Simplicial Complexes and Matroids with Vanishing $T^2

We investigate quotients by radical monomial ideals for which $T^2$, the second cotangent cohomology module, vanishes. The dimension of the graded components of $T^2$, and thus their vanishing, depends only on the combinatorics of the corresponding simplicial complex. We give both a complete charact...

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Published inThe Electronic journal of combinatorics Vol. 32; no. 2
Main Authors Constantinescu, Alexandru, Klein, Patricia, Nguyễn, Thái Thành, Singh, Anurag, Venturello, Lorenzo
Format Journal Article
LanguageEnglish
Published 25.04.2025
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Summary:We investigate quotients by radical monomial ideals for which $T^2$, the second cotangent cohomology module, vanishes. The dimension of the graded components of $T^2$, and thus their vanishing, depends only on the combinatorics of the corresponding simplicial complex. We give both a complete characterization and a full list of one dimensional complexes with $T^ 2 = 0$. We characterize the graded components of $T^ 2$ when the simplicial complex is a uniform matroid. Finally, we show that $T^2$ vanishes for all matroids of corank at most two and conjecture that all connected matroids with vanishing $T^2$ are of corank at most two.
ISSN:1077-8926
1077-8926
DOI:10.37236/13236