Colocalizing subcategories of singularity categories (Q6639810)

Let \(R\) be a commutative noetherian ring. The singularity category \(\mathcal{S}(R) = K_{\operatorname{ac}}\left(\operatorname{Inj}(R)\right)\), i.e., the homotopy category of acyclic complexes of injective \(R\)-modules, was introduced and studied by Krause who showed that \(\mathcal{S}(R)\) is a compactly generated triangulated category and there is a stabilization functor \(\mathcal{D}(R)\rightarrow \mathcal{S}(R)\) which can be used to describe the compact objects of \(\mathcal{S}(R)\). A classic problem regarding such categories is the classification of the localizing and colocalizing subcategories, e.g., for the derived category \(\mathcal{D}(R)\). The modern approach is via the machinery of tensor-triangular geometry and the theory of stratification and costratification provided that the given category is tensor-triangulated. However, \(\mathcal{S}(R)\) is not tensor-triangulated, so a different approach should be adopted. Stevenson developed a theory of actions of rigidly compactly generated tensor-triangulated categories on compactly generated triangulated categories. In particular, he utilized the action of \(\mathcal{D}(R)\) on \(\mathcal{S}(R)\) to classify the localizing subcategories of \(\mathcal{S}(R)\) for a locally hypersurface ring \(R\), and then generalized this classification result to the singularity category \(\mathcal{S}(X)\) of a noetherian separated scheme \(X\) with hypersurface singularities.\N\NIn this paper, the author applies the theory of costratification in order to classify the colocalizing subcategories of \(\mathcal{S}(R)\) for a locally hypersurface ring \(R\), and then generalizes his result to the singularity category of a scheme with hypersurface singularities. Specifically, for the case of rings, using the action of \(\mathcal{D}(R)\) on \(\mathcal{S}(R)\), he obtains a notion of cosupport for the objects of \(\mathcal{S}(R)\).