On \(K_1\) of curves over global fields. Appendix by C. Weibel
Let \(X\) be a curve over a global field \(k\). In this paper we study conjectures of Bloch and Vaserstein [\textit{L. N. Vaserstein} [in: Quadratic and Hermitian forms, Conf. Hamilton/Ont. 1983, CMS Conf. Proc. 4, 131--140 (1984; Zbl 0577.20033)] and \textit{S. Bloch} [Ann. Math. (2) 114, 229--265 (1981; Zbl 0512.14009)] on the structure of the group \(K_1(X)\). When \(X=\mathrm{Spec}(A)\) is affine (not necessarily regular) Vaserstein's conjecture says that the group \(SK_1(A)=: \operatorname{Ker}[K_1(A)\to A^*]\) is a torsion group. More concretely, this conjecture says that given a \(3\times 3\) matrix \(M\) of determinant 1 with coefficients in \(A\), some power of \(M\) is a product of elementary matrices. Bloch's conjecture predicts that if \(X\) is smooth and projective then the group \(V(X)=: \operatorname{Ker}[SK_1(X)\to k^*]\) is a torsion group. In the appendix to this paper (p. 191--193), \textit{C. Weibel} shows that the conjectures of Bloch and Vaserstein are equivalent. This is proved in the paper when \(A\) is regular. This being the case, we focus on the case of a smooth, projective, geometrically connected curve \(X\) over a global field \(k\). The main result of this paper is that when \(k\) is a number field then \(V(X)\) is an extension of a uniquely divisible group by a torsion group. The uniquely divisible part is the set of Galois invariants \(V(\bar X)^G\), where \(\bar k\) is an algebraic closure of \(k\), \(\bar X=X\times_k\bar k\) and \(G=\mathrm{Gal}(\bar k/k)\). Thus either \(V(\bar X)^G=0\) and Bloch's conjecture is true or the conjecture fails badly. The proof of the main theorem uses Saito's local class field theory of curves [\textit{S. Saito}, J. Number Theory 21, 44--80 (1985; Zbl 0599.14008)] and a theorem of Jannsen on the Galois cohomology of number fields [\textit{U. Jannsen}, in: Galois groups over \(\mathbb{Q}\), Proc. Workshop, Berkeley/CA 1987, Publ., Math. Sci. Res. Inst. 16, 315--359 (1989; Zbl 0703.14010)]. It is surprising that such an innocent statement has a proof which is not very innocent. For a smooth, projective curve \(X\) over a global field of finite characteristic, one can easily show that \(V(X)\) is either a torsion group or it is quite large. Because we know so much more about \(K_1\) of surfaces over finite fields than arithmetic surfaces, one can prove the conjecture in some cases using results of \textit{C. Soulé} [Math. Ann. 268, 317-345 (1984; Zbl 0573.14001)].
scientific article; zbMATH DE number 4163042
scientific article