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Home > Art, Film & Photography > Morphisms of Schemes: Etale Morphism, Rational Function, Proper Morphism, Finite Morphism, Quasi-Finite Morphism, Flat Morphism
Morphisms of Schemes: Etale Morphism, Rational Function, Proper Morphism, Finite Morphism, Quasi-Finite Morphism, Flat Morphism

Morphisms of Schemes: Etale Morphism, Rational Function, Proper Morphism, Finite Morphism, Quasi-Finite Morphism, Flat Morphism


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About the Book

Chapters: Etale Morphism, Rational Function, Proper Morphism, Finite Morphism, Quasi-Finite Morphism, Flat Morphism, Radicial Morphism, Isogeny. Source: Wikipedia. Pages: 35. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In algebraic geometry, a field of mathematics, an etale morphism (pronounced ) is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not necessarily local isomorphisms. Despite this, etale maps retain many of the properties of local analytic isomorphisms, and are useful in defining the algebraic fundamental group and the etale topology. Let be a ring homomorphism. This makes an -algebra. Choose a monic polynomial in and a polynomial in such that the derivative of is a unit in the localization . We say that is standard etale if and can be chosen so that is isomorphic as an -algebra to . Geometrically, this represents as an open subset of a covering space. Let be a morphism of schemes. We say that is etale if it has any of the following equivalent properties: The equivalence of these properties is difficult and relies heavily on Zariski's main theorem. Assume that is locally noetherian. For in, let and let be the induced map on completed local rings. Then the following are equivalent: If in addition all the maps on residue fields are isomorphisms, or if is separably closed, then is etale if and only if Any open immersion is an etale map, by the description of etale maps in terms of standard etale maps. Finite separable field extensions are etale. Any ring homomorphism of the form, where all the are polynomials, and where the Jacobian determinant is a unit in, is etale. Expanding upon the previous ex...More: http: //booksllc.net/?id=3022771


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Product Details
  • ISBN-13: 9781157472940
  • Publisher: Books LLC
  • Publisher Imprint: Books LLC
  • Height: 152 mm
  • Sub Title: Etale Morphism, Rational Function, Proper Morphism, Finite Morphism, Quasi-Finite Morphism, Flat Morphism
  • Width: 229 mm
  • ISBN-10: 115747294X
  • Publisher Date: 15 Sep 2010
  • Binding: Paperback
  • Spine Width: 2 mm
  • Weight: 68 gr


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