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Hasse principles for higher-dimensional fields. (English) Zbl 1346.14057

The paper is devoted to the proof of several seminal conjectures on local-global principles for higher-dimensional fields. These conjectures, proposed by K. Kato [J. Reine Angew. Math. 366, 142–183 (1986; Zbl 0576.12012)], are formulated in terms of Galois cohomology of a function field \(F\) in \(d\) variables over a global field \(K\) under the assumption that \(K/F\) is primary, i.e., \(K\) is separably closed in \(F\).
Conjecture 1 states that the restriction map \[ \alpha_n: H^{d+2}(F,\mathbb Z/n\mathbb Z(d+1)) \to \bigoplus_v H^{d+2}(F_v,\mathbb Z/n\mathbb Z(d+1)) \] is injective (here the sum is taken over all places of \(K\), \(F_v=K_v(V\times_K K_v)\), where \(V\) is a geometrically integral model of the field \(F\)). It was proved by Kato (loc. cit.) for \(d=1\).
The first principal result of the paper under review states that this conjecture holds provided \(n\) is invertible in \(K\). Here are the main steps of the proof. First, the assertion of the conjecture is reduced to another one, where finite coefficients \(\mathbb Z/n\mathbb Z(j)\) are replaced with infinite ones \(\mathbb Q_\ell/\mathbb Z_\ell (j)\). A similar step was made by Kato using the Merkurjev-Suslin theorem whereas Jannsen uses the Bloch-Kato conjecture (now the Voevodsky-Rost theorem). To prove the injectivity theorem for the restriction map with infinite coefficients, one has to use weights, i.e., Deligne’s proof of the Weil conjectures, and resolution of singularities to control the weights. In the case of positive characteristic, some problems are to overcome in absence of Hironaka’s theorem. For \(\ell\) invertible in \(K\), the author uses a weaker form of resolution, namely the de Jong alteration in a refined version established by Gabber.
Similarly to the cases \(d=1\) (Colliot-Thélène’s appendix to Kato’s paper quoted above), and \(d=2\) (J.-L. Colliot-Thélène and U. Jannsen [C. R. Acad. Sci. Paris Sér. I Math. 312, 759–762 (1991; Zbl 0743.11020)]), natural applications to quadratic forms arise, more precisely to sum of squares. Namely, the author proves that if \(F\) is a finitely generated field of characteristic zero, then the Pythagoras number of \(F\) is finite, and if \(F\) is of transcendence degree \(d\) over \(\mathbb Q\), then any sum of squares in \(F\) is a sum of \(2^{d+1}\) squares, provided \(d\geq 2\). The proof uses a weaker form of Conjecture 1, whose proof only needs the Milnor conjecture (now the Voevodsky theorem). (More elementary methods give the finiteness and the weaker bound \(2^{d+2}\), see A. Pfister [Jahresber. DMV 102, 15–41 (2000; Zbl 1140.19300)].)
The next Kato’s conjecture refers to the cokernel of the above restriction map. Kato proved it for \(d=1\). It is proved unconditionally in the case where \(K\) is a number field, and conditionally on resolution of singularities for \(n\) invertible in \(K\). (Unconditional proofs were given by M. Kerz and S. Saito [Publ. Math. IHES 115, 123–183 (2012; Zbl 1263.14026)] and by the author [arXiv:0910.2803].)
Yet another Kato’s conjecture, for schemes over finite fields, is proved conditionally on resolution of singularities.

MSC:

14G05 Rational points
11R34 Galois cohomology
14F20 Étale and other Grothendieck topologies and (co)homologies
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