PreprintCC BY 4.0arXiv
Regularity and v-number of syzygies of graded maximal Cohen-Macaulay modules
Abstract
Let A=Q/(f), where Q is a polynomial ring over an infinite field k and f=f1,…,fc is a homogeneous Q-regular sequence. For 1≤ j≤ c, set wj=deg(fj)>0. Let M be a finitely generated graded A-module of infinite projective dimension. Set Mi=SyziA(M). We prove that, for each l∈{0,1}, the sets AssA(M2i+l) eventually stabilize, and that reg(M2i+l) is eventually bounded above and below by linear polynomials. If wj=w for all 1≤ j≤ c, then the functions ilongmapsto reg(M2i+l), ilongmapsto v(M2i+l), and ilongmapsto vp(M2i+l), p∈𝒜l, are eventually linear with leading coefficient w, where 𝒜l denotes the stable set of associated primes.
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BibTeX
@misc{sahoo2026regularity,
title = {{Regularity and $v$-number of syzygies of graded maximal Cohen-Macaulay modules}},
author = {Samarendra Sahoo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29855},
eprint = {2610.29855},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29855},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Regularity and v-number of syzygies of graded maximal Cohen-Macaulay modules
AU - Samarendra Sahoo
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29855
UR - https://arxiv.org/abs/2610.29855
AB - Let A=Q/(f), where Q is a polynomial ring over an infinite field k and f=f_1,…,f_c is a homogeneous Q-regular sequence. For 1≤ j≤ c, set w_j=deg(f_j)>0. Let M be a finitely generated graded A-module of infinite projective dimension. Set M_i=Syz_i^A(M). We prove that, for each l∈{0,1}, the sets Ass_A(M_2i+l) eventually stabilize, and that reg(M_2i+l) is eventually bounded above and below by linear polynomials. If w_j=w for all 1≤ j≤ c, then the functions ilongmapsto reg(M_2i+l), ilongmapsto v(M_2i+l), and ilongmapsto v_p(M_2i+l), p∈𝒜_l, are eventually linear with leading coefficient w, where 𝒜_l denotes the stable set of associated primes.
ER -