Local and global fundamental classes for multiquadratic extensions
Fundamental classes play a dominant role in algebraic number theory and particularly in class field theory. Given an extension \(L/K\) with Galois group \(G\), the fundamental class is given in the local case by an element of H\(^2(G,L^\times)\) and in the global case by an element of H\(^2(G,C_L)\), with \(C_L\) the idele class group of \(L\). In both cases the fundamental class is realised by a certain 2-extension, in other words, a 4-term exact sequence whose two middle terms are \(G\)-cohomologically trivial (c.t.). These classes and sequences are hard to determine explicitly. The local case with cyclic Galois group \(G\) is fairly transparent, since then the H\(^2\) term can be replaced by a H\(^0\) term (\(G\)-fixed elements modulo norms), which is way easier to handle. The paper under review successfully deals with a case which is simple in some way but very far from being cyclic: \(L/K\) is a compositum of \(r\) quadratic extensions. The starting point of this very nice paper is a description of the relevant 4-term sequence in the local case, due to Serre. One has two short exact sequences, both with c.t. middle term \((\hat K_{ur}\otimes L)^\times\); the left hand term of the former is \(L^\times\) and the right hand term of the latter is \(\mathbb Z\). Moreover, the right hand term \(V\) of the former exact sequence agrees with the left hand term of the latter, so that one may concatenate them, to form a 2-extension (four-term exact sequence) leading from \(L^\times\) to \(\mathbb Z\), the two modules in the middle being identical and c.t. (There is a harmless typo in the paper: it is not \(V\) which is c.t. but the module \((\hat K_{ur}\otimes L)^\times\), as is clearly indicated by the context.) This 2-extension is reworked somewhat before being used in the determination of the fundamental class. We do not go into detail and only give the final outcome in the simpler of two cases (\(L/K\) is totally ramified): Let \(G\) be 2-elementary of order \(2^r\) with generators \(\sigma_1,\ldots,\sigma_r\). For every subset \(I\subset \{1,\ldots,r\}\) let \(\sigma^{\vphantom 1}_I=\prod_{i\in I} \sigma_i\). This defines a bijection between \(G\) and the set of subsets of \(\{1,\ldots,r\}\). Let \(\pi\) be a uniformizer of \(L\). For each \(I\) one may choose \(\eta^{\vphantom \phi}_I\in \hat L_{ur}\) with \(\eta_I^{\phi_L-1}=\pi^{\sigma_I-1}\), where \(\phi_L\) is the Frobenius attached to the base field \(L\). Then the fundamental class of \(L/K\) is represented by the inverse of the 2-cocycle \(g: G\times G \to L^\times\) given by: \(g(\sigma_I,\sigma_J) = \eta_J^{\sigma_I}\eta_I^{\vphantom 1} \eta_{I+J}^{-1}\). (The sum \(I+J\) in the index means \((I\cup J)\setminus (I\cap J)\).) Appearances notwithstanding, \(g\) is not a coboundary over \(L^\times\), since the elements \(\eta\) cannot be found in \(L\) but only in the completion of its maximal unramified extension. This is pointed out clearly in the paper. The author also succeeds in determining the global fundamental class of a \(G\)-extension of number fields \(L/K\), under the assumption that 2 splits completely in \(L/\mathbb Q\). In particular, the local dyadic extensions corresponding to \(L/K\) must all be trivial. Thus, the global case is not as tightly linked to the local case as one might think, since the assumption just mentioned ``trivialises'' the situation over 2, and locally at non-dyadic primes there are no ``big'' multiquadratic extensions, biquadratic being the maximum. Nevertheless, the global case is just as impressive a feat as the local case. For details of the outcome, we refer to the paper. As an application, the reciprocity map is described very explicitly both in the local and the global setting.
scientific article; zbMATH DE number 6124186