Linear forms on Sinnott's module (Q402653)

\textit{W. Sinnott} [Invent. Math. 62, 181--234 (1980; Zbl 0465.12001)] introduced a \(\mathbb Z [G]\)-module \(U \subset \mathbb Q [G]\), where \(G\) is the Galois group of an abelian number field \(K\), which is connected with both the Stickelberger ideal and the circular units of \(K\).NEWLINENEWLINEIn the present paper, modified versions \(U\), \(U'\) of Sinnott's module are defined, needing no reference to an abelian number field. The rather technical main result (Theorem 1.1) concerns \(\mathbb Z [G]\)-homomorphisms from \(U\), \(U'\), resp., to \(\mathbb Z [G]\). This result resembles, but is stronger than a former one by \textit{A. Hayward} [Compos. Math. 140, No. 1, 99--129 (2004; Zbl 1060.11075)] (cp. Remarks 1.2 and 4.2).NEWLINENEWLINEThe main result will be needed by the authors in a subsequent paper, where they prove some analogue of Gras' conjecture using the ``Euler-Kolyvagin machinery''.