Comparison of symbolic and ordinary powers of ideals

By M. Hochster, Craig Huneke. 2002

274 citations recorded by the source corpus.

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 RW to R, where RW 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 here is closely related to a proof given in [Hu, 5.14–16, p. 45] that regular local rings in characteristic p are UFDs,

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/acbf2481bd68f64135ef250d09c9536bb61e2d2e

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Citations in the corpus

A survey of test ideals (2011, background): “given by a(n) = (anW−1R)∩R, where W⊆Ris the compliment of the union of the associated primes of a. In the case that ais a prime ideal, a(n) coincides with the a-primary component of an. Theorem 6.23. [HH02], cf. [ELS01, HH07, TY08] Let (R,m) be a regular local ring with inﬁnite residue ﬁeld R/m. Let”
Asymptotic resurgences for ideals of positive dimensional subschemes of projective space (2012, background): “Even showing that ρ(I) is finite is not easy, but it follows from [7, 15] that ρ(I) ≤ N , and more generally that ρ(I) ≤ hI where hI is the maximum height of an associated prime of I.”
Symbolic Powers of Ideals (2015, methodology): “powers over regular rings containing ﬁelds, which was surprising at the time, is the theorem of Ein, Lazarfeld, and Smith [ELS01] in the geometric case in characteristic zero, and Hochster and Huneke [HH02] in general for d-dimensional regular rings containing a ﬁeld, which establishes that (1.0.2) ”
A canonical linear system associated to adjoint divisors in characteristic $p > 0$ (2011, background): “r all x∈ S. Then Slies on a hypersurface of degree ⌊de l ⌋. This result is not new to experts in characteristic p>0. Indeed, this bound has been recently observed by B. Harbourne as a corollary of [HH02], see [Har09, Slide 17] and cf. [HH11, ELS01]. However, it still demonstrates that global stat”
Star configurations in Pn (2013, background): “It follows that 1 ≤ ρ(I), and by applying the main result of [18] we know that ρ(I) ≤ n.”
Symbolic Powers of Monomial Ideals (2013, background): “Among the known results is a celebrated containment of Ein-Lazarsfeld-Smith [9] and Hochster-Huneke [12].”
Containment results for ideals of various configurations of points in PN (2011, unknown): “riterion 2, [BH2]). Let I ⊆ K[PN] be a homogeneous ideal (not necessarily saturated) deﬁning a 0-dimensional subscheme. If rreg(I) ≤ α(I(m)), then I(m) ⊆ Ir. We also recall one of the main results of [HoHu]. The containment I(Nt) ⊆ It is the special case for which m = 1. Theorem 2.2. Let I ⊆ K[PN] b”
On the containment problem (2016, background): “It was discovered by Ein, Lazarsfeld and Smith in characteristic zero, see [15] and by Hochster and Huneke in positive characteristic, see [20].”
A primer on Seshadri constants (2008, background): “In fact the results of [EiLaSm01] and [ HocHun02 ] imply the slightly stronger result that I(m) Ir if m > rN. As a next step, we can ask for the largest integer dN such that I(m) Ir whenever m > rN dN . The fact that I(2) is not always contained in I2, as we saw above, shows that d2 < 2, and hence e”