Refining thick subcategory theorems

Abstract

We use a K-theory recipe of Thomason to obtain classifications of triangulated subcategories via refining some standard thick subcategory theorems. We apply this recipe to the full subcategories of finite objects in the derived categories of rings and the stable homotopy category of spectra. This gives, in the derived categories, a complete classification of the triangulated subcategories of perfect complexes over some commutative rings. In the stable homotopy category of spectra we obtain only a partial classification of the triangulated subcategories of the finite p-local spectra. We use this partial classification to study the lattice of triangulated subcategories. This study gives some new evidence to a conjecture of Adams that the thick subcategory C2 can be generated by iterated cofiberings of the Smith-Toda complex. We also discuss several consequences of these classification theorems. Classifying various subcategories of triangulated categories like the derived cate- gories and the homotopy category of spectra has been an active area and has proved to be extremely useful in the study of global problems in (stable) homotopy theory. Several mathematicians brought to light many amazing a priori different theories by classifying various subcategories of triangulated categories. Following the sem- inal work of Devinatz, Hopkins, and Smith (DHS88) in stable homotopy theory, this line of research was initiated by Hopkins in the 80s. In his famous 1987 pa- per (Hop87), Hopkins classified the thick subcategories (triangulated subcategories that are closed under retractions) of the finite p-local spectra and those of perfect complexes over a noetherian ring. He showed that thick subcategories of the finite spectra are determined by the Morava K-theories and those of perfect complexes by the prime spectrum of the ring. These results have had tremendous impacts in their respective fields. The thick subcategory theorem for finite spectra played a vital role in the study of nilpotence and periodicity. For example, using this theo- rem Hopkins and Smith (HS98) were able to settle the class-invariance conjecture of Ravenel (Rav84) which classified the Bousfield classes of finite spectra. Similarly the thick subcategory theorem for the derived category establishes a surprising connec- tion 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). Mo- tivated by the work of Hopkins (Hop87), Neeman (Nee92) classified the Bousfield