Hochschild cohomology and derived Picard groups

By B. Keller. 2003

127 citations recorded by the source corpus.

Abstract We interpret Hochschild cohomology as the Lie algebra of the derived Picard group and deduce that it is preserved under derived equivalences.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/91607fd9e721eb773bdaf7548b77da6e392812d2

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In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

DERIVED INVARIANCE OF HIGHER STRUCTURES ON THE HOCHSCHILD COMPLEX (2004, background): “is an equivalence, then there is a canonical algebra isomorphism fX : HH (B,B) ! HH (A,A) associated with X. It was shown in [ 16 ] that fX also respects the Lie algebra structures given by the Gerstenhaber bracket.”
Gerstenhaber duality in Hochschild cohomology (2005, background): “The fact thatHH∗(f ) preserves the multiplication was already proved by Keller[10].”
The homotopy theory of dg-categories and derived Morita theory (2004, background): “nctors 6 are involved, and that some sort of higher categorical structures are then often needed (see e.g. [T2, §1]). A relation between the derived Picard group and Hochschild cohomology is given in [Ke2], and is somehow close to our Corollary 8.4. An interpretation of Hochschild cohomology as ﬁrst”
Derived Categories (2016, methodology): “Keller [81] used an abstraction of this idea – essentially viewing the derived Picard group as a derived group stack (evaluating it on commutative artinian DG rings) – to prove that the Hochschild cohomology of A is the Lie algebra, in the derived sense, of DPicK(A).”
Morita theory in abelian, derived and stable model categories (2003, methodology): “ still obtain an equivalence of Ktheory spaces; for more details we refer to [DuSh]. Similarly, Hochschild homology and cohomology (see [Ric91, Prop. 2.5], or, including the Gerstenhaber bracket, see [Kel03]) and cyclic homology (see [Kel96, Kel98]) are isomorphic for derived equivalent rings which ”
DERIVED CATEGORIES AND TILTING (2004, result): “Moreover, this isomorphism is compatible with the Gerstenhaber brackets, cf. [ 47 ].”