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Journal of K-Theory Journal of K-Theory

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Motivic zeta functions in additive monoidal categories

Published online by Cambridge University Press:  08 December 2011

Kenichiro Kimura ,
Shun-ichi Kimura  and
Nobuyoshi Takahashi
Kenichiro Kimura
Affiliation:
Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki, 305-8571 Japankimurak@math.tsukuba.ac.jp
Shun-ichi Kimura
Affiliation:
Department of Mathematics, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526 Japankimura@math.sci.hiroshima-u.ac.jp
Nobuyoshi Takahashi
Affiliation:
Department of Mathematics, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526 Japantakahasi@math.sci.hiroshima-u.ac.jp
Corresponding
E-mail address:
kimurak@math.tsukuba.ac.jp
kimura@math.sci.hiroshima-u.ac.jp
takahasi@math.sci.hiroshima-u.ac.jp
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Abstract

Let C be a pseudo-abelian symmetric monoidal category, and X a Schur-finite object of C. We study the problem of rationality of the motivic zeta function ζx(t) of X. Since the coefficient ring is not a field, there are several variants of rationality — uniform, global, determinantal and pointwise rationality. We show that ζx(t) is determinantally rational, and we give an example of C and X for which the motivic zeta function is not uniformly rational.

Keywords

Motivic zeta function monoidal category rationality
Type
Research Article
Information
Journal of K-Theory , Volume 9 , Issue 3 , June 2012 , pp. 459 - 473
Copyright
Copyright © ISOPP 2011

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References

1.Atiyah, M.: Power Operations in K-theory, Quart. J. ath. Oxford. Ser. 17 (1966) 165–193CrossRefGoogle Scholar
2.Bourbaki, N.: Éléments de mathématique. Part I. Les structures fondamentales de l'analyse. Livre III. Topologie générale. Chapitres I et II. Actual. Sci. Ind. 858. Hermann & Cie., Paris, 1940. viii+132+II pp.Google Scholar
3.Bourbaki, N.: Éléments de mathématique. Algèbre. Chapitres 1 à 3. Hermann, Paris 1970 xiii+635 pp.Google Scholar
4.Deligne, P.: Catégories tensorielles, Mosc. Math. J. 2(2002), no. 2, 227–248.Google Scholar
5.Del Padrone, A., Mazza, C.: Schur-finite motives and trace identities, Boll. Unione Mat. Ital. 2(2009), 37–44.Google Scholar
6.Fulton, W., Harris, J.: Representation Theory. Springer GTM 129.Google Scholar
7.Kapranov, M.: The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups, preprint(arXiv: math.AG/0001005).Google Scholar
8.Kimura, S.: Chow groups are finite dimensional, in some sense, Math. Ann. 331(2005), no. 1, 173–201.CrossRefGoogle Scholar
9.Kimura, S.: On the characterization of Alexander schemes, Compos. Math. 92(1994), no. 3, 273–284.Google Scholar
10.Larsen, M., Lunts, V. A.: Motivic measures and stable birational geometry, Mosc. Math. J. 3(2003), no. 1, 85–95.Google Scholar
11.Larsen, M., Lunts, V. A.: Rationality criteria for motivic zeta functions, Compos. Math. 140(2004), no. 6, 1537–1560.CrossRefGoogle Scholar
12.Macdonald, I.G.: The Poincaré polynomial of a symmetric product, Proc. Cambridge Philos. Soc. 58(1962), 563–568.CrossRefGoogle Scholar
13.Mazza, C.: Schur functors and motives, K-Theory 33(2004), no. 2, 89–106.CrossRefGoogle Scholar
14.Vistoli, A.: Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97(1989), no. 3, 613–670.CrossRefGoogle Scholar

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