Traces, Schubert calculus, and Hochschild cohomology of category $\mathcal{O}$
Abstract
We discuss how the Hochschild cohomology of a dg category can be computed as the trace of its Serre functor. Applying this approach to the principal block of the Bernstein--Gelfand--Gelfand category $\mathcal{O}$, we obtain its Hochschild cohomology as the compactly supported cohomology of an associated space. Equivalently, writing $\mathcal{O}$ as modules over the endomorphism algebra $A$ of a minimal projective generator, this is the Hochschild cohomology of $A$. In particular our computation gives the Euler characteristic of the Hochschild cohomology of $\mathcal{O}$ in type A.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2020
- DOI:
- 10.48550/arXiv.2012.02744
- arXiv:
- arXiv:2012.02744
- Bibcode:
- 2020arXiv201202744K
- Keywords:
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- Mathematics - Representation Theory;
- Mathematics - Algebraic Geometry
- E-Print:
- 9 pages