We study the triangulated subcategories of compact objects in stable homotopycategories such as the homotopy category of spectra, the derived categories ofrings, and the stable module categories of Hopf algebras. In the first part ofthis thesis we use a K-theory recipe of Thomason to classify thesesubcategories. This recipe when applied to the category of finite p-localspectra gives a refinement of the ``chromatic tower''. This refinement has someinteresting consequences. In particular, it gives new evidence to a conjectureof Frank Adams that the thick subcategory C_2 can be generated by iteratedcofiberings of the Smith-Toda complex V(1). Similarly by applying this K-theoryrecipe to derived categories, we obtain a complete classification of thetriangulated subcategories of perfect complexes over some noetherian rings.Motivated by these classifications, in the second part of the thesis, we studyKrull-Schmidt decompositions for thick subcategories. More precisely, we showthat the thick subcategories of compact objects in the aforementioned stablehomotopy categories decompose uniquely into indecomposable thick subcategories.Some consequences of these decompositions are also discussed. In particular, itis shown that all these decompositions respect K-theory. Finally in the lastchapter we mimic some of these ideas in the category of R-modules. Here weconsider abelian subcategories of R-modules that are closed under extensionsand study their K-theory and decompositions.
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