Abstract

We study Schlichting’s $K$-theory groups of the Buchweitz–Orlov singularity category ${\mathcal{𝒟}}^{sg}\left(X\right)$ of a quasiprojective algebraic scheme $X∕k$ with applications to algebraic K-theory.

We prove for isolated quotient singularities over an algebraically closed field of characteristic zero that $K_0\left({\mathcal{𝒟}}^{sg}\left(X\right)\right)$ is finite torsion, and that $K_1\left({\mathcal{𝒟}}^{sg}\left(X\right)\right)=0$. 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.

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Mathematical Subject Classification 2010

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