Abstract
All rings considered in this paper are unitary. commutative and noetherian: the notations are those of N. Bourbaki, A. Grothendieck and H. Matsumura; in particular, for a semi-local ring A, 2 means the radical adic completion of A. Let R be a property of a local noetherian ring and P the following property (associated to R) of a noetherian algebra A over a field k IS, 7.3 I: “The k-algebra A satisfies P if and on1.v $ for any finite extension k’ of k. the local rings of A mj), k’ sati@ the property R”. A ring homomorphism q~: A --f R is called a P-homomorphism if it is flat and if, for any prime ideal p of A, the k(p)-algebra B 0, k(p) satisfies P: by a P-ring, we mean a ring A such that, for any prime ideal p of A, the canonical homomorphism A p ---) a p is a P-homomorphism. This paper is devoted, for a nice property R, to the following problems asked by A. Grothendieck (8, 7.4, 7.5 I:
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