Hereditary crossed product orders over discrete valuation rings. (Q1952143)

Weak crossed products, given by 2-cocycles with non-invertible values, have been considered in 1982 by D. E. Haile. For a tamely ramified extension \(S|R\) of discrete valuation domains with Galois group \(G\), and a cocycle \(f\colon G\times G\to S\setminus 0\), the associated crossed product order \(A_f=\sum_{\sigma\in G}Sx_\sigma\) need not be hereditary. The so-called inertial subgroup \(H\) of \(G\) consists of the \(\sigma\in G\) with \(f(\sigma,\sigma^{-1})\in S^\times\). One can form the crossed product \(B_f=\sum_{\sigma\in H}Sx_\sigma\). The author proves first that the radical of \(A_f\) is given by \(\pi_SB_f\oplus\sum_{\sigma\notin H}Sx_\sigma\). Then he provides equivalent conditions for \(A_f\) to be hereditary. One of his characterizations states that the \(f(\sigma,\sigma^{-1})\) are ``almost invertible'': they do not belong to the squared radical of \(S\).