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On Fitting ideals of certain étale \(K\)-groups. (English) Zbl 1083.11073

The author computes the first Fitting ideal of \(K^{\text{ét}}_{2i-2} (O^S_F) (\phi)\), showing it is principal and generated by a Brumer-Stickelberger element, where \(O^S_F\) is the \(S\)-integer ring of the abelian number field \(F/\mathbb{Q},\;\) \(S\) is a set of primes of \(F\) tamely or wildly ramified over the odd prim number \(p\), \(\phi\) is a character of Gal\((F/\mathbb{Q})\) of order prime to \(p\) different from the \(i\)th power of the Teichmüller character, and \(H(\phi)\) means \(e_\phi H\) with \(e_\phi\) the usual orthogonal idempotent.

MSC:

11R70 \(K\)-theory of global fields
11R23 Iwasawa theory
19D50 Computations of higher \(K\)-theory of rings
19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
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