Remark

Tag 0DXV, permanent. Cite it as [Bourbaki, Tag 0DXV].

Algebra, chapter X, § 6. Utilisation de résolutions non canoniques

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 $\alpha_1, \beta_1, \gamma_1$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

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Cites

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