Definition 1

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

Let $M$ and $N$ be two $A$ -modules. The module of extensions of $N$ by $M$ is the graded $k$ -module

\text{(4)} \quad \operatorname{Ext}_A(M, N) = H(\operatorname{Homgr}_A(L(M), I(N))) .

The homogeneous components of $\operatorname{Ext}_A(M, N)$ are denoted

\text{(5)} \quad \operatorname{Ext}_A^n(M, N) = H^n(\operatorname{Homgr}_A(L(M), I(N))) .

As $L(M)$ (resp. $I(N)$ ) is zero to the right (resp. to the left), we have

(6)

\operatorname{Ext}^{n}_{A}(M,N)=0\qquad\text{pour }n<0.

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

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