Deformations of complexes for finite dimensional algebras

TL;DR

This paper establishes the existence of versal deformation rings for complexes over finite dimensional algebras and explores their invariance under certain derived equivalences, with applications to specific algebra families.

Contribution

It proves the well-definedness of versal deformation rings for complexes and shows their invariance under tilting complexes and stable equivalences, extending deformation theory in representation theory.

Findings

Versal deformation rings exist for all bounded complexes of finitely generated modules.

Two-sided tilting complexes preserve these deformation rings.

Applications to derived equivalence classes of dihedral type algebras.

Abstract

Let $k$ be a field and let $\Lambda$ be a finite dimensional $k$-algebra. We prove that every bounded complex $V^\bullet$ of finitely generated $\Lambda$-modules has a well-defined versal deformation ring $R(\Lambda,V^\bullet)$ which is a complete local commutative Noetherian $k$-algebra with residue field $k$. We also prove that nice two-sided tilting complexes between $\Lambda$ and another finite dimensional $k$-algebra $\Gamma$ preserve these versal deformation rings. Additionally, we investigate stable equivalences of Morita type between self-injective algebras in this context. We apply these results to the derived equivalence classes of the members of a particular family of algebras of dihedral type that were introduced by Erdmann and shown by Holm to be not derived equivalent to any block of a group algebra.