Gluing compactly generated t-structures over stalks of affine schemes

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 \(R_{\frak{m}}\) $$R_{\mathfrak{m}}$$ where \(\frak{m}\) $$\mathfrak{m}$$ runs through the maximal ideals in the Zariski spectrum Spec(R). The families are precisely those satisfying a gluing condition for the associated sequence of Thomason subsets of 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 ⊗-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 \(R_{\frak{m}}\) $$R_{\mathfrak{m}}$$ at maximal primes.