The sieve topology on the arithmetic site (Q2788560)

The paper under review discusses an interesting non-trivial topology on the set of points for the arithmetic topos (which is the presheaf topos associated to the monoid \(\mathbb{N}_+^\times\)), as introduced in [\textit{A. Connes} and \textit{C. Consani}, Adv. Math. 291, 274--329 (2016; Zbl 1368.14038)] to study the Riemann hypothesis from a noncommutative point of view. In the cited paper the set of points is shown to be in bijection with the finite adèle classes and hence the set of points inherits the topology from the finite adèle ring. One can argue that this topology is too coarse, in particular it has a countable basis of opens.NEWLINENEWLINEIn this paper the sieve topology is introduced, using that multiplicative submonoids correspond to sieves and that each submonoid gives rise to another presheaf topos, which are used to define a basis of open subsets. One can explicitly study by using superparticular numbers, which are in bijection with the finite adèle classes. It is shown that this topology is quasicompact, does not admit a countable basis of opens, nonempty opens are dense and one can separate incomparable points. These are properties one expects (or desires) from a candidate object playing the role of the compactification of the spectrum of the integers over the field with one element.