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$p$-anisotropy on the moment curve for homology manifolds and cycles

Karim Adiprasito, Kaiying Hou, Daishi Kiyohara, Daniel Koizumi, Monroe Stephenson

TL;DR

This work extends $p$-anisotropy results to characteristic $p$ for face rings of simplicial cycles by leveraging Gorensteinification and a degree map framework. The authors construct moment-curve based linear systems of parameters and show the existence of Artinian reductions that are $pm$-anisotropic, with nonvanishing $pm$-th powers for all nonzero elements up to a degree bound. They further adapt the results to rational coefficients, relate bases across characteristics, and apply the theory to homology manifolds. The paper also clarifies questions from prior work and proposes conjectures about total anisotropy and generic reductions in the Gorenstein setting, pointing to broader implications for Lefschetz-type properties in positive characteristic. Overall, the results deepen the understanding of anisotropy phenomena in face rings and provide concrete, geometrically meaningful parameter choices via the moment curve.

Abstract

We prove that the Gorensteinification of the face ring of a cycle is totally $p$-anisotropic in characteristic $p$. In other words, given an appropriate Artinian reduction, it contains no nonzero $p$-isotropic elements. Moreover, we prove that the linear system of parameters can be chosen corresponding to a geometric realization with points on the moment curve. In particular, this implies that the parameters do not have to be chosen very generically.

$p$-anisotropy on the moment curve for homology manifolds and cycles

TL;DR

This work extends -anisotropy results to characteristic for face rings of simplicial cycles by leveraging Gorensteinification and a degree map framework. The authors construct moment-curve based linear systems of parameters and show the existence of Artinian reductions that are -anisotropic, with nonvanishing -th powers for all nonzero elements up to a degree bound. They further adapt the results to rational coefficients, relate bases across characteristics, and apply the theory to homology manifolds. The paper also clarifies questions from prior work and proposes conjectures about total anisotropy and generic reductions in the Gorenstein setting, pointing to broader implications for Lefschetz-type properties in positive characteristic. Overall, the results deepen the understanding of anisotropy phenomena in face rings and provide concrete, geometrically meaningful parameter choices via the moment curve.

Abstract

We prove that the Gorensteinification of the face ring of a cycle is totally -anisotropic in characteristic . In other words, given an appropriate Artinian reduction, it contains no nonzero -isotropic elements. Moreover, we prove that the linear system of parameters can be chosen corresponding to a geometric realization with points on the moment curve. In particular, this implies that the parameters do not have to be chosen very generically.

Paper Structure

This paper contains 8 sections, 10 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Gorensteinification
  4. Degree Map
  5. Lemma on Annihilator
  6. Establishing full $p$-anisotropy
  7. Application to rational coefficients
  8. Open problems

Key Result

Theorem 1.1

Consider any field $\mathbbm{k}$ of characteristic $p$, any positive integer $m$, and any $(d-1)$-dimensional simplicial cycle $|\mu|$ over $\mathbbm{k}$. Assume that the following condition is satisfied: Then, for some field extension $\mathbbm{k}'$ of $\mathbbm{k}$, we have an Artinian reduction $\mathcal{A}^*(|\mu|)$ that is $pm$-anisotropic, i.e., for every element $u\in \mathcal{B}^k(\mu)$,

Theorems & Definitions (22)

  • Theorem 1.1: Total $pm$-anisotropy on the moment curve
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Lemma 2.4: Lee
  • Lemma 3.1
  • proof
  • ...and 12 more