Geometry of chain complexes and outer automorphisms under derived equivalence

2000/12/15 by Birge Huisgen-Zimmermann, Huisgen-Zimmermann, Birge, Manuel Saorin +2
Mathematics · #16E05 #16G10 #16P10 #18E30 #18G35 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.RA #math.RT #msc:16E05 #msc:16G10 #msc:16P10 #msc:18E30 #msc:18G35

paper · pdf · doi:10.48550/arxiv.math/0012119

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

arxiv created 2000/12/15 · openalex publication_date 2000/12/15 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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∈ Comp^A_{\bold d}, there exists a canonical vector space embedding T_X(Comp^A_{\bold d}) / T_X(G.X) \longrightarrow Hom_{D^b (A-Mod)}(X, X[1]), where G is the pertinent product of general linear groups acting on 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-Mod).

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