Refining thick subcategory theorems

Abstract

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 for a conjecture of Adams that the thick subcategory C 2 can be generated by iterated cofiberings of the Smith-Toda complex. We also discuss several consequences of these classification theorems.

Krull–Schmidt decompositions for thick subcategories

“…Towards this, the above theorem tells us that we can restrict ourselves to indecomposable thick subcategories. For the interested reader, we cite [7] where we use this brilliant recipe of Thomason to study classifications of triangulated subcategories of finite spectra and perfect complexes.…”