I am trying to proof the following:
Let $R$ be a Cohen-Macaulay local ring with $\operatorname{dim}(R)=d$ and let $$ 0\to M\to F_{n-1}\to F_{n-2}\to\cdots\to F_1\to F_0 $$ be an exact sequence of $R$-modules where each $F_i$ is finitely generated free. If $n\geq d$, then $M$ is a maximal Cohen-Macaulay module.
I am pretty sure that I have to use the fact, that the following conditions are equivalent for a noetherian, local ring $(R,\mathfrak{m})$ and an $R$-module $M$:
(i) $M$ is a maximal Cohen-Macaualy module over $R$
(ii) $\operatorname{Ext}^{i}_R(R/\mathfrak{m},M)=0\quad (i<d)$
(iii) $\operatorname{H}^{i}_{\mathfrak{m}}(M)=0\quad (i\not=d)$
where $\operatorname{H}^{i}_{\mathfrak{m}}(M)$ denotes the $i$-th cohomology functor with support on $\{\mathfrak{m}\}$.
I actually hope that it works with condition (ii) but I am not able to see the right useage of it yet, especially because the $F_i$ are free, so they don't kill $\operatorname{Ext}_R^{i}(R/\mathfrak{m},-)$ a priori. Any hints would be much appreciated.