Lifting (co)stratifications between tensor triangulated categories

TL;DR

This paper establishes criteria for the transfer of stratification and costratification properties between tensor-triangulated categories and applies these to non-positive DG-rings and ring spectra, aiding classification and understanding of derived categories.

Contribution

It provides necessary and sufficient conditions for descending stratification along tensor-exact functors and applies these to classify subcategories in derived categories of non-positive DG-rings.

Findings

Support-theoretic classification of localizing subcategories.

Formal justification for classical scheme support of derived schemes.

Reduction of finiteness questions to simpler categories.

Abstract

We give necessary and sufficient conditions for stratification and costratification to descend along a coproduct preserving, tensor-exact $R$-linear functor between $R$-linear tensor-triangulated categories which are rigidly-compactly generated by their tensor units. We then apply these results to non-positive commutative DG-rings and connective ring spectra. In particular, this gives a support-theoretic classification of (co)localizing subcategories, and thick subcategories of compact objects of the derived category of a non-positive commutative DG-ring with finite amplitude, and provides a formal justification for the principle that the space associated to an eventually coconnective derived scheme is its underlying classical scheme. For a non-positive commutative DG-ring $A$, we also investigate whether certain finiteness conditions in $\mathsf{D}(A)$ (for example, proxy-smallness) can be reduced to questions in the better understood category $\mathsf{D}(H^0A)$.