Grothendieck ring of pretriangulated categories

TL;DR

This paper introduces a Grothendieck ring structure on pretriangulated DG categories, enabling algebraic manipulation and applications in derived categories and motivic measures.

Contribution

It defines a new ring structure on classes of pretriangulated DG categories and explores its applications in derived categories and motivic measures.

Findings

The abelian group of classes becomes a commutative ring with a new multiplication.

Applications include representability of functors between derived categories.

Constructs a motivic measure based on the ring structure.

Abstract

We consider the abelian group $PT$ generated by quasi-equivalence classes of pretriangulated DG categories with relations coming from semi-orthogonal decompositions of corresponding triangulated categories. We introduce an operation of "multiplication" $\bullet$ on the collection of DG categories which makes this abelian group into a commutative ring. A few applications are considered: representability of "standard" functors between derived categories of coherent sheaves on smooth projective varieties and a construction of an interesting motivic measure.