Gluing compactly generated t-structures over stalks of affine schemes

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 Şahinkaya 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.

27 pages