Regular Analytic Independence and Extensions of Analytic Spread

Diagana, Youssouf M.

doi:10.1081/agb-120003986

Cited by 4 publications

(3 citation statements)

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“…[4] that λ(I ) Ϲ dim A − β where λ(I ) is the analytic spread of the ideal I and β is the asymptotic value of the sequence (depth A ( A/I n )). Since f is a noetherian filtration, there exists an integer m such that I nm = I n m for all n. Moreover we know from [6] that λ( f ) = λ(I m ) = λ(I n m ) for all n. Then the conclusion is obvious from Corollary 10. E x a m p l e. Let A = k[[X 3 , X 4 , X 5 ]] be the formal powers series on the field k and P be the prime ideal defining the monomial curve X 3 = t 3 , X 4 = t 4 , X 5 = t 5 .…”

Section: Theorem 7 Let (A M ) Be a Local Noetherian Ring J An Ideamentioning

confidence: 55%

Quasi-polynomials associated with Bass and Betti numbers of filtered modules

Dichi

2001

Arch. Math.

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It is shown that if M is a finite module on a local noetherian ring A which is filtered by an f -good filtration Φ = (M n ) where f is a noetherian filtration on A, then the i-th Betti and the i-th Bass numbers of the modules (M n ) and (M/M n ) define quasipolynomial functions whose period does not depend on i but only of the Rees ring of f . It is proved that the projective and injective dimension of the modules M/M n are perodic for large n. In the particular case where f is a good filtration or a strongly AP filtration it is shown that the projective and injective dimension as well as the depth stabilize. As an application, using a result proved by Brodmann, we give an upper bound of the analytic spread of f = (I n ) in terms of the limes inferior of depth(A/I n ). Introduction. Let( A, M ) be a local noetherian ring, I an ideal of A and M a finite A-module. We consider the graded homogeneous ring G I ( A) = n∈N I n /I n+1 and the finite graded G I ( A)-module G I (M) = n∈N I n M/I n+1 M associated with the pair (I, M). It is well known that for each integer i, the i-th Betti number and the i-th Bass number of the modules I n M/I n+1 M and M/I n M are polynomial functions in n with degree at most the Krull dimension of G I (M) ⊗ A/M minus 1 or at most the Krull dimension of G I (M) ⊗ A/Mrespectively. Moreover, a classical result asserts, that the projective and injective dimension, as well as the depth of the module M/I n M stabilize for large n. This paper generalizes the above results to the following situation: let f = (I n ) be a noetherian filtration of A that is a decreasing sequence of ideals of A which satisfies I 0 = A and I n I m ֤ I n+m for all n, m such that its Rees ring R( f ) =

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Section: Theorem 7 Let (A M ) Be a Local Noetherian Ring J An Ideamentioning

confidence: 55%

Quasi-polynomials associated with Bass and Betti numbers of filtered modules

Dichi

2001

Arch. Math.

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“…Under these assumptions, we have the following Theorem as in [3]: Let R be a ring and A be a subring of R. Let a 1 , . .…”

Section: Other Properties Of Independencementioning

confidence: 99%

Quasi-bigraduations of modules, slow analytic independence

Diagana¹

2018

imf

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When f = I (m,n) (m,n)∈(Z 2)∪{∞} is a + quasi-bigraduation on a ring R, a f +quasi-bigraduation of an R−module M is a family g = (G (m,n)) (m,n)∈(Z 2)∪{∞} of subgroups of M such that G ∞ = (0) and I (m,n) G (p,q) ⊆ G (m+p,n+q) , for all (m, n) and (p, q) ∈ (N×N) ∪ {∞}. We will show that r elements of R are slowly J−independent of order k with respect to a + quasi-bigraduation g on an R−module M if and only if the two property which follow hold: they are J−independent of order k with respect to the + quasi-bigraduation f 2 (A, I) and there exists a relation of compatibility between the + quasi-graduation on R deduced from g and g I where I is the sub-A−module of R constructed by these elements. We give criteria of slow J−independence of + quasi-bigraduations on an R−module in term of isomorphisms of graded algebras.

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“…Then the following numbers are known in the literature to be extensions to filtrations of the analytic spread, the last one being due to Y.M. Diagana [3] : -the maximum number J (f, k) of elements of the ideal J which are J−independent of order k with respect to f and -the maximum number a J (f, k) of elements of the ideal J which are regularly J−independent of order k with respect to f.…”

Section: Introductionmentioning

confidence: 99%

Quasi-bigraduations of rings, criteria of generalized analytic independence

Diagana¹

2018

ija

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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.

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Regular Analytic Independence and Extensions of Analytic Spread

Cited by 4 publications

References 4 publications

Quasi-polynomials associated with Bass and Betti numbers of filtered modules

Quasi-polynomials associated with Bass and Betti numbers of filtered modules

Quasi-bigraduations of modules, slow analytic independence

Quasi-bigraduations of rings, criteria of generalized analytic independence

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