Geometry of chain complexes and outer automorphisms under derived equivalence

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 $\operatorname{Comp}^{A}_{{\mathbf d}}$ of the family of finite $A$-module complexes with fixed sequence ${\mathbf{d}}$ of dimensions and an ``almost projective'' complex $X\in \operatorname{Comp}^{A} _{{\mathbf d}}$, there exists a canonical vector space embedding

where $G$ is the pertinent product of general linear groups acting on $\operatorname{Comp}^{A}_{{\mathbf{d}}}$, tangent spaces at $X$ are denoted by $T_{X}(-)$, and $X$ is identified with its image in the derived category $D^{b} (A{\operatorname{\text{-}Mod}})$.