Grothendieck ring of pretriangulated categories
A. I. Bondal, M. Larsen, V. A. Lunts

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 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" 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.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology · Algebraic Geometry and Number Theory
