K-theory and the singularity category of quotient singularities

Research output: Contribution to journal › Article › Research › peer review

Authors

  • Nebojsa Pavic
  • Evgeny Shinder

Research Organisations

External Research Organisations

  • The University of Sheffield
View graph of relations

Details

Original languageEnglish
Pages (from-to)381-424
Number of pages44
JournalAnnals of K-Theory : a journal of the K-Theory Foundation
Volume6
Issue number3
Publication statusPublished - 11 Sept 2021

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.

Keywords

    math.AG, math.KT, Singularity category, K-theory of singular varieties, Quotient singularity, Derived category

ASJC Scopus subject areas

Cite this

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, Vol. 6, No. 3, 11.09.2021, p. 381-424.

Research output: Contribution to journal › Article › Research › peer review

Pavic N, Shinder E. K-theory and the singularity category of quotient singularities. Annals of K-Theory : a journal of the K-Theory Foundation. 2021 Sept 11;6(3):381-424. doi: 10.2140/akt.2021.6.381
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 ; Vol. 6, No. 3. pp. 381-424.
Download
@article{d5a756f9f047478691aae5ae0ef1ad6f,
title = "K-theory and the singularity category of quotient singularities",
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.",
keywords = "math.AG, math.KT, Singularity category, K-theory of singular varieties, Quotient singularity, Derived category",
author = "Nebojsa Pavic and Evgeny Shinder",
note = "Funding Information: We would like to thank A. Betina, T. Bridgeland, J. Greenlees, A. Efimov, M. Kalck, J. Karmazyn, J. Kass, A. Kuznetsov, N. Pagani, D. Pomerleano, R. Potter, M. Schlichting, P. Sechin, V. Srinivas, P. Stellari, G. Stevenson, B. Totaro, A. Vishik, V. Vologodsky and M. Wemyss for helpful discussions and e-mail communication, and the referee for their comments about the paper. Evgeny Shinder was partially supported by Laboratory of Mirror Symmetry NRU HSE, RF government grant, ag. N 14.641.31.0001.",
year = "2021",
month = sep,
day = "11",
doi = "10.2140/akt.2021.6.381",
language = "English",
volume = "6",
pages = "381--424",
journal = "Annals of K-Theory : a journal of the K-Theory Foundation",
issn = "2379-1691",
publisher = "Mathematical Science Publishers",
number = "3",

}

Download

TY - JOUR

T1 - K-theory and the singularity category of quotient singularities

AU - Pavic, Nebojsa

AU - Shinder, Evgeny

N1 - Funding Information: We would like to thank A. Betina, T. Bridgeland, J. Greenlees, A. Efimov, M. Kalck, J. Karmazyn, J. Kass, A. Kuznetsov, N. Pagani, D. Pomerleano, R. Potter, M. Schlichting, P. Sechin, V. Srinivas, P. Stellari, G. Stevenson, B. Totaro, A. Vishik, V. Vologodsky and M. Wemyss for helpful discussions and e-mail communication, and the referee for their comments about the paper. Evgeny Shinder was partially supported by Laboratory of Mirror Symmetry NRU HSE, RF government grant, ag. N 14.641.31.0001.

PY - 2021/9/11

Y1 - 2021/9/11

N2 - 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.

AB - 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.

KW - math.AG

KW - math.KT

KW - Singularity category

KW - K-theory of singular varieties

KW - Quotient singularity

KW - Derived category

UR - http://www.scopus.com/inward/record.url?scp=85126313075&partnerID=8YFLogxK

U2 - 10.2140/akt.2021.6.381

DO - 10.2140/akt.2021.6.381

M3 - Article

VL - 6

SP - 381

EP - 424

JO - Annals of K-Theory : a journal of the K-Theory Foundation

JF - Annals of K-Theory : a journal of the K-Theory Foundation

SN - 2379-1691

IS - 3

ER -