On Deformations of Gorenstein-Projective Modules over Monomial Algebras with no Overlaps

TL;DR

This paper studies the deformation rings of indecomposable Gorenstein-projective modules over monomial algebras without overlaps, showing these rings are either trivial or isomorphic to a dual number algebra.

Contribution

It proves that for monomial algebras without overlaps, the versal deformation ring of such modules is always universal and explicitly classifies these rings.

Findings

Deformation rings are either trivial or dual numbers.

Universal deformation rings are classified for these modules.

The results apply to monomial algebras without overlaps.

Abstract

Let $\mathbf{k}$ be a field of arbitrary characteristic, let $\Lambda$ be a finite dimensional $\mathbf{k}$-algebra, and let $V$ be an indecomposable Gorenstein-projective $\Lambda$-module with finite dimension over $\mathbf{k}$. It follows that $V$ has a well-defined versal deformation ring $R(\Lambda, V)$, which is complete local commutative Noetherian $\mathbf{k}$-algebra with residue field $\mathbf{k}$, and which is universal provided that the stable endomorphism ring of $V$ is isomorphic to $\mathbf{k}$. We prove that if $\Lambda$ is a monomial algebra without overlaps, then $R(\Lambda,V)$ is universal and isomorphic either to $\mathbf{k}$ or to $\mathbf{k}[[t]]/(t^2)$