Reconstructing function fields from Milnor \(K\)-theory (Q2070897)

Reconstructing fields from their absolute Galois groups is the classical problem that has initiated the broad field of anabelian Geometry. The paper under review is motivated by the analogous question whether the Milnor \(K\)-ring \(K_*^M(F)\) determines the isomorphism class of the field \(F\). The paper's main outcome is that the Milnor \(K\)-ring modulo the ideal of divisible elements (resp., of torsion elements) determines in a functorial way finitely generated regular field extensions of transcendence degree \(\ge 2\) over algebraically closed fields (resp., over finite fields). The proof is split into two steps. First, the \(n=2\) case of the Bloch-Kato conjecture [\textit{A. S. Merkur'ev} and \textit{A. A. Suslin}, Math. USSR, Izv. 21, 307--340 (1983; Zbl 0525.18008); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No. 5, 1011--1046 (1982)] (resp., of the Bass-Tate conjecture [\textit{J. Tate}, in: Actes Congr. internat. Math. 1970, 1, 201--211 (1971; Zbl 0229.12013)]) implies that the Milnor \(K\)-ring detects algebraic dependence. Second, the paper's main result is that, for perfect fields \(k\) and \(k'\) (of any characteristic) and for finitely generated regular field extensions \(F|k\) and \(F'|k'\) of transcendence degree at least~\(2\), the canonical map \[ \mathrm{Isom}(F|k, F'|k') \rightarrow \overline{\mathrm{Isom}}^{\equiv}(F^\times/k^\times, F'^\times/k'^\times) \] from the set of isomorphisms \(F \overset{\sim}{\rightarrow} F'\) inducing an isomorphism \(k \overset{\sim}{\rightarrow} k'\) to the set of group isomorphisms \(\bar{\psi} \colon F^\times/k^\times \overset{\sim}{\rightarrow} F'^\times/k'^\times\) preserving algebraic dependence (and identifying \(\bar{\psi}\) and \(\bar{\psi}^{-1}\)) is bijective. The fundamental theorem of projective geometry (extended in the paper to infinite field extensions) and a Bertini theorem reduce the proof of this main result to showing that any \(\bar{\psi}\) as above (or \(\bar{\psi}^{-1}\)) maps every line in \(F^\times/k^\times\) whose image contains a (suitably defined) good pair of points isomorphically to a line \(F'^\times/k'^\times\). The proof of the latter is intricate and arguably the most original part of the paper (in positive characteristic). Variants of the main result have previously been proved by \textit{F. Bogomolov} and \textit{Y. Tschinkel} [Surv. Differ. Geom. 13, 223--244 (2009; Zbl 1247.19002)] in characteristic \(0\) and by \textit{A. Topaz} [Math. Ann. 366, No. 1--2, 337--385 (2016; Zbl 1388.12004)] in positive characteristic.