# Equivalence of definitions of Cohen-Macaulay type

I know that the Cohen-Macaulay type has these two definitions:

• Let $$(R,\mathfrak{m},k)$$ be a Cohen-Macaulay (noetherian) local ring and $$M$$ a finite $$R$$-module of depth $$t$$. The number $$r(M) = \dim_k \mathrm{Ext}_R^t(k,M)$$ is called the Cohen-Macaulay type of $$M$$.
• Denote by $$\beta_i(M)$$ the Betti numbers in a minimal free resolution of $$M$$ ($$M$$ is an $$R$$-module as before). Then the Cohen-Macaulay type of $$M$$ is the last non zero Betti number, that is, $$r(M) = \beta_s(M)$$.

So I would ask how to prove the equivalence of these two definitions. There are some books in which I can find this proof?

For the equivalence you need two more assumptions: (a) $$M$$ to have finite projective dimension and (b) $$R$$ to be Gorenstein. (a) is implicitly required in the second condition, otherwise you don't have the last nonzero Betti number. For necessity of (b), take $$M$$ to be $$R$$. Then the last betti is $$1$$, and the type of $$R$$ must be $$1$$ as well.
Assuming (a) and (b), one can prove the equivalence by induction on the projective dimension of $$M$$. It must be somewhere in the literature, perhaps Bruns-Herzog, but I do not have access to a precise reference right now.