  We investigate to what extent we can descend the classification of
localizing, smashing and thick ideals in a presentably symmetric monoidal
stable $\infty$-category $\mathscr{C}$ along a descendable commutative algebra
$A$. We establish equalizer diagrams relating the lattices of localizing and
smashing ideals of $\mathscr{C}$ to those of $\mathrm{Mod}_{A}(\mathscr{C})$
and $\mathrm{Mod}_{A\otimes A}(\mathscr{C})$. If $A$ is compact, we obtain a
similar equalizer for the lattices of thick ideals which, via Stone duality,
yields a coequalizer diagram of Balmer spectra in the category of spectral
spaces. We then give conditions under which the telescope conjecture and
stratification descend from $\mathrm{Mod}_{A}(\mathscr{C})$ to $\mathscr{C}$.
The utility of these results is demonstrated in the case of faithful Galois
extensions in tensor triangular geometry.
