[Paper Review] On a zeta-function of a dg-category

Sergey Galkin, Evgeny Shinder|arXiv (Cornell University)|Jun 18, 2015

Algebraic structures and combinatorial models2 references3 citations

TL;DR

This paper introduces a zeta-function for pretriangulated differential graded (dg) categories, establishing a categorical analogue of Kapranov's motivic zeta function. It proves that for smooth projective varieties with certain geometric properties, the categorical zeta-function satisfies a product formula involving the motivic zeta-function via a motivic measure, linking derived categories to symmetric power operations in Grothendieck rings.

ABSTRACT

We define a zeta-function of a pre-triangulated dg-category and investigate its relationship with the motivic zeta-function in the geometric case.

Motivation & Objective

• To define a zeta-function for pretriangulated dg-categories using symmetric power operations in the Grothendieck ring of dg-categories.
• To investigate the relationship between this categorical zeta-function and Kapranov's motivic zeta-function for varieties.
• To establish a precise formula relating the two zeta-functions under geometric assumptions such as rationality or low dimension.
• To explore the implications of this relation for semi-orthogonal decompositions and derived categories of varieties.
• To clarify the role of the motivic measure μ_dg as a ring homomorphism that does not preserve λ-operations, necessitating a new categorical framework.

Proposed method

• Define the categorical zeta-function as $ Z_{cat}(Χ,t) = \sum_{n \geq 0} [\mathrm{Sym}^n(\mathcal{C})] \, t^n $ in $ K_0(\mathrm{dg\text{-}cat}/k)[[t]] $, where $ \mathcal{C} $ is a pretriangulated dg-category.
• Utilize symmetric power operations from Ganter-Kapranov to define $ \lambda $-operations on $ K_0(\mathrm{dg\text{-}cat}/k) $, enabling the construction of the zeta-function.
• Employ the motivic measure $ \mu_{dg}: K_0(\mathrm{Var}/k) \to K_0(\mathrm{dg\text{-}cat}/k) $, sending a variety $ X $ to its dg-enhancement $ I(X) $, to relate geometric and categorical zeta-functions.
• Prove that for smooth projective $ X $ with $ [X] \in K_0(\mathrm{Var}/k) $ a polynomial in $ [\mathbb{A}^1] $, or $ \dim X \leq 2 $, the identity $ Z_{mot}(\mu_{dg}(X),t) = \prod_{k \geq 1} \mu_{dg}(Z_{mot}(X,t^k)) $ holds.
• Use known results from Göttsche, Bridgeland-King-Reid, and Haiman to verify the formula in the surface case, and extend it via induction and the multiplicative property of $ Z_{cat} $.
• Apply Möbius inversion to the transformation $ f(t) \mapsto \prod_{k \geq 1} f(t^k) $, showing it is invertible via the Möbius function $ \mu(k) $, to derive inverse relations between zeta-functions.

Experimental results

Research questions

• RQ1 How can a zeta-function be defined for a pretriangulated dg-category in a way that generalizes Kapranov's motivic zeta-function for varieties?
• RQ2 What is the precise relationship between the motivic zeta-function of a smooth projective variety and the categorical zeta-function of its derived category?
• RQ3 Does the motivic measure $ \mu_{dg} $, which is a ring homomorphism, preserve the structure of zeta-functions under symmetric power operations?
• RQ4 Can the categorical zeta-function be used to recover or reinterpret known results in enumerative geometry, such as those of Göttsche or Haiman, in the derived category setting?
• RQ5 To what extent does the formula $ Z_{mot}(\mu_{dg}(X),t) = \prod_{k \geq 1} \mu_{dg}(Z_{mot}(X,t^k)) $ extend beyond varieties with polynomial classes in $ K_0(\mathrm{Var}/k) $?

Key findings

• The categorical zeta-function $ Z_{cat} $ is multiplicative under semi-orthogonal decompositions, satisfying $ Z_{cat}(\langle \mathcal{A}, \mathcal{B} \rangle, t) = Z_{cat}(\mathcal{A}, t) \cdot Z_{cat}(\mathcal{B}, t) $.
• For a point $ \mathrm{Spec}(k) $, the categorical zeta-function is $ \prod_{k \geq 1} \frac{1}{1 - t^k} $, the generating function for integer partitions.
• For smooth projective varieties $ X $ with $ [X] $ a polynomial in $ [\mathbb{A}^1] $, or $ \dim X \leq 2 $, the identity $ Z_{mot}(\mu_{dg}(X), t) = \prod_{k \geq 1} \mu_{dg}(Z_{mot}(X, t^k)) $ holds.
• The formula for surfaces is established using the McKay correspondence (BKR99), Göttsche's formula for punctual Hilbert schemes, and Haiman's work on symmetric products of curves.
• The transformation $ f(t) \mapsto \prod_{k \geq 1} f(t^k) $ is invertible via Möbius inversion, with inverse $ g(t) \mapsto \prod_{k \geq 1} g(t^k)^{\mu(k)} $, enabling reconstruction of zeta-functions from their iterates.
• The result implies a weak version of a derived category isomorphism: $ [I_{S_n}(C^n)] = \sum_\alpha [I(C^{(\alpha)})] $ in $ K_0(\mathrm{dg\text{-}cat}/k)[1/2] $, where $ C $ is a smooth projective curve and $ \alpha $ runs over partitions of $ n $.