Example 2

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

Algebra, chapter X, § 7. Produit de composition

If $u \in \mathrm{Hom}_A(N, P)$ , $v \in \mathrm{Ext}_A(M, N)$ , then

u \circ v = \mathrm{Ext}_A(l_M, u)(v) \in \mathrm{Ext}_A(M, P):

analogously, if $u \in \mathrm{Ext}_A(N, P)$ , $v \in \mathrm{Hom}_A(M, N)$ , then

u \circ v = \mathrm{Ext}_A(v, l_P)(u) \in \mathrm{Ext}_A(M, P).

This follows from the definitions and the remarks of X. p. 88.

If $Q, M, N, P$ are four left $A$ -modules, and if

u \in \mathrm{Ext}_A(N, P), \quad v \in \mathrm{Ext}_A(M, N), \quad w \in \mathrm{Ext}_A(Q, M),

then $(u \circ v) \circ w = u \circ (v \circ w)$ : the composition product is associative; we will therefore denote the composition of several elements without parentheses. In particular, by Example 2:

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

Cited by

Nothing.

Cites

Nothing.