Constructible topology on schemes - MathOverflow

Constructible topology on schemes

2013-01-25T10:03:36Z

In EGA IV Grothendieck introduced notion of constructible topology. Is it only interesting gadget or can it be use for some practical purposes in algebraic geometry?

Answer by pz for Constructible topology on schemes

2013-01-25T15:17:10Z

In Hochster's paper 'Prime Ideal Structure in Commutative Rings' the author uses it to characterize spectral spaces. This in turn is used in Huber's work on Adic spaces, cf. 'Huber - Étale cohomology of Rigid Analytic Varieties and Adic Spaces' and 'Scholze - Perfectoid spaces'.

Answer by Matthieu Romagny for Constructible topology on schemes

2013-03-25T08:43:20Z

Julien Sebag tells me that the constructible topology is useful for the study of the Grothendieck ring of varieties. More precisely, it is relevant to the following question: "if $k$ is a field and $X$ is a $k$ -variety with a birational endomorphism $X--\to X$ inducing an isomorphism between open subsets $U$ and $V$ , are $X\setminus U$ and $X\setminus V$ piecewise isomorphic? You may read about this in this paper.

Another place where the constructible topology is essential is in motivic integration, where constructible sets play the role of the measurable sets of usual integration theory.