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 ${\mathcal{D}}^{sg}(X)$ 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.

Erroneous Lemma 2.1 from the first version removed (see Remark 2.2 in this version), and the computation for $A^n/G$ now relies on a cdh topology argument (Proposition 2.1). This did not affect the main results except that we now have to assume that the base field has characteristic zero. Exposition improved, several typos fixed

References in corpus (2)

• Conservative descent for semi-orthogonal decompositions
• The KH-Theory of Complete Simplicial Toric Varieties and the Algebraic K-Theory of Weighted Projective Spaces

Cited by in corpus (7)

• Categorical absorptions of singularities and degenerations
• Descent conditions for generation in derived categories
• A new equivalence between singularity categories of commutative algebras
• Categorical Donaldson-Thomas theory for local surfaces: $\mathbb{Z}/2$-periodic version
• Derived categories of nodal del Pezzo threefolds
• Classifying dg-categories of matrix factorizations
• Singularity categories via higher McKay quivers with potential