Varieties and localizations of module categories Original Research Article

Categorical localization is a technique that has recently been exploited to study modules for a finite group G over an algebraically closed field k of prime characteristic. A kG-moduleM determines a subvariety VG(M) of the maximal ideal spectrum of H*(G, k), and the dimension of VG(M) is called the complexity of M. Let imagec denote the subcategory of the stable module category consisting of all modules of complexity at most c. We show that for any module M in the localized quotient category imagec = imagec/imagec−1 the module M circled plus Ω(M) has a direct sum decomposition in which the summands correspond to c-dimensional components of VG(M). We then fix a component V of VG(k) and consider the quotient of the stable category by the subcategory of all modules M such that image. The structure of the endomorphism ring of the trivial module in this category is analyzed, and the quotient category of G is related to that of the normalizer of a certain elementary abelian p-subgroup.