In locally cohomologically stratified tensor triangular categories with noetherian spectrum, the dualizable localizing ideals are exactly the localizing ideals supported on convex subsets of the Balmer spectrum.
Spectra, spectra, spectra -- tensor triangular spectra versus Z ariski spectra of endomorphism rings
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Convexity in tensor triangular geometry
In locally cohomologically stratified tensor triangular categories with noetherian spectrum, the dualizable localizing ideals are exactly the localizing ideals supported on convex subsets of the Balmer spectrum.