Universal Deformation Rings for Complexes over Finite-Dimensional Algebras

TL;DR

This paper studies the deformation rings of complexes over finite-dimensional algebras, proving conditions under which these rings are universal and showing their invariance under certain singular equivalences.

Contribution

It establishes that the versal deformation ring is universal for complexes with specific Hom conditions in the singularity category and proves invariance under singular equivalences of Morita type.

Findings

Versal deformation rings are universal under certain Hom conditions.

Singular equivalences of Morita type preserve deformation ring isomorphism classes.

Conditions for universality of deformation rings in the derived category.

Abstract

Let $\mathbf{k}$ be field of arbitrary characteristic and let $\Lambda$ be a finite dimensional $\mathbf{k}$-algebra. From results previously obtained by F.M Bleher and the author, it follows that if $V^\bullet$ is an object of the bounded derived category $\mathcal{D}^b(\Lambda\textup{-mod})$ of $\Lambda$, then $V^\bullet$ has a well-defined versal deformation ring $R(\Lambda, V^\bullet)$, which is complete local commutative Noetherian $\mathbf{k}$-algebra with residue field $\mathbf{k}$, and which is universal provided that $\textup{Hom}_{\mathcal{D}^b(\Lambda\textup{-mod})}(V^\bullet, V^\bullet)=\mathbf{k}$. Let $\mathcal{D}_\textup{sg}(\Lambda\textup{-mod})$ denote the singularity category of $\Lambda$ and assume that $V^\bullet$ is a bounded complex whose terms are all finitely generated Gorenstein projective left $\Lambda$-modules. In this article we prove that if $\textup{Hom}_{\mathcal{D}_\textup{sg}(\Lambda\textup{-mod})}(V^\bullet, V^\bullet)=\mathbf{k}$, then the versal deformation ring $R(\Lambda, V^\bullet)$ is universal. We also prove that certain singular equivalences of Morita type (as introduced by X. W. Chen and L. G. Sun) preserve the isomorphism class of versal deformation rings of bounded complexes whose terms are finitely generated Gorenstein projective $\Lambda$-modules.