Support theory via actions of tensor triangulated categories

We give a definition of the action of a tensor triangulated category T on a triangulated category K. In the case that T is rigidly-compactly generated and K is compactly generated we show this gives rise to a notion of supports which categorifies work of Benson, Iyengar, and Krause and extends work of Balmer and Favi. We prove that a suitable version of the local-to-global principle holds very generally. A relative version of the telescope conjecture is formulated and we give a sufficient condition for it to hold.Comment: 33 pages, to appear in Journal f\"ur die reine und angewandte Mathemati

Filtrations via tensor actions

We extend the work of Balmer, associating filtrations of essentially small tensor triangulated categories to certain dimension functions, to the setting of actions of rigidly-compactly generated tensor triangulated categories on compactly generated triangulated categories. We show that the towers of triangles associated to such a filtration can be used to produce filtrations of Gorenstein injective quasi-coherent sheaves on Gorenstein schemes. This extends and gives a new proof of a result of Enochs and Huang. In the case of local complete intersections, a further refinement of this filtration is given and we comment on some special properties of the associated spectral sequence in this case

Strong generators in tensor triangulated categories

We show that in an essentially small rigid tensor triangulated category with connected Balmer spectrum there are no proper non-zero thick tensor ideals admitting strong generators. This proves, for instance, that the category of perfect complexes over a commutative ring without non-trivial idempotents has no proper non-zero thick subcategories that are strongly generated.Comment: 9 pages, comments welcom

The derived category of a graded Gorenstein ring

We give an exposition and generalization of Orlov's theorem on graded Gorenstein rings. We show the theorem holds for non-negatively graded rings which are Gorenstein in an appropriate sense and whose degree zero component is an arbitrary non-commutative right noetherian ring of finite global dimension. A short treatment of some foundations for local cohomology and Grothendieck duality at this level of generality is given in order to prove the theorem. As an application we give an equivalence of the derived category of a commutative complete intersection with the homotopy category of graded matrix factorizations over a related ring.Comment: To appear in the MSRI publications volume "Commutative Algebra and Noncommutative Algebraic Geometry (II)

Homotopy invariants of singularity categories

We present a method for computing $\mathbb{A}^1$-homotopy invariants of singularity categories of rings admitting suitable gradings. Using this we describe any such invariant, e.g. homotopy K-theory, for the stable categories of self-injective algebras admitting a connected grading. A remark is also made concerning the vanishing of all such invariants for cluster categories of type $A_{2n}$ quivers.Comment: final revisio