Annales Mathématiques Africaines

Filtrations, analytic spread, and integral dependence

Dramane Bio SALIFOU
107 31 Ave NE, Calgary, Alberta, T2E 2E8, Canada

Mathematics Subject Classification: (2010) 13A30, 13E05
Key words: nœtherian filtration, Rees ring, dimension, integral dependence, analytic spread.

Abstract:

Given a positive integer k which may be infinite, an ideal J of a commutative ring A and a filtration f of A, the purpose of this note is firstly, to prove that the (Krull) dimension of the generalized Rees quotient ring , where ℜ(A,f) is the generalized Rees quotient ring of the filtration f, and that is denoted by λ_J(f,k), extends the usual Analytic Spread of a nœtherian filtration f of a nœtherian ring A. Secondly, if g is another filtration of the nœtherian ring A which is integral over f, then the equality λ_J(g,k) = λ_J(f,k) holds.