Static modules and Clifford theories
Let \(A\) be a ring and denote by Mod-\(A\) the category of right \(A\)-modules. Assume that \(M\) is a self-small \(A\)-module and let \(D=\text{End}_AM\). Then the category of \(A\)-modules which divide a direct sum of copies of \(M\) is equivalent to the category of projective \(D\)-modules. Let further \(\rho\colon A\to B\) be a ring homomorphism, \(K=M\otimes_AB\), \(E=\text{End}_B(K)\) and \(F=\text{End}_A(K)\). Denote by \(\text{Mod}(B|A\text{-split})\) the category of \(B\)-modules having a \(K\)-presentation which splits upon restriction to \(A\), and by \(\text{Mod}(E|\text{Res}_F)\) the category of \(E\)-modules \(W\) for which \(W\otimes_EF\) is a projective \(F\)-module. By a result of \textit{J. Alperin} [J. Aust. Math. Soc., Ser. A 49, No. 3, 347-353 (1990; Zbl 0739.16008)], the restrictions of the functors \(\text{Hom}_B(K,-)\) and \(-\otimes_E K\) determine an equivalence between these categories. The authors prove that the above mentioned theorems are in fact equivalent, since it turns out that \(\text{Mod}(B|A\text{-split})\) coincides with the category of \(B\)-modules which divide a direct sum of copies of \(K\), while \(\text{Mod}(E|\text{Res}_F)\) coincides with the category of projective \(E\)-modules.
scientific article; zbMATH DE number 1051582