Resolutions over strict complete intersections

TL;DR

This paper investigates the structure of minimal free resolutions over strict complete intersections, establishing bounds on the order of differentials and constructing modules with maximal order, along with a simplified proof of periodicity of minors in resolutions.

Contribution

It provides new bounds on the order of differentials in resolutions over strict complete intersections and constructs modules achieving these bounds, also simplifying existing proofs of periodicity of minors.

Findings

Bound on order of differentials:  \, ord(\u03b4_i)  \, ord_Q(f_1) - 1

Existence of MCM modules with maximal order differentials

Simplified proof of periodicity of minors in resolutions

Abstract

Let $(Q, \mathfrak{n})$ be a regular local ring and let $f_1, \ldots, f_c \in \mathfrak{n}^2$ be a $Q$-regular sequence. Set $(A, \mathfrak{m}) = (Q/(\mathbf{f}), \mathfrak{n}/(\mathbf{f}))$. Further assume that the initial forms $f_1^*, \ldots, f_c^*$ form a $G(Q) = \bigoplus_{n \geq 0}\mathfrak{n}^i/\mathfrak{n}^{i+1}$-regular sequence. Without loss of any generality assume $ord_Q(f_1) \geq ord_Q(f_2) \geq \cdots \geq ord_Q(f_c)$. Let $M$ be a finitely generated $A$-module and let $(\mathbb{F}, \partial)$ be a minimal free resolution of $M$. Then we prove that $ord(\partial_i) \leq ord_Q(f_1) - 1$ for all $i \gg 0$. We also construct an MCM $A$-module $M$ such that $ord(\partial_{2i+1}) = ord_Q(f_1) - 1$ for all $i \geq 0$. We also give a considerably simpler proof regarding the periodicity of ideals of minors of maps in a minimal free resolution of modules over arbitrary complete intersection rings (not necessarily strict).