Refinements of chromatic towers and Krull-Schmidt decompositions in stable homotopy categories

Abstract

We study the triangulated subcategories of compact objects in stable homotopy categories such as the homotopy category of spectra, the derived categories of rings, and the stable module categories of Hopf algebras. In the first part of this thesis we use a K-theory recipe of Thomason to classify these subcategories. This recipe when applied to the category of finite p-local spectra gives a refinement of the ``chromatic tower''. This refinement has some interesting consequences. In particular, it gives new evidence to a conjecture of Frank Adams that the thick subcategory C_2 can be generated by iterated cofiberings of the Smith-Toda complex V(1). Similarly by applying this K-theory recipe to derived categories, we obtain a complete classification of the triangulated subcategories of perfect complexes over some noetherian rings. Motivated by these classifications, in the second part of the thesis, we study Krull-Schmidt decompositions for thick subcategories. More precisely, we show that the thick subcategories of compact objects in the aforementioned stable homotopy categories decompose uniquely into indecomposable thick subcategories. Some consequences of these decompositions are also discussed. In particular, it is shown that all these decompositions respect K-theory. Finally in the last chapter we mimic some of these ideas in the category of R-modules. Here we consider abelian subcategories of R-modules that are closed under extensions and study their K-theory and decompositions.