The representations of finite groups in projective linear groups, that is, projective representations, correspond to modules over twisted group rings. For representations over algebraically closed fields the theory of Schur multipliers provides a very satisfactory tool that may be used to reduce projective representations of a finite group to the usual representations of a finite central extension of that group. Over arbitrary fields, and certainly over commutative rings, other methods have to be used. In this paper we focus mainly on twisted group rings \(R *_ \alpha G\) over connected commutative rings \(R\), which are separable \(R\)-algebras, that is, we assume that the order of the group \(G\) is invertible in \(R\).
The first section is of rather general nature; the objective here is to extend the representation theory of finite-dimensional algebras over fields; that is, we study finitely generated projective indecomposable modules over separable algebras over connected commutative rings and, in particular, over twisted group rings. Throughout, the general version of the Brauer splitting theorem for twisted group rings plays an important role, because it allows the splitting by some Galois extension and the use of Galois-theoretic methods for connected commutative rings. A good example of the use of that Galois theory is Theorem 1.14, one of the main results of §1, describing the decomposition of the module induced by an indecomposable module in terms of Galois actions. In combination with the Brauer splitting theorem (generalized version) the latter result shows how an indecomposable left \(R *_ \alpha G\)-module decomposes when it is induced up to \(S *_ \alpha G\), where \(S\) is a Galois extension of \(R\) and a splitting ring for \(R *_ \alpha G\). Here \(S\) is a splitting ring if \(S *_ \alpha G \cong \text{End}_ S(P_ 1) \oplus \dots \oplus \text{End}_ S(P_ t)\), where \(P_ 1, \dots, P_ t\) are \(S\)- progenerators; in fact \(t\) is the number of \(\alpha\)-ray classes in \(G\).
Let \(A\) be any \(R\)-algebra; to a left \(A\)-module \(M\), which is finitely generated and projective as an \(R\)-module, we may associate a trace \(t_ M\) from \(A\) to \(R\). In §2 we present some general theory concerning such traces (or module characters). Then we apply this to obtain concrete expressions for primitive central idempotents of \(S *_ \alpha G\) in terms of the traces afforded by the progenerators \(P_ i\) appearing in the decomposition of \(S *_ \alpha G\), and we derive suitable orthogonality relations for the \(t_{P_ i}\). Replacing the 2-cocycle \(\alpha\) by a suitable one equivalent to it (such that the condition in Proposition 3.3 holds) and taking into account the corresponding change of the trace afforded by a module \(M\), one gets characters that are class functions in the sense that \(t_ M(u_ g) = t_ M(u_{xgx^{-1}})\) if \(g\) is \(\alpha\)-regular and \(x\) is arbitrary in \(G\), whereas \(t_ M(u_ g) = 0\) if \(g\) is not \(\alpha\)-regular; cf. §3. In this situation calculations using ray class sums appear. Moreover, we obtain calculations of centres of blocks of \(R *_ \alpha G\) in terms of the traces afforded by the progenerators appearing in the decomposition of \(S *_ \alpha G\).