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Minimal cellular resolutions of complementary edge ideals
Abstract
We construct an explicit minimal cellular resolution for every squarefree monomial ideal generated in degree n−2 in the polynomial ring S=K[x1,…,xn]. Such an ideal is the complementary edge ideal Ic(G) of a simple graph G on [n], and the resolution is supported on a labeled regular CW complex XG of dimension at most $2$. This yields closed formulas for the differentials and for the multigraded Betti numbers of Ic(G) in terms of graph invariants. We then determine the a-invariant of S/Ic(G) in terms of the vertex connectivity of G. Finally, when S/Ic(G) is Cohen--Macaulay, we describe the canonical module and its Hilbert series, and recover the known characterizations of the Gorenstein and level cases.
Related
- Minimal Resolutions of k-divisible and (k−1)-dominant Ideals
- Minimal resolutions of monomial ideals
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- Cartwright--Sturmfelsness of complementary determinantal edge ideals
- Regularity of symbolic powers of complementary edge ideals
BibTeX
@misc{moradi2026minimal,
title = {{Minimal cellular resolutions of complementary edge ideals}},
author = {Somayeh Moradi},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02168},
eprint = {2611.02168},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02168},
}
RIS
TY - UNPB
TI - Minimal cellular resolutions of complementary edge ideals
AU - Somayeh Moradi
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.02168
UR - https://arxiv.org/abs/2611.02168
AB - We construct an explicit minimal cellular resolution for every squarefree monomial ideal generated in degree n−2 in the polynomial ring S=K[x_1,…,x_n]. Such an ideal is the complementary edge ideal I_c(G) of a simple graph G on [n], and the resolution is supported on a labeled regular CW complex X_G of dimension at most $2$. This yields closed formulas for the differentials and for the multigraded Betti numbers of I_c(G) in terms of graph invariants. We then determine the a-invariant of S/I_c(G) in terms of the vertex connectivity of G. Finally, when S/I_c(G) is Cohen--Macaulay, we describe the canonical module and its Hilbert series, and recover the known characterizations of the Gorenstein and level cases.
ER -