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étale morphism
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(Definition)
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This is the appropriate generalization of “local homeomorphism” from topology or “local isomorphism” from real differential geometry. Equivalently,  is étale if and only if any of the following conditions hold:
A morphism  of varieties over an algebraically closed field is étale at a point  if it induces an isomorphism between the completed local rings
 and
 . If  and  are over an arbitrary field  , then the required condition becomes that  is a separable algebraic extension of  , where  , and  induces an isomorphism between
 and
 .
A morphism  of nonsingular varieties over an algebraically closed field is étale if and only if  induces an isomorphism on the tangent spaces. In the differentiable category, the implicit function theorem implies that such a function is actually an isomorphism on some small neighborhood. On schemes, of course, the Zariski topology is too coarse for this to be the case. One way to define a finer “topology”, making the scheme into a site, is by using étale maps.
The word étale comes from French, where it can be used to describe a calm or slack sea.
- 1
- Jean Dieudonné, A Panorama of Pure Mathematics, Academic Press, 1982.
- 2
- Robin Hartshorne, Algebraic Geometry, Springer-Verlag, 1977 (GTM 52).
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"étale morphism" is owned by mps. [ full author list (2) | owner history (3) ]
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(view preamble)
Cross-references: slack, site, finer, Zariski topology, schemes, neighborhood, function, implies, implicit function theorem, category, differentiable, tangent spaces, nonsingular varieties, algebraic extension, separable, local rings, isomorphism, induces, point, field, algebraically closed, varieties, morphism, Jacobian, dimension, smooth, vanishes, sheaf, finite type, differential geometry, real, topology, unramified, flat, morphism of schemes
There are 14 references to this entry.
This is version 11 of étale morphism, born on 2004-02-10, modified 2006-02-09.
Object id is 5559, canonical name is EtaleMorphism.
Accessed 5303 times total.
Classification:
| AMS MSC: | 14A15 (Algebraic geometry :: Foundations :: Schemes and morphisms) | | | 14F20 (Algebraic geometry :: homology theory :: Étale and other Grothendieck topologies and cohomologies) |
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Pending Errata and Addenda
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![$ A[x_1,\ldots,x_n]/(p_1,\ldots,p_n)$](http://images.planetmath.org:8080/cache/objects/5559/l2h/img7.png "$ A[x_1,\ldots,x_n]/(p_1,\ldots,p_n)$"){kind=small}