Decompositions modulo projectives of lattices over finite groups (Q1969111)

Let \(G\) be a finite group and for a \(\mathbb{Z} G\)-lattice \(A\), let \(\pi(A)\) be the set of primes \(p\) such that \(A_{(p)}\) is not \(\mathbb{Z}_{(p)}G\)-projective, where \(\mathbb{Z}_{(p)}\) is the localization of \(\mathbb{Z}\) at \(p\) and \(A_{(p)}=\mathbb{Z}_{(p)}\otimes_\mathbb{Z} A\). Let \(\sigma_1,\dots,\sigma_k\) be pairwise disjoint subsets of the set of prime divisors \(\pi(G)\) of the order \(|G|\); \(A\) is said to have a \((\sigma_1,\dots,\sigma_k)\)-decomposition, if \(A=L_1\oplus\cdots\oplus L_k\) with \(\pi(L_i)=\sigma_i\), \(i=1,\dots,k\), and \(A\) is said to have a \((\sigma_1,\dots,\sigma_k)\)-decomposition modulo projectives, if there exists an integer \(t\geq 0\) such that \(A\oplus\mathbb{Z} G^{(t)}\) has a \((\sigma_1,\dots,\sigma_k)\)-decomposition. The author characterizes these modulo projectives decompositions of \(\mathbb{Z} G\)-lattices in terms of their rational character; the possible \(\sigma_i\) in a \((\sigma_1,\dots,\sigma_k)\)-decomposition modulo projectives of a \(\mathbb{Z} G\)-lattice \(A\) turn out to be closed subsets of a graph called \(\Pi(A)\), with vertices the set \(\pi(A)\), and which depends on the rational character \(\chi_A\) of \(A\). In the context of decompositions of lattices, these graphs generalize the notion of the prime graph \(\Pi(A)\) of \(G\) defined by \textit{K. W. Gruenberg} and \textit{K. W. Roggenkamp} [Proc. Lond. Math. Soc., III. Ser. 31, 149-166 (1975; Zbl 0313.20004)].