Consider a stable homotopy category, such as the stable category of spectra or the derived category of some ring, and let F denote the full subcategory of its finite objects. Now one can ask, ``What are all the triangulated subcategories of F?'' Following Thomason and thick subcategory theorems of Hopkins, Smith, and Neeman, one can try to answer this question by computing Grothendieck groups of the thick subcategories of F. In this talk, I will take this approach to address this question by focusing on the two stable homotopy categories mentioned above. I will also report a few related results and end with some questions I have in mind.
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John H. Palmieri, Department of Mathematics, University of Washington, palmieri@math.washington.edu