Cotangent cohomology and higher-order obstructions of matroids
Abstract
Matroid Stanley-Reisner rings admit essential deformation obstructions of arbitrarily high order over every field, already at corank three. For each m≥3, we construct a connected cosimple matroid and m fine-homogeneous first-order classes whose proper subcollections integrate jointly and whose pairwise primary cup products vanish. A specified joint (m−1)-jet nevertheless has no lift to order m, even without preserving the grading. The construction rests on formulas for every multigraded component of the second cotangent cohomology T2 and of its primary cup image, expressed through flats and parallel classes of the dual matroid. These formulas show that T2 vanishes exactly when every connected component has corank at most two, or equivalently when there is no U1,4-minor. This proves a conjecture of Constantinescu, Klein, Nguyen, Singh, and Venturello (2025) and characterizes formal smoothness of the affine deformation functor. For the duals of rank-three truncations of U2,m⊕ U2,n, m,n≥3, each nonzero homogeneous class modulo the primary cup image is realized as a higher-order obstruction, and the quotient has an explicit presentation with m+n−2 minimal homogeneous generators.
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BibTeX
@misc{wang2026cotangent,
title = {{Cotangent cohomology and higher-order obstructions of matroids}},
author = {Qihang Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20649},
eprint = {2610.20649},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20649},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cotangent cohomology and higher-order obstructions of matroids
AU - Qihang Wang
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.20649
UR - https://arxiv.org/abs/2610.20649
AB - Matroid Stanley-Reisner rings admit essential deformation obstructions of arbitrarily high order over every field, already at corank three. For each m≥3, we construct a connected cosimple matroid and m fine-homogeneous first-order classes whose proper subcollections integrate jointly and whose pairwise primary cup products vanish. A specified joint (m−1)-jet nevertheless has no lift to order m, even without preserving the grading. The construction rests on formulas for every multigraded component of the second cotangent cohomology T^2 and of its primary cup image, expressed through flats and parallel classes of the dual matroid. These formulas show that T^2 vanishes exactly when every connected component has corank at most two, or equivalently when there is no U_1,4-minor. This proves a conjecture of Constantinescu, Klein, Nguyen, Singh, and Venturello (2025) and characterizes formal smoothness of the affine deformation functor. For the duals of rank-three truncations of U_2,m⊕ U_2,n, m,n≥3, each nonzero homogeneous class modulo the primary cup image is realized as a higher-order obstruction, and the quotient has an explicit presentation with m+n−2 minimal homogeneous generators.
ER -