Lifting (co)stratifications between tensor triangulated categories (Q6588715)

Given a triangulated category \(\mathcal{T}\), it is generally hopeless to ask for a complete classification of its objects up to isomorphism. Instead, one can ask for a classification of its objects up to extensions, retracts, and (co)products. In this regard, developing a support theory for \(\mathcal{T}\) allows one to approach such classifications systematically. The idea of classifying thick, localizing, and colocalizing subcategories of triangulated categories began in chromatic homotopy theory with the classification of thick subcategories of compact objects in the stable homotopy category by Hopkins and Smith. Hopkins then transported this idea into algebra, providing a classification of thick subcategories of perfect complexes in the derived category \(\mathcal{D}(R)\) of a commutative noetherian ring \(R\), and Neeman extended it further to a classification of (co)localizing subcategories of \(\mathcal{D}(R)\). On the other hand, Benson, Iyengar, and Krause developed an abstract support theory for triangulated categories with the action of a ring \(R\), leading to a notion of (co)stratification which in turn provides a classification of thick, localizing, and colocalizing subcategories, together with many more important consequences.\N\NIn this paper, the authors 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. Then they 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 \(A\), they also investigate whether certain finiteness conditions in \(\mathcal{D}(A)\) (for example, proxy-smallness) can be reduced to questions in the better understood category \(\mathcal{D}(H^{0}(A))\).