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Stanley depth of monomial ideals with small number of generators

100%
Cimpoeaş M.
Open Mathematics
|
2009
|
tom 7
|
nr 4
629-634
EN
For a monomial ideal I ⊂ S = K[x 1...,x n], we show that sdepth(S/I) ≥ n − g(I), where g(I) is the number of the minimal monomial generators of I. If I =νI′, where ν ∈ S is a monomial, then we see that sdepth(S/I) = sdepth(S/I′). We prove that if I is a monomial ideal I ⊂ S minimally generated by three monomials, then I and S/I satisfy the Stanley conjecture. Given a saturated monomial ideal I ⊂ K[x 1,x 2,x 3] we show that sdepth(I) = 2. As a consequence, sdepth(I) ≥ sdepth(K[x 1,x 2,x 3]//I) +1 for any monomial ideal in I ⊂ K[x 1,x 2,x 3].
2
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One-fibered ideals in 2-dimensional rational singularities that can be desingularized by blowing up the unique maximal ideal

76%
Lierde V.
Open Mathematics
|
2011
|
tom 9
|
nr 6
1349-1353
EN
Let (R;m) be a 2-dimensional rational singularity with algebraically closed residue field and whose associated graded ring is an integrally closed domain. Göhner has shown that for every prime divisor v of R, there exists a unique one-fibered complete m-primary ideal A v in R with unique Rees valuation v and such that any complete m-primary ideal with unique Rees valuation v, is a power of A v. We show that for v ≠ ordR, A v is the inverse transform of a simple complete ideal in an immediate quadratic transform of R, if and only if the degree coefficient d(A v; v) is 1. We then give a criterion for R to be regular.
3
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A procedure to compute prime filtration

64%
Rauf A.
Open Mathematics
|
2010
|
tom 8
|
nr 1
26-31
EN
Let K be a field, S = K[x 1, … x n] be a polynomial ring in n variables over K and I ⊂ S be an ideal. We give a procedure to compute a prime filtration of S/I. We proceed as in the classical case by constructing an ascending chain of ideals of S starting from I and ending at S. The procedure of this paper is developed and has been implemented in the computer algebra system Singular.
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