Lattices over integral group rings and abelian subquotients (Q1188165)

Let \(O\) be a Dedekind domain with quotient field \(K\) of characteristic zero. For a finite group \(\Gamma\), the author considers invariants of \(O\Gamma\)-lattices \(M\) with \(KM\) free over \(K\Gamma\), which measure the deviation from projectivity. For abelian \(\Gamma\), \textit{A. Fröhlich} [Q. J. Math., Oxf. II. Ser. 16, 193--232 (1965; Zbl 0192.14002)] introduced an \(O\)-ideal \(b(M,O\Gamma)\) which coincides with \(O\) exactly if \(M\) is projective over \(O\Gamma\). For non-abelian \(\Gamma\), the author studies the ideals \(b(M)(\Delta,\Sigma) = b(M^\Sigma,O[\Delta/\Sigma])\), where \(\Delta/\Sigma\) runs through the abelian subquotients of \(\Gamma\). He shows first that \(M\) is projective if and only if all the \(M^\Sigma\) are \(O[\Delta/\Sigma]\)-projective. The map \(b(M)\) naturally extends to a ring \(A_\Gamma\) generated by the \(\Gamma\)-conjugacy classes of abelian subquotients \(\Delta/\Sigma\). The author then proves that \(M\) is \(O\Gamma\)-projective exactly if the homomorphism \(b(M)\) factors through the ring of virtual \(\Gamma\)-representations. Generalizing this, he studies those \(O\Gamma\)-lattices \(M\) for which some power of \(b(M)\) factors through the ring of monomial representations of \(\Gamma\). Examples are provided in a final section.