  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.
