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A Tate sequence for global units. (English) Zbl 0948.11041

Consider a \(G\)-Galois extension \(K/k\) of number fields and choose a finite \(G\)-stable set \(S\) of places of \(K\) which contains all archimedean places. The object of study are four-term exact sequences \(0 \to E_S \to A \to B \to \nabla_S \to 0\), in which \(A\) and \(B\) are \(G\)-cohomologically trivial and \(\nabla_S\) is as explicit as possible. Of course this puts strong limitations on \(\nabla_S\), since the sequence must induce isomorphisms from \(G\)-cohomology of \(E_S\) to \(G\)-cohomology of \(\nabla_S\), with a dimension shift of 2. This type of sequence was constructed by Tate in the case where all ramified primes in \(K/k\) belong to \(S\) and the \(S\)-class groups of \(K\) is zero. Here the module \(\nabla_S\) is just \(X_S=ker(Y_S\to {\mathbb Z})\), with \(Y_S\) the free abelian group on the basis \(S\) and the map \(Y_S\to {\mathbb Z}\) being augmentation. (The authors of the paper under review write \(\Delta S\) instead of \(X_S\).) It has been known for some time that the construction works almost as well if the \(S\)-class group is just supposed to be \(G\)-cohomologically trivial, not necessarily zero. The main ingredient in the construction is the compatibility between local and global fundamental classes.
In the paper under review, this kind of exact sequence is established without assuming Tate’s hypotheses. The price to pay is that one inevitably needs a more involved module \(\nabla_S\), and more cumbersome uniqueness statements, since the choice of a \(G\)-transversal in \(S\) is not eliminated as easily as before. The module \(\nabla_S\) is an extension of its torsion part, the \(S\)-class group of \(K\), by a certain lattice \(\bar\nabla\). This lattice “contains” a contribution \(W_P^0\) for each ramified prime \(P\) not included in \(S\), where the exponent 0 means \(\mathbb Z\)-dual and the “inertial lattice” \(W_P\) is explicitly given in terms of the ramification data at \(P\). In particular, the cohomology of \(W_P\) is given by the cohomology of \(U_P\) (the units in the completion \(K_P\)) with a dimension shift of one.
The proofs are fairly technical. A considerable part is needed for arguments showing the independence of the outcome from various choices that have to be made. As an application, the multiplicative (= 3rd) Chinburg invariant is expressed in terms of a Tate sequence for arbitrary \(S\). This was an important ingredient in the author’s subsequent work on the Stark conjecture: see J. Am. Math. Soc. 10, No. 3, 513-552 (1997; Zbl 0885.11059).
(The reviewer would like to mention in passing that he is not responsible for the delay with which the review of the present article appears.).

MSC:

11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R37 Class field theory
11R34 Galois cohomology

Citations:

Zbl 0885.11059
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References:

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