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The two main theorems proved here are as follows: If $A$ is a finitedimensional algebra over an algebraically closed field, the identity componentof the algebraic group of outer automorphisms of $A$ is invariant under derivedequivalence. This invariance is obtained as a consequence of the followinggeneralization of a result of Voigt. Namely, given an appropriategeometrization $\text{Comp}^A_{\bold d}$ of the family of finite $A$-modulecomplexes with fixed sequence $\bold d$ of dimensions and an ``almostprojective'' complex $X\in \text{Comp}^A_{\bold d}$, there exists a canonicalvector 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 thepertinent 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 itsimage in the derived category $D^b (A\text{-Mod})$.

Geometry of chain complexes and outer automorphisms under derived equivalence