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A power structure over the Grothendieck ring of geometric dg categories

Part of: Families, fibrations Commutative algebra: Homological methods (Co)homology theory

Published online by Cambridge University Press:  05 December 2024

Adam Gyenge Open the ORCID record for Adam Gyenge [Opens in a new window]
Adam Gyenge*
Affiliation:
Budapest University of Technology and Economics, Department of Algebra and Geometry, Institute of Mathematics, Budapest H-1111, Hungary
*
[email protected]
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Abstract

We prove the existence of a power structure over the Grothendieck ring of geometric dg categories. We show that a conjecture by Galkin and Shinder (proved recently by Bergh, Gorchinskiy, Larsen and Lunts) relating the motivic and categorical zeta functions of varieties can be reformulated as a compatibility between the motivic and categorical power structures. Using our power structure, we show that the categorical zeta function of a geometric dg category can be expressed as a power with exponent the category itself. We give applications of our results for the generating series associated with Hilbert schemes of points, categorical Adams operations and series with exponent a linear algebraic group.

Keywords

power structuredg categoryGrothendieck ringzeta function

MSC classification

Primary: 13D15: Grothendieck groups, $K$-theory
Secondary: 14D23: Stacks and moduli problems 14F05: Sheaves, derived categories of sheaves and related constructions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 68 , Issue 1 , February 2025 , pp. 300 - 318
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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