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Complementary vectors of simplicial complexes

Matt Larson, Alan Stapledon

TL;DR

The paper classifies the complementary vectors of doubly Cohen–Macaulay simplicial complexes by translating the problem to level algebras arising from generic artinian reductions of Stanley–Reisner rings. Central to the approach is establishing Hodge–Riemann anisotropy for the Gorenstein quotients $ar{H}( riangle, u)$ and constructing a suitable $ u$ (over $ ield{F}_2$ and, when possible, over the base field) so that the Lefschetz maps $oldsymbol{ell}^{d-2q}$ induce isomorphisms for $q o d/2$, yielding a precise description of $ar{h}$ and $ar{g}$-vectors. A key consequence is that the complementary vector $ar{c}( riangle)$ is a sum of $M$-vectors, and every such sum arises from complexes with convex ear decompositions, providing a complete numerical and constructive framework. The results give new bounds for independent sets in matroids, extend to Buchsbaum* and balanced settings, and illuminate connections between combinatorial invariants and graded module theory, while also offering counterexamples to certain symmetric decomposition questions. The work highlights the power of combining differential-operator methods with perturbation techniques to bridge combinatorics and commutative algebra.

Abstract

We classify the complementary vectors of doubly Cohen-Macaulay complexes. This proves a conjecture of Swartz, negatively answers a question of Athanasiadis and Tzanaki, and gives new bounds on the number of independent sets in a matroid. Our technique works more generally for certain level quotients of Stanley-Reisner rings, giving new bounds on the face numbers of Buchsbaum* complexes.

Complementary vectors of simplicial complexes

TL;DR

The paper classifies the complementary vectors of doubly Cohen–Macaulay simplicial complexes by translating the problem to level algebras arising from generic artinian reductions of Stanley–Reisner rings. Central to the approach is establishing Hodge–Riemann anisotropy for the Gorenstein quotients and constructing a suitable (over and, when possible, over the base field) so that the Lefschetz maps induce isomorphisms for , yielding a precise description of and -vectors. A key consequence is that the complementary vector is a sum of -vectors, and every such sum arises from complexes with convex ear decompositions, providing a complete numerical and constructive framework. The results give new bounds for independent sets in matroids, extend to Buchsbaum* and balanced settings, and illuminate connections between combinatorial invariants and graded module theory, while also offering counterexamples to certain symmetric decomposition questions. The work highlights the power of combining differential-operator methods with perturbation techniques to bridge combinatorics and commutative algebra.

Abstract

We classify the complementary vectors of doubly Cohen-Macaulay complexes. This proves a conjecture of Swartz, negatively answers a question of Athanasiadis and Tzanaki, and gives new bounds on the number of independent sets in a matroid. Our technique works more generally for certain level quotients of Stanley-Reisner rings, giving new bounds on the face numbers of Buchsbaum* complexes.
Paper Structure (8 sections, 20 theorems, 14 equations, 1 figure)

This paper contains 8 sections, 20 theorems, 14 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Cycles and Lefschetz
  3. Generic cycles
  4. Constructions and M-vectors
  5. Generalizations and further questions
  6. Arbitrary characteristic
  7. Balanced complexes
  8. Pure $O$-sequences

Key Result

Theorem 1.2

Let $\Delta$ be a simplicial complex of dimension $d-1$, and let $\mu \colon \overline{H}^d(\Delta) \to K$ be a nonzero map which is defined over $\mathbb{F}_2$. Let $\ell = x_1 + \dotsb + x_n$. For each $q \le d/2$, multiplication by $\ell^{d - 2q}$ induces an isomorphism from $\overline{H}^q(\Delt

Figures (1)

  • Figure 1: Some counterexamples to possible extensions of Theorem \ref{['thm:gorensteinhilbert']}.

Theorems & Definitions (41)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Definition 1.5
  • Theorem 1.6
  • Corollary 1.7
  • proof : Proof of Corollary \ref{['cor:complementaryvector']}
  • Proposition 2.1
  • proof
  • ...and 31 more