Let I be a homogeneous ideal of a polynomial ring over a field, v(I) the number of elements of an... more Let I be a homogeneous ideal of a polynomial ring over a field, v(I) the number of elements of any minimal basis of I, e = e(I) the multiplicity or degree of R/I, h = h(I) the height or codimension of I, i = indeg (I) the initial degree of J, i.e. the minimal degree of non zero elements of I.This paper is mainly devoted to find bounds for v(I) when I ranges over large classes of ideals. For instance we get bounds when I ranges over the set of perfect ideals with preassigned codimension and multiplicity and when I ranges over the set of perfect ideals with preassigned codimension, multiplicity and initial degree. Moreover all the bounds are sharp since they are attained by suitable ideals. Now let us make some historical remarks.
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2006
Let R be a Cohen–Macaulay local ring, and let I ⊂ R be an ideal with minimal reduction J. In this... more Let R be a Cohen–Macaulay local ring, and let I ⊂ R be an ideal with minimal reduction J. In this paper we attach to the pair (I, J) a non-standard bigraded module ΣI, J. The study of the bigraded Hilbert function of ΣI, J allows us to prove an improved version of Wang's conjecture and a weak version of Sally's conjecture, both on the depth of the associated graded ring grI(R). The module ΣI, J can be considered as a refinement of the Sally module introduced previously by Vasconcelos.
Let I be a homogeneous ideal of a polynomial ring over a field, v(I) the number of elements of an... more Let I be a homogeneous ideal of a polynomial ring over a field, v(I) the number of elements of any minimal basis of I, e = e(I) the multiplicity or degree of R/I, h = h(I) the height or codimension of I, i = indeg (I) the initial degree of J, i.e. the minimal degree of non zero elements of I.This paper is mainly devoted to find bounds for v(I) when I ranges over large classes of ideals. For instance we get bounds when I ranges over the set of perfect ideals with preassigned codimension and multiplicity and when I ranges over the set of perfect ideals with preassigned codimension, multiplicity and initial degree. Moreover all the bounds are sharp since they are attained by suitable ideals. Now let us make some historical remarks.
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2006
Let R be a Cohen–Macaulay local ring, and let I ⊂ R be an ideal with minimal reduction J. In this... more Let R be a Cohen–Macaulay local ring, and let I ⊂ R be an ideal with minimal reduction J. In this paper we attach to the pair (I, J) a non-standard bigraded module ΣI, J. The study of the bigraded Hilbert function of ΣI, J allows us to prove an improved version of Wang's conjecture and a weak version of Sally's conjecture, both on the depth of the associated graded ring grI(R). The module ΣI, J can be considered as a refinement of the Sally module introduced previously by Vasconcelos.
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