Geometry of chain complexes and outer automorphisms under derived equivalence

The two main theorems proved here are as follows: If A is a finite dimensio nal algebra over an algebraically closed field, the identity component of t he algebraic group of outer automorphisms of A is invariant under derived e quivalence. This invariance is obtained as a consequence of the following g eneralization of a result of Voigt. Namely, given an appropriate geometriza tion Comp(d)(A) of the family of finite A-module complexes with fixed seque nce d of dimensions and an "almost projective" complex X is an element of C omp(d)(A), there exists a canonical vector space embedding TX (Comp(d)(A))/TX (G.X)--> Hom(D b(A-Mod)) (X; X[1]), where G is the pertinent product of general linear groups acting on Comp(d) (A), tangent spaces at X are denoted by TX (-), and X is identified with it s image in the derived category D-b (A-Mod).