Descent in tensor triangular geometry

We investigate to what extent we can descend the classification of localizing, smashing and thick ideals in a presentably symmetric monoidal stable $\mathrm{\infty }$-category $\mathcal{C}$ along a descendable commutative algebra $A$. We establish equalizer diagrams relating the lattices of localizing and smashing ideals of $\mathcal{C}$ to those of ${\mathrm{M}\mathrm{o}\mathrm{d}}_A(\mathcal{C})$ and ${\mathrm{M}\mathrm{o}\mathrm{d}}_{A\otimes A}(\mathcal{C})$. 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 ${\mathrm{M}\mathrm{o}\mathrm{d}}_A(\mathcal{C})$ to $\mathcal{C}$. The utility of these results is demonstrated in the case of faithful Galois extensions in tensor triangular geometry.