Geometry of chain complexes and outer automorphisms under derived equivalence

Details Geometry of chain complexes and outer automorphisms under derived equivalence Geometry of chain complexes and outer automorphisms under derived equivalence

2000-12-15

arxiv.org/abs/math/0012119v1

Mathematics

Representation Theory

21 pages. To appear in Trans. Amer. Math. Soc

Scientific paper

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 $D^b (A\text{-Mod})$.

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