Schemic Grothendieck Rings I: Motivic Sites

Schemic Grothendieck Rings and Motivic Rationality

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 i...

Schemic Grothendieck Rings Ii: Jet Schemes and Motivic Integration

We generalize the notion of a jet scheme (truncated arc space) to arbitrary fat points via adjunction, and show that this yields for each fat point, an endomorphism on each schemic Grothendieck ring as defined in [17]. We prove that some of the analogues for linear jets still hold true, like locally trivial fibration over the smooth locus. In this formalism, we can define several generating zet...

Motivic Classes of Commuting Varieties via Power Structures

We prove a formula, originally due to Feit and Fine, for the class of the commuting variety in the Grothendieck group of varieties. Our method, which uses a power structure on the Grothendieck group of stacks, allows us to prove several refinements and generalizations of the Feit-Fine formula. Our main application is to motivic DonaldsonThomas theory.

Grothendieck rings of Z-valued fields

We prove the triviality of the Grothendieck ring of a Z-valued field K under slight conditions on the logical language and on K. We construct a definable bijection from the plane K to itself minus a point. When we specialize to local fields with finite residue field, we construct a definable bijection from the valuation ring to itself minus a point. At the Edinburgh meeting on the model theory ...

Motivic Integration and the Grothendieck Group of Pseudo-Finite Fields