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 \(\infty \) -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 \(\mathrm {Mod}_{A}({\mathcal {C}})\) and \(\mathrm {Mod}_{A\otimes A}({\mathcal {C}})\) . 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 \(\mathrm {Mod}_{A}({\mathcal {C}})\) to \({\mathcal {C}}\) . The utility of these results is demonstrated in the case of faithful Galois extensions in tensor triangular geometry.