Layered resolutions of Cohen–Macaulay modules

Abstract

Let $S$ be a Gorenstein local ring and suppose that $M$ is a finitely generated Cohen–Macaulay $S$-module of codimension $c$. Given a regular sequence $f_1, \ldots, f_c$ in the annihilator of $M$ we set $R = S/(f_1, \ldots, f_c)$ and construct layered $S$-free and $R$-free resolutions of $M$. The construction inductively reduces the problem to the case of a Cohen–Macaulay module of codimension $c-1$ and leads to the inductive construction of a higher matrix factorization for $M$. In the case where $M$ is a sufficiently high $R$-syzygy of some module of finite projective dimension over $S$, the layered resolutions are minimal and coincide with the resolutions defined from higher matrix factorizations we described in [EP]. Our results provide a characterization of all MCM modules over a complete intersection in terms of higher matrix factorizations.