Schemic Grothendieck rings and motivic rationality

Details Schemic Grothendieck rings and motivic rationality Schemic Grothendieck rings and motivic rationality

2009-10-05

arxiv.org/abs/0910.0684v1

Mathematics

Algebraic Geometry

Scientific paper

We propose a suitable substitute for the classical Grothendieck ring of an algebraically closed field, in which any quasi-projective scheme is represented, while maintaining its non-reduced structure. This yields a more subtle invariant, called the schemic Grothendieck ring, in which we can formulate a form of integration resembling Kontsevich's motivic integration via arc schemes. In view of its more functorial properties, we can present a characteristic-free proof of the rationality of the geometric Igusa zeta series for certain hypersurfaces, thus generalizing the ground-breaking work on motivic integration by Denef and Loeser. The construction uses first-order formulae, and some infinitary versions, called formularies.

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