Grothendieck ring of pretriangulated categories

By A. Bondal, Michael Larsen, V. Lunts. 2004

143 citations recorded by the source corpus.

We consider the abelian group PT generated by quasi-equivalence classes of pretriangulated DG categories with relations coming from semiorthogonal 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/2633a08fb4ee62f683b00cb1478a94280a9d4d78

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Citations in the corpus

The homotopy theory of dg-categories and derived Morita theory (2004, background): “orks The fact that dg-categories provide natural and interesting enhancement of derived categories has been recognized for some times, and in particular in [B-K]. They have been used more recently in [B-L-L] in which a very special case of our theorem 8.9 is proved for smooth projective varieties. T”
On differential graded categories (2006, background): “ments of the Grothendieck group of the triangulated category rep(A,B). The composition is induced from that of Hmo. The category Hmo 0 is additive and endowed with a canonical functor Hmo →Hmo 0 (cf. [23] for a related construction). One can show [145] that a functor F deﬁned on Hmo with values in a”
Chow motives versus non-commutative motives (2011, methodology): “Making use of Bittner-Looijenga’s [8, 30] presentation of K0(Vark) in the case of a field k of characteristic zero, Bondal-Larsen-Lunts constructed in [10] a motivic measure μBLL with values in a certain ring of pre-triangulated dg categories.”
Higher traces, noncommutative motives, and the categorified Chern character (2015, background): “In that case, Bondal, Larsen, and Lunts [12] have constructed a motivic measure on k-varieties with value in the ring K((2)) 0 (k) (under a different name, see [31]): they showed that the assignment X → Perf(X), for X a smooth projective variety, induces a ring homomorphism K0(Vark)/(L − 1) → K (2) ”
Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry (2008, result): “els Funk(QC(X),QC(X′)) ≃QC(X ×k X′). For dg categories of perfect (equivalently, bounded coherent) complexes on smooth projective varieties, an analogous result was proved by Bondal, Larsen and Lunts [BLL] as well as by To¨en [To1] (generalizing Orlov’s theorem [O] characterizing equivalences as Fou”
A tour about existence and uniqueness of dg enhancements and lifts (2016, background): “On a smooth projective scheme, one can consider the dg category Inj(X) of bounded below complexes of injective quasi-coherent sheaves on X with bounded coherent cohomologies (see, for example, [9]).”
Hochschild cohomology of abelian categories and ringed spaces (2004, background): “Hence eDb(A) is a DG-“enhancement” [7,8] for the triangulated category Db(A).”
ON EQUIVARIANT TRIANGULATED CATEGORIES (2014, background): “We refer to [4] or [9] for the definitions and basic facts concerning DG-categories.”
The Character Theory of a Complex Group (2009, result): “tegories of quasicoherent sheaves, and in particular proved a 10 result identifying functors on quasicoherent sheaves with integral transforms, generalizing results of Orlov, Bondal, Larsen and Lunts [BLL] and To¨en [To]. Namely, we introduced the class of perfect stacks as recalled in the following”