In this last chapter we put together the various ideas we have been developing in the week’s lectures. The goal, as has been stated often enough, is a classification of the localising subcategories of the stable module category of a finite group, over a field of characteristic p. The strategy of the proof was described in Section 4.1.6, and we begin this chapter at the last step, which is also where the whole story begins, namely, Neeman’s classification of the localising subcategories of the derived category of a commutative noetherian ring. Using a (version of) this result, and a variation of the Bernstein-Gelfand-Gelfand correspondence, we explain how to tackle the case of the homotopy category of complexes of injective modules over an elementary abelian two group. This is the content of Section 5.2. Finally, in the last section, we use Quillen’s results to describe how to pass from arbitrary groups to elementary abelian ones. If the dust settles down, the reader should be able to see a fairly complete proof of our main results for the case p = 2.
KeywordsPrime Ideal Triangulate Category Local Cohomology Homotopy Category Exterior Algebra
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