of \( L \) is closed under infimums and is a complete lattice. | The proof is an exercise. | No |
Proposition 2.4. If \( \left( {\alpha ,\beta }\right) \) is a Galois connection between two partially ordered sets \( X \) and \( Y \), then \( \alpha \) and \( \beta \) induce mutually inverse, order reversing bijections between \( \operatorname{Im}\alpha \) and \( \operatorname{Im}\beta ;\alpha \circ \beta \) and \( ... | The proof is an exercise. | No |
Theorem 2.5 (MacNeille [1935]). Every partially ordered set can be embedded into a complete lattice so that all existing infimums and supremums are preserved. | Proof. Let \( \left( {X, \leqq }\right) \) be a partially ordered set. For every subset \( S \) of \( X \) let\n\n\[ L\left( S\right) = \{ x \in X \mid x \leqq s\text{ for all }s \in S\} ,\]\n\n\[ U\left( S\right) = \{ x \in X \mid x \geqq s\text{ for all }s \in S\} \]\n\nbe the sets of all lower and upper bounds of \(... | Yes |
Proposition 3.1. The lattice of submodules of any module is modular. | Proof. We saw that the submodules of a module constitute a lattice, in which \( A \land B = A \cap B \) and \( A \vee B = A + B \) . Let \( A, B, C \) be submodules such that \( A \subseteq C \) . Then \( A + \left( {B \cap C}\right) \subseteq \left( {A + B}\right) \cap C \) (this holds in every lattice). Conversely, i... | Yes |
Theorem 3.2. A lattice is modular if and only if it contains no sublattice that is isomorphic to \( {N}_{5} \) . | Proof. A sublattice of a modular lattice is modular and not isomorphic to \( {N}_{5} \) .\n\nConversely, a lattice \( L \) that is not modular contains elements \( a, b, c \) such that \( b \leqq c \) and \( u = b \vee \left( {a \land c}\right) < \left( {b \vee a}\right) \land c = v \) . Then \( v \leqq b \vee a, a \le... | Yes |
Theorem 3.3. In a modular lattice, any two finite maximal chains have the same length. | Proof. In a partially ordered set \( X \), an element \( b \) covers an element \( a \) when \( a < b \) and there is no \( x \in X \) such that \( a < x < b \) . We denote this relation by \( b \succ a \), or by \( a \prec b \) . In a lattice, a finite chain \( {x}_{0} < {x}_{1} < \cdots < {x}_{n} \) is maximal if and... | No |
Lemma 3.4. In a modular lattice, \( x \land y \prec x \) if and only if \( y \prec x \vee y \) . | Proof. Suppose that \( x \land y \prec x \) but \( y \nprec x \vee y \) . Then \( x \land y < x, x \nleqslant y \) , \( y < x \vee y \), and \( y < z < x \vee y \) for some \( z \) . Then \( x \vee y \leqq x \vee z \leqq x \vee y \) and \( x \vee z = x \vee y \) . Also \( x \land y \leqq x \land z \leqq x \), and \( x ... | Yes |
Lemma 3.5. In a modular lattice, if \( n \geqq 1,0 \prec {y}_{1} \prec \cdots \prec {y}_{n} \), and \( x \prec {y}_{n} \) , then \( 0 \prec {x}_{1} \prec \cdots \prec {x}_{n - 1} = x \) for some \( {x}_{1},\ldots ,{x}_{n - 1} \) . | Proof. By induction on \( n \) . There is nothing to prove if \( n = 1 \) . Let \( n > 1 \) . If \( x = {y}_{n - 1} \), then \( {y}_{1},\ldots ,{y}_{n - 1} \) serve. Now, assume that \( x \neq {y}_{n - 1} \) . Then \( x \nleqslant {y}_{n - 1},{y}_{n - 1} < {y}_{n - 1} \vee x \leqq {y}_{n} \), and \( {y}_{n - 1} \vee x ... | Yes |
Proposition 4.1. In a lattice \( L \), the distributivity conditions\n\n(1) \( x \vee \left( {y \land z}\right) = \left( {x \vee y}\right) \land \left( {x \vee z}\right) \) for all \( x, y, z \in L \) ,\n\n(2) \( x \land \left( {y \vee z}\right) = \left( {x \land y}\right) \vee \left( {x \land z}\right) \) for all \( x... | Proof. Assume (1). Then \( x \leqq z \) implies \( x \vee \left( {y \land z}\right) = \left( {x \vee y}\right) \land \left( {x \vee z}\right) = \) \( \left( {x \vee y}\right) \land z \) . Hence \( L \) is modular. Then \( x \land z \leqq x \) yields \( \left( {x \land z}\right) \vee \left( {y \land x}\right) = \) \( x ... | Yes |
Lemma 4.3. In a lattice \( L \) that satisfies the descending chain condition, every element of \( L \) is the supremum of a set of irreducible elements of \( L \) . | Proof. Assume that there is an element of \( L \) that is not the supremum of a set of irreducible elements of \( L \) . By the d.c.c., there is an element \( m \) of \( L \) that is minimal with this unsavory property. Then \( m \) is not a minimal element of \( L \) : otherwise, \( m \) is the least element of \( L \... | Yes |
Proposition 4.4. The order ideals of a partially ordered set \( S \), partially ordered by inclusion, constitute a distributive lattice \( \operatorname{Id}\left( S\right) \) . | Proof. First, \( S \) is an order ideal of itself, and every intersection of order ideals of \( S \) is an order ideal of \( S \) . By 2.1, \( \operatorname{Id}\left( S\right) \) is a complete lattice, in which infimums are intersections. Moreover, every union of order ideals of \( S \) is an order ideal of \( S \), so... | Yes |
Theorem 4.5. A finite lattice \( L \) is distributive if and only if \( L \cong \operatorname{Id}\left( S\right) \) for some finite partially ordered set \( S \), namely \( S = \operatorname{Irr}\left( L\right) \) . | Proof. Let \( L \) be distributive. For every \( x \in L \) , \[ \theta \left( x\right) = \{ i \in \operatorname{Irr}\left( L\right) \mid i \leqq x\} \] is an order ideal of \( \operatorname{Irr}\left( L\right) \) . By 4.3, \( x \) is the supremum of a set \( J \) of irreducible elements of \( L \) ; then \( J \subsete... | Yes |
Proposition 4.6. Let \( A \) and \( B \) be ideals of a distributive lattice \( L \) . In the lattice of ideals of \( L, A \vee B = \{ a \vee b \mid a \in A, b \in B\} \) . | Proof. Let \( C = \{ a \vee b \mid a \in A, b \in B\} \) . An ideal of \( L \) that contains both \( A \) and \( B \) also contains \( C \) . Hence it suffices to show that \( C \) is an ideal. If \( x \leqq y \in C \), then \( y = a \vee b \) for some \( a \in A \) and \( b \in B \), and \( x = x \land \left( {a \vee ... | Yes |
Lemma 4.7. Let \( I \) be an ideal of a distributive lattice \( L \) . For every \( a \in L \) , \( a \notin I \) there exists a prime ideal \( P \) of \( L \) that contains \( I \) but not a. | Proof. The union of a chain of ideals of \( L \) is an ideal of \( L \) ; hence the union of a chain of ideals of \( L \) that contain \( I \) but not \( a \) is an ideal of \( L \) that contains \( I \) but not \( a \) . By Zorn’s lemma, there is an ideal \( P \) of \( L \) that contains \( I \) but not \( a \) and is... | Yes |
Theorem 4.8 (Birkhoff [1933]). A lattice \( L \) is distributive if and only if it is isomorphic to a sublattice of \( {2}^{X} \) for some set \( X \) . | Proof. Let \( X \) be the set of all prime ideals of \( L \) . Define \( V : L \rightarrow {2}^{X} \) by\n\n\[ V\left( x\right) = \{ P \in X \mid x \notin P\} . \]\n\nBy 4.9 below (stated separately for future reference), \( V \) is a lattice homomorphism, so that \( \operatorname{Im}V \) is a sublattice of \( {2}^{X} ... | No |
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