Local Bezout estimates and multiplicities of parameter and primary ideals

Boďa, Eduard; Schenzel, Peter

doi:10.1016/j.jalgebra.2017.06.011

Cited by 5 publications

(7 citation statements)

References 8 publications

(19 reference statements)

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“…Proof. Since c·e 0 (q; M ) ≤ e(a; M ) (see [2]). Therefore claim in (1) follows from previous theorem 5.1.…”

Section: Applicationsmentioning

confidence: 98%

“…where t denotes the number of common tangents to f, g at origin when counted with multiplicities. Note that ℓ A ([f ⋆ B : B g ⋆ /f ⋆ B] n ) = t for all n ≫ 0 (see [2]).…”

Section: Applicationsmentioning

confidence: 99%

“…, t as in Lemma 4.2, there is an isomorphism L d−t (a, q, M ; n) ∼ = ∩ d i=1 (b, q n+β−c i )M : M a i /(b, q n+β−c )M. We refer Section 3 and 4 for the detail discussion about above theorem. In [2] and [8], authors posed the problem to study the Euler characteristic χ A (a, q, M ) of the complex K • (a, q, M ; n), see def. 2.3, independently of its value which is equal to e 0 (a; M ) − c 1 · .…”

Section: Introductionmentioning

confidence: 99%

“…· c d · e 0 (q; M ) ≥ 0 for n ≫ 0, cf. [2]. In section 6, we discuss a few properties of this Euler characteristic.…”

Section: Introductionmentioning

confidence: 99%

“…First Bydzovský [5] and most recently Bo da-Schenzel [2] presented an improvement of the local Bézout inequality. More precisely,…”

Section: Introductionmentioning

confidence: 99%

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On Regular Sequences in the Form Module with Applications to Local Bézout Inequalities

Khadam

2018

Bull. Iran. Math. Soc.

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Let q denote an ideal in a Noetherian local ring (A, m). Let a = a1, . . . , a d ⊂ q denote a system of parameters in a finitely generated A-module M . This note investigate an improvement of the inequality c1 · . . . · c d · e0(q; M ) ≤ ℓA(M/a M ), where ci denote the initial degrees of ai in the form ring GA(q). To this end, there is an investigation of regular sequences in the form module GM (q) by homology of a factor complex of the Koszul complex. In a particular case, there is a discussion of classical local Bézout inequality in the affinewhere µ(P ; C, D) denotes the local intersection multiplicity of P in C ∩ D. In a particular case when P is the origin, it follows that µ(P ; C, D) = ℓ A (A/(f, g)A), where A = k[x, y] (x,y) and f, g denote the equations in A. Note that ℓ A (A/(f, g)A) = e 0 (f, g; A) as A is a regular local ring.Since C, D have no component in common, {f, g} forms a system of parameters in A. Then e 0 (f, g; A) ≥ c · d · e 0 (m; A) = c · d, since e 0 (m; A) = 1, called the local Bézout inequality in the affine plane A 2 k . Here c, d denote the initial degree of f, g respectively. This estimate is well-known (see for instance [3] or [6]) and proved by resultants or Puiseux expansions. Moreover, equality holds if and only if C, D intersect transversally at the origin. In other words f ⋆ , g ⋆ , the initial forms of f, g in the form ring G A (m) ∼ = k[X, Y ], is a homogeneous system of parameters.First Bydzovský [5] and most recently Boda-Schenzel [2] presented an improvement of the local Bézout inequality. More precisely,where t is the number of common tangents of f, g at origin when counted with multiplicities. We generalized their result to an arbitrary situation. To this end, let q denote an ideal in a Noetherian local ring (A, m, k) such that ℓ A (M/qM ) is finite for a finitely generated A-module M . Let a = a 1 , . . . , a d ⊆ q denote a system of parameters of M such that a i ∈ q c i \ q c i +1 , c i > 0, for i = 1, . . . , d. Then we have the following result.2010 Mathematics Subject Classification. Primary: 13H15; Secondary: 13D40.

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“…Proof. Since c·e 0 (q; M ) ≤ e(a; M ) (see [2]). Therefore claim in (1) follows from previous theorem 5.1.…”

Section: Applicationsmentioning

confidence: 98%

“…where t denotes the number of common tangents to f, g at origin when counted with multiplicities. Note that ℓ A ([f ⋆ B : B g ⋆ /f ⋆ B] n ) = t for all n ≫ 0 (see [2]).…”

Section: Applicationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…· c d · e 0 (q; M ) ≥ 0 for n ≫ 0, cf. [2]. In section 6, we discuss a few properties of this Euler characteristic.…”

Section: Introductionmentioning

confidence: 99%

“…First Bydzovský [5] and most recently Bo da-Schenzel [2] presented an improvement of the local Bézout inequality. More precisely,…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations