Resolutions over strict complete intersections

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).