On the representation type of local twisted group rings (Q798405)

Let R be a complete discrete valuation ring with quotient field K. Let L/K be a finite Galois extension with Galois group G. Let S be the valuation ring of L with maximal ideal P. Let A be the trivial crossed- product algebra \((L/K,1)=\oplus_{\sigma\in G}L\sigma\) and let \(\Lambda\) be the twisted group ring \(S\circ G=\oplus_{\sigma\in G}S\sigma\) in which multiplication is given by \((a\sigma)(b\tau)=a\sigma (b)\sigma\tau \) (a,\(b\in L\); \(\sigma\),\(\tau \in G)\). Then A is a central simple K- algebra isomorphic to the algebra of all \(n\times n\) matrices over K with \(n=| L:K| =| G|\) and \(\Lambda\) is an R-order in A. The paper investigates the representation type of \(\Lambda\). Letting \(G_ i\) denote the ith ramification group of P in L/K it is shown that \(\Lambda\) is of finite representation type if and only if either (i) \(| G_ 1|\leq 2\) or (ii) \(| G_ 1| =3\) and \(G_ 1\neq G_ 2\).