In this talk, we establish a bijection between compactly generated t-structures in the derived category of a commutative ring R and specific families of compactly generated t-structures over the local rings Rm where m runs through the maximal ideals in the Zariski spectrum Spec(R). These families are characterized by a gluing condition on their associated sequences of Thomason subsets of Spec(R). As an application, we generalize results of Trlifaj and Sahinkaya and provide an explicit bijection between cosilting objects of cofinite type over R and compatible families of cosilting objects of cofinite type over all localizations Rm at maximal primes. Furthermore, we demonstrate that the compact generation of a homotopically smashing t-structure can be verified locally at maximal ideal localizations. Combining this with a result of Balmer and Favi, we conclude that the ⊗-Telescope Conjecture for a quasi-coherent and quasi-separated scheme is a stalk-local property. This talk presents a joint work with Michal Hrbek and Rongmin Zhu.