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Abstract
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We show that the restricted Lie algebra structure on Hochschild cohomology is
invariant under stable equivalences of Morita type between self-injective algebras.
Thereby, we obtain a number of positive characteristic stable invariants, such as the
-toral
rank of
.
We also prove a more general result concerning Iwanaga–Gorenstein algebras, using a
generalization of stable equivalences of Morita type. Several applications are given to
commutative algebra and modular representation theory.
These results are proven by first establishing the stable invariance of the
-structure
of the Hochschild cochain complex. In the appendix, we explain how the
-power
operation on Hochschild cohomology can be seen as an artifact of this
-structure.
In particular, we establish well-definedness of the
-power
operation, following some—originally topological—methods due to May, Cohen and
Turchin, using the language of operads.
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Keywords
Hochschild cohomology, Gerstenhaber bracket, restricted Lie
algebra, B-infinity algebra, stable equivalence of Morita
type, singularity category
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Mathematical Subject Classification
Primary: 16E40, 16D90
Secondary: 17B50, 13D03
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Milestones
Received: 25 April 2022
Revised: 8 September 2022
Accepted: 24 September 2022
Published: 3 March 2023
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