On equivalences between blocks of group algebras: Reduction to the simple components
Let \(p\) be a prime and let \(\mathcal O\) be a complete discrete valuation ring with quotient field \(K\) of characteristic zero and with residue field \(k={\mathcal O}/J({\mathcal O})\) of characteristic \(p\). All \(\mathcal O\)-algebras and \(\mathcal O\)-algebra modules will be assumed to be unitary, finitely generated and \(\mathcal O\)-free. Let \(G\) be a finite group and let \(R=\bigoplus_{g\in G}R_g\) and \(S=\bigoplus_{g\in G}S_g\) be \(G\)-crossed product \(\mathcal O\)-algebras. The opposite ring \(S^{\text{op}}\) is a \(G^{\text{op}}\)-crossed product \(\mathcal O\)-algebra and \(A=R\otimes_{\mathcal O}S^{\text{op}}\) is a \(G\times G^{\text{op}}\)-crossed product \(\mathcal O\)-algebra. Let \(\delta:G\to G\times G^{\text{op}}\) be defined by: \(g\mapsto(g,g^{-1})\) for all \(g\in G\). Then \(\Delta\equiv A_{\delta(G)}=\bigoplus_{g\in G}(R_g\otimes_{\mathcal O}S^{\text{op}}_{g^{-1}})\) is a \(\delta(G)\)-crossed product \(\mathcal O\)-subalgebra of \(A\). One of the main points of this paper is that there are three naturally isomorphic \(\mathcal O\)-linear functors \(A\otimes_\Delta-\), \(R\otimes_{R_1}-\), \(-\otimes_{S_1}S:\Delta\text{-mod}\to A\text{-mod}\) (Section 3). Let \(M\) be an \((R_1,S_1)\)-bimodule. Then its \(\mathcal O\)-dual \(M^*=\text{Hom}_{\mathcal O}(M,\mathcal O)\) is an \((S_1,R_1)\)-bimodule. One of the main results of this paper is: Theorem 3.4. (a) Assume that \((M,M^*)\) defines a Morita equivalence between \(R_1\) and \(S_1\) and that \(M\) is \(\delta(G)\)-invariant and extends to a \(\Delta\)-module. Then \(R\otimes_{R_1}M\) (resp. \(S\otimes_{S_1}M^*\)) is a naturally \(G\)-graded \((R,S)\)-bimodule (resp. \((S,R)\)-bimodule) and \((R\otimes_{R_1}M\), \(S\otimes_{S_1}M^*)\) defines a Morita equivalence between \(R\) and \(S\). (b) Assume that \(G\) is a \(p'\)-group and that \((C,C^*)\) defines a derived equivalence between \(R_1\) and \(S_1\). If \((C,d)\) extends to a complex of \(\Delta\)-modules, then \((R\otimes_{R_1}C\), \(S\otimes_{S_1}C^*\)) defines a derived equivalence between \(R\) and \(S\). These ideas are applied, in this paper, to modular group representation theory as follows. Assume also that \(K\) and \(k\) are ``big enough'' for all of the finite groups under consideration. Let \(\widetilde X\), \(\widetilde Y\) be finite groups with normal subgroups \(X\trianglelefteq\widetilde Y\), \(Y\trianglelefteq\widetilde Y\) and with an isomorphism \(\alpha:\widetilde X/X\to\widetilde Y/Y\). Let \(e\in Z({\mathcal O}X)\), \(f\in Z({\mathcal O}Y)\) be \(\widetilde X,\widetilde Y\)-invariant block idempotents, resp. and set \(R=e{\mathcal O}\widetilde X\) and \(S=f{\mathcal O}\widetilde Y\). Set \(G=\widetilde X/X\) and identify \(G\) with \(\widetilde Y/Y\) via \(\alpha\). Then \(R\) and \(S\) are \(G\)-crossed product \(\mathcal O\)-algebras with \(R_1=e{\mathcal O}X\) and \(S_1=f{\mathcal O}Y\). In this context, results on Puig equivalences and Rickard tilting complexes are obtained (Corollary 3.9). Section 4 presents applications to wreath products and Section 5 contains applications to isomorphic blocks, groups with cyclic Sylow \(p\)-subgroups, the principal 3-block of \(A_5\) and symmetric groups, etc. The basic idea of examining the consequences of the existence of an extension to the ``diagonal'' is related to earlier work of P. Fong and the reviewer on the character level. (Also submitted to MR).
groups with cyclic Sylow \(p\)-subgroups
scientific article; zbMATH DE number 930147