Quasi-bigraduations of rings, criteria of generalized analytic independence

Abstract

A + quasi-bigraduation of a commutative ring R is a family g = (G n,m) (n,m)∈(Z×Z)∪{+∞} of subgroups of R such that A = G 0,0 is a subring of R, G ∞ = (0) and G n,p G m,q ⊆ G n+m,p+q , for all (n, p), (m, q) ∈ (N × N). Here we show that r elements of R are J−independent of order k with respect to a + quasi-bigraduation g if and only if the two property which follow hold: they are J−independent of order k and there exists a relation of compatibility between g and g I where I is the sub-A−module of R constructed by these elements. The concept of J−independence will be extended to + quasi-bigraduations and we give criteria of J−independence in term of isomorphisms of graded algebras.

Quasi-bigraduations of modules, slow analytic independence

“…Properties of independence.Under the previous hypotheses we show the following results as in[6]: Under the notations and hypotheses of 2.2.3 and with the assumption that G k ∩ A ⊆ J + I k ∩ A the following assertions are equivalent : (i) a 1 , . .…”

“…In [6] we studied the notion of generalized analytic independence for a + quasi-bigraduation of a ring R.…”