Finite group torsors for the qfh topology
Let \(S\) be a Noetherian scheme. The qfh topology for \(S\) is generated by the finite families of finite morphisms such that their sum is a universal topological epimorphism. This paper compares the qhf and étale torsors over \(S\) for certain group schemes. \textit{V. Voevodsky} [Sel. Math., New Ser. 2, No. 1, 111--153 (1996; Zbl 0871.14016)] showed that for each \(n\geq 0\) there are isomorphisms \(H_{et}^{n}( S,A) \cong H_{qfh}^{n}( S,A) \) for abelian schemes \(A\) which are locally constant for the étale topology. In the work under review, a non-abelian version of this result is obtained in the case \(n=1\). Specifically, it is shown that for \(G\) a finite étale group scheme over \(S\) that the canonical map \(H_{et}^{1}( S,G) \to H_{qfh}^{1}( S,G) \) is a bijection. The main result is equivalent to the statement that the qfh stack completion map \( BG\to BSt_{qfh}G\) is a local weak equivalence in the étale topology. First it is established that finite constant simplicial sheaves with finitely many simplices of each degree satisfy a qfh descent condition on strict local Hensel rings and that the inverse image of a qfh sheaf over such a ring can be computed using filtered colimits. By restricting to the case where \(G\) is constant and finite (which can be done since the problem is a local one in the étale topology) the author quickly proves the main result.
scientific article; zbMATH DE number 1983501
scientific article