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Submersion et descente. (Submersion and descent). (French) Zbl 0626.14014

This paper is an algebraic study of the submersions of affine schemes, i.e. of the surjective morphisms Spec(A’)\(\to Spec(A)\) such that the topology of Spec(A) is a quotient of that Spec(A’). The author establishes some general properties of the submersions and a classification of them. He introduces a class of morphisms that he calls “subtrusive”. A subtrusive morphism is a morphism from Spec(A’) to Spec(A) such that every couple \(P\subset Q\) of prime ideals of A lifts to a couple P’\(\subset Q'\) of prime ideals of A’. It is proved that the pure morphisms are subtrusive. Universally subtrusive morphisms are defined and characterized and many examples are dicussed. The author also studies the stability of the submersions at a change of basis and at inductive limits.
From the applications we mention the examples of morphisms that descends the flatness and the results about the complete faithful modules.
Reviewer: D.Ştefănescu

MSC:

14B25 Local structure of morphisms in algebraic geometry: étale, flat, etc.
14A15 Schemes and morphisms
13C11 Injective and flat modules and ideals in commutative rings
14E05 Rational and birational maps
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