Geometry of chain complexes and outer automorphisms under derived equivalence

Abstract

The two main theorems proved here are as follows: If $A$ is a finite dimensional algebra over an algebraically closed field, the identity component of the algebraic group of outer automorphisms of $A$ is invariant under derived equivalence. This invariance is obtained as a consequence of the following generalization of a result of Voigt. Namely, given an appropriate geometrization $\text{Comp}^A_{\bold d}$ of the family of finite $A$-module complexes with fixed sequence $\bold d$ of dimensions and an ``almost projective'' complex $X\in \text{Comp}^A_{\bold d}$, there exists a canonical vector space embedding $$T_{X}(\text{Comp}^A_{\bold d}) / T_{X}(G.X) \ \longrightarrow \text{Hom}_{D^b (A\text{-Mod})}(X, X[1]),$$ where $G$ is the pertinent product of general linear groups acting on $\text{Comp}^A_{\bold d}$, tangent spaces at $X$ are denoted by $T_X(-)$, and $X$ is identified with its image in the derived category $D^b (A\text{-Mod})$.