On a Theorem of B. Teissier on Multiplicities of Ideals in Local Rings

Rees, D. Andrew S.; Sharp, Rodney Y.

doi:10.1112/jlms/s2-18.3.449

Cited by 66 publications

(76 citation statements)

References 3 publications

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“…We shall indicate the required reduction as painlessly as possible (by assuming familiarity with [8,9 and 10]). and J defined in [7]. As Teissier shows in [10], the proposition reduces to proving the following assertion.…”

Section: Corollary 12 (See [5 Theorem 23 and 7 Theorem 43]) Letmentioning

confidence: 75%

“…If eo = • • • = ed, then I = J. The proof now proceeds by induction on d, the case d = 2 being proved in [7]. We indicate how the result follows from the d -1 case in several steps.…”

Section: Corollary 12 (See [5 Theorem 23 and 7 Theorem 43]) Letmentioning

confidence: 99%

“…The first is to involve the so-called mixed multiplicities of / and J and the second is to provide a reduction procedure (really a strengthened superficial element argument) whereby the theorem can be reduced to the two dimensional case. In [7] Rees and Sharp give a purely algebraic proof of Teissier's theorem showing that (1) holds for any two dimensional local ring and (2) holds for any two dimensional quasi-unmixed local ring. They observe that Teissier's reduction carries over for (1) so that the inequality remains valid for any local ring.…”

Section: Corollary 12 (See [5 Theorem 23 and 7 Theorem 43]) Letmentioning

confidence: 99%

“…The second is a classical theorem of Rees which guarantees that if J Ç J are M-primary ideals in a quasi-unmixed local ring then I reduces J if and only if / and J have the same multiplicity. In a brief second section we complete the generalization of a theorem of B. Teissier concerning multiplicities begun by Rees and Sharp in [7].…”

Section: Introductionmentioning

confidence: 99%

“…As corollaries we quickly deduce two theorems of D. Rees concerning multiplicity. The first of these, implicit in [5] and proven in [7] under unnecessarily restrictive hypotheses, asserts that if say R is a domain, then there exist finitely many integers <¿¿ > 0 and discrete valuations Vi on the quotient field of R such that the multiplicity of / equals ^2diVi(I). The second is a classical theorem of Rees which guarantees that if J Ç J are M-primary ideals in a quasi-unmixed local ring then I reduces J if and only if / and J have the same multiplicity.…”

Section: Introductionmentioning

confidence: 99%

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Note on multiplicity

Katz¹

1988

Proc. Amer. Math. Soc.

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ABSTRACT. Let (R, M) be a local ring with infinite residue field and / = (xi,... ,Xd)R an ideal generated by a system of parameters. It is shown that the multiplicity of / equals the multiplicity of IT where and R = R/(0: x%),N\axge. Introduction.Let (R, M) be a local ring with infinite residue field. A device commonly employed in studying the multiplicity of an M-primary ideal is to go mod a superficial element. The effect is to reduce the dimension of the ring yet preserve the multiplicity. This technique is particularly useful in proving theorems about multiplicity by induction on the dimension of R. There are however, occasions when this process can be ineffective-certain properties do not lift from homomorphic images. Here we show that this difficulty can sometimes be circumvented by preserving the multiplicity upon passing to subrings of the total quotient ring of R having smaller dimension. To wit, we prove that if / = (xi,..., x<¡)R is an ideal generated by a system of parameters, then the multiplicity of / is the same

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Section: Corollary 12 (See [5 Theorem 23 and 7 Theorem 43]) Letmentioning

confidence: 75%

“…If eo = • • • = ed, then I = J. The proof now proceeds by induction on d, the case d = 2 being proved in [7]. We indicate how the result follows from the d -1 case in several steps.…”

Section: Corollary 12 (See [5 Theorem 23 and 7 Theorem 43]) Letmentioning

confidence: 99%