§ 6. Use of noncanonical resolutions

We retain the conventions of § 4.

1. Calculation of the modules $\mathrm{Tor}^A(P, M)$ and $\mathrm{Ext}_A(M, N)$

Let $M, N$ be left $A$ -modules, $P$ a right $A$ -module. Let moreover $a : R \to M$ be a left resolution of $M$ , $b : S \to P$ a left resolution of $P$ and $c : N \to E$ a right resolution of $N$ .

By $X$ , p. 49, prop. 3 and 3 bis, there exist morphisms of complexes $\alpha : L(M) \to R$ , $\beta : L(P) \to S$ , $\gamma : E \to I(N)$ such that $a \circ \alpha = p_M$ , $b \circ \beta = p_P$ , $\gamma \circ C = e_N$ , and the homotopy classes of $\alpha, \beta, \gamma$ depend only on the given resolutions. By $X$ , p. 64, prop. 3 and $X$ , p. 84, prop. 3, the homotopy classes of the morphisms

\beta \otimes \alpha : L(P) \otimes_A L(M) \to S \otimes_A R,

\mathrm{Homgr}_A(\alpha, \gamma) : \mathrm{Homgr}_A(R, E) \to \mathrm{Homgr}_A(L(M), I(N))

depend only on the given resolutions, whence by passing to homology graded $k$ -homomorphisms of degree 0

\psi(S, R) : \mathrm{Tor}^A(P, M) \to H(S \otimes_A R),

\varphi(R, E) : H(\mathrm{Homgr}_A(R, E)) \to \mathrm{Ext}_A(M, N),

independent of the choice of $\alpha, \beta, \gamma$ .

For example, taking for $a, b, c$ the identity mappings of $M, P, N$ respectively, one finds the homomorphisms $\psi(P, M) : \mathrm{Tor}^A(P, M) \to P \otimes_A M$ and $\varphi(M, N) : \mathrm{Hom}_A(M, N) \to \mathrm{Ext}_A(M, N)$ introduced in $X$ , p. 68, remark 2) and $X$ , p. 87, remark 2).

Theorem 1

a) If one of the resolutions $R$ or $S$ is flat, then $\psi(S, R)$ is an isomorphism of graded $k$ -modules.

b) If $R$ is projective or if $E$ is injective, $\varphi(R, E)$ is an isomorphism of graded $k$ -modules.

a) Suppose for example $\mathbf R$ flat, and choose $\alpha$ and $\beta$ as above. The homomorphism $\beta\otimes\alpha$ is the composite of the morphisms

L(P)\otimes_A L(M)\xrightarrow{1_{L(P)}\otimes\alpha}L(P)\otimes_A\mathbf R\xrightarrow{\beta\otimes1_{\mathbf R}}S\otimes_A\mathbf R.

Since $L(P)$ (resp. $\mathbf R$ ) is flat and $\alpha$ (resp. $\beta$ ) is a homomorphism, $1_{L(P)}\otimes\alpha$ (resp. $\beta\otimes1_{\mathbf R}$ ) is one, by prop. 4 of X, p. 67. Therefore $\beta\otimes\alpha$ is a homomorphism and $\psi(S,R)=H(\beta\otimes\alpha)$ is bijective.

b) One reasons analogously, using prop. 4 of X, p. 86.

Corollary

If $R$ is a flat resolution of $M$ , the homomorphism

\psi(P,R):\operatorname{Tor}^A(P,M)\longrightarrow H(P\otimes_A R)

is bijective. If $R$ is a projective resolution of $M$ , the homomorphism

\varphi(R,N):H(\operatorname{Homgr}_A(R,N))\longrightarrow \operatorname{Ext}_A(M,N)

is bijective. If $E$ is an injective resolution of $N$ , the homomorphism

\varphi(M,E):H(\operatorname{Homgr}_A(M,E))\longrightarrow \operatorname{Ext}_A(M,N)

is bijective.

Remark

The diagram of $k$ -modules

\begin{array}{ccc} \operatorname{Tor}^A(P,M)&\xrightarrow{\psi(S,R)}&H(S\otimes_A R)\\ \downarrow{\sigma_{P,M}}&&\downarrow{H(\sigma(S,R))}\\ \operatorname{Tor}^{A^\circ}(M^\circ,P^\circ)&\xrightarrow{\psi(R^\circ,S^\circ)}&H(R^\circ\otimes_{A^\circ}S^\circ), \end{array}

where $\sigma_{P,M}$ and $\sigma(S,R)$ are the commutation isomorphisms (X, p. 71 and 63), is commutative: it is obtained in fact, by passing to homology, from the commutative diagram of complexes

\begin{array}{ccc} L(P)\otimes_A L(M)&\xrightarrow{\beta\otimes\alpha}&S\otimes_A R\\ \downarrow{\sigma(L(P),L(M))}&&\downarrow{\sigma(S,R)}\\ L(M^\circ)\otimes_{A^\circ}L(P^\circ)&\xrightarrow{\alpha\otimes\beta}&R^\circ\otimes_{A^\circ}S^\circ \end{array}

("the morphisms $\psi$ are compatible with the commutation isomorphisms").

Analogously, let $a_1:R_1\to M$ , $b_1:S_1\to P$ , $c_1:N\to E_1$ be morphisms of complexes, where $R_1$ and $S_1$ are projective and right zero and $E_1$ injective and left zero. By X, p. 49, prop. 3 and 3 bis, there exist morphisms of complexes

\alpha_1:R_1\to L(M),\qquad \beta_1:S_1\to L(P),\qquad \gamma_1:I(N)\to E_1

such that $p_M \circ \alpha_1 = a_1, p_P \circ \beta_1 = b_1, \gamma_1 \circ e_N = c_1$ , whence morphisms of complexes

\beta_1 \otimes \alpha_1 : S_1 \otimes_A R_1 \to L(P) \otimes_A L(M),

\operatorname{Homgr}_A (\alpha_1, \gamma_1) : \operatorname{Homgr}_A (L(M), l(N)) \to \operatorname{Homgr}_A (R_1, E_1),

and by passing to homology, graded k-linear mappings of degree 0:

\psi'(S_1, R_1) : H(S_1 \otimes_A R_1) \to \operatorname{Tor}^A (P, M)

\varphi'(R_1, E_1) : \operatorname{Ext}_A (M, N) \to H(\operatorname{Homgr}_A (R_1, E_1))

which one verifies, as above, to be independent of the choice of $\alpha_1, \beta_1, \gamma_1$ .

Proposition 1

If* $a_1, b_1, c_1$ are homologisms, $\psi'(S_1, R_1)$ and $\varphi'(R_1, E_1)$ are the reciprocal bijections of the bijections $\psi(S_1, R_1)$ and $\varphi(R_1, E_1)$ *respectively.

In fact, $f = (\beta \otimes \alpha) \circ (\beta_1 \otimes \alpha_1)$ is a morphism of the complex $S_1 \otimes_A R_1$ into itself, and one has $(h_1 \circ \alpha_1) \circ f = f$ . By X, p. 49, prop. 3 and 3 bis, $f$ is a homotopism, hence $H(f) = 1$ and

\psi(S_1, R_1) \circ \psi'(B_1, R_1) = H(\beta \otimes \alpha) \circ H(\beta_1 \otimes \alpha_1) = H(f) = 1;

analogously $\psi'(S_1, R_1) \circ \psi(S_1, R_1) = 1$ . One reasons in an analogous manner for the mappings $\varphi$ and $\varphi'$ .

Example 1

Let $a$ be an element of $A$ such that the mapping $\varphi : x \mapsto xa$ of $A$ into itself is injective (" $a$ is not a right zero divisor "). Using the resolution

0 \to A_s \xrightarrow{\varphi} A_s \to A/Aa \to 0

one sees that for every right $A$ -module $M$ , one has

\operatorname{Tor}_i^A (M, A/Aa) = 0 \quad \text{for } i > 1

and that the $k$ -module $\operatorname{Tor}_1^A (M, A/Aa)$ is isomorphic to $\operatorname{Ker} (a_M)$ .

Analogously, for every left $A$ -module $M$ , one has

\operatorname{Ext}_A^i (A/Aa, M) = 0 \quad \text{for } i > 1

and the $k$ -module $\operatorname{Ext}_A^1 (A/Aa, M)$ is isomorphic to $M/aM$ .

Example 2

Suppose that $A$ is an integral domain; let $K$ be the field of fractions of $A$ and $M$ an $A$ -module. Using the flat resolution

0 \to A \to K \to K/A \to 0

(X, p. 9, example 5), we see that $\operatorname{Tor}_i^A (K/A, M) = 0$ for $i > 1$ ; moreover, in view of II, p. 116, prop. 26, (ii), the $A$ -module $\operatorname{Tor}_1^A (K, A, M)$ is isomorphic to the torsion submodule of $M$ .

Example 3

Suppose that $A$ is a local noetherian ring, let $m$ denote its maximal ideal, and put $\kappa = A/m$ . Let $M$ be a finitely generated $A$ -module and $P$ a minimal projective resolution of $M$ (X, p. 54). For every $n \geqslant 0$ , the $\kappa$ -vector spaces $\mathrm{Tor}_n^A(\kappa, M)$ and $\mathrm{Ext}_A^n(M, \kappa)$ are finite-dimensional, of dimension equal to the rank of the free $A$ -module $P_n$ ; in fact, the complexes $\kappa \otimes_A P$ and $\mathrm{Homgr}_A(P, \kappa)$ have zero differential.

2. Computation of the mappings $\mathrm{Tor}^A(g, f)$ and $\mathrm{Ext}_A(f, h)$

Let $f : M \to M', h : N' \to N$ be homomorphisms of left $A$ -modules, $g : P \to P'$ a homomorphism of right $A$ -modules, $a : R \to M, a' : R' \to M'$ , $b : S \to P, b' : S' \to P'$ , left resolutions of $M, M', P, P'$ , respectively, $c : N \to E, c' : N' \to E'$ right resolutions of $N$ and $N'$ , $\tilde{f} : R \to R', \tilde{g} : S \to S', \tilde{h} : E' \to E$ morphisms of complexes such that

a' \circ \tilde{f} = f \circ a, \quad b' \circ \tilde{g} = g \circ b, \quad \tilde{h} \circ c' = c \circ h .

Proposition 2

The following two diagrams are commutative*:

\begin{array}{ccc} \mathrm{Tor}^A(P, M) & \xrightarrow{\psi(S, R)} & \mathrm{H}(S \otimes_A R) \\ \mathrm{Tor}^A(g, f) \downarrow & & \downarrow \mathrm{H}(\tilde{g} \otimes \tilde{f}) \\ \mathrm{Tor}^A(P', M') & \xrightarrow{\psi(S', R')} & \mathrm{H}(S' \otimes_A R') , \end{array}

\begin{array}{ccc} \mathrm{H}(\mathrm{Homgr}_A(R', E')) & \xrightarrow{\varphi(R', E')} & \mathrm{Ext}_A(M', N') \\ \mathrm{H}(\mathrm{Homgr}_A(\tilde{f}, \tilde{h})) \downarrow & & \downarrow \mathrm{Ext}_A(f, h) \\ \mathrm{H}(\mathrm{Homgr}_A(R, E)) & \xrightarrow{\varphi(R, E)} & \mathrm{Ext}_A(M, N) . \end{array}

Let $\alpha : L(M) \to R, \alpha' : L(M') \to R', \gamma : L(P) \to S, \gamma' : L(P') \to S'$ be morphisms of complexes such that

a \circ \alpha = p_M, \quad a' \circ \alpha' = p_{M'}, \quad b \circ \gamma = p_P, \quad b' \circ \gamma' = p_{P'} .

By definition, $\mathrm{H}(\tilde{g} \otimes \tilde{f}) \circ \psi(S, R)$ is equal to

\mathrm{H}(\tilde{g} \otimes \tilde{f}) \circ \mathrm{H}(\gamma \otimes \alpha) = \mathrm{H}((\tilde{g} \circ \gamma) \otimes (\tilde{f} \circ \alpha)) ,

whereas $\psi(S', R') \circ \mathrm{Tor}^A(g, f)$ is equal to

\mathrm{H}(\gamma' \otimes \alpha') \circ \mathrm{H}(L(g) \otimes L(f)) = \mathrm{H}((\gamma' \circ L(g)) \otimes (\alpha' \circ L(f))) .

On the other hand, $\alpha' \circ L(f)$ and $\tilde{f} \circ \alpha$ are two morphisms from $L(M)$ into $R'$ such that $a' \circ (\alpha' \circ L(f)) = p_{M'} \circ L(f) = f \circ p_M = f \circ a \circ \alpha = a' \circ (\tilde{f} \circ \alpha)$ . By X, p. 49, prop. 3, $\alpha' \circ L(f)$ and $\tilde{f} \circ \alpha$ are homotopic; analogously $\gamma' \circ L(g)$ and $\tilde{g} \circ \gamma$ are homotopic, and consequently so are $(\gamma' \circ L(g)) \otimes (\alpha' \circ L(f))$ and $(\tilde{g} \circ \gamma) \otimes (\tilde{f} \circ \alpha)$ by prop. 3 of X, p. 64. Therefore

H(\tilde{g} \otimes \tilde{f}) \circ \psi(S, R) = H((\tilde{g} \circ \gamma) \otimes (\tilde{f} \circ \alpha)) = H((\gamma' \circ L(g)) \otimes (\alpha' \circ L(f)))

= \psi(S', R') \circ \mathrm{Tor}^A(g, f) .

One reasons analogously for the second diagram.

Remark

Similarly, consider commutative diagrams of morphisms of complexes

\begin{array}{ccc} R_1 & \xrightarrow{a_1} & M \\ \tilde{f} \downarrow & & \downarrow f \\ R'_1 & \xrightarrow{a'_1} & M' \end{array}

\begin{array}{ccc} S_1 & \xrightarrow{b_1} & P \\ \tilde{g} \downarrow & & g \downarrow \\ S'_1 & \xrightarrow{b'_1} & P' \end{array}

\begin{array}{ccc} N' & \xrightarrow{c'_1} & E' \\ h \downarrow & & \tilde{h} \downarrow \\ N & \xrightarrow{c_1} & E_1 \end{array}

where the complexes $R_1, R'_1, S_1, S'_1$ are projective and zero on the right and where the complexes $E_1, E'_1$ are injective and zero on the left. Then

\mathrm{Tor}^A(g, f) \circ \psi'(S_1, R_1) = \psi'(S'_1, R'_1) \circ H(\tilde{g} \otimes \tilde{f})

\varphi'(R_1, E_1) \circ \mathrm{Ext}_A(f, h) = H(\mathrm{Homgr}_A(\tilde{f}, \tilde{h})) \circ \varphi'(R'_1, E'_1)

as is proved analogously to prop. 2.

3. Calculation of the connecting homomorphisms

Let us consider a commutative diagram

(1)

\begin{array}{ccccccccc} 0 & \longrightarrow & R' & \xrightarrow{\tilde{u}} & R & \xrightarrow{\tilde{v}} & R'' & \longrightarrow & 0 \\ & & a' \downarrow & & a \downarrow & & a'' \downarrow & & \\ 0 & \longrightarrow & M' & \xrightarrow{u} & M & \xrightarrow{v} & M'' & \longrightarrow & 0 \end{array}

(2)

where the first row (1) is an exact sequence of complexes of left A-modules, the second row (2) is an exact sequence of left A-modules, and the vertical arrows are left resolutions.

Proposition 3

a) Let $P$ be an A-module, $b : S \to P$ a left resolution of $P$ ; suppose that the sequence of complexes of k-modules

(3)

0 \to S \otimes_A R' \xrightarrow{1_S \otimes \tilde{u}} S \otimes_A R \xrightarrow{1_S \otimes \tilde{v}} S \otimes_A R'' \to 0

is exact. Then the following diagram is commutative:

\begin{array}{ccc} \operatorname{Tor}^{A}(P,M'')&\xrightarrow{\partial(P,(2))}&\operatorname{Tor}^{A}(P,M')\\ \Big\downarrow\vcenter{\rlap{$\psi(S,R'')$}}&&\Big\downarrow\vcenter{\rlap{$\psi(S,R')$}}\\ H(S\otimes_A R'')&\xrightarrow{\partial(3)}&H(S\otimes_A R'). \end{array}

b) Let $N$ be a left $A$ -module, $c:N\to E$ a right resolution of $N$ ; suppose that the sequence of complexes of $k$ -modules

(4)

0\longrightarrow \operatorname{Homgr}_A(R'',E) \xrightarrow{\operatorname{Homgr}_A(\tilde{\imath},1)} \operatorname{Homgr}_A(R,E) \xrightarrow{\operatorname{Homgr}_A(\tilde{u},1)} \operatorname{Homgr}'_A(R',E)\longrightarrow 0

is exact. Then the following diagram is commutative:

\begin{array}{ccc} H(\operatorname{Homgr}_A(R',E)) &\xrightarrow{\delta((4))} &H(\operatorname{Homgr}_A(R'',E)) \\ \Big\downarrow\vcenter{\rlap{$\varphi(R',E)$}} && \Big\downarrow\vcenter{\rlap{$\varphi(R'',E)$}} \\ \operatorname{Ext}_A(M',N) &\xrightarrow{\partial((2),N)} &\operatorname{Ext}_A(M'',N). \end{array}

Let us prove a), for example. Let $\beta:L(P)\to S$ be a morphism of complexes such that $t\circ\beta=p_P$ ; let us consider the diagram of $k$ -complexes

\begin{array}{ccccccccc} 0&\longrightarrow&S\otimes_A R'&\xrightarrow{1\otimes\tilde{u}}&S\otimes_A R&\xrightarrow{1\otimes\tilde{v}}&S\otimes_A R''&\longrightarrow&0\\ &&\Big\uparrow\vcenter{\rlap{$\beta\otimes1_R$}}&&\Big\uparrow\vcenter{\rlap{$\beta\otimes1_R$}}&&\Big\uparrow\vcenter{\rlap{$\beta\otimes1_{R'}$}}&&\\ 0&\longrightarrow&L(P)\otimes_A R'&\xrightarrow{1\otimes\tilde{u}}&L(P)\otimes_A R&\xrightarrow{1\otimes\tilde{v}}&L(P)\otimes_A R''&\longrightarrow&0\\ &&\Big\downarrow\vcenter{\rlap{$1\otimes a'$}}&&\Big\downarrow\vcenter{\rlap{$1\otimes a$}}&&\Big\downarrow\vcenter{\rlap{$1\otimes a''$}}&&\\ 0&\longrightarrow&L(P)\otimes_A M'&\xrightarrow{1\otimes u}&L(P)\otimes_A M&\xrightarrow{1\otimes v}&L(P)\otimes_A M''&\longrightarrow&0. \end{array}

This is a commutative diagram with exact rows (by the hypothesis for the first row, and by the fact that $L(P)$ is flat for the other two). We therefore have a commutative diagram (X, p. 31, prop. 2, and X, p. 72, def. 2)

\begin{array}{ccc} H(S\otimes_A R'')&\xrightarrow{\partial(3)}&H(S\otimes_A R')\\ \Big\uparrow\vcenter{\rlap{$H(\beta\otimes1)$}}&&\Big\uparrow\vcenter{\rlap{$H(\beta\otimes1)$}}\\ H(L(P)\otimes_A R'')&\longrightarrow&H(L(P)\otimes_A R')\\ \Big\downarrow\vcenter{\rlap{$H(1\otimes a'')$}}&&\Big\downarrow\vcenter{\rlap{$H(1\otimes a')$}}\\ H(L(P)\otimes_A M'')&\longrightarrow&H(L(P)\otimes_A M')\\ \Big\uparrow\vcenter{\rlap{$\psi_P(M'')$}}&&\Big\uparrow\vcenter{\rlap{$\psi_P(M')$}}\\ \operatorname{Tor}(P,M'')&\xrightarrow{\partial(P,(2))}&\operatorname{Tor}(P,M'). \end{array}

By X, p. 67, prop. 4, H(1 ⊗ a'') and H(1 ⊗ a') are bijective; on the other hand, by definition of the homomorphisms ψ, we have H(β ⊗ 1) ◦ ψ(L(P), R'') = ψ(S, R'') and H(1 ⊗ a'') ◦ ψ(L(P), R'') = ψ(L(P), M'') = ψ_p(M''), therefore

\psi(S, R'') = H(\beta \otimes 1) ◦ H(1 \otimes a'')^{-1} ◦ \psi_p(M'');

analogously, $\psi(S, R') = H(\beta \otimes 1) ◦ H(1 \otimes a')^{-1} ◦ \psi_p(M')$ , and the required assertion $\partial((3)) ◦ \psi(S, R'') = \psi(S, R') ◦ \partial(P, (2))$ follows from the commutativity of the preceding diagram.

Remark 1

Using the commutation isomorphisms, one deduces from a) the analogous statement obtained by exchanging the roles of the two arguments of the tensor product.

Remark 2

With the notation of a), suppose either that S is flat, or that R, R', R'' are flat; then on the one hand sequence (3) is exact (X, p. 72, cor. 2) and one may apply prop. 3; on the other hand $\psi(S, R')$ is bijective (th. 1), therefore

\partial(P, (2)) = \psi(S, R')^{-1} ◦ \partial((3)) ◦ \psi(S, R'').

Remark 3

With the notation of b), suppose either that E is injective, or that R, R', R'' are projective; then on the one hand sequence (4) is exact (X, p. 83, prop. 2) and one may apply prop. 3; on the other hand, $\varphi(R', E)$ is bijective (th. 1) ; therefore

\delta((2), N) = \varphi(R'', E) ◦ \partial((4)) ◦ \varphi(R', E)^{-1}.

Let us now consider a commutative diagram

(5)

\begin{array}{ccccccccc} 0 & \rightarrow & N' & \xrightarrow{r} & N & \xrightarrow{s} & N'' & \rightarrow & 0 \\ & & c' & & & & c'' & & \end{array}

(6)

\begin{array}{ccccccccc} 0 & \rightarrow & E' & \xrightarrow{\tilde{r}} & E & \xrightarrow{\tilde{s}} & E'' & \rightarrow & 0 \end{array}

whose first row (5) is an exact sequence of left A-modules, the second row (6) an exact sequence of complexes of left A-modules, and in which the vertical arrows are right resolutions. Reasoning as in Prop. 3, one proves the following proposition:

Proposition 4

Let M be a left A-module, $a : R \to M$ a left resolution of M such that the sequence

(7)

0 \rightarrow \mathrm{Homgr}_A(R, E') \xrightarrow{\mathrm{Homgr}(\tilde{s}, 1)} \mathrm{Homgr}_A(R, E) \xrightarrow{\mathrm{Homgr}(\tilde{r}, 1)} \mathrm{Homgr}_A(R, E'') \rightarrow 0

is exact. Then the following diagram is commutative.

\begin{array}{ccc} \mathrm{H}(\mathrm{Homgr}_A(R, E'')) & \xrightarrow{\partial((7))} & \mathrm{H}(\mathrm{Homgr}_A(R, E')) \\ \varphi(R, E'') \downarrow & & \downarrow \varphi(R, E') \\ \mathrm{Ext}_A(M, N'') & \xrightarrow{\delta(M, (5))} & \mathrm{Ext}_A(M, N') \end{array}

Remark 4

If R is projective or if E, E', E'' are injective, the sequence (7) is exact ( $\lambda$ , p. 83, Prop. 2); moreover, $\varphi(R, E'')$ is bijective (Th. 1); hence

\delta(M, (5)) = \varphi(R, E') \circ \partial((7)) \circ \varphi(R, E'')^{-1}.

Remark 5

We leave it to the reader to state and prove the propositions analogous to Props. 3 and 4 and concerning the homomorphisms $\psi_1$ and $\varphi_1$ .

4. Finiteness of extension and torsion modules

Let M be a left A-module, I a filtered preordered set, $(N_i, u_{ji})$ an inductive system of left A-modules relative to I, $N = \lim N_i$ its inductive limit, $u_i : N_i \to N,\ i \in I$ , the canonical mapping. Then $(\mathrm{Ext}_A(M, N_i), \mathrm{Ext}_A(l_M, u_{ji}))$ is an inductive system of $k$ -modules and $(\mathrm{Ext}_A(l_M, u_i))$ an inductive system of mappings, whose inductive limit is a homomorphism of graded $k$ -modules, called canonical

\lim_{i \in I} \mathrm{Ext}_A(M, N_i) \to \mathrm{Ext}_A(M, \lim_{i \in I} N_i).

Proposition 5

If A is left noetherian and if M is a finitely generated A-module, the canonical homomorphism (8) is bijective.

For let (X, p. 53, Prop. 6) $p : L \to M$ be a free resolution of M such that $L_n$ is of finite type for each $n$ . The canonical morphism of $k$ -complexes

u : \lim \mathrm{Homgr}_A(L, N_i) \to \mathrm{Homgr}_A(L, \lim N_i)

is bijective, hence so also is the homomorphism

\lim \mathrm{H}(\mathrm{Homgr}_A(L, N_i)) \to \mathrm{H}(\mathrm{Homgr}_A(L, \lim N_i))

deduced from the homomorphisms $\mathrm{H}(\mathrm{Homgr}(l, u_i))$ (X, p. 28, Prop. 1). One then concludes by Prop. 2 (X, p. 103) and Th. 1 (X, p. 100).

Proposition 6

Let B be a ring and N an (A, B)-bimodule, which is a noetherian B-module (resp. of finite length).

a) Suppose A right noetherian and let M be a finitely generated right A-module. Then the B-modules (X, p. 81) $\mathrm{Tor}_n^A(M, N)$ are noetherian (resp. of finite length).

b) Suppose A left noetherian and let M be a finitely generated left A-module. Then the B-modules (X, p. 98) $\mathrm{Ext}_A^n(M, N)$ are noetherian (resp. of finite length).

Choose a free resolution $p : L \to M$ such that each of the A-modules $L_n$ is of finite type (X, p. 53, Prop. 6), and let C be the complex of B-modules $L \otimes_A N$ in case $a$ , $\mathrm{Homgr}_A(L, N)$ in case $b$ . Each of the B-modules $C_n$ is isomorphic to a product of a finite number of copies of N, hence is noetherian (resp. of finite length); the same is therefore true of the modules $\mathrm{H}_n(C)$ . But, by X, p. 100, Th. 1, these are isomorphic to the $\mathrm{Tor}_n^A(M, N)$ in case $a$ , to the $\mathrm{Ext}_A^{-n}(M, N)$ in case $b$ .

Corollary

Let $\rho : A \to B$ be a homomorphism of noetherian commutative rings, $M$ a finitely generated $A$ -module, $N$ a $B$ -module. If $N$ is a finitely generated $B$ -module (resp. of finite length), the same is true of the $B$ -modules $\operatorname{Tor}^{A}_{n}(M,N)$ and $\operatorname{Ext}^{n}_{A}(M,N)$ .

5. The homomorphisms $\operatorname{Tor}^{B}(P,N) \otimes_{A} Q \to \operatorname{Tor}^{B}(P,N \otimes_{A} Q)$ and $\operatorname{Ext}_{B}(M,N) \otimes_{A} Q \to \operatorname{Ext}_{B}(M,N \otimes_{A} Q)$

Let $B$ be a ring, $N$ a $(B,A)$ -bimodule, $M$ a left $B$ -module, $P$ a right $B$ -module, $Q$ a left $A$ -module. By X, p. 62, one has a homomorphism

\gamma_{1} : H(L(P) \otimes_{B} N) \otimes_{A} Q \to H(L(P) \otimes_{B} N \otimes_{A} Q) \,;

moreover (X, p. 69, Prop. 5), one has isomorphisms

\psi_{1}(N) : \operatorname{Tor}^{B}(P,N) \otimes_{A} Q \to H(L(P) \otimes_{B} N) \otimes_{A} Q \,,

\psi_{1}(N \otimes_{A} Q) : \operatorname{Tor}^{B}(P,N \otimes_{A} Q) \to H(L(P) \otimes_{B} N \otimes_{A} Q) \,.

The graded homomorphism of degree 0, called canonical

(9)

\operatorname{Tor}^{B}(P,N) \otimes_{A} Q \to \operatorname{Tor}^{B}(P,N \otimes_{A} Q)

is defined as the composite $\psi_{P}(N \otimes_{A} Q)^{-1} \circ \gamma_{1} \circ (\psi_{P}(N) \otimes 1_{Q})$ .

Analogously, from the canonical morphism of complexes

\alpha : \operatorname{Homgr}_{B}(L(M),N) \otimes_{A} Q \to \operatorname{Homgr}_{B}(L(M),N \otimes_{A} Q)

one deduces a homomorphism $H(\alpha)$ ; one has the canonical homomorphism (X, p. 62)

\gamma_{2} : H(\operatorname{Homgr}_{B}(L(M),N)) \otimes_{A} Q \to H(\operatorname{Homgr}_{B}(L(M),N \otimes_{A} Q)) \,,

and the isomorphisms (X, p. 88, prop. 5)

\varphi_{M}(N) : H(\operatorname{Homgr}_{B}(L(M),N)) \otimes_{A} Q \to \operatorname{Ext}_{B}(M,N) \otimes_{A} Q \,,

\varphi_{M}(N \otimes_{A} Q) : H(\operatorname{Homgr}_{B}(L(M),N \otimes_{A} Q)) \to \operatorname{Ext}_{B}(M,N \otimes_{A} Q) \,.

The graded homomorphism of degree 0, called canonical

(10)

\operatorname{Ext}_{B}(M,N) \otimes_{A} Q \to \operatorname{Ext}_{B}(M,N \otimes_{A} Q)

is defined as the composite

\varphi_{M}(N \otimes_{A} Q) \circ H(\alpha) \circ (\varphi_{M}(N) \otimes 1_{Q})^{-1} \,.

Proposition 7

a) If the $A$ -module $Q$ is flat, homomorphism (9) is bijective. b) If the $A$ -module $Q$ is finitely generated projective, homomorphism (10) is bijective.

c) If the $A$ -module $Q$ is flat, the ring $B$ noetherian and the $B$ -module $M$ finitely generated, homomorphism (10) is bijective.

a) If $Q$ is flat, $\gamma_1$ is bijective ( $X$ , p. 66, cor. 2).

b) If $Q$ is finitely generated projective, it is flat, hence $\gamma_2$ is bijective, and moreover $\alpha$ is bijective (II, p. 75, prop. 2, a)).

c) Under the hypotheses of c), $\gamma_2$ is bijective since $Q$ is flat. Moreover ( $X$ , p. 53, prop. 6), there exists a resolution $L$ of $M$ such that each $L_n$ is finitely generated free; let $u : L(M) \to L$ be a homotopism ( $X$ , p. 49, cor. to prop. 3); in the commutative diagram,

\begin{array}{ccc} \mathrm{Homgr}_B(L(M), N) \otimes_A Q & \xrightarrow{\alpha} & \mathrm{Homgr}_B(L(M), N \otimes_A Q) \\ \uparrow & & \uparrow \\ \mathrm{Homgr}_B(L, N) \otimes_A Q & \xrightarrow{\bar{\alpha}} & \mathrm{Homgr}_B(L, N \otimes_A Q). \end{array}

the vertical arrows deduced from $u$ are homotopisms ( $X$ , p. 64, prop. 3 and p. 83, prop. 3) and $\bar{\alpha}$ is bijective (II, p. 75, prop. 2 (ii)); hence $H(\alpha)$ is bijective, and homomorphism (10) is bijective.

6. The homomorphisms $\mathrm{Tor}^B(P, N \otimes_A Q) \to \mathrm{Tor}^A(P \otimes_B N, Q)$ and $\mathrm{Ext}_A(Q, \mathrm{Hom}_B(N, M)) \to \mathrm{Ext}_B(N \otimes_A Q, M)$

Let us retain the preceding notations, and suppose that $N$ is flat over $A$ . Then the morphism $N \otimes_A L(Q) \xrightarrow{1 \otimes p_Q} N \otimes_A Q$ is a homologism ( $X$ , p. 67, prop. 4), whence homomorphisms

\psi(P, N \otimes_A L(Q)) : \mathrm{Tor}^B(P, N \otimes_A Q) \to H(P \otimes_B N \otimes_A L(Q))

\varphi(N \otimes_A L(Q), M) : H(\mathrm{Homgr}_B(N \otimes_A L(Q), M)) \to \mathrm{Ext}_B(N \otimes_A Q, M).

Using then the isomorphisms

\overline{\psi}_Q(P \otimes_B N) : \mathrm{Tor}^A(P \otimes_B N, Q) \to H(P \otimes_B N \otimes_A L(Q)),

\beta : \mathrm{Homgr}_A(L(Q), \mathrm{Hom}_B(N, M)) \to \mathrm{Homgr}_B(N \otimes_A L(Q), M),

\varphi_Q(\mathrm{Hom}_B(N, M)) : H(\mathrm{Homgr}_A(L(Q), \mathrm{Hom}_B(N, M))) \to \mathrm{Ext}_A(Q, \mathrm{Hom}_B(N, M)),

one deduces from them graded homomorphisms of degree 0 called canonical:

(11)

\mathrm{Tor}^B(P, N \otimes_A Q) \to \mathrm{Tor}^A(P \otimes_B N, Q)

(12)

\mathrm{Ext}_A(Q, \mathrm{Hom}_B(N, M)) \to \mathrm{Ext}_B(N \otimes_A Q, M).

Proposition 8

a)* If $N$ is flat over $A$ and over $B$ , homomorphism (11) is bijective.

b) If $N$ is flat over $A$ and projective over $B$ , homomorphism (12) is bijective.

Indeed $N \otimes_A L(Q)$ is isomorphic to a direct sum of copies of $N$ , hence is a flat (resp. projective) $B$ -module when the $B$ -module $N$ is flat (resp. projective); one then applies Theorem 1 (X, p. 100).

7. The homomorphisms $B \otimes_A \mathrm{Tor}^A(E, F) \to \mathrm{Tor}^B(E \otimes_A B, B \otimes_A F)$ and $B \otimes_A \mathrm{Ext}_A(E, F) \to \mathrm{Ext}_B(B \otimes_A E, B \otimes_A F)$

In this No., we suppose that $A$ is commutative; given a ring homomorphism $\rho : A \to B$ such that $\rho(A)$ is contained in the center of $B$ and two $A$ -modules $E$ and $F$ . There is a canonical isomorphism of complexes of $A$ -modules

u : B \otimes_A (L(E) \otimes_A L(F)) \to (L(E) \otimes_A B) \otimes_B (B \otimes_A L(F));

on the other hand, since $L(E) \otimes_A B$ and $B \otimes_A L(F)$ are free $B$ -complexes, there is a canonical homomorphism of graded $A$ -modules (X, p. 102)

\psi'(L(E) \otimes_A B, B \otimes_A L(F)) : H((L(E) \otimes_A B) \otimes_B (B \otimes_A L(F)))

\to \mathrm{Tor}^B(E \otimes_A B, B \otimes_A F)

finally, there is a homomorphism (X, p. 62)

\gamma : B \otimes_A \mathrm{Tor}^A(E, F) \to H(B \otimes_A L(E) \otimes_A L(F)).

The canonical homomorphism of graded $B$ -modules

(13)

B \otimes_A \mathrm{Tor}^A(E, F) \to \mathrm{Tor}^B(E \otimes_A B, B \otimes_A F)

is defined as the composite $\psi'(L(E) \otimes_A B, B \otimes_A L(F)) \circ H(u) \circ \gamma$ .

Proposition 9

If $B$ is flat over $A$ , homomorphism (13) is bijective.* Indeed $\psi'(L(E) \otimes_A B, B \otimes_A L(F))$ is bijective (X, p. 102, prop. 1) and $\gamma$ is bijective (X, p. 66, cor. 2).

Suppose $B$ flat over $A$ . Substituting $E$ for $Q$ , $B$ for $N$ and $B \otimes_A F$ for $M$ in homomorphism (12), we obtain a homomorphism

\mathrm{Ext}_A(E, B \otimes_A F) \to \mathrm{Ext}_B(B \otimes_A E, B \otimes_A F)

which is bijective by Proposition 8. Substituting $E$ for $M$ , $F$ for $N$ , $A$ for $B$ , $B$ for $Q$ , in homomorphism (10), and interchanging the factors in the tensor products, we obtain a homomorphism of $B$ -modules

B \otimes_A \mathrm{Ext}_A(E, F) \to \mathrm{Ext}_A(E, B \otimes_A F),

whence by composition a homomorphism called *canonical

(14)

B \otimes_A \mathrm{Ext}_A(E, F) \to \mathrm{Ext}_B(B \otimes_A E, B \otimes_A F).

Proposition 10

Homomorphism (14) is bijective in the following cases: a) B is a finitely generated projective A-module; b) B is a flat A-module, A is noetherian, and E is a finitely generated A-module. This follows from Proposition 7 (X, p. 108).

8. Application: homology and cohomology of groups

Let G be a group, $\mathbf{Z}^{(G)}$ its algebra over $\mathbf{Z}$ (III, p. 19). Recall (cf. III, p. 20, example) that if M is a commutative group, it comes to the same thing to give an action of G on M (that is to say a homomorphism $\tau : G \to \mathrm{Aut}(M)$ ), or a structure of left $\mathbf{Z}^{(G)}$ -module on the additive group M. In particular, we shall consider the group $\mathbf{Z}$ as a left $\mathbf{Z}^{(G)}$ -module by endowing it with the trivial action.

Definition 1

Let M be a left $\mathbf{Z}^{(G)}$ -module (resp. right), n an integer $\geqslant 0$ . The group $\mathrm{Ext}_{\mathbf{Z}^{(G)}}^n(\mathbf{Z}, M)$ (resp. $\mathrm{Tor}_n^{\mathbf{Z}^{(G)}}(M, \mathbf{Z})$ ) is denoted by $H^n(G, M)$ (resp. $H_n(G, M)$ ) and is called the nth cohomology group (resp. homology group) of G with coefficients in M.

The standard resolution (X, p. 58) $B(\mathbf{Z}^{(G)}, \mathbf{Z})$ is a free resolution of the $\mathbf{Z}^{(G)}$ -module $\mathbf{Z}$ ; it follows that the groups $H^n(G, M)$ (resp. $H_n(G, M)$ ) identify with the homology groups of the complex:

\mathrm{Hom}_{\mathbf{Z}^{(G)}}(B(\mathbf{Z}^{(G)}, \mathbf{Z}), M) \quad (\text{resp. } M \otimes_{\mathbf{Z}^{(G)}} B(\mathbf{Z}^{(G)}, \mathbf{Z})) .

Using the canonical isomorphism of $(\mathbf{Z}^{(G)})^{\otimes n}$ onto $\mathbf{Z}^{(G^n)}$ (III, p. 36) and the properties of extension of scalars (II, p. 82), one concludes that $H^n(G, M)$ is canonically isomorphic to the homology group of ascending degree $n$ of the complex $C(G, M)$ defined as follows: $C^n(G, M) = 0$ for $n < 0$ ; for $n \geqslant 0$ , $C^n(G, M)$ is the $\mathbf{Z}$ -module of mappings of $G^n$ into M; for $n \geqslant 0$ , the differential $d^n : C^n(G, M) \to C^{n+1}(G, M)$ is given by

(d^n f)(g_0, ..., g_n) = g_0 \cdot f(g_1, ..., g_n) + \sum_{i=0}^{n-1} (-1)^{i+1} f(g_0, ..., g_i g_{i+1}, ..., g_n)

+ (-1)^{n+1} f(g_0, ..., g_{n-1})

for any $f$ in $C^n(G, M)$ and $g_0, ..., g_n$ in G.

Analogously, $H_n(G, M)$ identifies with the homology group of degree $n$ of the complex $C'(G, M)$ , where $C'_n(G, M) = M \otimes_{\mathbf{Z}} \mathbf{Z}^{(G^n)}$ for $n \geqslant 0$ , $C'_n(G, M) = 0$ for $n < 0$ , the differential $d_n : C'_n(G, M) \to C'_{n-1}(G, M)$ being defined by:

d_n(m \otimes e_{g_1, ..., g_n}) = m \cdot g_1 \otimes e_{g_2, ..., g_n} + \sum_{i=1}^{n-1} (-1)^i m \otimes e_{g_1, ..., g_i g_{i+1}, ..., g_n}

+ (-1)^n m \otimes e_{g_1, ..., g_{n-1}}

for any $n \geqslant 1$ , $m$ in M and $g_1, ..., g_n$ in G.

Example 1

It follows directly from the definition that $H^0(G, M)$ is isomorphic to the submodule of elements of $M$ invariant under the action of $G$ , and $H_0(G, M)$ to the quotient module of $M$ by the submodule generated by the elements $m.g-m$ for $m\in M$ , $g\in G$ .

Example 2

It follows from the preceding that $H^1(G, M)$ is isomorphic to the $\mathbf{Z}$ -module $Z^1(G, M)/B^1(G, M)$ , where $Z^1(G, M)$ is the $\mathbf{Z}$ -module of mappings $f$ of $G$ into $M$ satisfying:

f(g_1,g_2)=g_1.f(g_2)+f(g_1)\quad\text{for all }g_1,g_2\text{ in }G,

and $B^1(G, M)$ is the sub- $\mathbf{Z}$ -module of $Z^1(G, M)$ consisting of the $f$ for which there exists an element $m$ of $M$ such that:

f(g)=g.m-m\quad\text{for every }g\in G.

One sometimes says that $Z^1(G, M)$ is the $\mathbf{Z}$ -module of crossed homomorphisms of $G$ into $M$ , and $B^1(G, M)$ the submodule of principal crossed homomorphisms.

Let us denote by $\iota:G\to \operatorname{Aut}(M)$ the homomorphism deduced from the action of $G$ ; consider the external semidirect product $M\times_{\iota}G$ and the extension $\xi_{\iota}:M\times_{\iota}G\to G$ (I, p. 64). Let $e:G\to M\times_{\iota}G$ be a mapping such that $\rho\circ e=1_G$ ; one has $e=(f,1_G)$ , where $f\in C^1(G,M)$ . In order that $e$ be a homomorphism (that is to say, a section of the extension $\xi_{\iota}$ ), it is necessary and sufficient that $f\in Z^1(G,M)$ . In order that two sections of $\xi_{\iota}$ be conjugate by an element of $i(M)$ , it is necessary and sufficient that the corresponding crossed homomorphisms have the same class in $H^1(G,M)$ .

When $G$ operates trivially on $M$ , one has $B^1(G,M)=0$ and $H^1(G,M)$ is isomorphic to the $\mathbf{Z}$ -module of group homomorphisms of $G$ into $M$ .

Example 3

Analogously $H^2(G,M)$ is isomorphic to the $\mathbf{Z}$ -module $Z^2(G,M)/B^2(G,M)$ , where $Z^2(G,M)$ is the $\mathbf{Z}$ -module of mappings $f$ of $G\times G$ into $M$ , satisfying:

g_1.f(g_2,g_3)-f(g_1g_2,g_3)+f(g_1,g_2g_3)-f(g_1,g_2)=0

whatever $g_1,g_2,g_3$ in $G$ may be, and $B^2(G,M)$ is the sub- $\mathbf{Z}$ -module of $Z^2(G,M)$ consisting of the $f$ for which there exists a mapping $h$ of $G$ into $M$ such that:

f(g_1,g_2)=g_1.h(g_2)-h(g_1g_2)+h(g_1)

whatever $g_1,g_2$ in $G$ may be.

Thus one recovers the definition of the group $H^2(G,M)$ given in VIII, App. II; in particular there exists a canonical isomorphism of $H^2(G,M)$ onto the group of extension classes of $G$ by $M$ (loc. cit.).

Example 4

Let $M$ be a $\mathbf{Z}$ -module, considered as a right $\mathbf{Z}^{(G)}$ -module by making $G$ operate trivially. The group $H_1(G,M)$ is isomorphic to the quotient of $M\otimes_{\mathbf{Z}}\mathbf{Z}^{(G)}$ by the sub- $\mathbf{Z}$ -module generated by the elements $m\otimes(e_{g_1g_2}-e_{g_1}-e_{g_2})$ for $m$ in $M$ , $g_1,g_2$ in $G$ ; it follows easily that $H_1(G,M)$ is isomorphic to $M\otimes_{\mathbf{Z}}(G/(G,G))$ .

Let us denote by $\sigma$ the anti-automorphism of $\mathbf{Z}^{(G)}$ defined by $\sigma(e_g) = e_{g^{-1}}$ for $g \in G$ . Every left $\mathbf{Z}^{(G)}$ -module may be considered as a right $\mathbf{Z}^{(G)}$ -module by means of $\sigma$ , and conversely. This makes it possible, for example, to define the groups $H_q(G, M)$ for a left $\mathbf{Z}^{(G)}$ -module $M$ , by setting $H_q(G, M) = H_q(G, \sigma_*(M)) = \mathrm{Tor}_q^{\mathbf{Z}^{(G)}}(\mathbf{Z}, M)$ .

Lemma 1

Let $M$ be a $\mathbf{Z}$ -module; let $M^G$ denote the group $\mathrm{Hom}_{\mathbf{Z}}(\mathbf{Z}^{(G)}, M)$ endowed with its natural structure of left $\mathbf{Z}^{(G)}$ -module. Then:

H^i(G, M^G) = 0 \quad \text{for} \quad i \geqslant 1 .

It follows indeed from Prop. 8, b) (X, p. 109), applied with $A = N = \mathbf{Z}^{(G)}$ and $B = Q = \mathbf{Z}$ , that one has a canonical isomorphism:

\mathrm{Ext}_{\mathbf{Z}^{(G)}}(\mathbf{Z}, M^G) \to \mathrm{Ext}_{\mathbf{Z}}(\mathbf{Z}, M)

whence the lemma.

Proposition 11

Let $L$ be a finite-degree galois extension of a commutative field $K$ , with galois group $G$ . a) One has $H^i(G, L) = 0$ for $i \geqslant 1$ . b) One has $H^1(G, L^*) = 0$ . c) The group $H^2(G, L^*)$ is canonically isomorphic to the group $\mathrm{Br}(K, L)$ (VIII, § 13).

The normal basis theorem (V, §10, No. 9, Theorem 6) shows that $L$ is isomorphic as a $\mathbf{Z}^{(G)}$ -module to $K^G = \mathrm{Hom}_{\mathbf{Z}}(\mathbf{Z}^{(G)}, K)$ ; assertion a) then follows from Lemma 1. In view of Example 2, assertion b) follows from V, §10, No. 5, Corollary 1 to Proposition 9: finally, assertion c) was proved in VIII, §13.

Exercises

See the exercises for § 6.