INTEGRALLY CLOSED FINITE-COLENGTH IDEALS

IN TWO-DIMENSIONAL REGULAR LOCAL RINGS

ARE MULTIPLIER IDEALS

Joseph Lipman and Kei-ichi Watanabe

Abstract. Multiplier ideals in commutative rings are certain integrally closed ideals with properties that lend themselves to highly interesting applications. But how special are they among integrally closed ideals in general? In this note we show that in a two-dimensional regular local ring with algebraically closed residue field, in the class of ideals containing a power of the maximal ideal there is in fact no difference between ”multiplier” and ”integrally closed” (or ”complete.”) However, among ”integral” multiplier ideals (or ”adjoint ideals”) the only simple complete ideals are those of order one.

Introduction. There has arisen in recent years a substantial body of work on “multiplier ideals” in commutative rings (see [La]). Multiplier ideals are integrally closed ideals with properties that lend themselves to highly interesting applications. One is tempted then to ask just how special multiplier ideals are among integrally closed ideals in general.

In this note we show that in a two-dimensional regular local ring R with maximal ideal 𝔪 such that the residue field R/𝔪 is algebraically closed^{1} there is in fact no difference between multiplier ideals and integrally closed ideals, at least when we deal with finite-colength ideals (i.e., those containing a power of 𝔪):

Main Result. Every integrally closed finite-colength R-ideal is a multiplier ideal.

We do not know at present whether such a statement holds true for arbitrary integrally closed R-ideals, let alone higher dimensions.

Throughout, (R, 𝔪) will be as above. For convenience we say “complete ideal” instead of “integrally closed finite-colength R-ideal.”

2000 Mathematics Subject Classification. 13B22, 13H05. First author partially supported by the National Security Agency. Second author partially

supported by Grants-in-Aid in Scientific Researches, 13440015, 13874006; and his stay at MSRI was supported by the Bunri Fund, Nihon University. Both authors are grateful to MSRI for providing the environment without which this work would not have begun. Research at MSRI is supported in part by NSF grant DMS-9810361.

1This last condition can most likely be dropped, but we don’t want to get involved with the resulting technicalities.

1. Geometric formulation of the problem. The goal of this section is to develop the geometric criterion Corollary 1.4.2 for an ideal to be a multiplier ideal, while laying the groundwork for the proof in the next section that every complete ideal satisfies that criterion.

We begin by recalling some preliminary definitions and known results. (For some historical pointers to the development of the theory of complete ideals see the second paragraph on the first page of [L3].)

For any complete ideal I there exists a log resolution, i.e., a proper birational map f : X → Spec(R) with X a regular scheme such that the O_{X}-ideal IO_{X} is invertible. A quick way to see this, with f a composition of maps obtained by blowing up closed points, is via the Hoskin-Deligne formula [L2, p. 222, Thm. 3.1], which shows that the length of the “transform” of I decreases strictly with each blowup—until it vanishes, at which point I generates an invertible ideal sheaf.

We denote by E^{1}, E^{2}, …, E^{s} the integral (i.e., reduced and irreducible) components of the closed fiber f^{−1}{𝔪}. They are isomorphic, as schemes, to the projective line ℙ_{R/𝔪}^{1}, and any two of them intersect transversally. (This can easily be shown by induction on the number of blowups making up f.) Each E^{i} gives rise to the discrete valuation v_{E^{i}} whose valuation ring is the local ring on X of the generic point of E^{i}.

We define the group Div_{e}(X) of (exceptional) f-divisors to be the free abelian group on the set {E^{1}, …, E^{s}}. The invertible sheaf O_{X}(E) associated to an fdivisor E is defined in the standard way. We will use repeatedly, without explicit mention, the fact that

Γ(X, O_{X}(a_{1}E^{1} + ⋯ a_{s}E^{s})) = { x ∈ R | v_{E^{i}}(x) ≥ −a_{i} for 1 ≤ i ≤ s },

an R-ideal containing some power of 𝔪.

An f-divisor E = a_{1}E^{1} + ⋯ a_{s}E^{s} is effective if all the integers a_{i} are ≥ 0, or equivalently, if O_{X}(−E) ⊂ O_{X}. Such an E can be regarded as a one-dimensional subscheme of X, projective over R/𝔪, with structure sheaf O_{E} fitting in a natural exact sequence

0 → O_{X}(−E) → O_{X} → O_{E} → 0.

For any invertible O_{X}-module L the intersection number L · E of L with an effective E is the degree of the invertible O_{E}-module L_{E}:= L ⊗ O_{E}, i.e.,

L · E = χ_{E}O_{E} − χ_{E} L_{E}^{−1}

where χ_{E} M denotes the Euler characteristic of a coherent O_{E}-module M (E being viewed as a projective curve over R/𝔪.)

For any f-divisor F set F ·E := O_{X}(F)·E. This version of intersection number extends uniquely to a ℤ-valued symmetric bilinear form on Div_{e}(X) (see e.g., [L1, §13]).

An f-divisor F is said to be numerically effective, nef for short, if F · E^{i} ≥ 0 for all i = 1, 2, …, s (⇒ F · E ≥ 0 for all effective E ∈ Div_{e}^{+}(X)). F is said to be antinef if −F is nef.

The following basic result is contained in [L1, p. 220, Thm. (12.1)].

Theorem 1.1. An f-divisor F is nef if and only if O(F) is generated by its global sections.

Corollary 1.1.1. If F is antinef then F is effective.

That’s because O_{X}(−F) = Γ(X, O_{X}(−F))O_{X} ⊂ O_{X}. (A simpler proof, not using Theorem 1.1, can be found in [L1, p. 238].)

If I is complete and IO_{X} is invertible then I = Γ(X, IO_{X}), whence:

Corollary 1.1.2. (Cf. [L1, §18].) Sending E to Γ(X, O_{X}(−E)) is an isomorphism from the (additive) monoid of antinef f-divisors to the (multiplicative) monoid of those complete R-ideals I such that IO_{X} is invertible.^{2}

It is simple to show, by induction on the number of blowups making up f, that the intersection matrix (E^{i} · E^{j}) has determinant ±1. Hence for each i there is a unique f-divisor G_{i} such that G_{i} · E^{j} = 0 unless j = i, in which case G_{i} · E^{j} = −1; and for any f-divisor E it holds that −E = ∑_{i=1}^{s}(E.E^{i})G_{i}. Thus the monoid of antinef divisors is freely generated by these G_{i}; in other words, “unique factorization” holds in this monoid—and therefore in the monoid of complete ideals to which, by Corollary 1.1.2, it is isomorphic.

A complete ideal P is simple if whenever P = IJ then either I or J is the unit ideal.

Corollary 1.1.3. (Zariski, [ZS, p. 386, Thm. 3].) Every complete ideal is, in a unique way, the product of simple complete ideals.

(For deducing this corollary it helps to note that for any product IJ of ideals, if IJO_{X} is invertible then IO_{X} and JO_{X} are both invertible.)

Corollary 1.1.4. If I is a complete ideal such that IO_{X} = O_{X}(−E) is invertible and if P_{i}:= Γ(X, O_{X}(−G_{i})) is the simple complete ideal corresponding to the above G_{i}, then I = ∏_{i=1}^{s} P_{i}^{E.E^{i}}. In particular, P_{i} divides I ⟺ E · E^{i} ≠ 0.

Moreover, the valuations v_{E^{i}} associated to those E^{i} such that E · E^{i} ≠ 0 are precisely the Rees valuations of I (i.e., those valuations whose valuation ring is the local ring of the generic point of some integral component of the closed fiber of the normalized blowup of I). (See [L4, p. 300, Prop. (4.4)].)

The following Lemma^{3} will be needed. We write ∑_{i} a_{i}E^{i} ≥ ∑_{i} b_{i}E^{i} if a_{i} ≥ b_{i} for all i = 1, 2, …, s.

Lemma 1.2. Let E = ∑_{i} a_{i}E^{i} and let I be the R-ideal Γ(X, O_{X}(−E)). Then IO_{X} is invertible. Equivalently, there exists an antinef E^{−} ≥ E such that I = Γ(X, O_{X}(−E^{−})).

Corollary 1.1.2 implies that this E^{−} must be the least antinef f-divisor ≥ E.

2It is a theorem of Zariski that a product of two complete ideals is still complete [ZS, p. 385, Thm. 2′], [L1, p. 209, Thm. (7.1)].

3related to Enriques’s “principle of discharge” [Z, p. 28],

Proof. One may assume X ≠ Spec(R). There exists some antinef F ≥ E: pick n such that 𝔪^{n} ⊂ I, then define F by 𝔪^{n}O_{X} = O_{X}(−F). Among all antinef F = ∑ b_{i}E^{i} ≥ E choose one—call it F_{E}—for which ∑_{i}(b_{i} − a_{i}) has minimal value, denoted σ_{E}.

Procede by induction on σ_{E}. Suppose σ_{E} > 0 (otherwise there is nothing to prove), and that the Lemma holds for all E′ with σ_{E′} < σ_{E}. With F := F_{E} as above, there is an i such that E · E^{i} > 0 ≥ F · E^{i}, and since E^{j} · E^{i} ≥ 0 when j ≠ i (clearly) and E^{i} · E^{i} < 0,^{4} therefore b_{i} > a_{i}, whence F ≥ E + E^{i}. So σ_{E+E^{i}} < σ_{E}, and therefore Γ(X, O_{X}(−E − E^{i}))O_{X} is invertible. It suffices then to verify that Γ(X, O_{X}(−E − E^{i})) = I, by applying the left-exact functor Γ(X, −) to the natural exact sequence

0 ⟶ O_{X}(−E − E^{i}) ⟶ O_{X}(−E) ⟶ O_{X}(−E) ⊗ O_{E^{i}}

and observing that since O_{X}(−E) ⊗ O_{E^{i}} has degree −E · E^{i} < 0, therefore

Γ(X, O_{X}(−E) ⊗ O_{E^{i}}) = Γ(E^{i}, O_{X}(−E) ⊗ O_{E^{i}}) = 0. □

(1.3). (Canonical divisors.) Let Y →^{g} X →^{f} Spec(R) be proper birational maps with Y and X regular schemes. By a theorem of Zariski and Abhyankar (see, e.g., [L1, p. 204, Thm. (4.1)]) both f and g are compositions of point blowups.

Let F^{1}, F^{2}, …, F^{t} be the integral components of (fg)^{−1}{𝔪}. For an fdivisor E, g^{∗}E denotes the fg-divisor with F^{i}-coefficient v_{F^{i}}(O_{X}(−E)_{x_{i}}), where x_{i} ∈ X is the g-image of the generic point of F^{i}.

Since, as before, the intersection matrix (F^{i} · F^{j}) has determinant ±1, there is a unique fg-divisor K_{g} such that

K_{g} · F^{i} = \{ \begin{matrix}−F^{i} ·F^{i} − 2 & if g(F^{i}) is a point \\ 0 & otherwise. \end{matrix}

This K_{g} is called the canonical divisor of g.^{5}

(1.3.1). The following easily-checked properties characterize K_{g} for all g:

• If g is the blowup of a closed point x ∈ X then K_{g} = g^{−1}{x}.

• If Z →^{h} Y →^{g} X →^{f} Spec(R) are proper birational maps with Z, Y and X regular schemes, then

K_{gh} = h^{∗}K_{g} + K_{h}.

(1.4). (Multiplier ideals.) For E = ∑_{i} a_{i}E^{i} ∈ Div_{e}(X) ⊗_{ℤ} ℝ set

[E] = ∑_{i} [a_{i}]E^{i} ∈ Div_{e}(X)

where [a_{i}] is the greatest integer ≤ a_{i}.

4It is well-known, going back to Du Val, that the intersection matrix (E^{i} · E^{j}) is negative definite, see e.g., [L1, p. 224, Lemma (14.1)].

5One has O(K_{g}) = H^{0}(g^{!}O_{X}) with g^{!} as in Grothendieck duality theory [LS, p. 206, (2.3)].

Definition 1.4.1. Let I be a complete R-ideal, h : Y → Spec(R) a log resolution of I, say IO_{Y} = O_{Y} (−G), and let c be a positive real number. The multiplier ideal J (cI) is defined to be

J (cI):= Γ(Y, O(K_{h} − [cG])).

Thus, by Lemma 1.2,

J (cI)O_{Y} = O_{Y} (−D)

where D:= ([cG] − K_{h})^{−} is the least antinef h-divisor ≥ [cG] − K_{h}.

For a point blowup h_{1} : Y_{1} → Y one finds via (1.3.1) that the log resolution h can be replaced by the log resolution h ∘ h_{1} without affecting J (cI). Since any two log resolutions are dominated by a third, obtained from each of the two by a sequence of blowups,^{6} it follows that J (cI) does not depend on the choice of the log resolution h.

Corollary 1.4.2. A complete R-ideal J satisfies J = J (cI) for some c, I iff for some log resolution h : Y → Spec(R) of J, say JO_{Y} = O_{Y} (−F), there is an antinef h-divisor G and a real c > 0 such that

(1.4.2.1)  F = ([cG] − K_{h})^{−} .

2. Proof of Main Result. Let J be a complete R-ideal. We will describe a log resolution h : Y → Spec(R) of J, and a G and c as in Corollary 1.4.2, such that if JO_{Y} = O_{Y} (−F) then (1.4.2.1) holds. (The number of suitable (h, G, c) will be enormous.)

Factor J as J = ∏_{ℓ=1}^{u} P_{ℓ}^{e_{ℓ}} (P_{ℓ} simple complete, e_{ℓ} > 0)—see Corollary 1.1.3 and the two paragraphs preceding it. Let f : X → Spec(R) be any log resolution of J, JO_{X} = O_{X}(−F^{0}). We construct a proper birational map g_{N} : Y_{N} → X, with Y_{N} regular, for each u-tuple N:= (n_{1}, n_{2}, …, n_{u}) of non-negative integers, as follows.

For convenience of expression, we say “blow up a closed point x ∈ X generically, n times” to mean “blow up x_{0} := x to get g_{1} : X_{1} → X, then blow up a closed point x_{1} on g_{1}^{−1}x_{0} but not on any other integral component of the closed fiber of X_{1} → Spec(R) to get g_{2} : X_{2} → X_{1}, then blow up a closed point x_{2} on g_{2}^{−1}x_{1} but not on … then blow up a closed point x_{n−1} on g_{n−1}^{−1}x_{n−2} but not on any other integral component of the closed fiber of X_{n−1} → Spec(R) to get g_{n} : X_{n} → X_{n−1}.”

Then with g:= g_{1} ∘ g_{2} ∘ ∘ ⋯ ∘ g_{n} it holds that: (2.1) g^{−1}x is a chain of n integral curves D_{1}, D_{2}, …, D_{n} such that for 0 < i < n, D_{i} · D_{i+1} = 1 and D_{i} · D_{i} = −2, while D_{n} · D_{n} = −1; and if |j − i| > 1 then D_{i} · D_{j} = 0.

6By the above-mentioned theorem of Zariski and Abhyankar, it suffices to principalize some ideal sheaf on one of the log resolutions by a sequence of point blowups (“elimination of indeterminacies”), which can be done e.g., via the Hoskin-Deligne formula, as before.

(For the proof one can use, e.g., [L1, p. 229, middle, and p. 227, α) and β)]. Here, and subsequently, the reader may find it useful to do some rough sketches.)

As before, there corresponds to each D_{i} a simple complete ideal P_{i}; and, we claim, these P_{i} form a strictly decreasing sequence P_{1} > P_{2} > ⋯ > P_{n}, with P_{1} strictly contained in each of the simple ideals corresponding to the (one or two) integral components E^{j} of f^{−1}{𝔪} passing through x.

Indeed, let G_{j} be the f-divisor such that G_{j} · E^{j} = −1 and G_{j} · E = 0 for every other integral component E of f^{−1}{𝔪}, and let Q_{j} be the corresponding simple complete ideal. It follows from, e.g., [L1, p. 227, α) and β)] that g_{1}^{∗}G_{j} is antinef; and the corresponding simple complete ideal is

Γ(X_{1}, O_{X_{1}}(−g_{1}^{∗}G_{j})) = Γ(X, O_{X}(−G_{j})) = Q_{j} .

Further, with E′ := g_{1}^{−1}x, let G′ be the fg_{1}-divisor such that G′ · E′ = −1 and G′·E′′ = 0 for every other integral component E′′ of (fg_{1})^{−1}{𝔪}. Then g_{1}^{∗}G_{j}+E′ has intersection number −1 with E′ and ≥ 0 with each E′′. So G′ − g_{1}^{∗}G_{j} − E′ is antinef, hence effective (Corollary 1.1.1); and consequently G′ > g_{1}^{∗}G_{j}. Thus the simple complete ideal Γ(X_{1}, O_{X_{1}}(−G′)) is strictly contained in Q_{j}.

Continuing in this way we establish the claim.

Now for each simple factor P_{ℓ} of J, there is a unique integral component E^{ℓ} of f^{−1}{𝔪} such that P_{ℓ}O_{X} · E^{ℓ} = 1 and P_{ℓ}O_{X} · E = 0 for any other integral component E; and hence e_{ℓ} = JO_{X} · E^{ℓ}. For each ℓ = 1, 2, …, u, pick e_{ℓ} distinct closed points x_{1}, …, x_{e_{ℓ}} which lie on E^{ℓ} but on no E ≠ E^{ℓ} and blow up all of these points generically, n_{ℓ} times. Then Y_{N} is the resulting surface, and g_{N} is the composition of all the blowups. It is easily seen that (Y_{N}, g_{N}) does not depend (up to isomorphism) on the order in which the chosen points are blown up—though that won’t really be used.^{7}

To simplify notation, fix N and set (Y, g):= (Y_{N}, g_{N}) and F:= g^{∗}F^{0}, so that JO_{Y} = O_{Y} (−F). Also, set h:= fg : Y → Spec(R).

For an f-divisor E, we denote by E^{∗} the proper transform of E on Y, i.e., if g^{∗}E = ∑_{i} a_{i}E′_{i} + ∑_{j} b_{j}E′′_{j} where each gE′_{i} is a curve on X while each gE′′_{j} is a closed point, then E^{∗}:= ∑_{i} a_{i}E′_{i}.

For each ℓ = 1, 2, …, u and x_{j_{ℓ}} ∈ E^{ℓ} (j_{ℓ} = 1, 2, …, e_{ℓ}) let

{ E_{j_{ℓ}k_{ℓ}}^{ℓ} | k_{ℓ} = 1, 2, …, n_{ℓ} }

be the family of integral curves on Y whose g-image is x_{j_{ℓ}}, the ordering of these curves by the index k_{ℓ} conforming to the ordering of the D’s in (2.1).

7The initial ∑_{ℓ} e_{ℓ} points could be taken to be the intersection of the closed fiber on X and a generic curve C in the linear system | − F | (i.e., a divisor—having no component in the closed fiber—of the form (j) − F with j a generic element of J). Then at each stage the points to be blown up could be taken to be closed points on the inverse image of C.

If a_{ℓ} is the E^{ℓ}-coefficient of the divisor F^{0}, and b_{ℓ} of the divisor K_{f}, then one finds (using (1.3.1)) that

F = g^{∗}F^{0} = F^{0}^{∗} + ∑_{ℓ} ∑_{j_{ℓ},k_{ℓ}} a_{ℓ}E_{j_{ℓ}k_{ℓ}}^{ℓ},
g^{∗}K_{f} = K_{f}^{∗} + ∑_{ℓ} ∑_{j_{ℓ},k_{ℓ}} b_{ℓ}E_{j_{ℓ}k_{ℓ}}^{ℓ},
K_{g} = ∑_{ℓ} ∑_{j_{ℓ},k_{ℓ}} k_{ℓ}E_{j_{ℓ}k_{ℓ}}^{ℓ}.

Set G:= F + K_{g}. Expanding F and K_{g} as above, noting that g^{∗}D · E_{j_{ℓ}k_{ℓ}}^{ℓ} = 0 for any f-divisor D [L1, p. 227, β)], and using (2.1), one calculates that for every integral component E of h^{−1}{𝔪}, G · E = 0 unless E is one of the curves E_{j_{ℓ}n_{ℓ}}^{ℓ} at the end of the chains emanating from the ∑_{ℓ} e_{ℓ} originally chosen points (i.e., g(E) is a point and E · E = −1), in which case G · E = −1. So G is antinef, and by Corollary 1.1.4,

I:= Γ(Y, O_{Y} (−G))

is the product of the simple complete ideals corresponding to these ∑_{ℓ} e_{ℓ} curves having self-intersection −1.

Here is the key technical point:

Lemma 2.2. For all sufficiently small ϵ > 0, there exists N such that

(2.2.1)  (1 + ϵ)G − K_{h} = F + A  ([A] ≤ 0)

where the coefficient of [A] at each E_{j_{ℓ}n_{ℓ}}^{ℓ} is 0.

Proof. Using (1.3.1), one transforms (2.2.1) into the equality

ϵ(F + K_{g}) − g^{∗}K_{f} = A.

More explicitly (see above)

ϵ(F^{0}^{∗} + ∑_{ℓ} ∑_{j_{ℓ},k_{ℓ}}(a_{ℓ} + k_{ℓ})E_{j_{ℓ}k_{ℓ}}^{ℓ}) − (K_{f}^{∗} + ∑_{ℓ} ∑_{j_{ℓ},k_{ℓ}} b_{ℓ}E_{j_{ℓ}k_{ℓ}}^{ℓ}) = A.

So to get (2.2.1) we can choose any ϵ > 0 such that the coefficients of ϵF^{0} − K_{f} are all < 1, and then look for n_{ℓ} such that ϵ(a_{ℓ} + k_{ℓ}) − b_{ℓ} < 1 for all ℓ and k_{ℓ} ≤ n_{ℓ}, while ϵ(a_{ℓ} + n_{ℓ}) − b_{ℓ} ≥ 0. These conditions mean precisely that n_{ℓ} satisfies the inequalities

1/ϵ + b_{ℓ}/ϵ − a_{ℓ} > n_{ℓ} ≥ b_{ℓ}/ϵ − a_{ℓ} (ℓ = 1, 2, …, u).

Clearly, such integers n_{ℓ} can be found if ϵ < 1. □

For c = 1 + ϵ and N satisfying Lemma 2.2, and with h : Y → Spec(R) and F, G, as before, we have

F′:= ([cG] − K_{h})^{−} ≤ F,

so that

J′:= Γ(Y , O(−F′)) ⊃ J = Γ(Y , O(−F)).

Let us verify that J′ = J, thereby proving the main result.

Recall that the valuations v_{ℓ}:= v_{E^{ℓ}} are just the Rees valuations of J. (See the remark following Corollary 1.1.4). So

J = { ρ ∈ R | v_{ℓ}(ρ) ≥ v_{ℓ}(J) for all ℓ = 1, 2, …, u }.

Thus we need only show that for each ℓ, the E^{ℓ}^{∗}-coefficient a′_{ℓ} of F′ is the same as that of F (namely a_{ℓ}). Let us say that ℓ is “good” if a′_{ℓ} = a_{ℓ} and “bad” if a′_{ℓ} < a_{ℓ}.

If ℓ is good then since F′ ≤ F therefore F′ · E^{ℓ}^{∗} ≤ F · E^{ℓ}^{∗}. Corollary 1.1.4 shows then that J′ is divisible by P_{ℓ}^{e_{ℓ}}.

Suppose ℓ is bad. For j ∈ [1, e_{ℓ}] and with a′_{jk} the E_{jk}^{ℓ}-coefficient of F′ it is easily seen that

a′_{ℓ} =: a′_{j0} ≤ a′_{j1} ≤ a′_{j2} ≤ ⋯ ≤ a′_{jn_{ℓ}} = a_{ℓ}.

So there is a k ∈ [1, n_{ℓ}] such that a′_{j,k−1} < a′_{jk} = a′_{j,k+1} = ⋯ = a_{ℓ}. Then

F′ · E_{kl}^{ℓ} = \{ \begin{matrix}a′_{j,k−1} − 2a′_{jk} + a′_{j,k+1} < 0 & if k < n_{ℓ}, \\ a′_{j,n_{ℓ}−1} − a′_{jn_{ℓ}} < 0 & if k = n_{ℓ}. \end{matrix}

From Corollary 1.1.4 and the remarks after 2.1, one deduces that J′ is divisible by a simple complete ideal P′_{ℓj} < P_{ℓ}. This being so for all j, and the P′_{ℓj} being distinct (Corollary 1.1.2), it follows from Corollary 1.1.3 that J′ is divisible by P′_{ℓ1}P′_{ℓ2} ⋯ P′_{ℓe_{ℓ}} < P_{ℓ}^{e_{ℓ}}.

Thus (by Corollary 1.1.3 again) the existence of a bad ℓ leads to a factorization of J′ which contradicts J′ ⊃ J. So every ℓ is good, and J′ = J. □

Remarks. 1. By the choice of ϵ, the E^{ℓ}-coefficient of ϵF^{0} − K_{f} is < 1, i.e., ϵa_{ℓ} − b_{ℓ} < 1, i.e., b_{ℓ}/ϵ − a_{ℓ} > −1/ϵ. It could happen that b_{ℓ}/ϵ − a_{ℓ} < 0 for all ℓ. In this case one can take N = (0, 0, …, 0), and then J = J ((1 + ϵ)J).

2. The proof shows that if J is a simple complete ideal then there is a simple complete ideal P ⊂ J and a c > 0 such that J = J (cP).

3. Since the c = 1 + ϵ we have considered can be arbitrarily close to 1, one might ask if it is possible for c actually to be 1. (This would be the case studied in [L5], where J (I) is called the adjoint ideal of I.)

For simple complete J, the answer is given by:

Proposition 2.3. A simple complete ideal J is of the form J (I) for some I ⟺ J ⊄ 𝔪^{2} ⟺ J = (a, b^{n})R for some integer n > 0 and a, b ∈ R such that (a, b)R = 𝔪.

Proof. The last ⟺ holds because J ⊄ 𝔪^{2} means that J contains an element a such that R/aR is a discrete valuation ring. Moreover, if (a, b)R = 𝔪 and z ∈ R is integral over J = (a, b^{n})R then the canonical image of z in R/aR is integral over—and hence is a multiple of—that of b^{n}, whence z ∈ J, and thus J is complete (and clearly simple). And it is a simple exercise to show that for such a J, J = J (J^{2}). (One could use [L5, p. 749, Prop. (3.1.2)].)

For any simple complete J, on the minimal log resolution f : X → Spec(R) there is, as we have seen, a unique integral component E of f^{−1}{𝔪} such that JO_{X} · E ≠ 0. This E must satisfy E · E = −1: for, there is at least one integral component E′ such that E′ · E′ = −1, namely the closed fibre f′^{−1}{x′} for the blowup f′ : X → X′ of x′ ∈ X′ coming last in some seqence of blowups composing to f; and by minimality of f, J′ := JO_{X′, x′} cannot be invertible, from which one sees, with 𝔪′ the maximal ideal of R′:= O_{X′, x′}, that J′ = d𝔪′^{t} for some d ∈ R′ and t > 0, whence JO_{X} · E′ = t. (Of course t = 1.)

We claim that if F · F = −2 for all integral components F ≠ E of f^{−1}{𝔪}, then J ⊄ 𝔪^{2}.^{8} Indeed, this condition on the F ’s means that among the base points of J no two are proximate to the same one, and the conclusion follows from [L5, p. 301, (3)].

Assuming now that our simple J = J (I), let h : Y → Spec(R) be the minimal log resolution of I, say IO_{Y} = O_{Y} (−G). As above, we find that for any integral component F of h^{−1}{𝔪}, F · F = −1 ⇒ G · F < 0. (We may assume that F is the closed fiber for the last blowup in some sequence of blowups composing to g.) Since K_{h} · F = −F · F − 2 = −1 for every such F, it follows that G − K_{h} is antinef, and hence O_{Y} (K_{h} − G) = JO_{Y}.

Since JO_{Y} is invertible (Lemma 1.2), the minimality of f implies that there exists a g : Y → X such that h = fg. Let F^{1}^{∗}, F^{2}^{∗}, …, F^{n}^{∗} be the proper transforms on Y of the integral components F^{1}, F^{2}, …, F^{n} of f^{−1}{𝔪} other than E (see above). Then F^{i}^{∗} · F^{i}^{∗} ≤ F^{i} · F^{i} ≤ −2 and so K_{h} · F^{i}^{∗} ≥ 0. But (by [L1, p. 227, β)])

K_{h} · F^{i}^{∗} ≤ (K_{h} − G) · F^{i}^{∗} = JO_{Y} · F^{i}^{∗} = JO_{X} · F^{i} = 0,

and thus K_{h} · F^{i}^{∗} = 0, i.e., F^{i}^{∗} · F^{i}^{∗} = −2, whence, finally, F^{i} · F^{i} = −2. The above claim shows then that J ⊄ 𝔪^{2}. □

8The converse is part of the exercise at the end of the first paragraph.

References

[La] R. Lazarsfeld, Positivity in Algebraic Geometry, Draft available at

     <http://www.math.lsa.umich.edu/~rlaz/>.

[L1] J. Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Publ. Math. IHES 36 (1969), 195–279.

[L2] , On complete ideals in regular local rings, Algebraic Geometry and Commutative Algebra, vol. I, in honor of Masayoshi Nagata, Kinokuniya, Tokyo, 1988, pp. 203–231.

[L3] , Adjoints and polars of simple complete ideals in two-dimensional regular local rings, Bull. Soc. Math. Belgique 45 (1993), 224–244.

[L4] , Proximity inequalities for complete ideals in two-dimensional regular local rings, Contemporary Mathematics 159 (1994), 293–306.

[L5] , Adjoints of ideals in regular local rings, Math. Research Letters 1 (1994), 739– 755.

[LS] and A. Sathaye, Jacobian ideals and a theorem of Briançon-Skoda, Michigan Math. J. 28 (1981), 199–222.

[Z] O. Zariski, Algebraic Surfaces, (2nd supplemented edition), Springer-Verlag, New York, 1971.

[ZS] and P. Samuel, Commutative Algebra, vol. 2, van Nostrand, Princeton, 1960.

Dept. of Mathematics, Purdue University, W. Lafayette IN 47907, USA

   E-mail address: lipman@math.purdue.edu

Dept. of Mathematics, Nihon University, Sakura Josui 3-25-40, Setagaya, Tokyo 156-8550, Japan

   E-mail address: watanabe@math.chs.nihon-u.ac.jp