The yoga of schemic Grothendieck rings, a topos-theoretical approach

TL;DR

This paper introduces a topos-theoretic framework for a schemic Grothendieck ring, providing a more refined invariant that captures non-reduced structures and enables a form of motivic integration similar to Kontsevich's approach.

Contribution

It develops a topos-theoretic construction of the schemic Grothendieck ring, extending classical invariants to include non-reduced schemes and enabling motivic integration.

Findings

Constructed a topos-theoretic version of the schemic Grothendieck ring.

Enabled formulation of motivic integration via arc schemes.

Provided a more subtle invariant capturing non-reduced structures.

Abstract

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. Whereas the original construction was via definability, we have translated in this paper everything into a topos-theoretic framework.