On the number of Elements Independent with respect to an Ideal

Bruns, Winfried

doi:10.1112/jlms/s2-22.1.57

Cited by 12 publications

(4 citation statements)

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“…As an application we obtain an upper bound for sup 00 /. Originally this was shown by Bruns [2,Proposition 3]. For the sake of completeness we include the proof.…”

Section: The Construction Of Independent Elementsmentioning

confidence: 85%

“…They demonstrate that it is in general impossible to compute sup / from invariants which do not change if / is replaced by a power of itself, see also Section 5. This phenomenon led Bruns [2] to define the asymptotic stabilization sup 00 / = min {sup I n :n ^ 1} of /. Furthermore, Ratliff [6] studied sup~ / = min {sup (I n ) a : n ^ 1}, where I a denotes the integral closure of an ideal /of/?, that is, I a is the ideal of R consisting of all elements xeR satisfying an equation x m + c l x m~l +... + c m -0, where c^e/*, for / = l,...,m. Formulae for sup 00 /and sup£°/ were given by Bruns [2,Theorem 2] and Ratliff [6, (2.12.2)] respectively (for another approach see also [10]).…”

Section: Introductionmentioning

confidence: 99%

“…The third section contains the main idea for constructing independent elements. This is inspired by [2]. Asymptotic formulae, among them those of (1.1), are shown in Section 4.…”

Section: Introductionmentioning

confidence: 99%

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The Construction of Ideal Independent Elements

Schenzel

1985

Journal of the London Mathematical Society

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“…As an application we obtain an upper bound for sup 00 /. Originally this was shown by Bruns [2,Proposition 3]. For the sake of completeness we include the proof.…”

Section: The Construction Of Independent Elementsmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Construction of Ideal Independent Elements

Schenzel

1985

Journal of the London Mathematical Society

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“…Conversely, suppose that (2) holds, and let R and B be as in (3). Now B satisfies Sn as an i?-module, and by the remarks above, rankft B < n. Let C = B/R and consider the exact sequence O^R^B^C^O.…”

Section: ])mentioning

confidence: 99%

On a theorem of Huneke concerning multiplicities

O’Carroll¹

1987

Proc. Amer. Math. Soc.

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ABSTRACT.We consider a theorem by Huneke on multiplicities, and show that the extension surmised by Huneke is a stronger form of the Syzygy Conjecture.Craig Huneke has proved the following result.THEOREM 1 [9]. Let (A,m) be a complete local ring containing afield, letn be a positive integer and let e(A) be the multiplicity of A. Suppose that A satisfies Serre 's condition Sn and that e(A) inf (n, height P) for all P G Spec A.As remarked on [11, p. 125], the condition Si is equivalent to A having no embedded associated primes. We use the term 'equidimensional' in the sense of Grothendieck's E.G.A., i.e. a local ring is said to be equidimensional when all its minimal primes have the same co-height. Recall from [7, (5.10.9)] that a catenary (and so a complete) local ring which satisfies S2 is equidimensional.A recent handy reference for much of the material in this note is the monograph [5], where a special case of Huneke's result is discussed on pp. 71-72 and on p. 75. As the latter shows, and as has been remarked by Huneke himself, in the case n = 1 it is essential in Theorem 1 to add the hypothesis of equidimensionality.From his theorem Huneke deduces results by Nagata [12, (40.6)] and by Ikeda, in the case where the rings involved contain a field. (As regards the former result, see the interesting aside on [6, p. 41].) Since these are known to hold without this restriction, Huneke surmises that the following is probably true:, .Theorem 1 is valid without the assumption that Â ' contain a field. if R is a regular local ring, and if K is the (2) jth syzygy of an R-module with rank A" < j, then K is a free R-module.

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Asymptotic Sequences and Prime Divisors of the Completion of Local Rings

Schenzel

1988

Mathematische Nachrichten

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On the number of Elements Independent with respect to an Ideal

Cited by 12 publications

References 13 publications

The Construction of Ideal Independent Elements

The Construction of Ideal Independent Elements

On a theorem of Huneke concerning multiplicities

Asymptotic Sequences and Prime Divisors of the Completion of Local Rings

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