Derived invariants and motives, Part I, Integral Grothendieck Riemann-Roch and non-commutative motives

TL;DR

This paper develops a non-commutative framework for algebraic geometry, proving integral Grothendieck Riemann-Roch and relating motives of varieties with exceptional collections, advancing understanding of motives and reductions.

Contribution

It proves an integral Grothendieck Riemann-Roch theorem and provides explicit formulas for motives of varieties with full exceptional collections, extending classical results.

Findings

Proved integral Grothendieck Riemann-Roch theorem.

Established an integral analogue of Kontsevich's comparison theorem.

Derived explicit formulas for motives of varieties with full exceptional collections.

Abstract

The goal of this series of papers is to give a new non-commutative approach to problems about the density of reductions such as the conjecture of Joshi-Rajan, and the generalization of the conjecture of Serre. In this paper, we prove integral Grothendieck Riemann-Roch which was proved by Papas in the case ch$(k)=0$. As a corollary we prove an integral analogue of Kontsevich's comparison theorem, and we show that if a smooth projective variety $X$ has a full exceptional collection then there is an explicit formula of the motive of $X$ up to bounded torsion.