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 $\operatorname {Comp}^{A}_{{\mathbf d}}$ 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 \[ T_{X}(\operatorname {Comp}^{A}_{{\mathbf {d}}}) / T_{X}(G.X) \ \longrightarrow \ \ \operatorname {Hom} _{D^{b}(A{\operatorname {\text {-}Mod}})}(X,X[1]), \] 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 $D^{b} (A{\operatorname {\text {-}Mod}})$.