Rings of integers of type \(K(\pi,1)\) (Q2469672)

Let \(p\) be a prime number, \(k\) a number field, \({\mathcal O}_ k\) its ring of integers and \(S\) a finite set of nonarchimedean primes of \(k\) such that no prime dividing \(p\) is in \(S\). This is the tame case. The purpose of this paper is to investigate the Galois group \(G_ S (p)\) of the maximal \(p\)-extension \(k_ S(p)\) of \(k\) unramified outside \(S\). A connected locally Noetherian scheme \(Y\) is said to be of type \(K(\pi,1)\) for \(p\) if the higher homotopy groups of the \(p\)-completion \(Y_ {et}^ {(p)}\) of its étale homotopy type \(Y_{et}\) vanish. We have \(G_ S(p)\cong \pi_ 1((\mathrm{Spec}({\mathcal O}_ k)\setminus S)_{et} ^ {(p)})\). In case that \(S\) contains the \(p\)-primes, it is known that \(\mathrm{Spec}({\mathcal O}_ k)\setminus S\) is of type \(K( \pi,1)\) for \(p\) and in particular \(G_ S(p)\) is of cohomological dimension less than or equal to \(2\). In the tame case, \(G_ S(p)\) is more complicated. For instance, there exist examples such that \(\mathrm{Spec}({\mathcal O}_ k) \setminus S\) is not of type \(K(\pi,1)\) and until recently not a single \(K(\pi,1)\) example was known. This paper studies systematically the \(K(\pi,1)\) properties in the tame case, and the author conjectures: Conjecture: Let \(p\neq 2\) or \(k\) totally imaginary, \(T\) a set of primes of Dirichlet density equal to \(1\). Then there exists a finite subset \(T_ 1\subseteq T\) such that \(\roman{Spec} ({\mathcal O}_ k)\setminus (S\cup T_ 1)\) is of type \(K(\pi,1)\) for \(p\). The main result of the paper is that the conjecture is true if \(k\) does not contain a primitive \(p\)-th root of unity and the class number of \(k\) is prime to \(p\). The other main results are the following: {\(\bullet\)} Let \(S\) be a finite nonempty set and such that the norms of the primes in \(S\) are congruent to \(1\) modulo \(p\). If \(X\setminus S\) is of type \(K(\pi,1)\) for \(p\) and \(G_ S(p)\neq 1\), then \(\mathrm{cd}(G_ S(p))=2\), \(\mathrm{scd}(G_ S(p))=3\) and \(G_ S(p)\) is a duality group. The dualizing module \(D\) of \(G_ S(p)\) is given by \(D=\mathrm{tor}_ p( C_ S(k_ S(p)))\), the subgroup of \(p\)-torsion elements in the \(S\)-idèle class group of \(k_ S(p)\). {\(\bullet\)} With the same conditions, \(k_ S(p)_ {\mathfrak p} =k_{\mathfrak p}(p)\) for every \({\mathfrak p}\in S\), that is, \(k_ S(p)\) realizes the maximal \(p\)-extension of the local field \(k_{\mathfrak p}\). {\(\bullet\)} With the same conditions, we have \[ \lim_{\begin{matrix} \longleftarrow\\ ^{K\subseteq k_ S(p)}\end{matrix}} {\mathcal O}_ K^ {\ast}\otimes {\mathbb Z}_ p=0=\lim_{\begin{matrix} \longleftarrow\\ ^{K\subseteq k_ S(p)}\end{matrix}} {\mathcal O}_ {K,S}^ {\ast}\otimes {\mathbb Z}_ p \] where \(K\) runs through all finite subextensions of \(k\) in \(k_ S(p)\). {\(\bullet\)} Let \(S'\) be a finite non-empty set of primes of \(k\) whose norms are congruent to \(1\) modulo \(p\), \(\emptyset \neq S\subseteq S'\). Assume that \(X\setminus S\) is of type \(K(\pi,1)\) for \(p\) and \(G_ S(p)\neq 1\). If each \({\mathfrak q}\in S'\setminus S\) does not split completely in \(k_ S(p)\), then \(X\setminus S'\) is of type \(K(\pi,1)\) for \(p\). In this case, the arithmetic form of Riemann existence theorem holds. The paper runs as follows. First the necessary definitions and some calculations of étale cohomology are given. In Section 4, the paper deals with the first obstruction against the \(K(\pi,1)\)-property, the \(h^ 2\) defect. Next, Labute's results on mild pro-\(p\)-groups are recalled, they are used in the proof of the main result given in Section 6. In the last three sections the author proves the rest of the results.