K-theory and the singularity category of quotient singularities

Pavic, Nebojsa; Shinder, Evgeny

doi:10.48550/arxiv.1809.10919

Cited by 3 publications

(7 citation statements)

References 51 publications

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“…Finally, as we have already shown that π * : A i → A i is essentially surjective, it remains to show that the induced functor π * : A i ( A i ∩ Ker π * ) → A i is fully faithful; to show this we use the argument from [PS18,Lemma 2.31] which goes as follows.…”

Section: (Iii) We Have An Equalitymentioning

confidence: 99%

Derived categories of singular surfaces

Karmazyn¹,

Кузнецов²,

Shinder³

2018

Preprint

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We develop an approach that allows to construct semiorthogonal decompositions of derived categories of surfaces with cyclic quotient singularities whose components are equivalent to derived categories of local finite dimensional algebras.We first explain how to induce a semiorthogonal decomposition of a surface X with rational singularities from a semiorthogonal decomposition of its resolution. In the case when X has cyclic quotient singularities, we introduce the condition of adherence for the components of the semiorthogonal decomposition of the resolution that allows to identify the components of the induced decomposition with derived categories of local finite dimensional algebras. Further, we present an obstruction in the Brauer group of X to the existence of such semiorthogonal decomposition, and show that in the presence of the obstruction a suitable modification of the adherence condition gives a semiorthogonal decomposition of the twisted derived category of X.We illustrate the theory by exhibiting a semiorthogonal decomposition for the untwisted or twisted derived category of any normal projective toric surface depending on whether its Weil divisor class group is torsion-free or not. For weighted projective planes we compute the generators of the components explicitly and relate our results to the results of Kawamata based on iterated extensions of reflexive sheaves of rank 1. JOSEPH KARMAZYN, ALEXANDER KUZNETSOV, EVGENY SHINDER 4.3. Explicit identification of the Brauer group 35 4.4. Resolutions of twisted derived categories 36 4.5. Grothendieck groups of twisted derived categories 38 4.6. Semiorthogonal decompositions of twisted derived categories 41 5. Application to toric surfaces 42 5.1. Notation 43 5.2. The Brauer group of toric surfaces 43 5.3. Minimal resolution 45 5.4. Adherent exceptional collections 46 5.5. Special Brauer classes 48 6. Reflexive sheaves 49 6.1. Criteria of reflexivity and purity 50 6.2. Extension of reflexive rank 1 sheaves 52 6.3. Toric case 54 Appendix A. Semiorthogonal decomposition of perfect complexes 56 References 59 no. 4, 583-598.

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Section: (Iii) We Have An Equalitymentioning

confidence: 99%

Derived categories of singular surfaces

Karmazyn¹,

Кузнецов²,

Shinder³

2018

Preprint

Self Cite

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“…We define the following Grothendieck groups of X with supports on Z; the first two are classical, and the last two are defined and studied in [38]. We define…”

Section: Gorenstein Varieties and Algebrasmentioning

confidence: 99%

“…K sg 0 (X)) for these groups when Z = X. Essentially from definitions (see [38,Remark 1.13]) we get a canonical exact sequence…”

Section: Gorenstein Varieties and Algebrasmentioning

confidence: 99%

“…This has already been done by Weibel for curves and surfaces [48], and our study concentrates on isolated threefold singularities, while reproving some of Weibel's results for curves and surfaces along the way. This relies on previous joint work of the second and third authors [38], where K-theory of Orlov's singularity category is studied. We recall geometric description of K −1 for curves in Proposition 3.1 and Corollary 3.3 and K −1 for surfaces can be computed using Proposition 3.…”

Section: Introductionmentioning

confidence: 99%

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Obstructions to semiorthogonal decompositions for singular threefolds I: K-theory

Kalck¹,

Pavic²,

Shinder³

2019

Preprint

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We investigate necessary conditions for Gorenstein projective varieties to admit semiorthogonal decompositions introduced by Kawamata, with main emphasis on threefolds with isolated compound A n singularities. We introduce obstructions coming from Algebraic K-theory and translate them into the concept of maximal nonfactoriality.Using these obstructions we show that many classes of nodal threefolds do not admit Kawamata type semiorthogonal decompositions. These include nodal hypersurfaces and double solids, with the exception of a nodal quadric, and del Pezzo threefolds of degrees 1 ≤ d ≤ 4 with maximal class group rank.We also investigate when does a blow up of a smooth threefold in a singular curve admit a Kawamata type semiorthogonal decomposition and we give a complete answer to this question when the curve is nodal and has only rational components.

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“…Gorenstein terminal) threefolds. Using nodal blow ups for studying derived categories can be traced back to Orlov's interpretation of the Knörrer periodicity [22] (see also [24,Proposition 1.30]).…”

Section: Introductionmentioning

confidence: 99%

Derived categories of nodal del Pezzo threefolds

Pavic,

Shinder

2021

Preprint

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We give a complete answer for the existence of Kawamata type semiorthogonal decompositions of derived categories of nodal del Pezzo threefolds. More precisely, we show that nodal non smooth V d with 1 ≤ d ≤ 4 have no Kawamata type decomposition and that all nodal V5 admit a Kawamata decomposition. For the proof we go through the classification of singular del Pezzo threefolds, compute divisor class groups of nodal del Pezzo threefolds of small degree and use projection from a line to construct Kawamata semiorthogonal decompositions for d = 5. A Kawamata decomposition of nodal V6 has been previously constructed in [13].

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K-theory and the singularity category of quotient singularities

Cited by 3 publications

References 51 publications

Derived categories of singular surfaces

Derived categories of singular surfaces

Obstructions to semiorthogonal decompositions for singular threefolds I: K-theory

Derived categories of nodal del Pezzo threefolds

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