Proposition 7

Tag 0DX0, permanent. Cite it as [Bourbaki, Tag 0DX0].

Let $I$ and $J$ be two sets, $(M_\alpha)_{\alpha\in I}$ and $(N_\beta)_{\beta\in J}$ two families of $A$ -modules; the homomorphism

\operatorname{Ext}_A\left(\bigoplus_{\alpha\in I}M_\alpha,\prod_{\beta\in J}N_\beta\right) \longrightarrow \prod_{\substack{\beta\in J,\ \alpha\in I}} \operatorname{Ext}_A(M_\alpha,N_\beta)

induced by the canonical homomorphisms

M_\alpha \to \bigoplus_{\alpha\in I} M_\alpha

and

\prod_{\beta\in J} N_\beta \to N_\beta

is bijective.

It suffices to prove that for every $A$ -module $M$ (resp. $N$ ), the homomorphisms

\operatorname{Ext}(M, \prod_\beta N_\beta) \to \prod_\beta \operatorname{Ext}(M,N_\beta)

(resp.

\operatorname{Ext}(\bigoplus M_\alpha,N)\to\prod_\alpha\operatorname{Ext}(M_\alpha,N)

) are bijective. This follows from what has already been shown, from Proposition 1 of X, p. 28, and the canonical isomorphisms

\operatorname{Homgr}_A(L(M), \prod N_\beta) \to \prod \operatorname{Homgr}_A(L(M),N_\beta)

and

\operatorname{Homgr}_A(\bigoplus M_\alpha, I(N)) \to \prod \operatorname{Homgr}_A(M_\alpha, I(N))

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

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