Tensor actions and locally complete intersections

Abstract

We introduce a relative version of Balmer's tensor triangular geometry by considering the action of a tensor triangulated category on another triangulated category. Several of Balmer's results are extended to this relative setting giving rise to, among other things, a theory of supports for objects of a category upon which a tensor triangulated category acts. In the case that a rigidly-compactly generated tensor triangulated category acts on a compactly generated category we describe a version of the local- to-global principle of Benson, Iyengar, and Krause, and a relative version of the telescope conjecture. We prove the local-to-global principle holds quite generally which is new even in the case that a tensor triangulated category acts on itself as in Balmer's theory. We are also able to give sufficient conditions for the relative telescope conjecture to hold. As an application we study the stable injective category of a noetherian separated scheme X, as introduced by Krause, in terms of an action of the derived category D(X). We give a complete classification of the localizing subcategories of this category in the case that X is the spectrum of a hypersurface ring and prove that the telescope conjecture holds. Our methods allow us to extend these results, suitably modified, to certain complete intersection schemes of arbitrary codimension.