Stratification and the smashing spectrum

We develop the theory of stratification for a rigidly-compactly generated tensor-triangulated category using the smashing spectrum and the small smashing support. Within the stratified context, we investigate connections between big prime ideals, objectwise-prime ideals and homological primes, and w...

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Bibliographic Details
Published in:Mathematische Zeitschrift 2023-12, Vol.305 (4), Article 54
Main Author: Verasdanis, Charalampos
Format: Article
Language:English
Subjects:
Context
Homology
Mathematics
Mathematics and Statistics
Stratification
Tensors
Topology
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Summary:We develop the theory of stratification for a rigidly-compactly generated tensor-triangulated category using the smashing spectrum and the small smashing support. Within the stratified context, we investigate connections between big prime ideals, objectwise-prime ideals and homological primes, and we show that the Telescope Conjecture holds if and only if the homological spectrum is T 0 and the homological support detects vanishing. We also reduce stratification to smashing localizations. Moreover, we study induced maps between smashing spectra and prove a descent theorem for stratification. Outside the stratified context, we prove that the Telescope Conjecture holds if and only if the smashing spectrum is T 0 with respect to the small topology.
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-023-03378-4