Proposition 1

Let $A$ be a ring, $J$ an ideal of $A$ and $0 \to M' \to M \to M'' \to 0$ an exact sequence of $A$-modules. Put

Then one is in one of the following three cases, which are mutually exclusive:

Consider the exact sequence of extension modules associated with $A/J$ and with the exact sequence above (A, X, p. 92, th. 2). Exclude the case $p = p' = p'' = +\infty$; there then exists in this sequence a first nonzero module, and the following module is also nonzero. This gives the following three possibilities:

a) The first nonzero module is $\operatorname{Ext}_A^{p'}(A/J, M')$. Then $p' = p \leqslant p''$.

b) The first nonzero module is $\operatorname{Ext}_A^p(A/J, M)$. Then $p = p'' < p'$.

c) The first nonzero module is $\operatorname{Ext}_A^{p''}(A/J, M'')$. Then $p'' + 1 = p' \leqslant p$.