Cohen-Macaulay higher conormal and Kähler differential modules of squarefree monomial ideals
Abstract
Let S=k[x1,…,xn] and let I=IΔsubsetneq S be a nonzero squarefree monomial ideal. Motivated by the classical higher-order Kähler differential modules and by the theory of higher conormal modules, we study not only the higher conormal quotients I/Iq, but more generally the shifted quotients Ir/Iq, 1≤ r<q, in the same I-adic conormal filtration, together with their symbolic analogues I(r)/I(q). We prove that, for every 1≤ r<q with q≥3, the module Ir/Iq is Cohen--Macaulay if and only if I is a complete intersection. In sharp contrast, I(r)/I(q) is Cohen--Macaulay if and only if Δ is a matroid, where loops are allowed. Thus, the Cohen--Macaulayness of a single nonexceptional window forces the Cohen--Macaulayness of every window in the corresponding filtration. The unique exceptional pair is (r,q)=(1,2): at this conormal level, we show that the Cohen--Macaulayness of I/I2 forces I2=I(2), and hence I/I(2) is Cohen--Macaulay.
Related
- One-periodic category of twisted complexes, Cohen-Macaulay modules and differential modules
- On the Hilbert depth of a special class of squarefree monomial ideals
- 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{ha2026cohenmacaulay,
title = {{Cohen-Macaulay higher conormal and Kähler differential modules of squarefree monomial ideals}},
author = {Tài Huy Hà and Nguyen Cong Minh},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18071},
eprint = {2609.18071},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18071},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cohen-Macaulay higher conormal and Kähler differential modules of squarefree monomial ideals
AU - Tài Huy Hà
AU - Nguyen Cong Minh
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18071
UR - https://arxiv.org/abs/2609.18071
AB - Let S=k[x_1,…,x_n] and let I=I_Δsubsetneq S be a nonzero squarefree monomial ideal. Motivated by the classical higher-order Kähler differential modules and by the theory of higher conormal modules, we study not only the higher conormal quotients I/I^q, but more generally the shifted quotients I^r/I^q, 1≤ r<q, in the same I-adic conormal filtration, together with their symbolic analogues I^(r)/I^(q). We prove that, for every 1≤ r<q with q≥3, the module I^r/I^q is Cohen--Macaulay if and only if I is a complete intersection. In sharp contrast, I^(r)/I^(q) is Cohen--Macaulay if and only if Δ is a matroid, where loops are allowed. Thus, the Cohen--Macaulayness of a single nonexceptional window forces the Cohen--Macaulayness of every window in the corresponding filtration. The unique exceptional pair is (r,q)=(1,2): at this conormal level, we show that the Cohen--Macaulayness of I/I^2 forces I^2=I^(2), and hence I/I^(2) is Cohen--Macaulay.
ER -