Grothendieck ring of pretriangulated categories

Details Grothendieck ring of pretriangulated categories Grothendieck ring of pretriangulated categories

2004-01-02

arxiv.org/abs/math/0401009v1

Mathematics

Algebraic Geometry

33 pages

Scientific paper

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.

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