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Grothendieck ring of pretriangulated categories

A. I. Bondal·M. Larsen·V. A. Lunts
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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.

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A. I. Bondal, M. Larsen, V. A. Lunts. 2004-01-02. Grothendieck ring of pretriangulated categories. https://arxiv.org/abs/math/0401009

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