PreprintCC BY 4.0arXiv
Bounds on Hilbert coefficients of Cohen-Macaulay modules having finite projective dimension
Abstract
Let (A,m) be a Gorenstein local ring with G(A) Cohen-Macaulay, and let M be a Cohen-Macaulay A-module of finite projective dimension. In \cite{Quasipure}, the authors proved that e1(M)≥ binomc+12, where c=regG(A) and ei(M) is the ith Hilbert coefficient of M. We first show that this bound remains valid when A is Cohen-Macaulay. We then study upper bounds for e2(M) when e1(M)=binomc+12+i for i=1,2, and investigate the consequences of equality. In particular, we obtain depth properties and explicit descriptions of the h-polynomial of G(M). Finally, we extend these results to strict complete intersection rings without assuming that M has finite projective dimension.
Related
- One-periodic category of twisted complexes, Cohen-Macaulay modules and differential modules
- Adjoint test modules along Cohen--Macaulay morphisms
- Local-Cohen-Macaulay DG-rings with unequal finitistic projective dimensions
- Sharp effective Nullstellensatz for projective Cohen-Macaulay schemes
- Derived functors and Hilbert polynomials over dominant Cohen-Macaulay rings
BibTeX
@misc{sahoo2026bounds,
title = {{Bounds on Hilbert coefficients of Cohen-Macaulay modules having finite projective dimension}},
author = {Samarendra Sahoo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10295},
eprint = {2609.10295},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10295},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bounds on Hilbert coefficients of Cohen-Macaulay modules having finite projective dimension
AU - Samarendra Sahoo
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10295
UR - https://arxiv.org/abs/2609.10295
AB - Let (A,m) be a Gorenstein local ring with G(A) Cohen-Macaulay, and let M be a Cohen-Macaulay A-module of finite projective dimension. In \cite{Quasipure}, the authors proved that e_1(M)≥ binomc+12, where c=regG(A) and e_i(M) is the ith Hilbert coefficient of M. We first show that this bound remains valid when A is Cohen-Macaulay. We then study upper bounds for e_2(M) when e_1(M)=binomc+12+i for i=1,2, and investigate the consequences of equality. In particular, we obtain depth properties and explicit descriptions of the h-polynomial of G(M). Finally, we extend these results to strict complete intersection rings without assuming that M has finite projective dimension.
ER -