Extensions of lattices over p-solvable groups (Q1112957)

Let G be a finite p-solvable group for a prime p and K be a finite extension of the p-adic number field \({\mathbb{Q}}_ p\) and \(k=R/\pi R\) its residue class field. Let U, V be RG-lattices \((=R\)-free RG-modules of finite R-rank). The main result of this paper is the following theorem: \(Ext^ 1_{RG}(U,V)=0\) if either (i) \(K\otimes_ RU\cong K\otimes_ RV\) is irreducible belonging to a block with cyclic defect group and K is unramified over \({\mathbb{Q}}_ p\), or (ii) \(k\otimes_ RU\cong k\otimes_ RV\) is irreducible and the ramification index of K over \({\mathbb{Q}}_ p\) is less than p-1 (and hence p odd). Both of these results rely on the Fong- Swan theorem and the first one generalizes a result of S. D. Berman.