Tame class field theory for singular varieties over algebraically closed fields (Q288652)

Let \(k\) be an algebraically closed field of characteristic \(p \geq 0\), \(X\) a separated scheme of finite type over \(k\) and \(m \in \mathbb N\). The authors construct a reciprocity homomorphism NEWLINE\[NEWLINE\mathrm{rec}_X: H^S_1(X,\mathbb Z/m\mathbb Z) \to \pi_1^{t,ab} (X)/mNEWLINE\]NEWLINE from the first mod \(m\) algebraic singular homology to the abelianized tame fundamental group of \(X\) mod \(m\) and prove its surjectivity. Furthermore, if \(p \nmid m\) or if ``resolution of singularities holds for schemes of dimension \(\leq \dim X +1\)'', \(\mathrm{rec}_X\) is an isomorphism.NEWLINENEWLINEThat the two above groups are isomorphic for \(p \nmid m\) is already known from the work of \textit{A. Suslin} and \textit{V. Voevodsky} [Invent. Math. 123, No. 1, 61--94 (1996; Zbl 0896.55002)], but now an explicit homomorphism is constructed. Imitating an idea from topology, the authors use torsors to construct a pairing between \(H^S_1(X,\mathbb Z/m\mathbb Z)\) and the first mod \(m\) tame étale cohomology group, which leads to an explicit definition of \(\mathrm{rec}_X\). Then the main result is first proved for smooth curves, and then via blow-ups for the general case.