Abstract:
We prove the 2-out-of-3 property for the rationality of motivic zeta function in distinguished triangles in Voevodsky's category $\mathscr{DM}$. As an application, we show the rationality of motivic zeta functions for all varieties whose motives are in the thick triangulated monoidal subcategory generated by motives of quasi-projective curves in $\mathscr{DM}$. Together with a result due to P. O'Sullivan, this also gives an example of a variety whose motive is not finite-dimensional while the motivic zeta function is rational.
Keywords:zeta function, motivic measure, finite-dimensional motives, triangulated category of motives over a field, homotopy category of motivic symmetric spectra, Grothendieck group of a triangulated category, $\lambda$-ring, rationality.
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