Corollary 2

Let

be an exact sequence of $A$-modules.

a) We have

Equality occurs as soon as $\mathrm{dp}_A(M'') \neq \mathrm{dp}_A(M') + 1$.

b) We have

Equality occurs as soon as $\mathrm{dp}_A(M) \neq \mathrm{dp}_A(M')$.

c) We have

Equality occurs as soon as $\mathrm{dp}_A(M) \neq \mathrm{dp}_A(M'')$.

Let us prove for example a), the proofs of b) and c) being analogous. If N is an arbitrary a-module and n an integer $\geq 0$, we have an exact sequence

If $\mathrm{dp}_A(M'), \mathrm{dp}_A(M'') \leq n$, then $\mathrm{Ext}_A^{n+1}(M', N) = 0$ and $\mathrm{Ext}_A^{n+1}(M'', N) = 0$ (prop. 1), hence $\mathrm{Ext}_A^{n+1}(M, N) = 0$ and $\mathrm{dp}_A(M) \leq n$ (prop. 1), so that

If $\mathrm{dp}_A(M)<\sup\bigl(\mathrm{dp}_A(M'),\mathrm{dp}_A(M'')\bigr)$, then necessarily $\mathrm{dp}_A(M)<+\infty$. For every $n>\mathrm{dp}_A(M)$ and every a-module N, we have

by the preceding exact sequence; by prop. 1, this immediately implies $\mathrm{dp}_A(M'')=\mathrm{dp}_A(M')+1$, since one of the quantities $\mathrm{dp}_A(M')$, $\mathrm{dp}_A(M'')$ is $>\mathrm{dp}_A(M)$.