Variation of Hilbert Coefficients

For a Noetherian local ring $(\RR, \m)$, the first two Hilbert coefficients, $e_0$ and $e_1$, of the $I$-adic filtration of an $\m$-primary ideal $I$ are known to code for properties of $\RR$, of the blowup of $\spec(\RR)$ along $V(I)$, and even of their normalizations. We give estimations for these coefficients when $I$ is enlarged (in the case of $e_1$ in the same integral closure class) for general Noetherian local rings.


Introduction
Let (R, m) be a Noetherian local ring of dimension d ≥ 1, and let I be an m-primary ideal. We will consider multiplicative, decreasing filtrations of R ideals, A = {I n | I 0 = R, I n+1 = II n , ∀n ≫ 0}, integral over the I-adic filtration, conveniently coded in the corresponding Rees algebra and its associated graded ring R(A) = n≥0 I n , gr A (R) = n≥0 I n /I n+1 .
Let R = n≥0 I n be the integral closure of the Rees algebra R = R(I) with I n = I n for all n ≥ 0, which we assume to be finite over R.
We will consider the Hilbert coefficients e i (I) associated to m-primary ideals I, for i = 0, 1. These integers play important roles in the corresponding blowup algebras. Some of these issues have a long tradition in the context of Cohen-Macaulay local rings, but others are of a recent vintage for general Noetherian local rings. From the several problem areas, we highlight the following: (i) The comparison between e 0 and e 1 ; (ii) e 1 and normalization; (iii) The structure of R associated to the values of e 1 ; (iv) Variation of e i , that is how e i (I) changes when I is enlarged.
We are concerned here with the last item but give brief comments on the others first.
AMS 2010 Mathematics Subject Classification. Primary 13A30; Secondary 13B22, 13H10, 13H15. The first author is partially supported by a grant from the City University of New York PSC-CUNY Research Award Program-41. The second author is partially supported by Grant-in-Aid for Scientific Researches (C) in Japan (19540054) and by a grant from MIMS (Meiji Institute for Advanced Study of Mathematical Sciences). The last author is partially supported by the NSF.
(i) For Cohen-Macaulay rings, an uniform bound for e 1 (I) first appeared for rings of dimension 1 in the work of D. Kirby ([K]), e 1 (m) ≤ e 0 (m) 2 .
Progressively, quadratic bounds of this type were developed for arbitrary m-primary ideals in all dimensions by several authors. As a basic source, [RV2] has a systematic development of these formulas along with a comprehensive bibliography. It also treats more general filtrations which were helpful to us. Among the formulas which more directly influenced the authors here, we single out two developed in the work of J. Elias ([E1,E2]) and M. E. Rossi and G. Valla ([RV1,RV2]).
For an d-dimensional Cohen-Macaulay local ring and an m-primary ideal I, the first bound asserts that if I is minimally generated by m = ν(I) elements, The other bound uses the m-adic order of I, that if I ⊂ m s and I = m s , then Recently, K. Hanumanthu and C. Huneke ([HH]) brought a new parameter to bear on these formulas with their proof that where k is the maximal length of chains of integrally closed ideals between I and m.
(ii) Since e 1 (I) ≤ e 1 (I) := e 1 (R), bounds with a different character arise. A baseline is the fact that when R is analytically unramified, but not necessarilly Cohen-Macaulay, one has e 1 (I) ≥ 0 ( [GHM]). An upper bound for e 1 (I) (see [PUV] for other bounds) is the following. Let (R, m) be a reduced Cohen-Macaulay local ring of dimension d, essentially of finite type over a perfect field, and let I be an m-primary ideal. Let δ be a regular element of the Jacobian ideal of R. Then where t is the Cohen-Macaulay type of R. In particular, if R is a regular local ring (iii) When R is not Cohen-Macaulay, the issues become less structured since the values of e 1 (I) may be negative. In fact, using the values of e 1 (I) for ideals generated by systems of parameters led to the characterization of several properties (Cohen-Macaulay, Buchsbaum, finite cohomology) of the ring R itself (see [GhGHOPV], [GhHV], [GO], [MV], [V2]).
(iv) We shall now outline the main results of this note.(We refer to [V1] for basic definitions and Rees algebras theory.) Sections 2 and 3 are organized around a list of questions about the changes that e 0 (I) and e 1 (I) undergo when I varies. An important case is e 0 (J), e 1 (J) −→ e 0 (I), e 1 (I), I = (J, x).
Clearly the optimal baseline is that of an ideal J generated by a system of parameters, but we will consider very general cases. As will be seen, some relationships involve the multiplicity f 0 (J) of the special fiber. To describe one of these estimates, let (R, m) be a Noetherian local ring of dimension d ≥ 1, let J be an m-primary ideal and let I = (J, h 1 , . . . , h m ) be integral over J of reduction number s = red J (I). Then Theorem 2.6 asserts that where f 0 (J) is the multiplicity of the special fiber of R(J) = n≥0 J n . We add a word of warning in reading some of the formulas with terms like e 1 (I) − e 1 (J). When J is a minimal reduction of I, e 1 (J) is always non-positive, according to [MSV], and vanishes when R is Cohen-Macaulay. In fact, for unmixed local rings the vanishing characterizes Cohen-Macaulayness ( [GhGHOPV]).
In Section 3, we address the need to link the value of red J (I) to other properties of J. This is a well-known fact when R is a Cohen-Macaulay ring, but we give a general formulation in Theorem 3.3: Let (R, m) be a Noetherian local ring of dimension d ≥ 1 and infinite residue field. For an m-primary ideal I and a minimal reduction J of I, there exists a minimal reduction Q of I such that red Q (I) ≤ max{d·λ(R/J) − 2d + 1, 0}.
2. Upper bounds for the variations of e 0 (I) and e 1 (I) In our calculations we make repeated use of the following elementary observation.
Since λ(M/L) = n − 1, the induction hypothesis shows Proof. For n ∈ N, consider the following filtration: Then we obtain For each r, M r /M r−1 is generated by the image of h r J n−r + M r−1 . Consider the natural surjection Using Lemma 2.1, we have It follows that The iterated Hilbert function Proof. Let s = red J (I). Then h s+1 ∈ JI s . For n ≥ s, we obtain the following filtration: Therefore is the difference of two polynomials of degree d and with same leading (binomial) coefficients e 0 (J) and e 0 (I), therefore it is at most a polynomial of degree d − 1 and leading coefficient e 1 (I) − e 1 (J). On the other hand, for n ≫ 0, we have which proves that e 1 (I) − e 1 (J) ≤ red J (I)·λ(R/(J : I))·f 0 (J). Moreover, if R is Gorenstein, then Proof. The first assertion follows from e 1 (Q) ≤ 0 [MSV] and f 0 (Q) = 1 for every parameter ideal Q. Suppose that R is Gorenstein. Then it is enough to show that This follows from Let R = k[x, y, z] (x,y,z) and let J and I be R-ideals such that > e 0 (J) − e 0 (I).
Let Q = (x a − z c , y b − z c , x α y β z γ ) and suppose that a > 3α, b > 3β, c > 3γ. Note that I = (Q, z c ). Then Q is a minimal reduction of I and the reduction number red Q (I) ≤ 2. We can estimate e 1 (I): e 1 (I) = e 1 (I) − e 1 (Q) ≤ 2λ(R/(Q : I)). where M r = (J n , HJ n−1 , . . . , H r−1 J n−r+1 , H r J n−r ). Note that M r /M r−1 is generated by the image of H r J n−r . More precisely, if I = (J, h 1 , . . . , h m ), then M r /M r−1 is generated by batches of elements, difficult to control. This filtration has been used by several authors when J is generated by a system of parameters. As ν(I n /J n ) is increasing, the method of iterating the assertion in Theorem 2.3 tends to induce a bigger upper bound for e 1 (I) − e 1 (J) than necessary. Instead, our formulation using the filtration above wraps it differently to accommodate our data. Proof. We have already given parts of the proof. The remaining part is to estimate the growth of the length of M r /M r−1 = [(h 1 , . . . , h m ) r J n−r + M r−1 ] /M r−1 . We note that this module is annihilated by J : I and is generated by the 'monomials' in the h i of degree r, with coefficients in J n−r . There is a natural surjection Proof. By tensoring the following exact sequence with R/m 0 −→ J n −→ I n −→ I n /J n −→ 0, we obtain J n /mJ n −→ I n /mI n −→ (I n /J n ) ⊗ R/m → 0. Therefore, using Lemma 2.1, we get This induces the inequalities of the leading coefficients (in degree d − 1) Using Theorem 2.6, we obtain Example 2.10. Let R = k[x, y] (x,y) where k[x, y] denotes the polynomial ring over an infinite field k. Let m = (x, y) and I = m n = (a 1 , . . . , a n , a n+1 ) for some n ≥ 2. We assume that Q = (a 1 , a 2 ) is a minimal reduction of I. Let J = (a 1 , a 2 , . . . , a n ). Then since Q ⊆ J, I is integral over J with red J (I) = 1, because I = J and red Q (I) = 1. Using e 1 (I) = e 1 (m n ) = 1 2 n(n − 1) and we obtain e 1 (J) ≥ 1 2 n(n − 1) − λ(R/(J : a n+1 ))f 0 (J).
One situation that may be amenable to further analysis is when I = J : m, or more generally I = J : m s for some values of s. We refer to I as a socle extension of J. Since r(R) ≥ 2, the ring R is not Gorenstein. We have e 1 (I) = λ(I/Q) + 1, so that e 1 (I) = e 0 (I) − λ(R/I) + 1.

The reduction number formula
In order to make use of Theorem 2.6, we need information about the reduction number of I in terms related to multiplicity. Let us recall [V1,Theorem 2.45 To establish such a result for arbitrary Noetherian local rings, we proceed differently. The version of the following lemma for Cohen-Macaulay rings can be found in [S1, Chapter 3, Theorem 1.1].
Lemma 3.2. Let (R, m) be a Noetherian local ring of dimension 1. Let x be a parameter of R. Let E be a finitely generated R-module and U an R-submodule of E. Then we have the following.
Consider the following two short exact sequences: Then we obtain (b) We may assume that the field R/m is infinite. Let yR be a minimal reduction of m. Then since x ∈ m s ⊆ y s R, we get λ(R/(x)) = e 0 (xR) ≥ e 0 (y s R) = e 0 (y s R) = s·e 0 (yR) = s·λ(R/(y)).

✷
Theorem 3.3. Let (R, m) be a Noetherian local ring of dimension d ≥ 1 with infinite residue field. For an m-primary ideal I and a minimal reduction J of I, there exists a minimal reduction Q of I such that red Q (I) ≤ max{d·λ(R/J) − 2d + 1, 0}.
We need to estimate ν(I n /L n ). Set Then we obtain the following series of exact sequences : where i = 1, . . . , n − 1. Note that for each i, M i is a submodule of L n−i /L n−i+1 as an R/L-module. Hence by Lemma 3.2, for each i = 1, . . . , n − 1, Since N n = I n + L L is a submodule of R/L, by Lemma 3.2, we get ν(N n ) ≤ λ(R/J).
Therefore we obtain Recall that if then there is a minimal reduction Q of I such that red Q (I) ≤ n − 1 ( [ES], [V1,Theorem 2.36]). Hence by solving the inequality we obtain the desired relation. ✷ Several open questions arise. Foremost whether it extends to higher dimensional Cohen-Macaulay rings (with a correction term depending on the dimension). Another question is which offsetting terms should be added in the non Cohen-Macaulay case. For instance, in dimension 2 whether the addition of −e 1 (Q), a term that can be considered a non Cohen-Macaulayness penalty, would give a valid bound.

Normalization
The following observation shows how the special fiber of the normalization impacts e 0 (I). Of course, more interesting issue would be to obtain relationships going the other way.
Proposition 4.1. Let (R, m) be a normal local domain and let I be an m-primary ideal. Suppose that R = ∞ n=0 C n is finite over R = R(I).
Proof. We first observe that C n+1 = IC n = IC n , for n ≫ 0. In particular, in that range, C n+1 ⊂ mC n . Consider now the corresponding exact sequence 0 → mC n /C n+1 −→ C n /C n+1 −→ C n /mC n → 0.
Counting multiplicities, we have