A power structure over the Grothendieck ring of geometric dg categories
Ádám Gyenge
TL;DR
This work constructs and analyzes a power-structure framework on the Grothendieck rings of varieties and geometric dg categories, establishing a compatibility between motivic and categorical zeta functions via a ring homomorphism $\phi:K_0(Var)\to K_0(gdg- cat)$. By proving that $\phi((A(t))^m)=(\phi(A(t)))^{\phi(m)}$, it recasts the Galkin–Shinder conjecture in terms of power structures and yields a compact expression for the categorical zeta function: $Z_{cat}(\mathcal{M},t)=\prod_{n=1}^{\infty}(1-t^n)^{-[\mathcal{M}]}$. The paper then derives explicit applications, including Hilbert scheme generating series, categorical Adams operations, and the behavior of zeta functions for linear algebraic groups, highlighting the deep interplay between motivic data and dg-categorical structures.
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.