Indecomposable endo-permutation modules over p-groups
Let \({\mathfrak O}\) be a valuation ring with residue field \({\mathfrak F}\) of characteristic \(p>0\) and P a p-group. For an \({\mathfrak O}P\)-lattice \({\mathfrak K}\), let \({\mathfrak E(K)}=End_{{\mathfrak O}P}({\mathfrak K})/J(End_{{\mathfrak O}P}(K))\). In the literature, there are two notions of ''absolutely indecomposable'' lattice. One, due to \textit{J. A. Green} [Math. Z. 70, 430- 445 (1959; Zbl 0086.024)] is that \({\mathfrak E(K)}\) should be a purely inseparable extension of \({\mathfrak F}\). The other, due to \textit{B. Huppert} [Arch. Math. 26, 242-249 (1975; Zbl 0304.20008)] is that \({\mathfrak E(K)}\) should be isomorphic to \({\mathfrak F}.\) In an earlier paper [Ann. Math., II. Ser. 107, 459-494 (1978; Zbl 0395.16007)], the author proved that any indecomposable endo-permutation \({\mathfrak O}P\)-lattice is absolutely indecomposable. Unfortunately, he only proved absolute indecomposability in the sense of Green, whereas he wanted it in the sense of Huppert. Here, he uses Green's methods to obtain the result he actually wanted in the earlier paper.
indecomposable endo-permutation \({\mathfrak O}P\)-lattice
scientific article; zbMATH DE number 3973133