Self-equivalences of stable module categories
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Let P be an abelian p-group, E a cyclic \(p'\)-group acting freely on P and k an algebraically closed field of characteristic \(p>0\). In this work, we prove that every self-equivalence of the stable module category of \(k[P\rtimes E]\) comes from a self-equivalence of the derived category of \(k[P\rtimes E]\). Work of Puig and Rickard allows us to deduce that if a block B with defect group P and inertial quotient E is Rickard equivalent to \(k[P\rtimes E]\), then they are splendidly Rickard equivalent. That is, Broué's original conjecture implies Rickard's refinement of the conjecture in this case. All of this follows from a general result concerning the self-equivalences of the thick subcategory generated by the trivial module.
KeywordsGeneral Result Module Category Stable Module Defect Group Trivial Module
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