Find a concept
Start typing to search the mathematical index.
A family of morphisms whose generated sieve is covering in the site's Grothendieck topology.
Let be a site and let be an object of . A family of morphisms
is a covering family if the sieve it generates belongs to . The generated sieve consists of all morphisms that factor through at least one .
This definition depends on the chosen Grothendieck topology. For the étale topology, the covering families are precisely the jointly surjective families of étale morphisms.
A site is a pair , where is a category and is a Grothendieck topology on .
The topology determines which sieves, or equivalently which covering families, may be used to formulate local definitions and gluing conditions. A site is categorical data: its objects are not required to be open subsets of one topological space.
Let be a category. A sieve on an object is a collection of morphisms with codomain that is closed under precomposition: if belongs to , then every composite belongs to .
A Grothendieck topology on assigns to every object a collection of sieves, called covering sieves, such that:
Thus a Grothendieck topology specifies what it means to work locally on the objects of a category; it need not arise from a topology on a set of points.
The étale topology is the Grothendieck topology on schemes in which a family
is a covering family exactly when every is étale and the family is jointly surjective on underlying points:
Unlike the Zariski topology, this is not merely a point-set topology on one scheme. It permits étale morphisms as local charts. Restricting it to schemes over a fixed scheme that are étale gives the small étale site of .
A morphism of schemes is étale if it is simultaneously
These conditions are local on both source and target, so étaleness may be checked on affine open covers. For an affine morphism , this says that is a flat, finitely presented -algebra with .
For a finite field extension , the morphism is étale exactly when is separable. Étale morphisms are therefore the algebraic-geometric analogue of local diffeomorphisms.