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Simplicial complexes and matroids with vanishing $T^2$

Alexandru Constantinescu, Patricia Klein, Thai Thanh Nguyen, Anurag Singh, Lorenzo Venturello

TL;DR

The paper analyzes the second cotangent cohomology $T^2$ of Stanley-Reisner rings $\mathbb{K}[\Delta]$ and shows that multigraded dimensions depend only on the combinatorics of $\Delta$, enabling a complete characterization of when $T^2=0$ and a full classification of 1-dimensional complexes with vanishing $T^2$. It characterizes $T^2$ for uniform matroids and proves vanishing precisely when the corank is at most two. Extending to general matroids, the authors prove corank-\leq 2 matroids are unobstructed, and conjecture that all connected unobstructed matroids are built as joins of corank-\leq 2 components, supported by theoretical results and computational evidence. These results illuminate when deformations of Stanley-Reisner schemes are unobstructed, with implications for smoothness in the Hilbert scheme and an explicit combinatorial description of the obstruction space.

Abstract

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.

Simplicial complexes and matroids with vanishing $T^2$

TL;DR

The paper analyzes the second cotangent cohomology of Stanley-Reisner rings and shows that multigraded dimensions depend only on the combinatorics of , enabling a complete characterization of when and a full classification of 1-dimensional complexes with vanishing . It characterizes for uniform matroids and proves vanishing precisely when the corank is at most two. Extending to general matroids, the authors prove corank-\leq 2 matroids are unobstructed, and conjecture that all connected unobstructed matroids are built as joins of corank-\leq 2 components, supported by theoretical results and computational evidence. These results illuminate when deformations of Stanley-Reisner schemes are unobstructed, with implications for smoothness in the Hilbert scheme and an explicit combinatorial description of the obstruction space.

Abstract

We investigate quotients by radical monomial ideals for which , the second cotangent cohomology module, vanishes. The dimension of the graded components of , 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 . We characterize the graded components of when the simplicial complex is a uniform matroid. Finally, we show that vanishes for all matroids of corank at most two and conjecture that all connected matroids with vanishing are of corank at most two.
Paper Structure (11 sections, 11 theorems, 64 equations, 2 figures)

This paper contains 11 sections, 11 theorems, 64 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Combinatorics
  4. Algebra
  5. Cotangent cohomology for simplicial complexes
  6. Remarks on $T^2(\Delta)$
  7. One-dimensional complexes with vanishing $T^2$
  8. Characterization in dimensions zero and one
  9. Complete classification in dimension one
  10. $T^2$ for uniform matroids
  11. Matroids with vanishing $T^2$

Key Result

Lemma 2.1

The modules $T^i_{\mathbf{a}-\mathbf{b}}$ vanish unless $0\neq\mathbf{b}\in\{0,1\}^n$, $\operatorname{supp}\mathbf{a}\in\Delta$ and $\operatorname{supp} \mathbf{b} \subseteq [\operatorname{link}_\Delta \operatorname{supp} \mathbf{a}]$Where $[\Delta]=\Set{v\in[n]~\,:\,~ v\in\Delta}$ denotes the set o

Figures (2)

  • Figure 1: The only two surviving candidates on at least 9 vertices.
  • Figure 2: The list of all one-dimensional simplicial complexes with vanishing $T^2$. Those simplicial complexes that are moreover matroids appear in blue. Note that all these matroids are joins of uniform matroids, with one exception: the one in the second row and fifth column.

Theorems & Definitions (35)

  • Lemma 2.1: altmann2000stanleyreisner
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5: altmann2000stanleyreisner
  • Remark 2.6
  • proof
  • Remark 2.7
  • proof
  • Remark 2.8
  • proof
  • ...and 25 more