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Antidiagonal Initial Complexes of Infinite Matrix Schubert Varieties are Cohen-Macaulay

Anna Natalie Chlopecki, Nathaniel Gallup, Jason Meintjes

TL;DR

The authors extend Cohen–Macaulayness to non-Noetherian rings by proving that the Stanley–Reisner ring of an infinite simplicial complex is CM in the sense of flat direct limits whenever it is built as a directed union of finite CM full subcomplexes. They formulate an equivalent ring-theoretic criterion using square-free monomial ideals and show that initial ideals with respect to an antidiagonal order preserve this CM property in the flat direct limit setting. The framework is then applied to initial ideals and, notably, to infinite matrix Schubert varieties, yielding new examples of non-Noetherian Cohen–Macaulay rings. This work provides a robust combinatorial–algebraic toolkit for understanding CM behavior in infinite-dimensional settings and connects infinite Schubert geometry with non-Noetherian CM theory.

Abstract

We show that, under certain constraints, the Stanley-Reisner ring of an infinite simplicial complex is Cohen-Macaulay in the sense of ideals and weak Bourbaki unmixed. We apply this result to prove the wanted claim -- that initial complexes of matrix Schubert varieties corresponding to infinite permutations in $S_{\infty}$ with respect to an antidiagonal term order are Cohen-Macaulay (in the same sense), giving rise to new examples of non-Noetherian Cohen-Macaulay rings.

Antidiagonal Initial Complexes of Infinite Matrix Schubert Varieties are Cohen-Macaulay

TL;DR

The authors extend Cohen–Macaulayness to non-Noetherian rings by proving that the Stanley–Reisner ring of an infinite simplicial complex is CM in the sense of flat direct limits whenever it is built as a directed union of finite CM full subcomplexes. They formulate an equivalent ring-theoretic criterion using square-free monomial ideals and show that initial ideals with respect to an antidiagonal order preserve this CM property in the flat direct limit setting. The framework is then applied to initial ideals and, notably, to infinite matrix Schubert varieties, yielding new examples of non-Noetherian Cohen–Macaulay rings. This work provides a robust combinatorial–algebraic toolkit for understanding CM behavior in infinite-dimensional settings and connects infinite Schubert geometry with non-Noetherian CM theory.

Abstract

We show that, under certain constraints, the Stanley-Reisner ring of an infinite simplicial complex is Cohen-Macaulay in the sense of ideals and weak Bourbaki unmixed. We apply this result to prove the wanted claim -- that initial complexes of matrix Schubert varieties corresponding to infinite permutations in with respect to an antidiagonal term order are Cohen-Macaulay (in the same sense), giving rise to new examples of non-Noetherian Cohen-Macaulay rings.
Paper Structure (7 sections, 24 theorems, 16 equations, 2 figures)

This paper contains 7 sections, 24 theorems, 16 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Potentially Infinite Simplicial Complexes and Stanley-Reisner Rings
  3. Compatible Systems of Parameters
  4. Flat Homomorphisms
  5. Proofs of Theorems \ref{['thm: main thm']} and \ref{['thm: main thm initial rings']}
  6. Applications to Initial Ideals
  7. Infinite Matrix Schubert Varieties

Key Result

Theorem A

Let $\Delta$ be a simplicial complex so that $V(\Delta)$ is countable. Suppose there exists an increasing sequence $\Delta_1 \subseteq \Delta_2 \subseteq \ldots$ of finite full subcomplexes of $\Delta$ such that $\Delta_n$ is Cohen-Macaulay and $\bigcup_{n \in \mathbb{N}} \Delta_n = \Delta$. Then, t

Figures (2)

  • Figure 1: Pictoral description of casework.
  • Figure 2: The $1$-skeletons for $\Delta_{\sigma_m}$ for $m\in[3]$ from Example \ref{['example:infiniteSchubIdeal2']}, respectively. A red edge in the complex denotes a missing edge (counterintuitively).

Theorems & Definitions (55)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Example 1
  • Example 2
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • ...and 45 more