  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.

Algebraic Geometry;K-Theory and Homology