Mathematische Annalen Positivity and complexity of ideal sheaves

Inventiones Mathematicae, 2002

All given rings in this paper are commutative, associative with identity, and Noetherian. Recently, L. Ein, R. Lazarsfeld, and K. Smith [ELS] discovered a remarkable and surprising fact about the behavior of symbolic powers of ideals in affine regular rings of equal characteristic 0: if h is the largest height of an associated prime of I, then I (hn) ⊆ I n for all n ≥ 0. Here, if W is the complement of the union of the associated primes of I, I (t) denotes the contraction of I t R W to R, where R W is the localization of R at the multiplicative system W. Their proof depends on the theory of multiplier ideals, including an asymptotic version, and, in particular, requires resolution of singularities as well as vanishing theorems. We want to acknowledge that without their generosity and quickness in sharing their research this manuscript would not exist. Our objective here is to give stronger results that can be proved by methods that are, in some ways, more elementary. Our results are valid in both equal characteristic 0 and in positive prime characteristic p, but depend on reduction to characteristic p. We use tight closure methods and, in consequence, we need neither resolution of singularities nor vanishing theorems that may fail in positive characteristic. For the most basic form of the result, all that we need from tight closure theory is the definition of tight closure and the fact that in a regular ring, every ideal is tightly closed. We note that the main argument The authors were supported in part by grants from the National Science Foundation. Version of July 25, 2001. 2 MELVIN HOCHSTER AND CRAIG HUNEKE here is closely related to a proof given in [Hu, 5.14-16, p. 45] that regular local rings in characteristic p are UFDs, which proceeds by showing that Frobenius powers of height one primes are symbolic powers. Our main results in all characteristics are summarized in the following theorem. Note that I * denotes the tight closure of the ideal I. The characteristic zero notion of tight closure used in this paper is the equational tight closure of [HH6] (see, in particular Definition (3.4.3) and the remarks in (3.4.4) of [HH6]). This is the smallest of the characteristic zero notions of tight closure, and therefore gives the strongest result. See §3.1 for a discussion of the Jacobian ideal J (R/K) utilized in part (c). Theorem 1.1. Let R be a Noetherian ring containing a field. Let I be any ideal of R, and let h be the largest height 1 of any associated prime of I. (a) If R is regular, I (hn+kn) ⊆ (I (k+1)) n for all positive n and nonnegative k. In particular, I (hn) ⊆ I n for all positive integers n. (b) If I has finite projective dimension then I (hn) ⊆ (I n) * for all positive integers n. (c) If R is finitely generated, geometrically reduced (in characteristic 0, this simply means that R is reduced) and equidimensional over a field K, and locally I is either 0 or contains a nonzerodivisor (this is automatic if R is a domain), then, with J = J (R/K), for every nonnegative integer k and positive integer n, we have that J n I (hn+kn) ⊆ ((I (k+1)) n) * and J n+1 I (hn+kn) ⊆ (I (k+1)) n. In particular, we have that J n I (hn) ⊆ (I n) * and J n+1 I (hn) ⊆ I n for all positive integers n. These results, when specialized to the case where R is regular, recover the cited result from [ELS]. The theorem above is a composite of Theorems 2.6, 3.7, and 4.4 below. We note that by results 2 of [Swsn] one expects, in many cases, to have results that assert that, given a fixed ideal I in a Noetherian ring, for some choice of positive integer h ′ (independent of n but depending on I) one has I (h ′ n) ⊆ I n for all positive integers n. What is not expected is the very simple choice of h ′ that one can make in a regular ring, and the extent to which it is independent of information about I. E.g., when d = dim R 1 The results stated here are all valid if one defines h instead to be the largest analytic spread of IR P for any associated prime P of I, which, in general, may be smaller: see Discussion 2.3 2 E.g., it is shown in [Swsn] that if I ⊆ J are ideals of a Noetherian ring, and we let I : J ∞ = t I : J t , then if the I-adic filtration is equivalent to the I n : J ∞ filtration, there exists an integer h ′ such that for all n, I h ′ n : J ∞ ⊆ I n .

ANNALI DELL UNIVERSITA DI FERRARA, 2003

A Note on Symbolic and Ordinary Powers of Homogeneous Ideals. ALESSANDRO ARSIE (*) -JON EIVIND VATNE (**) SUNTO -In questa nota siamo interessati ai moduli graduati M k : I(k)/I k e Nk := Ik/I k, dove I ~ un ideale saturato nelranello delle coordinate omogenee S := K[x0, ..., Xn] di F n, I (k) ~ la potenza simbolica e I k ~ la saturazione della potenza ordinaria. Poco noto su questi moduli e qui viene fornito un limite superiore ai loro diametri. Ne calcoliamo inoltre le funzioni di Hilbert e studiamo alcuni sottomoduli caratteristici nel cast speciale di n + 1 punti in posizione generale in pn. ABSTRAC_T -In this note we are interested in the graded modules Mk = I(k)/I k and Nk := := Ik/I k, where I is a saturated ideal in the homogeneous coordinate ring S := , and I is the saturation of the ordi-:= K[xo .... xn] of F n, I (k) is the symbolic power k nary power. Very little is known about these modules, and we provide a bound on their diameters, we compute the Hilbert functions and we study some characteristic submodules in the special case of n + 1 general points in Fn. MSC(2002) Classification: 13C99, 14M10.

Mathematical Research Letters, 1999

Inventiones Mathematicae, 1999

The purpose of this paper is to present a geometric theorem which clarifies and extends in several directions work of Brownawell, Kollár and others on the effective Nullstellensatz. Specifically, we work on an arbitrary smooth complex projective variety X, with the previous "classical" results corresponding to the case when X is projective space. In this setting we prove a local effective Nullstellensatz for ideal sheaves, and a corresponding global division theorem for adjoint-type bundles. We also make explicit the connection with the intersection theory of Fulton and MacPherson. Finally, constructions involving products of prime ideals that appear in earlier work are replaced by geometrically more natural conditions involving orders of vanishing along subvarieties.

Acta Mathematica Vietnamica

For an ideal I in a local ring (R, m), we prove that the integer-valued function ℓ R (H 0 m (R/I n+1)) is a polynomial for n big enough if either I is a principle ideal or I is generated by part of an almost p-standard system of parameters. Furthermore, we are able to compute the coefficients of this polynomial in terms of length of certain local cohomology modules and usual multiplicity if either the ideal is principal or it is generated by part of a standard system of parameters in a generalized Cohen-Macaulay ring. We also give an example of an ideal generated by part of a system of parameters such that the function ℓ R (H 0 m (R/I n+1)) is not a polynomial for n ≫ 0.

Applicable Algebra in Engineering, Communication and Computing, 2017

In this paper, we study first the relationship between Pommaret bases and Hilbert series. Given a finite Pommaret basis, we derive new explicit formulas for the Hilbert series and for the degree of the ideal generated by it which exhibit more clearly the influence of each generator. Then we establish a new dimension depending Bézout bound for the degree and use it to obtain a dimension depending bound for the ideal membership problem.

Contemporary Mathematics, 2013

The representation of polynomials by arithmetic circuits evaluating them is an alternative data structure which allowed considerable progress in polynomial equation solving in the last fifteen years. We present a circuit based computation model which captures all known symbolic elimination algorithms in effective algebraic geometry and show the intrinsically exponential complexity character of elimination in this complexity model.

Journal of Algebra, 1983

1999

Modern computer algebra systems allow the computation of complicated examples in Commutative Algebra, Algebraic Geometry and Arithmetic Geometry. During the last couple of years such computations have helped to predict and check many theorems. Vice versa, inspired by complicated examples coming from theory, computer algebra developers have refined their algorithms and implementations.

Mathematische Zeitschrift, 1998

In the first part of this paper we show that the Castelnuovo-Mumford regularity of a monomial ideal is bounded above by its arithmetic degree. The second part gives upper bounds for the Castelnuovo-Mumford regularity and the arithmetic degree of a monomial ideal in terms of the degrees of its generators. These bounds can be formulated for an arbitrary homogeneous ideal in terms of any Gröbner basis.