The \'etale cohomology ring of a punctured arithmetic curve

Abstract

We compute the cohomology ring $H^*(U,\mathbb{Z}/n\mathbb{Z})$ for $U=X\setminus S$ where $X$ is the spectrum of the ring of integers of a number field $K$ and $S$ is a finite set of finite primes. As a consequence, we obtain an efficient way to compute presentations of $Q_2(G_S)$, where $G_S$ is Galois group of the maximal extension of $K$ unramified outside of a finite set of primes $S$, for varying $K$. This includes the following cases (for $p$ any prime dividing $n$): $\mu_p(\overline{K}) \not\subseteq K$; $S$ does not contain the primes above $p$; and $p=2$ with $K$ admitting real archimedean places. We also show how to recover the classical reciprocity law of the Legendre symbol from the graded commutativity of the cup product.