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 \(\mathscr{C}\) along a descendable commutative algebra \(A\). We establish equalizer diagrams relating the lattices of localizing a...

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Bibliographic Details
Published in:arXiv.org 2023-05
Main Authors: Barthel, Tobias, Castellana, Natalia, Heard, Drew, Naumann, Niko, Pol, Luca, Sanders, Beren
Format: Article
Language:English
Subjects:
Equalizers
Lattices
Mathematical analysis
Tensors
Online Access:Get full text
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Summary: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.
ISSN:2331-8422