A power structure over the Grothendieck ring of geometric dg categories

Gyenge, Ádám (2024) A power structure over the Grothendieck ring of geometric dg categories. PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY. ISSN 0013-0915

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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.

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