A power structure over the Grothendieck ring of geometric dg categories

TL;DR

This paper establishes a power structure over the Grothendieck ring of geometric dg categories, linking motivic and categorical zeta functions, and applies this to various geometric and algebraic series.

Contribution

It introduces a new power structure over the Grothendieck ring of geometric dg categories, enabling reformulation of conjectures and new expressions for categorical zeta functions.

Findings

Categorical zeta function can be expressed as a power with the category itself as exponent.

Reformulation of Galkin-Shinder conjecture as compatibility between motivic and categorical power structures.

Applications to Hilbert schemes, Adams operations, and algebraic group series.

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.