Multiplicities and mixed multiplicities of arbitrary filtrations

Mixed multiplicities of divisorial filtrations

Let $(R,\m,k)$ be a local (Noetherian) ring of positive prime characteristic $p$ and dimension $d$. Let $G_\dt$ be a minimal resolution of the residue field $k$, and for each $i\ge 0$, let $\gothic t_i(R) = \lim_{e\to \8} {\length(H_i(F^e(G_\dt)))}/{p^{ed}}$. We show that if $\gothic t_i(R) = 0$ for some $i>0$, then $R$ is a regular local ring. Using the same method, we are also able to show that if $R$ is an excellent local domain and $\Tor_i^R(k,R^+) = 0$ for some $i>0$, then $R$ is regular (where $R^+$ is the absolute integral closure of $R$). Both of the two results were previously known only for $i = 1$ or $2$ via completely different methods.

The Minkowski equality of filtrations

Completions of uncountable local rings with countable spectra

This is essentially an expository paper about m-primary ideals I in a local ring (R, m) which have a single exceptional prime, i. e., whose closed fiber in the normalization \(\overline {R\left( I \right)} \) of the blow-up R(I) is irreducible. Stated slightly differently, it is about m-primary ideals which have a single associated Rees valuation. If R is analytically unramified then the existence of such an ideal in R implies that R is analytically irreducible. It is conjectured that, conversely, every analytically irreducible local domain has such an ideal.

The local ring of a point on a curve is a one-dimensional local Cohen-Macaulay ring; in this chapter we study this class of rings. After proving some results on transversal elements in section 1, our main interest in section 2 is the integral closure of a one-dimensional local Cohen-Macaulay ring; we use Manis valuations in describing the integral closure. In section 3 we give necessary and sufficient conditions in order to ensure that the completion of a one-dimensional local Cohen-Macaulay ring which is a domain (resp. has no nilpotent elements) again is a domain (resp. has no nilpotent elements). Here the reader is supposed to be acquainted with the notion of the completion of a local ring and its properties.KeywordsPrime IdealLocal RingMaximal IdealValuation RingRegular ElementThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let Q be an analytically unramified Cohen-Macaulay local ring of dimension 2, with maximal ideal m and infinite residue field k. If a is an m-primary ideal of Q, a∗ will denote its integral closure, λ(a) = l(Q/a∗), e(a) will denote its multiplicity, θ(a) = 2. λ(a)−e(a) and θ ¯ (a) = lim θ(an)/n. This paper uses explicit formulae for θ(ar), θ(arbs) related to earlier results of Narita to prove that θ ¯ (ab) = θ ¯ (a) + θ ¯ (b), and that the identity θ(ab) = θ(a) + θ(b) holding for all m-primary ideals a, b characterises the pseudo-rational local rings of J. Lipman among normal Q. This is used to prove a number of results concerning the normal genus of an ideal and pseudo-rational local rings. In the last section, an expression for θ(a) is obtained in the case where Q is regular which is related to the theory of infinitely near points of D. G. Northcott.

In this article, we extend the notion of multiplicity for weakly graded families of ideals which are bounded below linearly. In particular, we show that the limit exists where is a bounded below linearly weakly graded family of ideals in a Noetherian local ring of dimension with . Furthermore, we prove that “volume = multiplicity” formula and Minkowski inequality hold for such families of ideals. We explore some properties of for weakly graded families of ideals of the form where is an ‐primary graded family of ideals. We provide a necessary and sufficient condition for the equality in Minkowski inequality for the weakly graded families of ideals of the form where is a bounded filtration. Moreover, we generalize a result of Rees characterizing the inclusion of ideals with the same multiplicities for the above families of ideals. Finally, we investigate the asymptotic behavior of the length function where is a filtration of ideals (not necessarily ‐primary).

Dualizing complexes and systems of parameters

Bounds on Annihilator Lengths in Families of Quotients of Noetherian Rings

The first section of this paper is devoted to the definition, and proof of the existence, of what are called complete and joint reductions of a set of ideals of a d-dimensional local ring Q with maximal ideal m and residue field k. The former is defined for a set of ideals a1,…, as of Q, and consists of a set of sd elements xij (i = l,…,s;j = 1, …,d) where xij ⊂ ai and the elements yj = xljx2j… xsj(j = 1,…, d) form a reduction of a1 a2… as. The definition of a joint reduction applies to a set of, not necessarily distinct, ideals a1,…, ad, and d s a set of elements xi (i = 1,…, d) such that ∑ d i = 1 xia1…ai=1ai+1…ad is a reduction of a1a2…ad. The second section commences with a treatment of Hilbert functions of several m-primary ideals based on deas of Buchsbaum and Auslander [1, corrections], and sketched in [8]. This is used to prove that the multiplicity of a joint reduction of d m-primary ideals a1,…,ad depends only on a1,…,ad and is a mixed multiplicity as defined by Teissier in [9]. The final part of Section 2 is devoted to a proof of the following result. Suppose that Q is analytically unramified, and that L(Q / (an)′), where (an)′ is the integral closure of an, with a m-primary, is equal to the polynomial (e(a)nd/d!−½(f(a)nd−1/(d−1)!)+… for large n. Then f(a) is a homogeneous polynomial over the set of m-primary ideals of Q in the sense that f ( a 1 r 1 ⋯ a s r s ) can be expressed as a homogeneous polynomial of degree d − 1 for r1,…, rs ⩾ 0(and not merely for r1,…, rs all positive). This extends a result proved in the case d = 2 in [7] to general d.

Given any local Noetherian ring (R, 𝔪), we study invariants, such as the dimension and multiplicity, of the Sally module S J (I) of any 𝔪-primary ideal I with respect to a minimal reduction J. As a by-product we obtain an estimate for the Hilbert coefficients of 𝔪 that generalizes a bound established by Elias and Valla in a local Cohen–Macaulay setting. We also find sharp estimates for the multiplicity of the special fiber ring ℱ(I) of I, which recover previous bounds established by Polini, Vasconcelos, and the author in the local Cohen–Macaulay case. A particular attention is also paid to Sally modules in local Buchsbaum rings.

Suppose that $R$ is a 2 dimensional excellent local domain with quotient field $K$, $K^*$ is a finite separable extension of $K$ and $S$ is a 2 dimensional local domain with quotient field $K^*$ such that $S$ dominates $R$. Suppose that $\nu^*$ is a valuation of $K^*$ such that $\nu^*$ dominates $S$. Let $\nu$ be the restriction of $\nu^*$ to $K$. The associated graded ring ${\rm gr}_{\nu}(R)$ was introduced by Bernard Teissier. It plays an important role in local uniformization. We show that the extension $(K,\nu)\rightarrow (K^*,\nu^*)$ of valued fields is without defect if and only if there exist regular local rings $R_1$ and $S_1$ such that $R_1$ is a local ring of a blow up of $R$, $S_1$ is a local ring of a blowup of $S$, $\nu^*$ dominates $S_1$, $S_1$ dominates $R_1$ and the associated graded ring ${\rm gr}_{\nu^*}(S_1)$ is a finitely generated ${\rm gr}_{\nu}(R_1)$-algebra. We also investigate the role of splitting of the valuation $\nu$ in $K^*$ in finite generation of the extensions of associated graded rings along the valuation. We will say that $\nu$ does not split in $S$ if $\nu^*$ is the unique extension of $\nu$ to $K^*$ which dominates $S$. We show that if $R$ and $S$ are regular local rings, $\nu^*$ has rational rank 1 and is not discrete and ${\rm gr}_{\nu^*}(S)$ is a finitely generated ${\rm gr}_{\nu}(R)$-algebra, then $\nu$ does not split in $S$. We give examples showing that such a strong statement is not true when $\nu$ does not satisfy these assumptions. We deduce that if $\nu$ has rational rank 1 and is not discrete and if $R\rightarrow R'$ is a nontrivial sequence of quadratic transforms along $\nu$, then ${\rm gr}_{\nu}(R')$ is not a finitely generated ${\rm gr}_{\nu}(R)$-algebra.

We find necessary and sufficient conditions for a complete local (Noetherian) ring to be the completion of an uncountable local (Noetherian) domain with a countable spectrum. Our results suggest that uncountable local domains with countable spectra are more common than previously thought. We also characterize completions of uncountable excellent local domains with countable spectra assuming the completion contains the rationals, completions of uncountable local unique factorization domains with countable spectra, completions of uncountable noncatenary local domains with countable spectra, and completions of uncountable noncatenary local unique factorization domains with countable spectra.

Let (A, m) be a Noetherian local ring such that the residue field A/m is infinite. Let I be arbitrary ideal in A, and M a finitely generated A-module. We denote by ℓ(I, M) the Krull dimension of the graded module ⊕n≥0InM/mInM over the associated graded ring of I. Notice that ℓ(I, A) is just the analytic spread of I. In this paper, we define, for 0 ≤ i ≤ ℓ = ℓ(I, M), certain elements ei(I, M) in the Grothendieck group K0(A/I) that suitably generalize the notion of the coefficients of Hilbert polynomial for m-primary ideals. In particular, we show that the top term eℓ (I, M), which is denoted by eI(M), enjoys the same properties as the ordinary multiplicity of M with respect to an m-primary ideal.