Dimension filtration, sequential Cohen--Macaulayness and a new polynomial invariant of graded algebras

TL;DR

This paper introduces the Bj"orner--Wachs polynomial, a new invariant of graded algebras derived from dimension filtration, and characterizes sequential Cohen--Macaulayness via its stability under generic initial ideals.

Contribution

It defines the Bj"orner--Wachs polynomial for graded algebras and links its stability to the sequential Cohen--Macaulay property, providing a new algebraic characterization.

Findings

The Bj"orner--Wachs polynomial generalizes the $h$-triangle of simplicial complexes.

Sequential Cohen--Macaulay algebras are characterized by the stability of this polynomial under generic initial ideals.

Connections are established between this polynomial and local cohomology Hilbert series.

Abstract

Let $\k$ be a field and let $A$ be a standard $\mathbb{N}$-graded $\k$-algebra. Using numerical information of some invariants in the primary decomposition of $0$ in $A$, namely the so called dimension filtration, we associate a bivariate polynomial $\BW(A;t,w)$, that we call the Bj\"{o}rner--Wachs polynomial, to $A$. It is shown that the Bj\"{o}rner--Wachs polynomial is an algebraic counterpart of the combinatorially defined $h$-triangle of finite simplicial complexes introduced by Bj\"{o}rner \& Wachs. We provide a characterisation of sequentially Cohen--Macaulay algebras in terms of the effect of the reverse lexicographic generic initial ideal on the Bj\"{o}rner--Wachs polynomial. More precisely, we show that a graded algebra is sequentially Cohen--Macaulay if and only if it has a stable Bj\"{o}rner--Wachs polynomial under passing to the reverse lexicographic generic initial ideal. We conclude by discussing connections with the Hilbert series of local cohomology modules.