The homological shift algebra of a monomial ideal

TL;DR

This paper introduces homological shift algebras associated with monomial ideals, studying their properties, asymptotic behavior, and conditions for linear resolutions, especially when the ideals have linear powers.

Contribution

It defines the homological shift algebras for monomial ideals, analyzes their structure, and establishes conditions for linear resolutions and Golodness in the asymptotic regime.

Findings

Homological shift algebras form finitely generated bigraded modules over the Rees algebra for ideals with linear powers.

Identifies families of monomial ideals with linear resolutions for all large powers.

Shows that these algebras are Golod for ideals with linear powers and large powers.

Abstract

Let $S=K[x_1,\dots,x_n]$ be the polynomial ring over a field $K$, and let $I\subset S$ be a monomial ideal. In this paper, we introduce the $i$th \textit{homological shift algebras} $\text{HS}_i(\mathcal{R}(I))=\bigoplus_{k\ge1}\text{HS}_i(I^k)$ of $I$. If $I$ has linear powers, these algebras have the structure of a finitely generated bigraded module over the Rees algebra $\mathcal{R}(I)$ of $I$. Hence, many invariants of $\text{HS}_i(I^k)$, such as depth, associated primes, regularity, and the $\text{v}$-number, exhibit well behaved asymptotic behavior. We determine several families of monomial ideals $I$ for which $\text{HS}_i(I^k)$ has linear resolution for all $k\gg0$. Finally, we show that $\text{HS}_i(I^k)$ is Golod for all monomial ideals $I\subset S$ with linear powers and all $k\gg0$.