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- Smooth_morphism abstract "In algebraic geometry, a morphism between schemes is said to be smooth if(i) it is locally of finite presentation(ii) it is flat, and(iii) for every geometric point the fiber is regular.(iii) means that for any the fiber is a nonsingular variety. Thus, intuitively speaking, a smooth morphism gives a flat family of nonsingular varieties. (iii) also means that for a morphism satisfying (i) and (ii) "smoothness" may be checked geometric-fiber-wise.If S is the spectrum of a field and f is of finite type, then one recovers the definition of a nonsingular variety. There are many equivalent definitions of a smooth morphism. Let be locally of finite presentation. Then the following are equivalent. f is smooth. f is formally smooth (see below). f is flat and the relative differential is locally free of rank equal to the relative dimension of . For any , there exists a neighborhood of s and a neighborhood of such that and the ideal generated by the m-by-m minors of is B. Locally, f factors into where g is étale. Locally, f factors into where g is étale.A morphism of finite type is étale if and only if it is smooth and quasi-finite.A smooth morphism is stable under base change and composition. A smooth morphism is locally of finite presentation.A smooth morphism is universally locally acyclic.".
- Smooth_morphism wikiPageExternalLink LEC.pdf.
- Smooth_morphism wikiPageID "7837041".
- Smooth_morphism wikiPageRevisionID "595347881".
- Smooth_morphism hasPhotoCollection Smooth_morphism.
- Smooth_morphism subject Category:Morphisms_of_schemes.
- Smooth_morphism comment "In algebraic geometry, a morphism between schemes is said to be smooth if(i) it is locally of finite presentation(ii) it is flat, and(iii) for every geometric point the fiber is regular.(iii) means that for any the fiber is a nonsingular variety. Thus, intuitively speaking, a smooth morphism gives a flat family of nonsingular varieties.".
- Smooth_morphism label "Smooth morphism".
- Smooth_morphism sameAs m.0km3q3d.
- Smooth_morphism sameAs Q7546425.
- Smooth_morphism sameAs Q7546425.
- Smooth_morphism wasDerivedFrom Smooth_morphism?oldid=595347881.
- Smooth_morphism isPrimaryTopicOf Smooth_morphism.