The homological shift algebra of a monomial ideal

TL;DR

The paper introduces the i-th homological shift algebra $HS_i(\mathcal{R}(I))$ for a monomial ideal $I$ and shows it is a finitely generated bigraded module over the Rees algebra when $I$ has linear powers, enabling robust asymptotic analysis of invariants such as reg, depth, associated primes and the $v$-number for $HS_i(I^k)$. It provides explicit descriptions for special families (two-variable monomial ideals, principal Borel ideals, and monomial complete intersections) and proves linearity of regularity in $k$ for large $k$ in many cases, along with stability results for depth and associated primes. The paper identifies several classes of ideals with eventual homological linear powers, including two-variable ideals with linear resolution, principal Borel and Hibi-type ideals, and whisker-graph cover ideals, while also giving counterexamples showing higher homological shifts may fail to have linear resolutions. A core achievement is the demonstration of asymptotic Golodness: if $I$ has linear powers, then $HS_i(I^k)$ is Golod for all $i$ and all sufficiently large $k$, via a general module-theoretic Golod criterion applicable to $HS_i(\mathcal{R}(I))$; however, strong Golodness may fail in general. These results illuminate the deep interaction between multigraded syzygies, asymptotic invariants, and Golod properties in monomial ideals.

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 $k\gg0$.

Theorems & Definitions (42)

• Example 1.1
• proof
• Theorem 1.2
• proof
• Proposition 1.3
• proof
• Theorem 1.4
• Proposition 1.5
• proof
• Corollary 1.6