Resolutions over strict complete intersections

Abstract

Let $(Q,\mathfrak{n})$ be a regular local ring and let $f_1,\dots ,f_c\in {\mathfrak{n}}^2$ be a $Q$-regular sequence. Set $(A,\mathfrak{m})=(Q/(\mathbf{f}),\mathfrak{n}/(\mathbf{f}))$. Further assume that the initial forms $f_1^{\ast },\dots ,f_c^{\ast }$ form a $G(Q)={⨁}_{n\ge 0}{\mathfrak{n}}^i/{\mathfrak{n}}^{i+1}$-regular sequence. Without loss of any generality assume $or{d}_Q(f_1)\ge or{d}_Q(f_2)\ge \cdots \ge or{d}_Q(f_c)$. Let $M$ be a finitely generated $A$-module and let $(\mathbb{F},\partial )$ be a minimal free resolution of $M$. Then we prove that $ord({\partial }_i)\le or{d}_Q(f_1)-1$ for all $i\gg 0$. We also construct an MCM $A$-module $M$ such that $ord({\partial }_{2i+1})=or{d}_Q(f_1)-1$ for all $i\ge 0$. 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).

arXiv admin note: text overlap with arXiv:1109.5326. Typo in title corrected