K-theory and the singularity category of quotient singularities

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

Autoren

  • Nebojsa Pavic
  • Evgeny Shinder

Organisationseinheiten

Externe Organisationen

  • The University of Sheffield
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Details

OriginalspracheEnglisch
Seiten (von - bis)381-424
Seitenumfang44
FachzeitschriftAnnals of K-Theory : a journal of the K-Theory Foundation
Jahrgang6
Ausgabenummer3
PublikationsstatusVeröffentlicht - 11 Sept. 2021

Abstract

In this paper we study Schlichting's K-theory groups of the Buchweitz-Orlov singularity category $\mathcal{D}^{sg}(X)$ of a quasi-projective algebraic scheme $X/k$ with applications to Algebraic K-theory. We prove that for isolated quotient singularities $\mathrm{K}_0(\mathcal{D}^{sg}(X))$ is finite torsion, and that $\mathrm{K}_1(\mathcal{D}^{sg}(X)) = 0$. One of the main applications is that algebraic varieties with isolated quotient singularities satisfy rational Poincare duality on the level of the Grothendieck group; this allows computing the Grothendieck group of such varieties in terms of their resolution of singularities. Other applications concern the Grothendieck group of perfect complexes supported at a singular point and topological filtration on the Grothendieck groups.

ASJC Scopus Sachgebiete

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K-theory and the singularity category of quotient singularities. / Pavic, Nebojsa; Shinder, Evgeny.
in: Annals of K-Theory : a journal of the K-Theory Foundation, Jahrgang 6, Nr. 3, 11.09.2021, S. 381-424.

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

Pavic, Nebojsa ; Shinder, Evgeny. / K-theory and the singularity category of quotient singularities. in: Annals of K-Theory : a journal of the K-Theory Foundation. 2021 ; Jahrgang 6, Nr. 3. S. 381-424.
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abstract = "In this paper we study Schlichting's K-theory groups of the Buchweitz-Orlov singularity category Dsg(X) of a quasi-projective algebraic scheme X/k with applications to Algebraic K-theory. We prove that for isolated quotient singularities K0(Dsg(X)) is finite torsion, and that K1(Dsg(X))=0. One of the main applications is that algebraic varieties with isolated quotient singularities satisfy rational Poincare duality on the level of the Grothendieck group; this allows computing the Grothendieck group of such varieties in terms of their resolution of singularities. Other applications concern the Grothendieck group of perfect complexes supported at a singular point and topological filtration on the Grothendieck groups.",
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