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Zeta functions in triangulated categories. (English. Russian original) Zbl 1223.14025

Math. Notes 87, No. 3, 345-354 (2010); translation from Mat. Zametki 87, No. 3, 369-381 (2010).
Let \({\mathcal T}'\) be the thick triangulated monoidal subcategory of compact objects in the homotopy category \({\mathcal T}=\text{Ho}({\mathcal C})\) of a simplicial model symmetric monoidal category \({\mathcal C} \). Assume that all Hom groups in \({\mathcal T}\) are vector spaces over \({\mathbb Q}\). The author proves that the wedge and symmetric powers induce two special \(\lambda\)-structures in the Grothendieck ring \(K_0({\mathcal T}')\), which are opposite to each other. From this it follows that the motivic zeta function is multiplicative with respect to distinguished triangles in \({\mathcal T}\). As an application, the motivic zeta functions of all varieties whose motives lie in a thick triangulated tensor subcategory of Voevodsky’s category \({\mathcal DM} \) with coefficients in \({\mathbb Q}\) generated by motives of quasi-projective curves over a field are rational. Together with a result due to P. O’Sullivan, this also gives an example of a variety whose motive is not finite-dimensional in the sense of Kimura and whose motivic zeta function is rational.

MSC:

14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14C15 (Equivariant) Chow groups and rings; motives
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
18F30 Grothendieck groups (category-theoretic aspects)
18E30 Derived categories, triangulated categories (MSC2010)
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References:

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