New approaches to bounding the multiplicity of an ideal

Francisco, Christopher A.

doi:10.1016/j.jalgebra.2005.06.004

Cited by 9 publications

(14 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In recent years, this conjecture has attracted attention from commutative algebra and combinatorics (see for example [10], [12], [14], [15], [18], [19], [20], [21], [22], [23]). Here we provide another link to combinatorics and add a large class of rings for which the conjecture holds.…”

Section: Introductionmentioning

confidence: 99%

The Multiplicity Conjecture for Barycentric Subdivisions

Kubitzke

Welker

2008

Communications in Algebra

View full text Add to dashboard Cite

Abstract. For a simplicial complex ∆ we study the effect of barycentric subdivision on ring theoretic invariants of its Stanley-Reisner ring. In particular, for Stanley-Reisner rings of barycentric subdivisions we verify a conjecture by Huneke and Herzog & Srinivasan, that relates the multiplicity of a standard graded k-algebra to the product of the maximal and minimal shifts in its minimal free resolution up to the height. On the way to proving the conjecture we develop new and list well known results on behavior of dimension, Hilbert series, multiplicity, local cohomology, depth and regularity when passing from the Stanley-Reisner ring of ∆ to the one of its barycentric subdivision.

show abstract

Section: Introductionmentioning

confidence: 99%

The Multiplicity Conjecture for Barycentric Subdivisions

Kubitzke

Welker

2008

Communications in Algebra

View full text Add to dashboard Cite

show abstract

“…We just remark here that Francisco [3], who already studied some cases of the Multiplicity Conjecture by looking at possible numerical cancelations among the Betti numbers, suggested an approach (from which he obtained some interesting results) to perform cancelations also when -like in the above cases -there could be more than one way to make them (basically, his choice was to give priority to the rightmost cancelation). However, Francisco's technique and results (which go in a different direction) will not be employed nor further discussed in this paper.…”

Section: New Conjectural Bounds For Codimension 3 Level Algebrasmentioning

confidence: 99%

“…We refer the reader to the recent works [3] and [4] for a comprehensive history of all the main results obtained to date on the MC.…”

Section: Introductionmentioning

confidence: 99%

Improving the bounds of the multiplicity conjecture: The codimension 3 level case

Zanello

2007

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

The Multiplicity Conjecture (MC) of Huneke and Srinivasan provides upper and lower bounds for the multiplicity of a Cohen-Macaulay algebra A in terms of the shifts appearing in the modules of the minimal free resolution (MFR) of A. All the examples studied so far have lead to conjecture (see [J. Herzog, X. Zheng, Notes on the multiplicity conjecture. Collect. Math. 57 (2006) 211-226] and [J. Migliore, U. Nagel, T. Römer, Extensions of the multiplicity conjecture, Trans. Amer. Math. Soc. (preprint: math.AC/0505229) (in press)]) that, moreover, the bounds of the MC are sharp if and only if A has a pure MFR. Therefore, it seems a reasonable -and useful -idea to seek better, if possibly ad hoc, bounds for particular classes of Cohen-Macaulay algebras.In this work we will only consider the codimension 3 case. In the first part we will stick to the bounds of the MC, and show that they hold for those algebras whose h-vector is that of a compressed algebra.In the second part, we will (mainly) focus on the level case: we will construct new conjectural upper and lower bounds for the multiplicity of a codimension 3 level algebra A, which can be expressed exclusively in terms of the h-vector of A, and which are better than (or equal to) those provided by the MC. Also, our bounds can be sharp even when the MFR of A is not pure.

show abstract

“…For extensions of this conjecture see [15,16,18,23,25]. For some new approaches to this problem see [1,5,7,15,16] . In our proof we show that in the class of ideals of Theorem 1.3 we have that the limit on the left hand side is < 1.…”

Section: Conjecture 12 E(a/i) ≤ U (I)mentioning

confidence: 99%

On the upper bound of the multiplicity conjecture

Puthenpurakal

2008

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. Let A = K[X 1 , . . . , X n ] and let I be a graded ideal in A. We show that the upper bound of the multiplicity conjecture of Herzog, Huneke and Srinivasan holds asymptotically (i.e., for I k and all k 0) if I belongs to any of the following large classes of ideals:(1) radical ideals, (2) monomial ideals with generators in different degrees, (3) zero-dimensional ideals with generators in different degrees. Surprisingly, our proof uses local techniques like analyticity, reductions, equimultiplicity and local results like Rees's theorem on multiplicities.

show abstract

New approaches to bounding the multiplicity of an ideal

Cited by 9 publications

References 24 publications

The Multiplicity Conjecture for Barycentric Subdivisions

The Multiplicity Conjecture for Barycentric Subdivisions

Improving the bounds of the multiplicity conjecture: The codimension 3 level case

On the upper bound of the multiplicity conjecture

Contact Info

Product

Resources

About