Base change for higher Stickelberger ideals (Q1277201)

Let \(K/k\) be a finite abelian extension with Galois group \(G\). It is well known how to attach, for each \(n \geq 0\), Stickelberger ideals \(I(n)\) to such extensions, and it is a classical result that the Stickelberger ideal \(I(0)\) attached to an abelian extension \(K/\mathbb Q\) annihilates the class group of \(K\), the corresponding conjecture for abelian extensions \(K/k\) of totally real fields \(k\) being known as the Brumer-Stark conjecture. In general, it is conjectured that \(I(n)\) annihilates the Quillen \(K\)-groups \(K_{2n}({\mathcal O}_K)\) [see \textit{J. Coates}, Algebraic Number Fields, Durham 1975, 269-353 (1977; Zbl 0393.12027)]). For an intermediate field \(F\) of \(K/k\), one can pose the following ``base change'' problem: does the truth of the said conjecture for \(K/k\) imply its truth for \(K/F\)? For \(n = 0\), such a result was proved by \textit{D. Hayes} [J. Reine Angew. Math. 497, 83-89 (1998)], and in this paper the author proves it for all \(n \geq 0\). The proof boils down to the verification of a number of integrality statements, the key ingredient being a deep result due to \textit{P. Deligne} and \textit{K. Ribet} [Invent. Math. 59, 227-286 (1980; Zbl 0434.12009)]; another ingredient is the theorem of Klingen-Siegel, but the rest of this well written paper is surprisingly elementary.