Gluing compactly generated t-structures over stalks of affine schemes

TL;DR

This paper establishes a correspondence between global and local t-structures in derived categories over rings, showing that certain properties can be verified locally at maximal ideals, with applications to the Telescope Conjecture.

Contribution

It introduces a bijection between compactly generated t-structures on a ring and compatible families over localizations, and proves local verification of the Telescope Conjecture for schemes.

Findings

Global t-structures are determined by local data at maximal ideals.

The Telescope Conjecture for schemes is a stalk-local property.

Explicit bijection between cosilting objects over rings and localizations.

Abstract

We show that compactly generated t-structures in the derived category of a commutative ring $R$ are in a bijection with certain families of compactly generated t-structures over the local rings $R_\mathfrak{m}$ where $\mathfrak{m}$ runs through the maximal ideals in the Zariski spectrum $\mathrm{Spec}(R)$. The families are precisely those satisfying a gluing condition for the associated sequence of Thomason subsets of $\mathrm{Spec}(R)$. As one application, we show that the compact generation of a homotopically smashing t-structure can be checked locally over localizations at maximal ideals. In combination with a result due to Balmer and Favi, we conclude that the $\otimes$-Telescope Conjecture for a quasi-coherent and quasi-separated scheme is a stalk-local property. Furthermore, we generalize the results of Trlifaj and \c{S}ahinkaya and establish an explicit bijection between cosilting objects of cofinite type over $R$ and compatible families of cosilting objects of cofinite type over all localizations $R_\mathfrak{m}$ at maximal primes.