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Title:  Hochschild Cohomology and Representationfinite Algebras 
Authors:  Buchweitz, RagnarOlaf Liu, Shiping 
Keywords:  Hochschild cohomology derivations mesh algebras simply connected algebras Auslander algebras 
Issue Date:  Mar2004 
Publisher:  Cambridge University Press 
Citation:  Proceedings of the London Mathematical Society, 88, (2), 355380 
Abstract:  Using Grothendieck's semicontinuity theorem for halfexact 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 finitedimensional and representationfinite algebra algebra $A$ over an algebraically closed field with Auslander algebra $\Lambda$ the following conditions are equivalent: (1) $A$ admits no outer derivation; (2) $\Lambda$ admits no outer derivations; (3) $A$ is simply connected; (4) $\Lambda$ is strongly simply connected. 
URI:  http://hdl.handle.net/1807/9847 
Appears in Collections:  Mathematics

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