Stratifying derived categories of cochains on certain spaces

In recent years, Benson, Iyengar and Krause have developed a theory of stratification for compactly generated triangulated categories with an action of a graded commutative Noetherian ring. Stratification implies a classification of localizing and thick subcategories in terms of subsets of the prime ideal spectrum of the given ring. In this paper two stratification results are presented: one for the derived category of a commutative ring-spectrum with polynomial homotopy and another for the derived category of cochains on certain spaces. We also give the stratification of cochains on a space a topological content.

💡 Research Summary

This paper applies the stratification theory developed by Benson, Iyengar, and Krause (BIK) to two new contexts: (1) the derived category of a commutative, coconnective S‑algebra whose homotopy groups form a polynomial ring on finitely many even‑degree generators, and (2) the derived category of cochains on a class of spaces called “soci” (spherically odd complete intersections).

The BIK framework starts with a compactly generated triangulated category T equipped with an action of a graded‑commutative Noetherian ring S. For each object X in T one defines a support supp_S X ⊂ Spec S via the collection of local cohomology functors Γ_p. If two conditions hold—(S1) the local‑global principle (every object is built from its local pieces Γ_p X) and (S2) each localizing subcategory Im Γ_p is either zero or minimal—then T is said to be stratified by S. In this situation the assignment X ↦ supp_S X yields a bijection between localizing subcategories of T and arbitrary subsets of supp_S T, and a bijection between thick subcategories of the compact objects T^c and specialization‑closed subsets of supp_S T.

First main result (Theorem 1.1 / 7.2).
Let R be a commutative, coconnective S‑algebra such that π_*R ≅ k