# Equivalence of Cohen-Macaulay type definitions

I knew that the type of Cohen-Macaulay had these two definitions:

• Let $$(R, mathfrak {m}, k)$$ to be a local group Cohen-Macaulay (noetherian); $$M$$ a finish $$R$$-module of depth t. The number $$r (M) = dim_k Ext_R ^ t (k, M)$$ is called Cohen-Macaulay type $$M$$.
• M & # 39; said $$beta_i (M)$$ Betti numbers in a minimal free resolution of $$M$$ ($$M$$ is a $$R$$as before) then the type of Cohen-Macaulay $$M$$ is the last non-zero Betti number, is it $$r (M) = beta_s (M)$$.

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