Asymptotic Castelnuovo-Mumford regularity of (co)homology modules involving powers of ideals
Abstract
Let R be a Noetherian standard ℕ-graded ring, and let I1,…,Ir be homogeneous ideals of R. Let L,M, and N be finitely generated ℤ-graded R-modules with N⊆ M. For underlinen :=(n1,…,nr)∈ℕr, set bf Iunderlinen:=I1n1⋯ Irnr. We study the asymptotic behaviour of the (Castelnuovo-Mumford) regularity of two families of Ext and Tor modules: those obtained from the quotients bf IunderlinenM/bf IunderlinenN, and those obtained from the quotients M/bf IunderlinenN, with L as the first argument in both the families. For each fixed homological degree, we prove that the regularity of the former family is eventually given by the supremum of finitely many linear functions of underlinen, where the coefficient of ni belongs to the set of degrees of generators of Ii. For the second family, when at least one of the ideals Ii is generated by homogeneous elements of positive degree, we establish, under suitable conditions, a dichotomy: the regularity is either eventually equal to the regularity of the Ext or Tor module of (L,M), or exhibits the same asymptotic behaviour as in the first family. These results follow from a theorem describing the asymptotic behaviour of the regularity of (U+bf IunderlinenV)/bf IunderlinenW, under suitable conditions, where U, V, and W are graded submodules of a finitely generated graded R-module with W⊆ V.
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BibTeX
@misc{ghosh2026asymptotic,
title = {{Asymptotic Castelnuovo-Mumford regularity of (co)homology modules involving powers of ideals}},
author = {Dipankar Ghosh and Ramakrishna Nanduri and Siddhartha Pramanik},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05803},
eprint = {2610.05803},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05803},
}
RIS
TY - UNPB
TI - Asymptotic Castelnuovo-Mumford regularity of (co)homology modules involving powers of ideals
AU - Dipankar Ghosh
AU - Ramakrishna Nanduri
AU - Siddhartha Pramanik
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.05803
UR - https://arxiv.org/abs/2610.05803
AB - Let R be a Noetherian standard ℕ-graded ring, and let I_1,…,I_r be homogeneous ideals of R. Let L,M, and N be finitely generated ℤ-graded R-modules with N⊆ M. For underlinen :=(n_1,…,n_r)∈ℕ^r, set bf I^underlinen:=I_1^n_1⋯ I_r^n_r. We study the asymptotic behaviour of the (Castelnuovo-Mumford) regularity of two families of Ext and Tor modules: those obtained from the quotients bf I^underlinenM/bf I^underlinenN, and those obtained from the quotients M/bf I^underlinenN, with L as the first argument in both the families. For each fixed homological degree, we prove that the regularity of the former family is eventually given by the supremum of finitely many linear functions of underlinen, where the coefficient of n_i belongs to the set of degrees of generators of I_i. For the second family, when at least one of the ideals I_i is generated by homogeneous elements of positive degree, we establish, under suitable conditions, a dichotomy: the regularity is either eventually equal to the regularity of the Ext or Tor module of (L,M), or exhibits the same asymptotic behaviour as in the first family. These results follow from a theorem describing the asymptotic behaviour of the regularity of (U+bf I^underlinenV)/bf I^underlinenW, under suitable conditions, where U, V, and W are graded submodules of a finitely generated graded R-module with W⊆ V.
ER -