Subcategories of singularity categories via tensor actions

We obtain, via the formalism of tensor actions, a complete classification of the localizing subcategories of the stable derived category of any affine scheme with hypersurface singularities and of any local complete intersection over a field; in particular this classifies the thick subcategories of the singularity categories of such rings. The analogous result is also proved for certain locally complete intersection schemes. It is also shown that from each of these classifications one can deduce the (relative) telescope conjecture.

💡 Research Summary

In this paper Greg Stevenson develops a comprehensive classification of the localizing subcategories of singularity categories, using the machinery of tensor actions and Balmer’s tensor triangular geometry. The central objects of study are the stable derived category (also called the singularity category) S(X) = K^ac(Inj X), the homotopy category of acyclic complexes of injective quasi‑coherent sheaves on a noetherian separated scheme X, and its compact part D_{sg}(X), which measures the singularities of X.

The key technical innovation is to exhibit an action of the unbounded derived category D(X) = D(QCoh X) on S(X). This action is a left tensor action in the sense of