Geometric aspects of representation theory for {DG} algebras: answering a question of Vasconcelos

TL;DR

This paper uses geometric representation theory techniques to analyze DG modules over graded algebras, proving a tangent space isomorphism and answering a longstanding question about finiteness of semidualizing complexes.

Contribution

It establishes a geometric interpretation of Ext groups for DG modules and resolves Vasconcelos's 1974 question on the finiteness of semidualizing complexes.

Findings

Proves a tangent space isomorphism for DG modules

Shows a local ring has finitely many semidualizing complexes up to shift-isomorphism

Applies geometric techniques to representation theory of DG algebras

Abstract

We apply geometric techniques from representation theory to the study of homologically finite differential graded (DG) modules $M$ over a finite dimensional, positively graded, commutative DG algebra $U$. In particular, in this setting we prove a version of a theorem of Voigt by exhibiting an isomorphism between the Yoneda Ext group $\operatorname{YExt}^1_U(M,M)$ and a quotient of tangent spaces coming from an algebraic group action on an algebraic variety. As an application, we answer a question of Vasconcelos from 1974 by showing that a local ring has only finitely many semidualizing complexes up to shift-isomorphism in the derived category $\mathcal{D}(R)$.