Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 42 (2001), No. 2, 443-450.

Rings of Global Sections in Two-dimensional Schemes

Holger Brenner

Fakultät für Mathematik, Ruhr-Universität Bochum 44780 Bochum, e-mail: brenner@cobra.ruhr-uni-bochum.de

Abstract. In this paper we study the ring of global sections Γ(U, O) of an open subset U = D(I) ⊆ Spec A, where A is a two-dimensional noetherian ring. The main concern is to give a geometric criterion when these rings are finitely generated, in order to correct an invalid statement of Schenzel in [7].

1. Introduction

Let A be a noetherian ring with an ideal I ⊆ A and U = D(I) ⊆ Spec A the corresponding open subset. If U is an affine scheme, then the ring of global sections B = Γ(U, O_{X}) – which is also called the ideal-transform T (I) – is of finite type over A. The converse is by no means true, in dimension two however we have the following result due to Eakin et. al. ([4], Theorem 3.2): Suppose A is a local excellent^{1} Cohen-Macaulay domain of dimension two, and let I be an ideal of height one. Then (among other characterizations) D(I) is affine if and only if B is noetherian if and only if B is of finite type over A. Schenzel states in [7], Theorem 4.1 and 4.2, that this holds more general for two-dimensional excellent local domains. However, this is not true, as the following example shows.^{2} Example. Let X = Spec A be an affine excellent irreducible surface which is regular outside one single closed point P and such that in the normalization two points P_{1}, P_{2} lie over P. Outside these points the normalization mapping X̃ ⟶ X is isomorphic. Let Y = V (I) be the image of an irreducible curve Y′ passing through P_{1}, but not through P_{2}. Then U = X −Y is not affine, since the preimage of Y consists of the curve Y′ and of the isolated point P_{2}. On the other hand, U = X − Y is normal and isomorphic to X̃ − Y′ − P_{2}, so the rings of global sections are identical. Since X̃ is normal, this ring equals also the ring of global sections of X̃ − Y′. X̃ is a normal excellent affine surface, thus the complement of a curve is affine, and B is finitely generated. For an explicit example see below.

1In fact the result was stated under the somewhat weaker conditions that the normalization is finite and the local rings of the normalization are analytically irreducible, instead of excellent.

2The mistake in [7] is at the end of the proof of Theorem 4.1, where the statement T ⊆ T_{N} is wrong.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2001 Heldermann Verlag

In this paper we give a criterion for two-dimensional local rings to decide the finiteness of the ring of global sections of U = D(I), I an ideal of height one. The criterion is based on the combinatoric of the components in the completion Â of A. It says that in case U is not affine the ring of global sections of U is not finitely generated if and only if there exists an irreducible component of Spec Â where U is affine and a component where U is not affine such that their intersection is one-dimensional. The criterion is (due to the connectedness theorem of Hartshorne) seen to be fulfilled in case A is Cohen-Macaulay, thus we recover the result of Eakin et. al. as a corollary (Cor. 2.4). Another consequence is that if D(I) is non-affine and connected, then T (I) is not noetherian (Cor. 2.3). In the third section we extend the result to the non-complete case and describe the conditions used in the criterion in terms of the normalization.

2. The complete case

Let X = Spec A be the spectrum of a local complete noetherian ring A of dimension 2, and let P denote the closed point. Let X_{j} = V (p_{j}) = Spec A/p_{j} be the irreducible components of X corresponding to the minimal primes p_{j}, j ∈ J. Let I be an ideal in A, Y = V (I) and U = D(I). U is affine if and only if U_{j} = U ∩ X_{j} is affine on every component, and this is due to the theorem of Lichtenbaum-Hartshorne (see [3], 8.2.1) the case if and only if ht I(A/p_{j}) ≤ 1 for every j ∈ J. Thus U is not affine if and only if there exists a two-dimensional component X_{j} where Y_{j} = Y ∩ X_{j} consists just of the single point P. We want to know for an ideal I of height one whether the ring of global sections of D(I) is finitely generated. If D(I) is affine, this is the case, so we suppose furtheron that D(I) is not affine. We divide J = J_{0} ∪ J_{1} in such a way, that for j ∈ J_{1} the open subsets U_{j} ⊆ X_{j} are affine and for j ∈ J_{0} not. Thus the X_{j}, j ∈ J_{0}, are the two-dimensional components of X where Y_{j} is just the closed point. The affineness of U is equivalent with J_{0} = ∅. Put a_{0} = ⋂_{j∈J_{0}} p_{j} and a_{1} = ⋂_{j∈J_{1}} p_{j} and X_{0} = Spec A/a_{0}, X_{1} = Spec A/a_{1}. We denote the structure sheaves on these closed subschemes of X with O_{i}, i = 0, 1. Furthermore we put U_{i} = U ∩X_{i}, i = 0, 1, considered as an open subset in X_{i} with the induced scheme structure, put B_{i} = Γ(U_{i}, O_{i}). U_{1} = Spec B_{1} ⊂ X_{1} is affine, U_{0} is not affine. The closed embedding X_{1} ↪ X yields a (closed) restriction map Γ(U, O_{X}) ⟶ Γ(U_{1}, O_{1}) = B_{1}.

Finally, let b = a_{0} + a_{1} ⊆ m and R = A/b. R is a zero- or one-dimensional local complete noetherian ring, let Z = Spec R und Z^{×} = D(m) ⊂ Z. The dimension of Z = V (b) = X_{0}∩X_{1} is the crucial point for Γ(U, O_{X}) to be noetherian or not.

[figure: Z; Y; X_{1}; X_{0}; Y; X_{1}; X_{0}]

For our proof we have to put on A the condition S_{1} of Serre, meaning that every associated prime of A is minimal, equivalently that every zero-divisor lies in a minimal prime or that every ideal of height one contains a non-zero-divisor. This is fulfilled for example if A is reduced.

Theorem 2.1. Let A be a two-dimensional complete local noetherian ring, fulfilling the condition S_{1}. Let I be an ideal of height one and suppose that U = D(I) is not affine. Then the following are equivalent.

(1) Γ(U, O_{X}) is not of finite type.

(2) Γ(U, O_{X}) is not noetherian.

(3) The image of Γ(U, O_{X}) ⟶ Γ(U_{1}, O_{1}) is not noetherian.

(4) The intersection Z of X_{0} and X_{1} is one-dimensional.

Proof. The implications (3) ⇒ (2) and (2) ⇒ (1) are clear. (1) ⇒ (4). Suppose Z = {P } is only the closed point. Then U is the disjoint union of U_{0} and U_{1} (both closed hence open in U). Thus we have

Γ(U, O_{X}) = Γ(U_{0}, O_{0}) ⊕ Γ(U_{1}, O_{1}) .

Since U_{1} is affine, the second component is of finite type. Since U_{0} = X_{0} − {P }, the mapping A/a_{0} ⟶ Γ(U_{0}, O_{X}) is also of finite type, see Lemma 2.2 (1). So we have to show (4) ⇒ (3). We denote the image of Γ(U, O_{X}) ⟶ Γ(U_{1}, O_{1}) by C. Let h ∈ A be an element such that in Z = Spec R we have V (h) = V (m) = {P }. Thus 1/h is a function defined on Z^{×} = Z − {P } = D(h). Since Z^{×} ↪ U_{1} is a closed embedding and since U_{1} is affine, there exists a function q ∈ B_{1} = Γ(U_{1}, O_{1}) with q |_{Z^{×}}= 1/h.

[figure: U_{0}; X_{0}; Z^{×}; Z; U_{1}; X_{1}; ↪; ↪; ↩; ↩; ∪; ∪; ∪; U; ∩]

Let a ∈ b ⊂ A be a regular element (i.e. a non-zero-divisor) inside the describing ideal of Z. The functions aq^{n} are defined on U_{1} and the restrictions to Z^{×} are zero, thus they are extendible to Z. Since Z ↪ X_{0} is closed and X_{0} is affine, these functions are also extendible to X_{0} and in particular to U_{0}. So we may assume that these functions are defined on U and we see that they lie in C. Consider in C the ideal (a, aq, aq^{2}, aq^{3}, ...) spanned by this functions, and suppose that it is finitely generated. Then we have an equation

aq^{n+1} = a_{n}aq^{n} + ... + a_{1}aq + a_{0}a

with a_{i} ∈ C ⊂ B_{1}. We may assume that a_{i} ∈ Γ(U, O_{X}). Since a is regular in A, it is also a regular in A/a_{i}. (For if ax ∈ a_{i} = ⋂_{j∈J_{i}} p_{j}, we have ax ∈ p_{j} for all j ∈ J_{i} and thus x ∈ p_{j} for all j ∈ J_{i}, so x = 0 mod a_{i}.) Since the restriction A/a_{1} = Γ(X_{1}, O_{1}) ⟶ Γ(U_{1}, O_{1}) is injective, a is also a regular element in B_{1}. This yields in B_{1} (on U_{1}) the equation q^{n+1} = a_{n}q^{n} + ... + a_{1}q + a_{0}. This equation restricted to Z^{×} ⊆ U_{1} yields an integral equation for q = 1/h over R[a′_{i}] ⊆ R_{h}, where the a′_{i} denote the restrictions of a_{i} on R_{h} = Γ(Z^{×}, O_{Z}). We claim that the a′_{i} are integral over R: Consider the elements a_{i} ∈ Γ(U, O_{X}) as functions on U_{0} – as elements of B_{0}. Since U_{0} = X_{0}−{P }, the a_{i} ∈ B_{0} are integral over A/a_{0} = Γ(X_{0}, O_{0}), see Lemma 2.2. The closed embeddings (Z^{×} ⊂ Z) ↪ (U_{0} ⊂ X_{0}) show that the a′_{i} are integral over R = Γ(Z, O_{Z}). It follows that q |_{Z}= 1/h would be integral over R, but this is not possible. □

Lemma 2.2. Let A be a local noetherian ring of dimension two fulfilling S_{1}. Let m be the maximal ideal and B = Γ(D(m), O) the ring of global sections. Then the following hold.

(1) A ⟶ B is of finite type.

(2) If furthermore all components of Spec A have dimension two, B is even finite over A.

Proof. We first prove the second part, using [6], 5.11.4 (or [2], 2.5.). A point x ∈ Ass O_{X} has height zero, for every ideal of bigger height contains a regular element. The closure x̄ is a two-dimensional component and therefore the point P has codimension two on it. The first part follows from the second part. The one-dimensional components of X meet the other components only in the closed point, thus the punctured curves are connected components of W = D(m). These are affine and of finite type. □

We deduce from the theorem two corollaries.

Corollary 2.3. Let A be a local complete noetherian ring of dimension two fulfilling S_{1}. Let I be an ideal of height one. If U = D(I) is connected and Γ(U, O_{X}) is of finite type, then U is affine.

Proof. Suppose U is not affine, then in the partition described above U_{0} is not empty, and U_{1} is not empty since I is of height one. Put Z = X_{0} ∩ X_{1}. Since U is connected, U_{0} and U_{1} are not disjoint, thus Z does not consist only of the closed point, it must be a curve. Then due to the theorem the ring of global sections can not be noetherian. □

We recover the result of Eakin et. al. in the Cohen-Macaulay case.

Corollary 2.4. Let A be a local complete noetherian Cohen-Macaulay ring of dimension two. Let I be an ideal of height one. Then U = D(I) is affine if and only if its ring of global sections is of finite type (or noetherian).

Proof. Again, suppose U to be not affine, put X = X_{0}∪X_{1} as before with the describing ideals a_{0} and a_{1}. Then a_{0} ∩ a_{1} is nilpotent, thus due to the connectedness theorem of Hartshorne (see [5], Theorem 18.12) the ideal a_{0} + a_{1} has height one. Since it describes the intersection, Z = X_{0} ∩ X_{1} is one-dimensional and Γ(U, O_{X}) is not noetherian. □

Example. Of course, U = D(I) can be affine without being connected. A = K[[x, y, z]]/(xy) is Cohen-Macaulay (K a field), the complement of the common axis V (x, y) is affine, but not connected.

Remark. We may associate to a complete local ring of dimension two a graph Γ in such a way, that for each irreducible two-dimensional component we associate a point, and two points are connected by an edge if and only if the intersection of the corresponding components is one-dimensional. Then an open subset as above yields a partition Γ = Γ_{0} ∪ Γ_{1}, and the ring of global sections is noetherian if and only if there is no edge between points of Γ_{0} and of Γ_{1}.

3. Interpretation in the normalization

We want to extend the result from the complete case to the general case. Suppose we are given a curve V (I) ⊆ Spec A where A is a two-dimensional noetherian domain. Then Γ(D(I), O) is of finite type if this is true in every (closed) point x ∈ Spec A, see [1]. Furthermore, we have the following lemma.

Lemma 3.1. Let A ⟶ A′ be faithfully flat and let U ⊆ Spec A denote an open subset with preimage U′. Then B = Γ(U, O) is of finite type over A if and only if B′ = Γ(U′, O′) is of finite type over A′.

Proof. We have B′ = B ⊗_{A} A′ due to flatness. This yields the first implication. If B′ is of finite type, we may assume that it is generated by finitely many elements of B, thus there is a surjection A′[T_{1}, …, T_{n}] ⟶ B′ = B ⊗_{A} A′ induced by A[T_{1}, …, T_{n}] ⟶ B. Due to faithfulness, this must also be surjective. □

Therefore the condition in the theorem that Γ(U, O) is of finite type is preserved by passing to the completion, and we may skip in Cor. 2.4 the assumption of completeness. So we take a look at the condition that the intersection of two components in the completion is one-dimensional, and we want to describe it in terms of the normalization of A. For this we recall some correspondences between normalization and completion, see [6], 7.6.1 and 7.6.2. Let X be the spectrum of a local excellent domain A with completion X̂ and normalization X̃. Then the normalization of X̂ equals the completion of X̃ (semilocal), and this consists of connected components being the normalizations of the irreducible components of X̂ and the completion of the localizations of X̃ as well. In particular, there is a correspondence between the irreducible components of X̂ and the closed points of X̃.

For a closed subset C ⊆ X the completion of C equals the preimage of C in X̂ yielding a canonical inclusion Ĉ ⊆ X̂. The irreducible components of Ĉ correspond again to closed points of C̃, but this is of course not the preimage of C in the normalization X̃.

Lemma 3.2.. Let A be an excellent local domain of dimension two, P_{0} ∈ X̃ the closed point on X̃ corresponding to the irreducible component X_{0} of the completion X̂. Let C ⊂ X be an irreducible curve and let D ⊂ X̃ be the preimage of C without the isolated points.

(1) There exists an irreducible component of Ĉ on X_{0} if and only if P_{0} is not an isolated point on φ^{−1}(C) (φ : X̃ ⟶ X normalization map ).

(2) The irreducible component C_{0} of Ĉ lies on X_{0} if and only if there exists a point R ∈ D̃ over P_{0} mapping to the point Q_{0} ∈ C̃ corresponding to C_{0}.

(3) The component C_{0} of Ĉ connects the irreducible components X_{1} and X_{2} of X̂ if and only if the corresponding point Q_{0} ∈ C̃ is reached by points R_{1}, R_{2} ∈ D̃ lying over P_{1} and P_{2}.

Proof. (1) We consider the mapping (completion) X̃_{0} ⟶ X̃_{P_{0}}, where X̃_{P_{0}} means the localization at P_{0}. The preimage of C ⊂ X in X̃_{P_{0}} is just the closed point if and only if this is true in X̃_{0}, and this is the case if and only if Ĉ is zero-dimensional on X_{0}.

(2) The preimage of Ĉ in ˜ X̂ without the isolated points equals D̂, being the preimage of D. The statement C_{0} ⊂ X_{0} is equivalent to the statement that there exists an irreducible component D_{0} ⊆ D̂ ⊂ ˜ X̂ over C_{0} lying on X̃_{0}. Let R be the point on D̃ corresponding to the component D_{0} ⊆ D̂. Suitable diagrams show that D_{0} dominates C_{0} is equivalent with R maps to Q_{0} and that D_{0} ⊆ X̃_{0} is equivalent with R maps to P_{0}.

(3) follows from (2). □

This motivates the following definition.

Definition. Let X denote a reduced irreducible noetherian scheme, φ : X̃ ⟶ X its normalization, P ∈ X a closed point and P_{1}, P_{2} ∈ X̃, φ(P_{1}) = φ(P_{2}) = P. We call an irreducible curve C ⊂ X a melting curve for the points P_{1} and P_{2} if and only if P_{1}, P_{2} are not isolated on φ^{−1}(C) and there exist points R_{1}, R_{2} ∈ D̃ (D as in Lemma 3.2) over P_{1}, P_{2} mapping to one common point Q ∈ C̃.

Theorem 3.3. Let X = Spec A, where A is an excellent local domain of dimension two. Then the intersection of the components X_{1} and X_{2} on X̂ is one-dimensional if and only if there exists a melting curve for P_{1}, P_{2} ∈ X̃.

Proof. If C is a melting curve for P_{1} and P_{2} with common point Q as in the definition, then the previous proposition says that the corresponding component C_{0} lies on X_{1} and X_{2}, thus the intersection is one-dimensional. For the converse, let C_{0} be an irreducible curve on X_{1} ∩ X_{2} with prime ideal q ⊂ Â of height one. Then p = q ∩ A is also of height one. For q is not a normal point of Â, since on the normalization there are at least two points above it. Then also p is not a normal point, because the normal locus commutes with completion under the condition of excellence (see [6], 7.8.3.1.) Thus ht p = 1, C = V (p) is a curve, C_{0} a component of its completion and we may apply the previous proposition. □

Proposition 3.4. Let P_{1}, P_{2} be two closed points in the normalization X̃ over P ∈ Spec A, where A is a two-dimensional noetherian domain. Then the following hold.

(1) If there exist two different irreducible curves C_{1}, C_{2} with P_{i} ∈ C_{i} = V (q_{i}) on X̃ such that q_{1} ∩ A = q_{2} ∩ A = p, then C = V (p) is a melting curve for P_{1}, P_{2}.

(2) If C is normal (or analytically irreducible) and P_{1} and P_{2} are not isolated on φ^{−1}(C), then C is a melting curve.

(3) If P_{1}, P_{2} ∈ C′ is irreducible and φ(C′) = C is a melting curve, then φ|_{C′} : C′ ⟶ C is not birational. A melting curve lies in the non-normal locus.

Proof. (1) Both mappings C_{1} ⟶ C and C_{2} ⟶ C are surjective, and this is then also true for the normalizations. Thus for any closed point Q ∈ C̃ there are points on C̃_{i} over P_{i} mapping to Q.

(2) If C is analytically irreducible, then any closed point of D̃ maps to the only closed point of C̃.

(3) Suppose C′ ⟶ C is birational. Then the normalizations of these curves are the same, and different points cannot be identified. If the generic point of a curve C is normal, then D consists just of one irreducible component, and D ⟶ C is birational. □

Examples. We give some typical examples of (non-)melting curves to illustrate the cases the previous proposition is talking about. They are given by mappings A_{K}^{2} ⟶ A_{K}^{n} such that the affine plane is the normalization of the image (K is a field).

(1) (x, y) ⟼ (x, y^{3} −y, y^{2} −1). This identifies the two different curves V (y −1) and V (y +1). The common image curve C is a melting curve.

[figure: ⟶; ^{........................}; ^{........................}]

(2) (x, y) ⟼ (x, y^{2}, xy). The line V (x) is melted with itself, identifying the points (0, 1) and (0, −1). V (x) ⟶ V (r, t) is not birational, C is a melting curve.

[figure: ⟶]

(3) (x, y) ⟼ (x, y^{2}, y((y − 1)^{2} + x^{2})((y + 1)^{2} + x^{2}), xy). This identifies only the two points. V (x) is birational with its image C, thus C is not a melting curve.

[figure: ⟶]

(4) Consider the mapping (x, y) ⟼ (x, y^{2}, y(x^{2} − y^{2}(y + 1)) followed by the identification of the points (0, 0, 0) and (−1, 0, 0). Then D = V (x^{2} − y^{2}(y + 1)) ⟼ C is not birational, but C (= the image of D) is not a melting curve for their common point. Thus the necessary condition in Prop. 3.4 (3) is not sufficient.

References

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[2] Bingener, J.; Storch, U.: Resträume zu analytischen Teilmengen in Steinschen Räumen. Math. Ann. 210 (1974), 33–53.

[3] Brodman, M.P.; Sharp, R.Y.: Local Cohomology. Cambridge 1998.

[4] Eakin, P.M.; Heinzer, W.; Katz, D.; Ratliff, L.J.: Notes on ideal-transforms, Rees rings and Krull rings. J. Algebra 110 (1987), 407–419.

[5] Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. New York 1995.

[6] Grothendieck, A.; Dieudonné, J.: Eléments de géométrie algébrique IV. Publ. Math. I.H.E.S. 20,24,28,32, 1964–1967.

[7] Schenzel, P.: Flatness and Ideal-Transforms of Finite Type. In: Commutative Algebra (ed. W. Bruns, A. Simis), 1988.

Received June 21, 2000