Hochschild Cohomology and Representation-Finite Algebras |
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Authors: | Buchweitz Ragnar-Olaf; Liu Shiping |
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Institution: | Department of Mathematics, University of Toronto, Toronto Ontario, Canada M5S 3G3; E-mail: ragnar{at}math.toronto.edu
Département de mathématiques, Université de Sherbrooke Sherbrooke, Québec, Canada J1K 2R1. E-mail: shiping{at}dmi.usherb.ca |
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Abstract: | 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 groupof the mesh algebra of a translation quiver over a domain vanishesif and only if the translation quiver is simply connected. Wethen establish an exact sequence relating the first Hochschildcohomology group of an algebra to that of the endomorphism algebraof a module and apply it to study the first Hochschild cohomologygroup of an Auslander algebra. Our main result shows that fora finite-dimensional and representation-finite algebra algebraA over an algebraically closed field with Auslander algebra the following conditions are equivalent: - (1)A admits no outer derivation;
- (2)
admits no outer derivations; - (3) A is simply connected;
- (4)
is strongly simply connected. . 2000 Mathematics Subject Classification 16E30, 16G30. |
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Keywords: | Hochschild cohomology derivations mesh algebras simply connected algebras Auslander algebras |
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