A local cohomology obstruction to small Cohen--Macaulay modules
Abstract
We propose a local-cohomology obstruction to the existence of small Cohen--Macaulay modules over completed section rings of surfaces. Let Y be a smooth connected complex projective surface, and let H be an ample globally generated divisor with KY≡4H numerically. Put c(Y,H)=15H2/8−χ(OOY). The main claim is that, when c(Y,H)>0, every nonzero finite reflexive module M over the completed vertex local ring of ⨁n≥0H0(Y,OOY(nH)) satisfies dimC H2mm(M)≥ c(Y,H)rk M. The argument combines elementary transformations of reflexive extensions, Harder--Narasimhan slopes, Bogomolov's inequality, and a two-section Koszul estimate. A fixed-source kernel estimate transfers the resulting cohomology to the punctured spectrum without requiring a grading on M. For the degree-six member of the six-line Hirzebruch--Kummer family, H2=1080 and χ(OOY)=1926, so the bound is 99rk M. We give an explicit complete-intersection model for this application and spell out the argument leading to the claimed nonexistence of nonzero finite maximal Cohen--Macaulay modules.
Related
- Homological Trimming and Regularity of Filtrations via Local Obstruction Modules
- One-periodic category of twisted complexes, Cohen-Macaulay modules and differential modules
- Adjoint test modules along Cohen--Macaulay morphisms
- Bounds on Hilbert coefficients of Cohen-Macaulay modules having finite projective dimension
- Regularity and v-number of syzygies of graded maximal Cohen-Macaulay modules
BibTeX
@misc{chen2026local,
title = {{A local cohomology obstruction to small Cohen--Macaulay modules}},
author = {Liang Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24142},
eprint = {2609.24142},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24142},
}
RIS
TY - UNPB
TI - A local cohomology obstruction to small Cohen--Macaulay modules
AU - Liang Chen
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24142
UR - https://arxiv.org/abs/2609.24142
AB - We propose a local-cohomology obstruction to the existence of small Cohen--Macaulay modules over completed section rings of surfaces. Let Y be a smooth connected complex projective surface, and let H be an ample globally generated divisor with K_Y≡4H numerically. Put c(Y,H)=15H^2/8−χ(OO_Y). The main claim is that, when c(Y,H)>0, every nonzero finite reflexive module M over the completed vertex local ring of ⨁_n≥0H^0(Y,OO_Y(nH)) satisfies dim_C H^2_mm(M)≥ c(Y,H)rk M. The argument combines elementary transformations of reflexive extensions, Harder--Narasimhan slopes, Bogomolov's inequality, and a two-section Koszul estimate. A fixed-source kernel estimate transfers the resulting cohomology to the punctured spectrum without requiring a grading on M. For the degree-six member of the six-line Hirzebruch--Kummer family, H^2=1080 and χ(OO_Y)=1926, so the bound is 99rk M. We give an explicit complete-intersection model for this application and spell out the argument leading to the claimed nonexistence of nonzero finite maximal Cohen--Macaulay modules.
ER -