Corollary

Tag 0E19, permanent. Cite it as [Bourbaki, Tag 0E19].

Algebra, chapter X, § 8. Dimension homologique

Let $A$ be a ring of homological dimension $\leq 1$ , $C$ a complex of projective a-modules, $\widetilde{C}$ a complex of projective right a-modules, $C'$ a complex of injective a-modules, $P$ a right a-module, $M$ a left a-module, and $n$ an integer. Then there are split exact sequences

0\to\bigoplus_{p+q=n}H_p(\widetilde C)\otimes_A H_q(C)\xrightarrow{\gamma}H_n(\widetilde C\otimes_A C)\xrightarrow{\alpha}\bigoplus_{p+q=n-1}\operatorname{Tor}_1^A(H_p(\widetilde C),H_q(C))\to0,

0\to\prod_p\operatorname{Ext}_A^1(H_p(C),H^{n-p-1}(C'))\xrightarrow{\beta}H^n(\operatorname{Homgr}_A(C,C'))\xrightarrow{\lambda}\prod_p\operatorname{Homgr}_A(H_p(C),H^{n-p}(C'))\to0,

0\to P\otimes_A H_n(C)\xrightarrow{\gamma}H_n(P\otimes_A C)\xrightarrow{\alpha}\operatorname{Tor}_1^A(P,H_{n-1}(C))\to0,

0\to\operatorname{Ext}_A^1(H_{n-1}(C),M)\xrightarrow{\beta}H^n(\operatorname{Homgr}_A(C,M))\xrightarrow{\lambda}\operatorname{Hom}_A(H_n(C),M)\to0.

Since $Z(C)$ , $B(C)$ , $Z(\widetilde C)$ , and $B(\widetilde C)$ are projective and $B(C')$ is injective, this follows from X, p. 78, Cor. 2, p. 96, Cor. 1, and p. 98, Cor. 2.

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

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Cites

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