Regularity and $v$-number of syzygies of graded maximal Cohen-Macaulay modules

Abstract

Let $A=Q/(\mathbf f)$, where $Q$ is a polynomial ring over an infinite field $k$ and $\mathbf f=f_1,\ldots,f_c$ is a homogeneous $Q$-regular sequence. For $1\leq j\leq c$, set $w_j=\operatorname{deg}(f_j)>0.$ Let $M$ be a finitely generated graded $A$-module of infinite projective dimension. Set $M_i=\operatorname{Syz}_i^A(M)$. We prove that, for each $l\in\{0,1\}$, the sets $\operatorname{Ass}_A(M_{2i+l})$ eventually stabilize, and that $\operatorname{reg}(M_{2i+l})$ is eventually bounded above and below by linear polynomials. If $w_j=w$ for all $1\leq j\leq c$, then the functions $i\longmapsto \operatorname{reg}(M_{2i+l})$, $i\longmapsto v(M_{2i+l}),$ and $i\longmapsto v_{\mathfrak p}(M_{2i+l}),\, \mathfrak p\in\mathcal A_l,$ are eventually linear with leading coefficient $w$, where $\mathcal A_l$ denotes the stable set of associated primes.