  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.
