Weil and Grothendieck approaches to adelic points

Abstract

In [We, Ch. 1], Weil defines a process of “adelization” of algebraic varieties over global fields. There is an alternative procedure, due to Grothendieck, using adelic points. One aim of this (largely) expository note is to prove that for schemes of finite type over global fields (i.e., without affineness hypotheses), and also for separated algebraic spaces of finite type over such fields, Weil’s adelization process naturally coincides (as a set) with the set of adelic points in the sense of Grothendieck (and that in the affine case the topologies defined by these two viewpoints coincide; Grothendieck’s approach doesn’t provide a topology beyond the affine case). The other aim is to prove in general that topologies obtained by Weil’s method satisfy good functorial properties, including expected behavior with respect to finite flat Weil restriction of scalars. The affine case suffices for most applications, but the non-affine case is useful (e.g., adelic points of G/P for connected reductive groups G and parabolic subgroups P ). We also discuss topologizing X(k) for possibly non-separated algebraic spaces X over locally compact fields k; motivation for this is given in Example 5.5. Although everything we prove (except perhaps for the case of algebraic spaces) is “well known” folklore, and [Oes, I, §3] provides an excellent summary in the affine case, some aspects are not so easy to extract from the available literature. Moreover, (i) some references that discuss the matter in the non-affine case have errors in the description of the topology on adelic points, and (ii) much of what we prove is needed in my paper [Con], or in arithmetic arguments in [CGP]. In effect, these notes can be viewed as an expanded version of [Oes, I, §3], and I hope they will provide a useful general reference on the topic of adelic points of algebro-geometric objects (varieties, schemes, algebraic spaces) over global fields. In §2 we carry out Grothendieck’s method in the affine case over any topological ring R, characterizing the topology on sets of R-points by means of several axioms. The generalization to arbitrary schemes of finite type via a method of Weil is developed in §3. We explore properties of these topologies in §4, especially for adelic points and behavior with respect to Weil restriction of scalars. Finally, in §5 everything is generalized to the case of algebraic spaces. Notation. We write AF to denote the adele ring of a global field F , and likewise A n F denotes Euclidean n-space over AF . There is no risk of confusion with the common use of such notation to denote affine n-space over SpecF since we avoid ever using this latter meaning for the notation.