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 Mat. Zametki, 2010, Volume 87, Issue 3, Pages 369–381 (Mi mz4117)

Zeta Functions in Triangulated Categories

V. I. Guletskii

University of Liverpool

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

DOI: https://doi.org/10.4213/mzm4117

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English version:
Mathematical Notes, 2010, 87:3, 345–354

Bibliographic databases:

UDC: 513.6
Revised: 17.09.2009

Citation: V. I. Guletskii, “Zeta Functions in Triangulated Categories”, Mat. Zametki, 87:3 (2010), 369–381; Math. Notes, 87:3 (2010), 345–354

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz4117
• https://doi.org/10.4213/mzm4117
• http://mi.mathnet.ru/eng/mz/v87/i3/p369

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This publication is cited in the following articles:
1. Mazza C., Weibel C., “Schur-Finiteness in Lambda-Rings”, J. Algebra, 374 (2013), 66–78
2. Biglari Sh., “On Lambda Operations on Mixed Motives”, J. K-Theory, 12:2 (2013), 381–404
3. Ramachandran N., Tabuada G., “Exponentiable Motivic Measures”, J. Ramanujan Math. Soc., 30:4 (2015), 349–360
4. Gorchinskiy S., Guletskii V., “Symmetric powers in abstract homotopy categories”, Adv. Math., 292 (2016), 707–754
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