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Transfer maps in Hochschild (co)homology and applications to stable and derived invariants and to the Auslander-Reiten conjecture

Authors: Steffen Koenig, Yuming Liu and Guodong Zhou
Journal: Trans. Amer. Math. Soc. 364 (2012), 195-232
MSC (2010): Primary 16G10, 16E40; Secondary 20C20
Published electronically: August 3, 2011
MathSciNet review: 2833582
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Abstract: Derived equivalences and stable equivalences of Morita type, and new (candidate) invariants thereof, between symmetric algebras will be investigated, using transfer maps as a tool. Close relationships will be established between the new invariants and the validity of the Auslander-Reiten conjecture, which states the invariance of the number of non-projective simple modules under stable equivalence. More precisely, the validity of this conjecture for a given pair of algebras, which are stably equivalent of Morita type, will be characterized in terms of data refining Hochschild homology (via Külshammer ideals) being invariant and also in terms of cyclic homology being invariant. Thus, validity of the Auslander-Reiten conjecture implies a whole set of ring theoretic and cohomological data to be invariant under stable equivalence of Morita type, and hence also under derived equivalence. We shall also prove that the Batalin-Vilkovisky algebra structure of Hochschild cohomology for symmetric algebras is preserved by derived equivalence. The main tools to be developed and used are transfer maps and their properties, in particular a crucial compatibility condition between transfer maps in Hochschild homology and Hochschild cohomology via the duality between them.

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Additional Information

Steffen Koenig
Affiliation: Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D-50931 Köln, Germany
Address at time of publication: Institut für Algebra und Zahlentheorie, Universtät Stuttgart, Pfaffenwaldring 57, D-70569, Stuttgart, Germany

Yuming Liu
Affiliation: School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China

Guodong Zhou
Affiliation: Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany
Address at time of publication: Ecole Polytechnique Fédérale de Lausanne, SB-MATHGEOM-CTG, Lausanne 1015, Switzerland

Keywords: Auslander–Reiten conjecture, derived equivalence, Hochschild (co)homology, stable equivalence of Morita type, transfer map.
Received by editor(s): January 29, 2010
Received by editor(s) in revised form: April 1, 2010
Published electronically: August 3, 2011
Additional Notes: The second author was supported by Marie Curie Fellowship IIF and by SRF for ROCS, SEM
The third author benefited from financial support via a postdoctoral fellowship from the network “Representation theory of algebras and algebraic Lie theory” and the DAAD. This research work was mainly done while the last two authors visited the University of Köln.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.