Theorem 3

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

Suppose the $A$ -modules $B(C)$ and $Z(C)$ are projective. There exists, for each $n$ , a unique homomorphism of $k$ -modules

\beta^n : \prod_{p+q=n-1} \mathrm{Ext}_A^1(H_p(C), H^q(C')) \to H^n(\mathrm{Homgr}_A(C, C'))

making the diagram

\begin{array}{ccc} \mathrm{Homgr}_A^{n-1}(B(C), H(C')) & \xleftarrow{\lambda^{n-1}(B(C), C')} & H^{n-1}(\mathrm{Homgr}_A(B(C), C')) \\ \downarrow \varphi^{n-1} & & \downarrow H^n(\mathrm{Homgr}(\delta, 1)) \\ \prod_{p+q=n-1} \mathrm{Ext}_A^1(H_p(C), H^q(C')) & \xrightarrow{\beta^n} & H^n(\mathrm{Homgr}_A(C, C')) . \end{array}

commutative.

The sequences of graded $k$ -modules

(11) $0 \to \prod_{p+q=n-1} \mathrm{Ext}_A^1(H_p(C), H^q(C')) \xrightarrow{\beta^n} H^n(\mathrm{Homgr}_A(C, C'))$

\xrightarrow{\lambda^n(C, C')} \prod_{p+q=n} \mathrm{Hom}_A(H_p(C), H^q(C')) \to 0

are exact.

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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