Group graded basic Morita equivalences and the Harris-Knörr correspondence
Let \(\mathcal{O}\) be a complete discrete valuation ring with algebraically closed residue field \(k\) of characteristic \(p>0\). Moreover, let \(H\) be a normal subgroup of a finite group \(G\), and let \(b\) be a \(G\)-stable block of the group algebra \(\mathcal{O}H\) with defect group \(Q\). The Clifford extension of \(b\) in \(G\) is a certain twisted group algebra of \(\overline{G}[b]\) over \(k\) where \(\overline{G}[b]\) is a normal subgroup of \(\overline{G} := G/H\) introduced by \textit{E. C. Dade} [in: Representation theory of finite groups and related topics. Proceedings of the symposium in pure mathematics of the American Mathematical Society. Held at the University of Wisconsin, Madison, Wisconsin, April 14--16, 1970. Providence, RI: American Mathematical Society (AMS). 33--36 (1971; Zbl 0297.20022); Ill. J. Math. 17, 198--272 (1973; Zbl 0352.20008)]. The paper under review investigates the behaviour of this Clifford extension under \(\overline{G}\)-graded basic Morita equivalences. Now, let \(b'\) be the Brauer correspondent of \(b\) in \(H' := N_H(Q)\). Then, \textit{M. E. Harris} and \textit{R. Knörr} [Commun. Algebra 13, 1213--1218 (1985; Zbl 0561.20002)] have shown that the Brauer map \(\mathrm{Br}_Q\) induces a bijection between the blocks of \(G\) covering \(b\) and the blocks of \(G' := N_G(Q)\) covering \(b'\). The author proves that this bijection coincides with the bijection between blocks induced by a \( \overline{G}\)-graded basic Morita equivalence between \(\mathcal{O}Gb\) and \(\mathcal{O}G'b'\) whenever such a Morita equivalence exists.
scientific article; zbMATH DE number 7216587
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