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Hochschild Cohomology and Representation‐Finite Algebras
Author(s) -
Buchweitz RagnarOlaf,
Liu Shiping
Publication year - 2004
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611503014394
Subject(s) - mathematics , quiver , cohomology , algebraically closed field , grothendieck group , group cohomology , algebra over a field , pure mathematics , endomorphism , functor , group (periodic table) , abelian group , chemistry , organic chemistry
Using Grothendieck's semicontinuity theorem for half‐exact functors, we derive two semicontinuity results on Hochschild cohomology. We apply these to show that the first Hochschild cohomogy group of the mesh algebra of a translation quiver over a domain vanishes if and only if the translation quiver is simply connected. We then establish an exact sequence relating the first Hochschild cohomology group of an algebra to that of the endomorphism algebra of a module and apply it to study the first Hochschild cohomology group of an Auslander algebra. Our main result shows that for a finite‐dimensional and representation‐finite algebra algebra A over an algebraically closed field with Auslander algebra Λ the following conditions are equivalent:(1) A admits no outer derivation; (2) Λ admits no outer derivations; (3) A is simply connected; (4) Λ is strongly simply connected. . 2000 Mathematics Subject Classification 16E30, 16G30.