Pureté, rigidité, et morphismes entiers

Abstract

Bousfield and Kan have shown that a ring morphism with domain Z {\mathbf {Z}} is rigid; we say that a ring morphism is rigid if it admits a factorization by an epimorphism, followed by a pure morphism. A ring A A is said to be rigid if every morphism with domain A A is a rigid one. Our principal results are: the rigid domains are the Prüferian rings A A , with Dim ⁡ ( A ) ≤ 1 \operatorname {Dim} (A) \leq 1 , and the Noetherian rigid rings are the Z.P.I. rings. The quasi-compact open sets of an affine rigid scheme, having as underlying ring a domain or a Noetherian ring, are affine and schematically dense if they contain the assassin of the ring. Every injective integral ring morphism with rigid domain is a pure morphism. We give two criteria of purity for integral injective morphisms. As a consequence of these results we obtain the following properties: if A A is a normal ring, containing the field of rationals, or is a regular ring, containing a field, every injective integral morphism with domain A A is a pure one. For a reduced ring, we define the category of reduced modules and show that any injective integral morphism is pure with respect to the category of the reduced modules.