Stable invariance of the restricted Lie algebra structure of Hochschild cohomology
File(s)
Author(s)
Briggs, Benjamin
Rubio y Degrassi, Lleonard
Type
Journal Article
Abstract
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
p
-toral rank of
HH
1
(
A
,
A
)
. 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
B
∞
-structure of the Hochschild cochain complex. In the appendix, we explain how the
p
-power operation on Hochschild cohomology can be seen as an artifact of this
B
∞
-structure. In particular, we establish well-definedness of the
p
-power operation, following some—originally topological—methods due to May, Cohen and Turchin, using the language of operads.
p
-toral rank of
HH
1
(
A
,
A
)
. 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
B
∞
-structure of the Hochschild cochain complex. In the appendix, we explain how the
p
-power operation on Hochschild cohomology can be seen as an artifact of this
B
∞
-structure. In particular, we establish well-definedness of the
p
-power operation, following some—originally topological—methods due to May, Cohen and Turchin, using the language of operads.
Date Issued
2022-11
Date Acceptance
2022-09-24
Citation
Pacific Journal of Mathematics, 2022, 321 (1), pp.45-71
ISSN
0030-8730
Publisher
Mathematical Sciences Publishers
Start Page
45
End Page
71
Journal / Book Title
Pacific Journal of Mathematics
Volume
321
Issue
1
Copyright Statement
© 2022 Mathematical Sciences Publishers.
Identifier
http://dx.doi.org/10.2140/pjm.2022.321.45
Publication Status
Published
Date Publish Online
2023-03-03