Abstract.
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.
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Received January 22, 1998; in final form June 23, 1998
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Carlson, J., Rouquier, R. Self-equivalences of stable module categories. Math Z 233, 165–178 (2000). https://doi.org/10.1007/PL00004789
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DOI: https://doi.org/10.1007/PL00004789