Vol. 6, No. 3, 2021

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$K\mkern-2mu$-theory and the singularity category of quotient singularities

Nebojsa Pavic and Evgeny Shinder

Vol. 6 (2021), No. 3, 381–424

We study Schlichting’s K-theory groups of the Buchweitz–Orlov singularity category 𝒟sg(X) of a quasiprojective algebraic scheme Xk with applications to algebraic K-theory.

We prove for isolated quotient singularities over an algebraically closed field of characteristic zero that 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 Poincaré 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.

$K\mkern-2mu$-theory of singular varieties, quotient singularity, derived category, singularity category
Mathematical Subject Classification 2010
Primary: 14J17, 18E30, 19E08
Received: 29 March 2019
Revised: 10 November 2020
Accepted: 6 December 2020
Published: 11 September 2021
Nebojsa Pavic
Leibniz Universität Hannover
Evgeny Shinder
School of Mathematics and Statistics
University of Sheffield
United Kingdom
National Research University Higher School of Economics