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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Geometry of chain complexes and outer automorphisms under derived equivalence

Authors: Birge Huisgen-Zimmermann and Manuel Saorín
Journal: Trans. Amer. Math. Soc. 353 (2001), 4757-4777
MSC (2000): Primary 16E05, 16G10, 16P10, 18E30, 18G35
Published electronically: July 25, 2001
MathSciNet review: 1852081
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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

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

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

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Additional Information

Birge Huisgen-Zimmermann
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106

Manuel Saorín
Affiliation: Departamento de Mátematicas, Universidad de Murcia, 30100 Espinardo-MU, Spain

Received by editor(s): November 6, 2000
Published electronically: July 25, 2001
Additional Notes: While carrying out this project, the first-named author was partially supported by an NSF grant, and the second-named author by grants from the DGES of Spain and the Fundación ‘Séneca’ of Murcia. The contents of this article were presented at the Conference on Representations of Algebras at Saõ Paolo in July 1999 and at the Workshop on Interactions between Algebraic Geometry and Noncommutative Algebra at the Mathematical Sciences Research Institute (Berkeley) in February 2000, by the second- and first-named authors, respectively.
Dedicated: The authors wish to dedicate this paper to Idun Reiten on the occasion of her sixtieth birthday
Article copyright: © Copyright 2001 American Mathematical Society