Compactly generated tensor t-structures on the derived category of a Noetherian scheme

TL;DR

This paper establishes a classification of tensor compatible t-structures on the derived category of a Noetherian scheme, extending previous results from rings to schemes and confirming a tensor version of the telescope conjecture.

Contribution

It introduces a tensor compatibility condition for t-structures and proves a bijective correspondence with filtrations of Thomason subsets for schemes.

Findings

Classified tensor compatible t-structures via Thomason filtrations

Extended classification from rings to schemes

Confirmed a tensor version of the telescope conjecture for schemes

Abstract

We introduce a tensor compatibility condition for t-structures. For any Noetherian scheme $X$, we prove that there is a one-to-one correspondence between the set of filtrations of Thomason subsets and the set of aisles of compactly generated tensor compatible t-structures on the derived category of $X$. This generalizes the earlier classification of compactly generated t-structures for commutative rings to schemes. Hrbek and Nakamura have reformulated the famous telescope conjecture for t-structures. As an application of our main theorem, we prove that a tensor version of the conjecture is true for separated Noetherian schemes.