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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.


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 (ukℜ,J()Aℜ,f()A,f), where (A,f) is the generalized Rees quotient ring of the filtration f, and that is denoted by λJ(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 λJ(g,k) = λJ(f,k) holds.

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