Note on multiplicity

Abstract

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

The Minkowski equality and inequality for multiplicity of ideals

“…The proof of Minkowski equality dimension 3 or higher was reduced to dimension 2 by D. Katz [15] by passing to certain subrings of the total quotient ring of R having smaller dimension than dim R. In this section we present an alternative proof using the specialization property of the integral closure of an ideal first proved by Shiroh Itoh [14, Theorem 1] if R is Cohen-Macaulay of dimension d ≥ 2 and it is analytically unramified. Hong and Ulrich [11] provided another proof for analytically unramified universally catenary local rings.…”

Hilbert functions of multigraded algebras, mixed multi- plicities of ideals and their applications

“…Rees and Sharp [48] investigated these conjectures for all local rings and proved: The converse was proved by Teissier [61] for Cohen-Macaulay normal complex analytic algebras by using mixed multiplicities. Then Katz [33] showed that in quasi-unmixed local rings, Minkowski equalities hold for m-primary ideals I and J if and only if they are projectively equivalent.…”

The Minkowski Equality for Filtrations

“…The development of the subject of mixed multiplicities and its connection to Teissier's work on equisingularity [37] can be found in [18]. A survey of the theory of mixed multiplicities of ideals can be found in [36,Chapter 17], including discussion of the results of the papers [31] of Rees and [35] of Swanson, and the theory of Minkowski inequalities of Teissier [37], [38], Rees and Sharp [34] and Katz [21]. Later, Katz and Verma [22], generalized mixed multiplicities to ideals that are not all m R -primary.…”

“…There is a beautiful characterization of when equality holds in the Minkowski inequality (7) by Teissier [39] (for Cohen-Macaulay normal two-dimensional complex analytic R), Rees and Sharp [34] (in dimension 2) and Katz [21] (in complete generality).…”

“…In this case, we have the (Noetherian) m R -filtrations I(1) = {I(1) i } and I(2) = {I(2) i }. Since the Minkowski equality holds, we have that there exists m, n ∈ Z + such that I(1) m = I(2) n where I(1) m and I(2) n are the respective integral closures of ideals by the Teissier, Rees and Sharp, Katz Theorem[39],[34],[21] recalled in Subsection 1.3. Now e(I(1) m ) = m d e(I(1)) = m d e(I(1)), e(I(2) n ) = n d e(I(2)) = n d e(I(2)) and mγ µ (I(1)) = mµ(I(1)) = µ(I(1) m ), nγ µ (I(2)) = nµ(I(2)) = µ(I(2) n ), giving the desired formula e(I(2)) γ µ (I(1)) = e(I(1)) γ µ (I(2)).…”