Filtrations Via Tensor Actions

Abstract

Abstract. We extend 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.

The local-to-global principle for triangulated categories via dimension functions

“…The proof is of a somewhat different flavour to the author's original proof and is based upon the inductive construction of objects from the filtrations provided by the dimension functions. It is closer in spirit to the proof of [BIK11, Theorem 3.4] and the constructions in [Ste12]. In fact it gives a slightly sharper statement, showing that one can build A from the Γ x 1 ⊗ A without using tensor products.…”

“…The theorem is, at first glance, perhaps somewhat abstruse. The intuition is that, given a rigidly-compactly generated tensor triangulated category S as in the theorem, the corresponding dimension function gives a filtration of Spc S c and hence of S (see [Bal07] and [Ste12] for more details on such filtrations). This filtration gives a functorial way of building any A ∈ S from the Γ x A by taking cones and homotopy colimits.…”

The derived category of the projective line

“…The subset V is discrete, in the sense that there are no specialization relations between any distinct pair of points in it. It follows from [19] that Γ V D(QCoh P 1 ) decomposes as Γ…”

Relative tensor triangular Chow groups for coherent algebras

“…The lower sequence is the end of the K-theory long exact localization sequence for the Serre localization There is also a local description of Z ∆ i (X , A). Abstractly, it follows from [39] and [19, proposition 2.18, lemma 2.19], that…”

“…There is also a local description of Z ∆ i (X , A). Abstractly, it follows from [39] and [19, proposition 2.18, lemma 2.19], that…”