The chromatic tower for D(R)

Refining thick subcategory theorems

“…We now recall some definitions and a theorem due to Hopkins and Neeman [Nee92] which classifies the thick subcategories of perfect complexes over a noetherian ring.…”

“…Similarly the thick subcategory theorem for the derived category establishes a surprising connection between stable homotopy theory and algebraic geometry; using this theorem one is able to recover the spectrum of a ring from the homotopy structure of its derived category! These ideas were later pushed further into the world of derived categories of rings and schemes by Neeman [Nee92] and Thomason [Tho97], and into modular representation theory by Benson, Carlson and Rickard [BCR97]. Motivated by the work of Hopkins [Hop87], Neeman [Nee92] classified the Bousfield classes and localising subcategories in the derived category of a noetherian ring.…”

“…These ideas were later pushed further into the world of derived categories of rings and schemes by Neeman [Nee92] and Thomason [Tho97], and into modular representation theory by Benson, Carlson and Rickard [BCR97]. Motivated by the work of Hopkins [Hop87], Neeman [Nee92] classified the Bousfield classes and localising subcategories in the derived category of a noetherian ring. In modular representation theory, the Benson-Carlson-Rickard classification of the thick subcategories of stable modules over group algebras has led to some deep structural information on the representation theory of finite groups.…”

Classifying finite localizations of quasicoherent sheaves

“…A thick triangulated subcategory T of D per (X) is said to be a tensor subcategory if for every E ∈ D per (X) and every object A ∈ T , the tensor product E⊗A is also in T . Thomason [6] established a classification similar to (1.1) for the tensor thick subcategories of D per (X) in terms of the topology of X. Hopkins and Neeman (see [7,8]) did the same in the case where X is affine and Noetherian.On the basis of Thomason's classification theorem, Balmer [9] reconstructed the Noetherian scheme X out of the tensor thick triangulated subcategories of D per (X). This result was generalized to quasicompact, quasiseparated schemes by Buan-Krause-Solberg [2].…”

“…A thick triangulated subcategory T of D per (X) is said to be a tensor subcategory if for every E ∈ D per (X) and every object A ∈ T , the tensor product E⊗A is also in T . Thomason [6] established a classification similar to (1.1) for the tensor thick subcategories of D per (X) in terms of the topology of X. Hopkins and Neeman (see [7,8]) did the same in the case where X is affine and Noetherian.…”