The embedded deformation problem for monomial ideals

TL;DR

This paper investigates the homological properties of rings defined by monomials on regular sequences, establishing a correspondence between embedded deformations and certain algebraic elements, with implications for cohomological support varieties.

Contribution

It provides a positive answer to Avramov's question by linking embedded deformations to degree two central elements and classifies cohomological support varieties for specific monomial ideals.

Findings

Embedded deformation corresponds to a degree two central element in the homotopy Lie algebra.

Established a lower bound for the dimension of cohomological support varieties.

Classified possible subvarieties of affine space for rings with up to five monomial relations.

Abstract

This article is concerned with homological properties of local or graded rings whose defining relations are monomials on some regular sequence. The main result of the article positively answers a question of Avramov for such a ring $R$. More precisely, we establish that an embedded deformation of $R$ corresponds exactly to a degree two central element in the homotopy Lie algebra of $R$, as well as a free summand of the conormal module of $R$. A major input in the proof is an analysis of cohomological support varieties. Other main results include establishing a lower bound for the dimension of the cohomological support variety of any complex over such rings, and classifying all possible subvarieties of affine $n$-space that are the cohomological support of rings defined by $n$ monomial relations where $n$ is five or less.