Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology

Table of Contents - June 2012 - Volume 9, Issue 03  

Research Article

Motivic zeta functions in additive monoidal categories

Kenichiro Kimuraa1, Shun-ichi Kimuraa2 and Nobuyoshi Takahashia3*

a1 Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki, 305-8571 Japan kimurak@math.tsukuba.ac.jp

a2 Department of Mathematics, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526 Japan kimura@math.sci.hiroshima-u.ac.jp

a3 Department of Mathematics, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526 Japan takahasi@math.sci.hiroshima-u.ac.jp

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.

(Received May 20 2010)

Key Words

  • Motivic zeta function;
  • monoidal category;
  • rationality

Footnotes

* K. K. was partially supported by JSPS Grant-in-Aid for Scientific Research (C), No. 22540008. S. K. was partially supported by JSPS Grant-in-Aid for Scientific Research (C), No. 21540038. N. T. was partially supported by JSPS Grant-in-Aid for Scientific Research (C), No. 23540050.