The blowup closure of a set of ideals with applications to T I closure

Sara Faridi ^{∗}

October 1, 2001

Abstract

We introduce a new closure operation on sets of ideals in a commutative Noetherian ring of characteristic p, called the blowup closure. We develop the theory of this operation and prove that given a set of ideals in a Noetherian ring of characteristic p, under mild conditions, the tight integral closure of this set agrees with the blowup closure of certain extensions of those ideals in an extension of the original ring. We then use properties of blowup closure to settle open questions on tight integral closure posed by Hochster in [Ho2]. In particular, we show that under mild conditions on the ring, tight integral closure persists under ring maps, and that it commutes with localization if and only if tight closure does.

In [Ho2] Hochster defined an operation on a set of ideals in a commutative Noetherian ring of positive characteristic : tight integral closure (or TI closure). This operation mixes the ideas of tight and integral closure in the sense that one can describe the tight closure or the integral closure of any given ideal as the T I closure of a certain set of ideals. Using this operation, Hochster was able to prove a Briançon-Skoda type theorem for sets of ideals in a regular ring which significantly improved the original tight closure version in [HH1]. However, it turned out to be very difficult to verify basic properties of TI closure, even though from the definition one would expect most of the properties of tight closure to generalize to T I closure.

In this paper we introduce the blowup closure of a set of ideals which has the advantage that one can reduce its study to the case where all the ideals are principal. In practice, this is the simplest case in which one can work with several ideals at the same time. Moreover, we show that under mild conditions, T I closure can be described as a blowup closure in a larger ring (Corollary 3.5).

Consequently, we show that the T I closure of a set of ideals can be expressed as the tight closure of one ideal in an extension ring of the original one. We apply this fact to address the questions on T I closure that were stated in [Ho2]. We settle the question of persistence of T I closure under ring maps (Theorem 3.4), and show (Theorem 3.6) that T I closure commutes with localization if and only if tight closure does.

We also develop a theory of test elements for T I closure in Section 3.3. Test elements are the key ingredients for tight closure arguments, and the existence of test elements is one of the most important results in tight closure theory (see [HH2] or [HH3]). However, this notion does not exist for integral closure, and so we alter the traditional definition of test elements in tight closure theory to make sense of test elements for T I closure. T I closure test elements, unlike those in tight closure, depend on the ideals that one works with. In Theorem 3.12, we describe specific test elements for the T I closure of a set of ideals in an affine algebra, via a similar result of [HH2] for tight closure.

∗Mathematics Department, George Washington University, Washington, DC 20052. E-mail: faridi@gwu.edu

This result produces an easy method for calculating T I closure test elements when the defining set of equations for the algebra is known.

In Section 3.4 we define test exponents for T I closure (see [HH4] for this notion in tight closure theory) and show that the existence of test exponents in T I closure is related to the problem of T I closure commuting with localization. The final section of the paper deals with T I closure in characteristic zero.

Acknowledgments. The contents of this paper are part of my PhD thesis at The University of Michigan. I am grateful to Karen Smith and Mel Hochster for their guidance during the preparation of the thesis.

1 Basic Definitions

In all the discussions, we assume that all rings are commutative with identity. When we refer to a ring of characteristic p, we mean p is a positive prime integer.

Notation. If R is a ring, then x ∈ R^{o} means that x is an element of R that is not in any minimal prime of R, and M(R) is the set of all minimal primes of R. By R′ we mean the normalization of R (see Definition 1.1). When R has prime characteristic p, q is a power of p, and I is an ideal of R, I^{[q]} denotes the ideal generated by the qth powers of the elements of I. In particular, if I is generated by x_{1}, …, x_{n}, then I^{[q]} is generated by x_{1}^{q}, …, x_{n}^{q}. For a graded ring S and f a homogeneous element of S, by S_{(f)} we mean the zeroth graded piece of the localized ring S_{f}, i.e., S_{(f)} = (S_{f})_{0}.

1.1 Integral Closure

Definition 1.1. Let R be a Noetherian ring, and let p_{1}, …, p_{m} be the minimal primes of R. Then we define the normalization of R, denoted by R′, to be

(R/p_{1})′ × … × (R/p_{m})′,

where for i = 1, …, m, (R/p_{i})′ is the integral closure of the domain R/p_{i} in the field of fractions of R/p_{i}.

We refer the reader to [M] or [BH] for basic properties of integral closure of rings and ideals. We state some well-known facts on integral closure that we will use later in this paper; for written proofs we refer the reader to [F].

Proposition 1.2. Let R be a reduced Noetherian ring, and let p_{1}, …, p_{m} be the minimal primes of R. Then:

(a) The minimal primes of R′ are p_{1}^{♯}, …, p_{m}^{♯}, where for 1 ≤ i ≤ m,

p_{i}^{♯} = (R/p_{1})′ × … × (R/p_{i−1})′ × (0) × (R/p_{i+1})′ × … × (R/p_{m})′;

(b) R′/p_{i}^{♯} ≃ (R/p_{i})′, for all i.

We next focus on the notion of integral closure of an ideal.

Definition 1.3. For a ring S and an ideal J of S, the integral closure of J in S, denoted by \overline{J}, is defined as the set of all all x ∈ S that satisfy an equation of the form x^{n}+a_{1}x^{n−1}+...+a_{n−1}x+a_{n} = 0 where a_{i} ∈ J^{i}, i = 1, ..., n; or equivalently, the set of all x ∈ S for which there exists a c ∈ S that does not belong to any minimal prime of S, such that cx^{n} ∈ J^{n} for all positive integers n ([Ho2] 1.2).

Below we record a useful feature of integral closure that we use often in the later sections.

Proposition 1.4. Let R be a domain, and let I = (g_{1}, …, g_{s}) be an ideal in R. For every i, let S_{i} be the normalization of R[g_{1}/g_{i}, …, g_{s}/g_{i}]. Then x ∈ \overline{I} if and only if x ∈ IS_{i} for all i = 1, …, s.

Proof. Suppose that x ∈ \overline{I}. Then it follows that x ∈ \overline{IS_{i}} = IS_{i} for all i, since IS_{i} = (g_{i})S_{i} is a principal ideal, and S_{i} is a normal ring, and so IS_{i} is integrally closed.

Now suppose that x ∈ IS_{i} for all i. Let V be a valuation domain containing R. Since I is a finitely generated ideal, the image of I in V will be generated by one of the g_{i}. It follows that for some i, S_{i} ⊆ V. Then x ∈ IS_{i} implies that x ∈ IV, and so x ∈ \overline{I} (see [ZS2] page 353).

The integral closure of an ideal I in a ring R is closely related to the normalization of the Rees ring of I, where the Rees ring of I is the graded subring of the polynomial ring R[t] of the form:

R[It] = R ⊕ It ⊕ I^{2}t^{2} ⊕ ….

A well known description for the integral closure of Rees rings, which follows from Proposition 20 in Chapter 5 of [B], is the following:

Theorem 1.5. Let R be a normal domain, and I an be ideal of R, and t be an indeterminate over R. Then the normalization of R[It] in its field of fractions is

R ⊕ \overline{I}t ⊕ \overline{I^{2}}t^{2} ⊕ …,

where \overline{J} denotes the integral closure of the ideal J in R.

1.2 Tight Closure

Tight closure is an operation on ideals in Noetherian rings of positive prime characteristic p and Noetherian rings that contain a field. The main idea for characteristic p arguments is that when R is a ring of prime characteristic p, the Frobenius map (the map from R to R that takes an element r to its pth power r^{p}) will be a ring homomorphism. The theory is extended to rings of characteristic zero via the method of reduction to characteristic p. For details and references on tight closure, and for proofs of the facts stated below, see [HH1], [Hu] or [BH].

Definition 1.6. Let R be a Noetherian ring of characteristic p > 0. Let I be an ideal of R. Then the tight closure of I , denoted by I^{∗}, is the collection of all z ∈ R for which there exists c ∈ R^{o} such that cz^{q} ∈ I^{[q]} for all large q = p^{e}. We say that an ideal I is tightly closed if I = I^{∗}.

It is easy to check that the tight closure of an ideal is itself an ideal. In general the integral closure of an ideal is much larger than its tight closure.

Example 1.7. Let R = k[x, y] be a polynomial ring over a field of characteristic p. Let I = (x^{2}, y^{2}). Then \overline{I} = (x^{2}, xy, y^{2}), while I^{∗} = I = (x^{2}, y^{2}) since R is a regular ring.

A key element in tight closure theory is the existence of test elements.

Definition 1.8. Let R be a Noetherian ring of characteristic p. An element c ∈ R^{o} is called a test element for R, if for every ideal I of R and every x ∈ I^{∗}, cx^{q} ∈ I^{[q]} for all q = p^{e}, e ≥ 0. The ideal generated by the test elements for R is called the test ideal of R, and is denoted by τ(R).

Under certain mild conditions on the ring, Hochster and Huneke proved that test elements exist. Before stating this result, we introduce some terminology.

Note 1.9. If R is a ring of characteristic p, R is essentially of finite type over a ring S if it is a localization of a finitely generated algebra over S. R is F-finite if the Frobenius map is a finite map. Below, as well as in the next sections, we frequently assume that our rings are F -finite or essentially of finite type over an excellent local ring. It is worth pointing out that these are not very restrictive conditions on a ring. For example, any reduced finitely generated algebra over a perfect field or any complete local ring with perfect residue field will satisfy these properties (see Chapter 10 of [BH] or [Hu]). It is straightforward to check that both these properties are preserved after localizing at a multiplicative set in R, and after taking finitely generated algebra extensions.

Theorem 1.10 (existence of test elements). Let R be a Noetherian reduced ring of positive characteristic p, and suppose that R is essentially of finite type over an excellent local ring, or that R is F -finite. Let c ∈ R^{o} be such that R_{c} is regular. Then c has a power that is a test element for R, and remains so after localizing or completing R.

A very useful property of tight closure follows from this theorem.

Theorem 1.11 (persistence of tight closure). Let R ⟶ S be homomorphism of Noetherian rings of positive characteristic p. Suppose that R is essentially of finite type over an excellent local ring, or that R_{red} (i.e. R modulo the nilradical of R) is F -finite. Then

I^{∗}S ⊆ (IS)^{∗}.

These theorems are [Hu] 2.1 and 2.3, respectively.

Remark 1.12. Persistence improves several features of tight closure. One property that we will use is the following: If tight closure persists for the map f : R ⟶ S of Noetherian rings, then the contraction of a tightly closed ideal of S will be a tightly closed ideal of R. To see this, let J ⊆ S be tightly closed in S, and take u ∈ (f^{−1}(J))^{∗}. Then f(u) ∈ (f^{−1}(J)S)^{∗} ⊆ J^{∗} = J (see Proposition 10.1.2 of [BH]). Hence f(u) ∈ J, and so u ∈ f^{−1}(J).

1.2.1 Tight Closure in Equal Characteristic Zero

The method of reduction to characteristic p allows one to extend tight closure theory to rings of equal characteristic zero, that is, rings of characteristic zero containing a field. Consequently, tight closure results in characteristic p find analogous statements in characteristic zero. For a brief but more detailed treatment of this topic see [Ho1] or [HH1]. The complete and general source on tight closure theory in equal characteristic zero is [HH2].

The definition of tight closure in equal characteristic zero is based on the existence of descent data, which provide us with a finitely generated subalgebra R_{D} of the ring over a finitely generated algebra D over the integers. We can then look at fibers of R_{D} over maximal ideals of D, which are rings of positive characteristic over finite (and therefore perfect) fields. We define tight closure for R via the (usual positive characteristic) definition of tight closure for these fibers.

Definition 1.13. Let R be a finitely generated algebra over a field K of characteristic zero, let I be an ideal of R and let x ∈ R. By descent data for R, I, and x we mean a triple (D, R_{D}, I_{D}), where D is a finitely generated Z-subalgebra of K, R_{D} is a finitely generated D-subalgebra of R, and I_{D} is an ideal of R_{D} such that:

(a) I_{D} and R_{D}/I_{D} are D-free.

(b) The canonical map K ⊗_{D} R_{D} → R induced by the inclusions of K and R_{D} in R is a Kalgebra isomorphism.

(c) I = I_{D}R.

(d) x ∈ R_{D}.

Descent data always exists: see [HH2] Section 2.1.

Definition 1.14. Let R be a finitely generated algebra over a field K of characteristic zero and let I be an ideal of R. We say that an element x of R is in the tight closure of I, denoted by I^{∗}, if there exist descent data (D, R_{D}, I_{D}) such that for every maximal ideal m of D, if k = D/m, then x_{k} ∈ I_{k}^{∗} in R_{k}, where the subscript k denotes images after applying k⊗_{D}. If I = I^{∗}, then we say that I is tightly closed.

In fact, by replacing D by a localization at a single element, one can see that it suffices to check that for almost all m ∈ MaxSpecD, i.e. for all m in a Zariski dense open subset of MaxSpecD, if k = D/m, then x_{k} ∈ I_{k}^{∗}.

The following theorem follows from [HH2] 2.5.2 and 2.5.3:

Theorem 1.15 (independence of choice of descent and uniform multipliers). Let K be a field of characteristic zero, let R be a finitely generated K-algebra, let I be an ideal of R and let u ∈ R. Let (D, R_{D}, I_{D}) be descent data for R, I, and u.

(a) If u ∈ I^{∗}, then for every maximal ideal m of D, if k = D/m, then x_{k} ∈ I_{k}^{∗} in R_{k}.

(b) There is an element c_{D} of R_{D}^{o} such that u ∈ I^{∗} iff for almost all maximal ideals m of D and k = D/m, c_{k}u_{k}^{q} ∈ I_{k}^{[q]} for all positive powers q of p.

Part (b) involves the existence of universal test elements; these are the characteristic zero analogues of test elements ([HH2] 2.4.2). When R is reduced and equidimensional, I ⊆ R, u ∈ I^{∗} and c_{D} ∈ R_{D} is a universal test element, then for almost all m ∈ MaxSpecD, if k = D/m, we have c_{k}u_{k}^{q} ∈ I_{k}^{[q]} for all positive powers q of p. We will state a theorem due to Hochster and Huneke which enables us to explicitly calculate universal test elements in Section 3.5 (Theorem 3.23).

1.3 Tight Integral Closure

Tight integral closure (or T I closure) is an operation on a set of ideals in a Noetherian ring that generalizes the ideas of tight and integral closure. This notion was introduced by Hochster in 1998 ([Ho2]). Below, we give a brief description of T I closure and refer the reader to [Ho2] for properties of this operation.

Definition 1.16. Let R be a Noetherian commutative ring of prime characteristic p, and let I_{1}, …, I_{n} be ideals in R. We define x ∈ R to be in (I_{1}, …, I_{n})\underline{∗}, called the tight integral closure (or TI closure) of I_{1}, …, I_{n}, if and only if there exists an element c ∈ R^{o} such that cx^{q} ∈ I_{1}^{q} + … + I_{n}^{q}, for all large powers q of p.

Note that if n = 1, (I)\underline{∗} = \overline{I}, and if I_{1}, …, I_{n} are all principal ideals, then (I_{1}, …, I_{n})\underline{∗} = (I_{1} + … + I_{n})^{∗}.

It turns out that T I closure has several of the basic properties of both tight and integral closure: its study reduces to the case of domains and it respects inclusions of ideals. In the case of a set of monomial ideals in a polynomial ring, their T I closure can be computed as the sum of their integral closures (see [Ho2] for details of these results).

Using T I closure, Hochster was able to prove an improved version of the Briançon-Skoda theorem ([Ho2] Theorem 2.3). Below we demonstrate by an example this result in a case where T I closure can be computed.

Example 1.17. Let R be the polynomial ring k[x, y, u, v] over a field k of characteristic p, and let I = (u^{2}, v^{2}) and J = (x^{2}, y^{2}) be ideals of R. Then it follows from the T I closure version of the Briançon-Skoda theorem ([Ho2] Theorem 2.3), that \overline{(I + J)^{2}} ⊆ (I, J)\underline{∗}. So

(u, v, x, y)^{4} = \overline{(u^{2}, v^{2}, x^{2}, y^{2})^{2}} ⊆ ((u^{2}, v^{2}), (x^{2}, y^{2}))\underline{∗} = \overline{(u^{2}, v^{2})} + \overline{(x^{2}, y^{2})} = (u^{2}, v^{2}, uv, x^{2}, y^{2}, xy).

This is an improvement of the tight closure Briançon-Skoda theorem ([HH1] Theorem 5.4), which would yield \overline{(I + J)^{4}} ⊆ (I + J)^{∗}, and hence

(u, v, x, y)^{8} = \overline{(u^{2}, v^{2}, x^{2}, y^{2})^{4}} ⊆ (u^{2}, v^{2}, x^{2}, y^{2})^{∗} = (u^{2}, v^{2}, x^{2}, y^{2}).

TI closure can be defined for affine algebras over fields of characteristic zero by the method of reduction to characteristic p, described in Section 1.2. In [Ho2], characteristic zero analogues of several of the results on T I closure are stated.

Definition 1.18. Let R be a finitely generated algebra over a field K of characteristic zero, and let I = {I_{1}, …, I_{n}} be a set of ideals of R. We say that an element x of R is in the tight integral closure of I, denoted by I\underline{∗}, if there exist descent data (D, R_{D}, I_{1,D}, …, I_{n,D}), such that for every maximal ideal m of D, if k = D/m, then x_{k} ∈ I_{k}\underline{∗}, where I_{k} denotes the set of ideals \{I_{1,k}, …, I_{n,k}\}, with I_{t,k} = k ⊗_{k} I_{t,D}.

1.4 Questions on T I closure

As can be seen above, T I closure is able to generalize several statements, and to tie up tight closure and integral closure into one definition. However, there are many useful properties that both tight and integral closure satisfy, but have been turned out to be difficult to verify for T I closure using Definition 1.16. In [Ho2], Hochster stated the following questions:

Question 1.19. Does T I closure persist? That is, if h : R ⟶ S is a homomorphism of Noetherian rings of characteristic p, and I_{1}, …, I_{n} are ideals of R, then is it true that (I_{1}, …, I_{n})\underline{∗} S ⊆ (I_{1}S, …, I_{n}S)\underline{∗} ?

This property holds for both tight closure (with mild conditions on the ring, see Theorem 1.11) and for integral closure. It is easy to show that it also holds for for T I closure if h(R^{o}) ⊆ S^{o}; this happens for example when h is a flat map, or if h is any injective map of domains. In general, the lack of a test element theory for T I closure makes the problem obscure in its original setting. This question will be answered in Section 3.2.

Question 1.20. If R is a Noetherian ring of characteristic p, I_{1}, …, I_{n} are ideals of R and c ∈ R^{o} and x ∈ R are such that

cx^{p^{e}} ∈ I_{1}^{p^{e}} + … + I_{n}^{p^{e}},

for infinitely many e (rather than all large e), then can one conclude that x ∈ (I_{1}, …, I_{n})\underline{∗} ?

This is again a property that holds for both tight and integral closure, and it is reasonable to expect it for T I closure. In Section 3.2 we give an affirmative answer to this question.

Question 1.21. Can one develop a theory of test elements for T I closure (see Definition 1.8)?

Such a theory exists for tight closure (see Section 1.2), but it is not possible to define such a notion for integral closure that does not depend on the ideal:

Example 1.22. Let R be the polynomial ring k[x, y], where k is a field. If c ∈ R − {0} were an integral closure test element for R, then one would have cz^{m} ∈ I^{m} for all ideals I of R, all z ∈ \overline{I} and all positive integers m.

Now take the family of ideals \{I_{n}\}_{n∈N}, where for each n, I_{n} = (x^{2n}, y^{2n}). Then x^{n}y^{n} ∈ \overline{I_{n}} for all n, since (x^{n}y^{n})^{2} = x^{2n}y^{2n} ∈ I_{n}^{2}. As c is a test element, this implies that cx^{n}y^{n} ∈ (x^{2n}, y^{2n}) for all n. Since this holds in the polynomial ring k[x, y], one can reduce to the case where c is a monomial in R = k[x, y], which is not possible, since the degree of c will have to grow larger when n gets large.

Still, one could hope to find elements c that work for Definition 1.16, but depend on the ideals I_{1}, …, I_{n}. Such specific elements of R are introduced in Section 3.3.

Question 1.23. Does T I closure commute with localization?

It is known that integral closure commutes with localization, and the same is conjectured for tight closure. In Section 3.2 we show that the question of T I closure commuting with localization is equivalent to the question of tight closure commuting with localization.

Question 1.24. Let R be an affine algebra over a field K of characteristic zero, and I be a set of ideals in R and x ∈ I\underline{∗}. Let (D, R_{D}, I_{D}) be descent data. Then can one find a c ∈ R_{D}^{o} such that for all m ∈ MaxSpecD and k = D/m, c_{k}x_{k}^{q} ∈ I_{k}^{[q]} for all positive powers q of p?

The last question will be addressed in Section 3.5.

2 Blowup Closure

We now explore a new notion: the blowup closure of a set of ideals. Here, we are motivated by the fact that the extension of an ideal I to its blowup scheme ProjR[It] is locally principal. For a set of ideals, we consider the blowup scheme of their product, and each one of the original ideals will be locally principal there. This is the simplest situation to handle several ideals at the same time.

Although the definition of blowup closure makes sense for any commutative ring of positive characteristic, we immediately restrict ourselves to rings R such that R is either essentially of finite type over an excellent local ring, or R_{red} is F -finite. The reason is that tight closure persists for maps from such rings (Theorem 1.11), and this property simplifies most blowup closure arguments. These conditions are not very restrictive, since the class of such rings includes most rings that one would normally encounter in commutative algebra and in algebraic geometry (see Note 1.9).

Definition 2.1. Let R be a Noetherian commutative ring of prime characteristic p, and let I_{1}, …, I_{n} be ideals in R. We define x ∈ R to be in (I_{1}, …, I_{n})^{∼}, called the blowup closure of I_{1}, …, I_{n}, if and only if for every affine open set of the blowup of the product ideal I = I_{1} … I_{n}, if S is the coordinate ring of that affine set, then x ∈ (JS)^{∗}, where J = I_{1} + … + I_{n}.

The following discussion shows how for certain rings, one can reduce the process of checking if an element x is in (I_{1}, …, I_{n})^{∼} to checking if it is in (JS_{i})^{∗}, 1 ≤ i ≤ m, for any fixed open affine cover SpecS_{1}, …, SpecS_{m} of the blowup of I.

Lemma 2.2. Let R be a ring of positive characteristic p, and let J be an ideal of R. Then for x ∈ R, x ∈ J^{∗} if and only if there are g_{1}, …, g_{s} ∈ R that generate the unit ideal in R, and for each j, 1 ≤ j ≤ s, x ∈ (J_{g_{j}})^{∗}.

Proof. If x ∈ J^{∗}, it is immediate that x ∈ (JR_{f})^{∗} for all f ∈ R, therefore one direction is clear. Suppose we are given a sequence of elements g_{1}, …, g_{s} ∈ R such that (g_{1}, …, g_{s}) = R, and for each j, 1 ≤ j ≤ s, x ∈ (J_{g_{j}})^{∗}.

For each j, let c_{j} be such that c_{j}x^{p^{e}} ∈ J_{g_{j}}^{[p^{e}]} for all large enough e. One can replace each c_{j} by its product with a large enough power of g_{j}, so that c_{j} ∈ R. Let c be the product of all the c_{j}’s for j = 1, …, s. Then c ∈ R and cx^{p^{e}} ∈ J_{g_{j}}^{[p^{e}]} for all j = 1, …, s, and all large enough e.

Fix p^{e}. Then for each j, there is some power N_{j} of g_{j} such that g_{j}^{N_{j}}cx^{p^{e}} ∈ J^{[p^{e}]}. Let q be a power of p that is larger than all the N_{j} for j = 1, …, s. Then for all j, g_{j}^{q}cx^{p^{e}} ∈ J^{[p^{e}]}. On the other hand, since (g_{1}, …, g_{s}) is the unit ideal, so is (g_{1}^{q}, …, g_{s}^{q}), and so it follows that cx^{p^{e}} ∈ J^{[p^{e}]}. Therefore x ∈ J^{∗}.

Now let X be a scheme, and let J be a sheaf of ideals on X. Suppose that there is an open affine cover U_{1}, …, U_{n} of X, such that for each i, U_{i} = SpecB_{i}, where B_{i} is essentially of finite type over an excellent local ring, or (B_{i})_{red} is F -finite. Suppose x ∈ J (U_{i})^{∗} in O_{X}(U_{i}), for i = 1, …, n. Let U be any open affine set of X. We will show that x ∈ J (U)^{∗}.

One can write U as (U ∩ U_{1}) ∪ … ∪ (U ∩ U_{n}). If U = SpecA, then one can refine this cover of U into

SpecA_{f_{1}} ∪ … ∪ SpecA_{f_{s}},

where the elements f_{1}, …, f_{s} of A generate the unit ideal of A. So for every j, we have an inclusion SpecA_{f_{j}} ⊆ U_{i}, for some i, 1 ≤ i ≤ n. This corresponds to a homomorphism of rings B_{i} ⟶ A_{f_{j}}. Since x ∈ J (SpecB_{i})^{∗}, from the persistence of tight closure it follows that x ∈ J (SpecA_{f_{j}})^{∗}. Since the f_{1}, …, f_{s} generate the unit ideal, Lemma 2.2 implies that x ∈ J (U)^{∗}.

We have thus proved that:

Theorem 2.3. Let R be a Noetherian commutative ring of prime characteristic p, such that R is either essentially of finite type over an excellent local ring, or R_{red} is F -finite. Let I_{1}, …, I_{n} be ideals in R. If x is an element of R, then x ∈ (I_{1}, …, I_{n})^{∼} if and only if for every non-nilpotent element f of a fixed set of generators for the product ideal I = I_{1} … I_{n},

x ∈ (JR[It]_{(ft)})^{∗},

where J = I_{1} + … + I_{n}, and R[It]_{(ft)} is the zeroth graded piece of the localized Rees ring R[It]_{ft}.

Note 2.4. Let R be a ring as above, and I_{1}, …, I_{n} be ideals of R with I = I_{1} … I_{n} and J = I_{1} + … + I_{n}. Let G be a fixed set of generators for the ideal I such that every f in G is of the form f = f_{1} … f_{n}, where each f_{i} is an element of a fixed set of generators f_{1}^{i}, …, f_{s_{i}}^{i} for I_{i}. For future reference we show what (JR[It]_{(ft)})^{∗} looks like in practice when f is not nilpotent.

(JR[It]_{(ft)})^{∗} = ((I_{1} + … + I_{n})R[I_{1} … I_{n}t]_{(f_{1}...f_{n}t)})^{∗}
≃ ((I_{1} + … + I_{n})R[\frac{I_{1}}{f_{1}} \frac{I_{2}}{f_{2}} … \frac{I_{n}}{f_{n}}])^{∗}
= ((f_{1}, …, f_{n})R[\frac{I_{1}}{f_{1}} \frac{I_{2}}{f_{2}} … \frac{I_{n}}{f_{n}}])^{∗}
= ((f_{1}, …, f_{n})R[\frac{f_{1}^{1}}{f_{1}} , …, \frac{f_{s_{1}}^{1}}{f_{1}} , …, \frac{f_{1}^{n}}{f_{n}} , …, \frac{f_{sn}^{n}}{f_{n}}])^{∗}
= ((f_{1}, …, f_{n})R[\frac{I_{1}}{f_{1}} , \frac{I_{2}}{f_{2}} , …, \frac{I_{n}}{f_{n}}])^{∗}.

2.1 Blowup Closure Can Be Tested Modulo Minimal Primes

Theorem 2.5. Let R be a Noetherian ring of positive characteristic such that R is either essentially of finite type over an excellent local ring, or R_{red} is F -finite, and let I_{1}, …, I_{n} be ideals in R. Then x ∈ (I_{1}, …, I_{n})^{∼} if and only if \overline{x} ∈ (I_{1}R/p, …, I_{n}R/p)^{∼}, for all minimal primes p of R, where \overline{x} is the image of x in R/p.

We first specify the structure of the minimal primes of Rees rings.

Proposition 2.6 (minimal primes of Rees rings). Let R be a Noetherian ring, and p_{1}, …, p_{m} be the minimal primes of R. If I is any ideal of R, then:

(a) The minimal primes of R[It] are p̂_{1}, …, p̂_{m}, where for 1 ≤ i ≤ m,

p̂_{i} = p_{i}R[t] ∩ R[It] = p_{i} ⊕ (I ∩ p_{i})t ⊕ (I^{2} ∩ p_{i})t^{2} ⊕ …;

(b) For all primes p of R, R[It]/p̂ ≃ (R/p)[I(R/p)t].

Proof. (a) See [Va], Proposition 1.1 part (iii).

(b) For p ∈ SpecR, construct the map

ϕ : R[It] ⟶ R/p[I(R/p)t]

with ϕ(xt^{n}) = \overline{x}t^{n}, where \overline{x} is the image of x in R/p.

This is a surjective homomorphism of graded rings. To find the kernel, we observe that ϕ(xt^{n}) = 0 if and only if \overline{x}t^{n} = 0. So xt^{n} is in the kernel of ϕ if and only if x ∈ I^{n} ∩ p. So the kernel of ϕ is equal to p̂, and hence we have an isomorphism R[It]/p̂ ≃ (R/p)[I(R/p)t].

Corollary 2.7. Let R be a Noetherian ring, and let I be an ideal of R. Suppose ft^{n} is a homogeneous element of R[It] for some positive integer n.

(a) There is a one to one correspondence between the minimal primes of R[It] that do not contain ft^{n}, and the minimal primes of R[It]_{(ft^{n})}.

(b) Suppose f ∈ I and p is a minimal prime of R not containing f. Let p̂ be the minimal prime of R[It] corresponding to p, and let p′ be the minimal prime of R[It]_{(ft)} corresponding to p̂. Then

\frac{R[It]_{(ft)}}{p′} ≃ (\frac{R[It]}{p̂})_{(ft)} ≃ (R/p[(IR/p)t])_{(ft)} .

Proof. The statement of part (a) is equivalent to saying that there is a one to one correspondence between M(R[It]_{ft^{n}}) and M(R[It]_{(ft^{n})}).

When n = 1, we have R[It]_{ft} ≃ R[It]_{(ft)}[u, u^{−1}]. This is because R[It]_{(ft)} ≃ R[g_{1}/f, …, g_{s}/f], where g_{1}, …, g_{s} is a fixed set of generators for I. We can then define the map

R[g_{1}/f, …, g_{s}/f][u, u^{−1}] ⟶ R[It]_{ft}

by sending u^{m} to (ft)^{m} for all nonzero integers m. It is easy to check that this map is an isomorphism, and it follows that all members of M(R[It]_{ft}) are extensions of those in M(R[It]_{(ft)}).

If n > 1, then R[It]_{(ft^{n})} ≃ R[\frac{I^{n}}{f}], where by \frac{I^{n}}{f} we mean all elements of the form \frac{x}{f}, where x ∈ I^{n}. This is isomorphic to (R[It]^{(n)})_{(ft)}, where R[It]^{(n)} = R[I^{n}t] is the nth Veronese subring of R[It]. By the previous paragraph, we know that that there is a one to one correspondence between M((R[It]^{(n)})_{ft}) and M((R[It]^{(n)})_{(ft)}). On the other hand, the homogeneous primes of R[It]^{(n)} are contractions of the homogeneous primes of R[It] (see [E]). Since all the minimal primes of R[It] are homogeneous by Proposition 2.6, it follows again that M(R[It]_{ft^{n}}) and M(R[It]_{(ft^{n})}) correspond. This settles part (a).

To prove part (b), from part (a) we notice that since R[It]_{ft} ≃ R[It]_{(ft)}[u, u^{−1}], we have

\frac{R[It]_{(ft)}}{p′} [u, u^{−1}] ≃ \frac{R[It]_{ft}}{p̂R[It]_{ft}} ≃ (\frac{R[It]}{p̂})_{ft} ≃ (\frac{R[It]}{p̂})_{(ft)} [u, u^{−1}],

where the second isomorphism is because localization is flat, and the third follows again from part (a) of this theorem along with part (b) of Proposition 2.6. Therefore

\frac{R[It]_{(ft)}}{p′} ≃ (\frac{R[It]}{p̂})_{(ft)} ,

and combining this with Proposition 2.6 part (b), we obtain the desired result.

Proof of Theorem 2.5. Let I = I_{1} … I_{n}, J = I_{1} + … + I_{n}, and G be a fixed set of generators for I. Take x ∈ (I_{1}, …, I_{n})^{∼}. Then for any f ∈ G,

x ∈ (JR[It]_{(ft)})^{∗}.  (1)

Since tight closure can be tested modulo minimal primes (see Section 1.2), we see that Equation 1 is equivalent to

\overline{x} ∈ (J \frac{R[It]_{(ft)}}{p′} )^{∗},

for every minimal prime p′ of R[It]_{(ft)}, which by part (a) of Corollary 2.7 and Proposition 2.6 corresponds to a minimal prime p of R that does not contain f. From Corollary 2.7 part (b) we see that

\overline{x} ∈ (J \frac{R[It]_{(ft)}}{p′} )^{∗} ≃ (J (\frac{R[It]}{p̂})_{(ft)})^{∗} ≃ (JR/p[I(R/p)t]_{(ft)})^{∗}.

This holds for all minimal primes p of R that do not contain f. Hence we equivalently have

\overline{x} ∈ (I_{1}R/p, …, I_{n}R/p)^{∼}

for all minimal primes p of R.

2.2 Basic Properties of Blowup Closure

The following fact, which follows from the contraction property of tight closure, makes the computation of blowup closure simpler in several cases.

Proposition 2.8. Let R be a Noetherian commutative ring of prime characteristic p, such that R is either essentially of finite type over an excellent local ring, or R_{red} is F -finite. Let I_{1}, …, I_{n} be ideals in R, I = I_{1} … I_{n}, J = I_{1} + … + I_{n}, f ∈ I and x ∈ R. Then

x ∈ (JR[It]_{(ft)})^{∗} iff x ∈ (JR[It]′_{(ft)})^{∗},

where R[It]′ is the normalization of R[It].

Proof. Let p_{1}, …, p_{m} be the set of minimal primes of R[It] that do not contain ft. These correspond to minimal primes p′_{1}, …, p′_{m} of R[It]_{(ft)} (Corollary 2.7) and minimal primes p_{1}^{♯}, …, p_{m}^{♯} of R[It]′_{(ft)} (Proposition 1.2 and [E] proposition 4.13).

Suppose x ∈ (JR[It]′_{(ft)})^{∗}. Since tight closure can be tested modulo minimal primes (see [BH] Proposition 10.1.2(e)), we see that for i = 1, …, m,

\overline{x} ∈ (J \begin{matrix}R[It] & {′ \atop (ft)} \\ p_{i} & ♯ \end{matrix} )^{∗},

where \overline{x} is the image of x in the ring R/p_{i}. But Proposition 1.2 implies that

\begin{matrix}R[It] & {′ \atop (ft)} \\ p_{i} & ♯ \end{matrix} ≃ (\frac{R[It]_{(ft)}}{p′_{i}} )′,

so \overline{x} (or rather the image of \overline{x} under this isomorphism) belongs to

(J(\frac{R[It]_{(ft)}}{p′_{i}} )′)^{∗}.

Therefore, by the contraction property of tight closure ([Hu] Theorem 1.7), for all i,

\overline{x} ∈ (J(\frac{R[It]_{(ft)}}{p′_{i}} )′)^{∗} ⋂ \frac{R[It]_{(ft)}}{p′_{i}} ⊆ (J (\frac{R[It]_{(ft)}}{p′_{i}} ))^{∗}.

Applying Proposition 10.1.2(e) of [BH] again, we see that x ∈ (JR[It]_{(ft)})^{∗}.

The reverse inclusion follows because of the inclusion of R[It]_{(ft)} in R[It]′_{(ft)}, and the persistence of tight closure (Theorem 1.11).

Theorem 2.9. Let R be a Noetherian ring of prime characteristic p,such that R is either essentially of finite type over an excellent local ring, or R_{red} is F -finite, and let I_{1}, …, I_{n} be ideals of R. Then:

(a) The ideal (I_{1}, …, I_{n})^{∼} is tightly closed, and

(I_{1}, …, I_{n})\underline{∗} ⊆ (I_{1}, …, I_{n})^{∼}.

(b) If all I_{1}, …, I_{n} are principal, then (I_{1}, …, I_{n})^{∼} = (I_{1} + … + I_{n})^{∗}.

(c) If n = 1, then (I_{1})^{∼} = \overline{I_{1}}.

(d) (\overline{I_{1}}, …, \overline{I_{n}})^{∼} = (I_{1}, …, I_{n})^{∼}.

Proof. Throughout the proof, we let G be a fixed set of generators for the ideal I = I_{1} … I_{n}, such that every f in G is of the form f = f_{1} … f_{n}, where each f_{i} is an element of a fixed set of generators for I_{i}. We let J = I_{1} + … + I_{n}.

(a) We can assume that R is a domain (Theorem 2.5 and [Ho2] Proposition 1.4).

For every non-nilpotent element f in G, tight closure persists under the map R ⟶ R[It]_{(ft)} , and so (JR[It]_{(ft)})^{∗}∩R is tightly closed in R (see Remark 1.12). On the other hand, the intersection of tightly closed ideals is tightly closed, and since (I_{1}, …, I_{n})^{∼} is the intersection of finitely many ideals of the form (JR[It]_{(ft)})^{∗} ∩ R, it follows that (I_{1}, …, I_{n})^{∼} is tightly closed in R.

Now let z ∈ (I_{1}, …, I_{n})\underline{∗}. Then cz^{q} ∈ I_{1}^{q} + … + I_{n}^{q}, for all large q = p^{e} and some nonzero c ∈ R. If f ∈ G, and f = f_{1} … f_{n} where f_{i} is a generator of I_{i} as above, then this equation can be extended to

cz^{q} ∈ I_{1}^{q}R[It]_{(ft)} + … + I_{n}^{q}R[It]_{(ft)}
= (f_{1})^{q}R[It]_{(ft)} + … + (f_{n})^{q}R[It]_{(ft)} (see Note 2.4)
= (f_{1}, …, f_{n})^{[q]}R[It]_{(ft)}
= J^{[q]}R[It]_{(ft)}
= (JR[It]_{(ft)})^{[q]}

for all large powers q of p. It follows that z ∈ (JR[It]_{(ft)})^{∗}. Since this holds for all f ∈ G, by 2.3 we conclude that z ∈ (I_{1}, …, I_{n})^{∼}.

(b) We can again assume that R is a domain. If I_{1}, …, I_{n} are principal, then so is their product I. So for each f ∈ G, R[It]_{(ft)} is the same as R. We therefore have

(I_{1}, …, I_{n})^{∼} = (I_{1} + … + I_{n})^{∗}.

(c) Assume that R is a domain. In this case, I = J = I_{1}. We use the “normalized” definition of blowup closure, following Proposition 2.8, and we obtain

(I)^{∼} = ⋂_{f∈G} IR[It]′_{(ft)} ∩ R = \overline{I},

since the extension of I to the normal ring R[It]′_{(ft)} is principal , and therefore tightly closed (see Proposition 1.4 and [HH1] Corollary 5.8).

(d) We use the “normalized” definition of blowup closure, following Proposition 2.8.

Let J′ = \overline{I_{1}} + … + \overline{I_{n}} and I′ = \overline{I_{1}} … \overline{I_{n}}. Then R[It]′ = R[I′t]′, since \overline{(I′)^{n}} = \overline{I^{n}}, and so ProjR[I′t]′ like ProjR[It]′ is covered by affines of the form SpecR[It]′_{(ft)} for f ∈ G. On the other hand, suppose f = f_{1} … f_{n} ∈ G where f_{i} ∈ I_{i}, i = 1, …, n.

Then

(J′R[It]′_{(ft)})^{∗} = (J′R[I′t]′_{(ft)})^{∗} = ((f_{1}, …, f_{n})R[I′t]′_{(ft)})^{∗} = (JR[It]′_{(ft)})^{∗},

as pointed in Note 2.4. It follows that (\overline{I_{1}}, …, \overline{I_{n}})^{∼} = (I_{1}, …, I_{n})^{∼}.

Theorem 2.10 (blowup closure from contractions). Suppose that R and S are Noetherian domains of prime characteristic p, such that R is either essentially of finite type over an excellent local ring, or R_{red} is F -finite. Let I_{1}, …, I_{n} be ideals of R. Suppose S is a module finite extension of R. Then

(I_{1}S, …, I_{n}S)^{∼} ∩ R ⊆ (I_{1}, …, I_{n})^{∼}

.

Proof. Let I and J be the product and the sum of I_{1}, …, I_{n}, respectively, and let G be a fixed set of generators for I in R.

Suppose S is generated as an R-module by elements u_{1}, …, u_{m}. Then u_{1}, …, u_{m} also generate S[(IS)t] as a module over R[It]. To see this, take a homogeneous element zt^{n} ∈ S[(IS)t]. Then z ∈ I^{n}S, and so if I^{n} = (v_{1}, …, v_{r}) in R, we can write z = a_{1}v_{1} + … + a_{r}v_{r} for a_{1}, …, a_{r} ∈ S. On the other hand for i = 1, …, r, a_{i} = a_{i1}u_{1} +…+a_{im}u_{m}, where a_{ij} ∈ R for j = 1, …, m. Rewriting the equation describing z above, we have z = b_{1}u_{1} + … + b_{m}u_{m}, where b_{j} ∈ I^{n} for j = 1, …, m. It follows that zt^{n} = (b_{1}t^{n})u_{1} + … + (b_{m}t^{n})u_{m}. So S[(IS)t] is module finite over R[It]. A similar argument shows that S[(IS)t]_{(ft)} is module finite over R[It]_{(ft)}.

Now let z ∈ (I_{1}S, …, I_{n}S)^{∼} ∩ R. Then by definition, z belongs to

(JS[(IS)t]_{(ft)})^{∗} ∩ R
= (JS[(IS)t]_{(ft)})^{∗} ∩ R[It]_{(ft)} ∩ R
⊆ (JR[It]_{(ft)})^{∗} ∩ R  by contraction of tight closure ([Hu] 1.7)

This holds for all f ∈ G, which implies that z ∈ (I_{1}, …, I_{n})^{∼}.

Theorem 2.11 (persistence of blowup closure). Let ϕ : R ⟶ S be a homomorphism of Noetherian rings of prime characteristic p, and let I_{1}, …, I_{n} be ideals of R. Suppose that either R is essentially of finite type over an excellent local ring, or that R_{red} is F-finite. Then:

(I_{1}, …, I_{n})^{∼}S ⊆ (I_{1}S, …, I_{n}S)^{∼}.

Proof. Let I = I_{1} … I_{n} and J = I_{1}+…+I_{n}. Fix a finite set of generators for I, and let f = f_{1} … f_{n}, f_{i} ∈ I_{i}, be in that set. If R is essentially of finite type over an excellent local ring or R_{red} is F -finite, then R[It]_{(ft)} will have the same property since it is an algebra of finite type over R.

On the other hand, if ϕ(f) is not nilpotent, then ϕ induces a map

R[It]_{(ft)} → S[(IS)t]_{(ϕ(f)t)} ,

under which tight closure persists (see Theorem 1.11). It follows that when ϕ(f) is not nilpotent,

(R ∩ (JR[It]_{(ft)})^{∗}) S ⊆ (JR[It]_{(ft)})^{∗}S[(IS)t]_{(ϕ(f)t)} ⊆ (JS[(IS)t]_{(ϕ(f)t)})^{∗}.

If g_{1}, …, g_{r} is a set of generators for I in R, then IS is generated by the ϕ(g_{i}), i = 1, …, r, that are nonzero. From the discussion above and Theorem 2.3, it follows that

(I_{1}, …, I_{n})^{∼}S ⊆ (I_{1}S, …, I_{n}S)^{∼}.

Blowup closure satisfies most properties that T I closure does. However, in many cases these two operations do not produce the same ideal. Here is an example of ideals in a polynomial ring for which these two operations are not the same:

Example 2.12. Let R = k[x, y] be a polynomial ring over a field k of characteristic p. Consider the ideals I = (y^{3}) and J = (x^{3}, x^{2}y). Notice that I and J are both integrally closed ideals. We know from [Ho2] Proposition 1.9 that

xy^{2} ∉ (I, J)\underline{∗} = \overline{I} + \overline{J} = (x^{3}, y^{3}, x^{2}y).

We show that, however, xy^{2} ∈ (I, J)^{∼}. To show this, we check the ideal I + J against two localized Rees rings. With notation as in Note 2.4, if we take f_{1} = y^{3} and f_{2} = x^{2}y, we have

xy^{2} ∈ (y^{3}, x^{2}y)k[x, y, \frac{x}{y}],

since xy^{2} = y^{3}(x/y).

If we take f_{1} = y^{3} and f_{2} = x^{3}, we have

xy^{2} ∈ (y^{3}, x^{3})k[x, y, \frac{y}{x}],

since xy^{2} = x^{3}(y/x)^{2}.

Therefore xy^{2} ∈ (I, J)^{∼}, but xy^{2} ∉ (I, J)\underline{∗}.

Moreover, blowup closure fails to respect inclusions: If J_{1}, …, J_{n} is a set of ideals such that I_{i} ⊆ J_{i}, for i = 1, …, n, then the inclusion (I_{1}, …, I_{n})^{∼} ⊆ (J_{1}, …, J_{n})^{∼} does not necessarily hold. Here is an example:

Example 2.13. Let R = k[x, y, u, v]. Let I_{1} = (x^{3}, x^{2}y), I_{2} = (y^{3}), J_{1} = (x^{3}, x^{2}y, u) and J_{2} = (y^{3}, v). Then xy^{2} ∈ (I_{1}, I_{2})^{∼} as was shown in the previous example. But xy^{2} ∉ (J_{1}, J_{2})^{∼}, because looking at the affine patch corresponding to the generators u and v of J_{1} and J_{2}, respectively, we can see that

xy^{2} ∉ (u, v)k[x, y, u, v, \frac{x^{3}}{u} , \frac{x^{2}y}{u} , \frac{y^{3}}{v}].

We show in the next section that T I closure can be described as the contraction of the blowup closure of some ideals in a larger ring. Before discussing this, we demonstrate a case where blowup closure can be directly calculated.

2.3 The Case of Monomial Ideals in a Polynomial Ring

Theorem 2.14. Let R = k[x_{1}^{1}, …, x_{m_{1}}^{1}, …, x_{1}^{n}, …, x_{m_{n}}^{n}] be a polynomial ring in distinct variables x_{j}^{i} , 1 ≤ i ≤ n and 1 ≤ j ≤ m_{i}, over an algebraically closed field k of prime characteristic p. Let I_{1}, …, I_{n} be monomial ideals in R, where the generators of I_{i} are monomials in the variables x_{1}^{i}, …, x_{m_{i}}^{i} for 1 ≤ i ≤ n. Then (I_{1}, …, I_{n})^{∼} = \overline{I_{1}} + … + \overline{I_{n}} .

To prove this theorem, we will show that (I_{1}, …, I_{n})^{∼} is a monomial ideal. Then, for a given monomial M ∈ (I_{1}, …, I_{n})^{∼}, we will show that for some 1 ≤ α ≤ n,

M ∈ R ∩ I_{α}R[It]′ = \overline{I_{α}},

and it will follow that (I_{1}, …, I_{n})^{∼} = \overline{I_{1}} + … + \overline{I_{n}} .

We begin by fixing the notation. For each i, let I_{i} = (f_{1}^{i}, …, f_{s_{i}}^{i}), where f_{j}^{i} is a monomial in the polynomial ring S_{i} = k[x_{1}^{i}, …, x_{m_{i}}^{i}]. Let J = I_{1} +…+I_{n} = (f_{1}^{1}, …, f_{s_{1}}^{1}, …, f_{1}^{n}, …, f_{s_{n}}^{n}), and I = I_{1} … I_{n} = (f_{i_{1}}^{1} … f_{i_{n}}^{n} : 1 ≤ i_{j} ≤ s_{j}, 1 ≤ j ≤ n).

Our first goal is to show that

(I_{1}, …, I_{n})^{∼} = ⋂_{1≤j≤n^{1≤i_{j} ≤s_{j}}}(f_{i_{1}}^{1}, …, f_{i_{n}}^{n})R[It]′_{(f_{i_{1}}^{1}...f_{in}^{n}t)} ∩ R.

By 2.4, for a fixed index set i_{1}, …, i_{n}, we are interested in the tight closure of the ideal (f_{i_{1}}^{1}, …, f_{i_{n}}^{n}) in the ring:

k[x_{1}^{1}, …, x_{m_{1}}^{1}, …, x_{1}^{n}, …, x_{m_{n}}^{n}, \frac{f_{1}^{1}}{f_{i_{1}}^{1}} , …, \frac{f_{s_{1}}^{1}}{f_{i_{1}}^{1}} , …, \frac{f_{1}^{n}}{f_{i_{n}}^{n}} , …, \frac{f_{s_{n}}^{n}}{f_{i_{n}}^{n}}].  (2)

Equivalently, by Theorem 2.8, we can study the tight closure of this ideal in the normalization of the ring described in 2. We claim that the tight closure of (f_{i_{1}}^{1}, …, f_{i_{n}}^{n}) in the normalization of the ring in 2 is equal to (f_{i_{1}}^{1}, …, f_{i_{n}}^{n}) itself.

To see this, let f = f_{i_{1}}^{1} … f_{i_{n}}^{n}. Since R[It] is a monomial subring of R[t], R[It]′ is also a monomial ring, and it is weakly F -regular (i.e., all ideals are tightly closed; see [Sm2]). Also, tight closure commutes with localization for R[It] ([Sm2]). Since normalization also commutes with localization ([E] Proposition 4.13) it follows that

(JR[It]′_{ft})^{∗} = JR[It]′_{ft}.

On the other hand,

(JR[It]′_{(ft)})^{∗} ⊆ ((JR[It]′_{ft})^{∗})_{0} = ((JR[It]′_{ft}))_{0} = JR[It]′_{(ft)},

and so (JR[It]′_{(ft)})^{∗} = JR[It]′_{(ft)}.

We have therefore shown that if I_{1}, …, I_{n} are monomial ideals in a polynomial ring R, then

(I_{1}, …, I_{n})^{∼} = ⋂_{1≤j≤n^{1≤i_{j} ≤s_{j}}}(f_{i_{1}}^{1}, …, f_{i_{n}}^{n})R[It]′_{(f_{i_{1}}^{1}...f_{in}^{n}t)} ∩ R.  (3)

Notice that this argument does not require the distinction of the sets of variables generating I_{1}, …, I_{n}.

Proposition 2.15. Let I_{1}, …, I_{n} be monomial ideals (not necessarily generated by distinct variables) in a polynomial ring R = k[u_{1}, …, u_{m}] where k is an infinite field. Then (I_{1}, …, I_{n})^{∼} is also a monomial ideal in R.

Proof. We use the fact that an ideal I in a polynomial ring R = k[u_{1}, …, u_{m}] over an infinite field k is generated by monomials, if and only if I is invariant under the action of the torus (k^{∗})^{m} (see [Ho3] page 319), where k^{∗} denotes k − {0}, and the torus action on R is defined as follows. If λ = (λ_{1}, …, λ_{m}) ∈ (k^{∗})^{m}, and x is a monomial u_{1}^{c_{1}} … u_{m}^{c_{m}} of R, then

λx = (λ_{1}u_{1})^{c_{1}} … (λ_{m}u_{m})^{c_{m}} = λ_{1}^{c_{1}} … λ_{m}^{c_{m}}x,

and if x is a sum of monomials M_{1} + … + M_{r}, then λx = λM_{1} + … + λM_{r}.

The action of the torus on monomials with negative powers is defined in a similar way. In the ring R[u_{1}^{−1}, …, u_{m}^{−1}], where R is as above, if λ = (λ_{1}, …, λ_{m}) ∈ (k^{∗})^{m} and u = u_{1}^{c_{1}} … u_{m}^{c_{m}}, where the c_{i} are integers, then

λu = (λ_{1}u_{1})^{c_{1}} … (λ_{m}u_{m})^{c_{m}} = λ_{1}^{c_{1}} … λ_{m}^{c_{m}}u,

and if u = M_{1} + … + M_{r}, where M_{i} are monomials in the u_{i} with integer powers, then λu = λM_{1} + … + λM_{r}.

We show that (I_{1}, …, I_{n})^{∼} is invariant under the action of the torus (k^{∗})^{m}. Suppose each I_{i} is generated by f_{1}^{i}, …, f_{s_{i}}^{i}. By the discussion preceding the theorem we only need to prove that for any given index set i_{1}, …, i_{n}, if

x ∈ (f_{i_{1}}^{1}, …, f_{i_{n}}^{n})R[It]′_{(f_{i_{1}}^{1}...f_{in}^{n}t)}

and λ ∈ (k^{∗})^{m}, then

λx ∈ (f_{i_{1}}^{1}, …, f_{i_{n}}^{n})R[It]′_{(f_{i_{1}}^{1}...f_{in}^{n}t)}.

Take λ = (λ_{1}, …, λ_{m}) and x as above. Then x can be written as

x = A_{1}f_{i_{1}}^{1} + … + A_{n}f_{i_{n}}^{n},

with A_{1}, …, A_{n} ∈ R[It]′_{(f_{i_{1}}^{1}...f_{in}^{n}t)}.

So

\begin{aligned}λx & = λ(A_{1}f_{i_{1}}^{1}) + … + λ(A_{n}f_{i_{n}}^{n}) \\ & = (λA_{1})(λf_{i_{1}}^{1}) + … + (λA_{n})(λf_{i_{n}}^{n}) \end{aligned}

Since each f_{i_{j}}^{j} is a monomial in the u_{i}, λf_{i_{j}}^{j} will be just some scalar times f_{i_{j}}^{j}, and will therefore still belong to (f_{i_{1}}^{1}, …, f_{i_{n}}^{n}).

As for the A_{j}, we claim that λA_{j} still remains in R[It]′_{(f_{i_{1}}^{1}...f_{in}^{n}t)} for j = 1, …, n. To see this, fix some j. One can write A_{j} as

A_{j} = \frac{B_{j}}{(f_{i_{1}}^{1} … f_{i_{n}}^{n})^{r}t^{r}} ,

where B_{j} ∈ R[It]′. We can then write B_{j} as

B_{j} = M_{1}t^{r} + … + M_{s_{j}}t^{r},

where M_{1}, …, M_{s_{j}} are monomials of R that belong to \overline{I^{r}} (see Theorem 1.5). So if we set f = f_{i_{1}}^{1} … f_{i_{n}}^{n}, then A_{j} = M_{1}f^{−r} + … + M_{s_{j}}f^{−r}, and so

λA_{j} = α_{1}M_{1}f^{−r} + … + α_{s_{j}}M_{s_{j}}f^{−r},

where α_{1}, …, α_{s_{j}} ∈ k are scalars. It follows that

λA_{j} = \frac{α_{1}M_{1} + … + α_{s_{j}}M_{s_{j}}}{f^{r}} ∈ R[It]′_{(ft)}.

Therefore λx ∈ (f_{i_{1}}^{1}, …, f_{i_{n}}^{n})R[It]′_{(ft)}, and so we are done.

Lemma 2.16. Let R = k[u_{1}, …, u_{m}, \frac{M_{1}}{N_{1}} , …, \frac{M_{r}}{N_{r}}], where u_{1}, …, u_{m} are distinct variables, and M_{1}, …, M_{r} and N_{1}, …, N_{r} are nonzero monomials in the polynomial ring k[u_{1}, …, u_{m}] over a field k. Suppose M, g_{1}, …, g_{s} are nonzero monomials in k[u_{1}, …, u_{m}] such that M ∈ (g_{1}, …, g_{s})R. Then for some i, 1 ≤ i ≤ s, M ∈ (g_{i})R.

Proof. Since M ∈ (g_{1}, …, g_{s})R, there are A_{1}, …, A_{s} in R such that M = A_{1}g_{1} + … + A_{s}g_{s}. After taking the common denominator A of the right hand side of the equation, and multiplying both sides of the equation by A, we end up with an equation of the form AM = p_{1}g_{1} + … + p_{s}g_{s}, where p_{1}, …, p_{s} are polynomials in k[u_{1}, …, u_{m}], and for each i, A_{i} = \frac{p_{i}}{A}. Now notice that AM is a monomial, and so when you add the polynomials p_{1}g_{1}, …, p_{s}g_{s} all terms that are not equal to a scalar multiple of AM cancel out with each other. So we can without loss of generality for each i, replace p_{i} with α_{i}Q_{i}, where α_{i} ∈ k is nonzero if the monomial AM appears as a term of p_{i}g_{i}, and Q_{i} is a monomial in k[u_{1}, …, u_{m}] for which Q_{i}g_{i} = AM. So AM = α_{1}Q_{1}g_{1} + … + α_{s}Q_{s}g_{s}, and α_{1} + … + α_{s} = 1. Now it is clear that least one of the α_{i} must be nonzero; say α_{1} is nonzero. Then α_{1}Q_{1} is a term of p_{1}, and since A_{1} = \frac{p_{1}}{A} ∈ R, \frac{Q_{1}}{A} ∈ R, and therefore M = \frac{Q_{1}}{A} g_{1} ∈ (g_{1})R.

A Noetherian ring R satisfies Serre’s condition (R_{n}) if R_{p} is a regular local ring for p ∈ SpecR with dim R_{p} ≤ n. We say that R satisfies Serre’s condition (S_{n}) if depthR_{p} ≥ min(n, dim R_{p}) for all p ∈ SpecR. For more on these conditions, see the first three chapters of [BH].

Proposition 2.17. Let R and S be two domains, which are finitely generated k-algebras, where k is an algebraically closed field. Then:

(a) For a positive integer n, if R and S satisfy (S_{n}), then so does R ⊗_{k} S.

(b) For a positive integer n, if R and S satisfy (R_{n}), then so does R ⊗_{k} S.

Proof. (a) This is Theorem 5.5.5 of [V].

(b) One can express R and S as

R = \frac{k[u_{1}, …, u_{m}]}{(g_{1}, …, g_{s})} and S = \frac{k[v_{1}, …, v_{r}]}{(h_{1}, …, h_{t})},

where u_{1}, …, u_{m} and v_{1}, …, v_{r} are distinct variables. Then

R ⊗_{k} S = \frac{k[u_{1}, …, u_{m}, v_{1}, …, v_{r}]}{(g_{1}, …, g_{s}, h_{1}, …, h_{t})};

see [ZS1] for the tensor product of two rings.

Suppose R and S satisfy (R_{n}), that means the defining ideals J_{1} and J_{2} of the singular loci R and S, respectively, have heights larger than n. The defining ideal J of the singular locus of R ⊗_{k} S, is the ideal generated by the d × d minors of the Jacobian matrix

( \begin{matrix}\frac{δg_{1}}{δu_{1}} & … & \frac{δg_{1}}{δu_{m}} & 0 & … & 0 \\ ⋮ & ... & ⋮ & ⋮ & ... & ⋮ \\ \frac{δg_{s}}{δu_{1}} & … & \frac{δg_{s}}{δu_{m}} & 0 & … & 0 \\ 0 & … & 0 & \frac{δh_{1}}{δv_{1}} & … & \frac{δh_{1}}{δv_{r}} \\ ⋮ & ... & ⋮ & ⋮ & ... & ⋮ \\ 0 & … & 0 & \frac{δh_{t}}{δv_{1}} & … & \frac{δh_{t}}{δv_{r}} \end{matrix} ) ,

where d is the height of the ideal (g_{1}, …, g_{s}, h_{1}, …, h_{t}) in the polynomial ring k[u_{1}, …, u_{m}, v_{1}, …, v_{r}] (see Corollary 16.20 of [E]). On the other hand, d = d_{1} + d_{2}, where d_{1} is the height of (g_{1}, …, g_{s}) in k[u_{1}, …, u_{m}], and d_{2} is the height of (h_{1}, …, h_{t}) in k[v_{1}, …, v_{r}] (see Chapter II of [Har]).

It is an easy exercise to see that the only time that the determinant of a square matrix of the form

( \begin{matrix}A & 0 \\ 0 & B \end{matrix} )

is nonzero is when A and B are both square matrices. So J is generated by determinants of the form

det ( \begin{matrix}A & 0 \\ 0 & B \end{matrix} ) = (det A)(det B),

where A and B are square matrices. In particular, it contains products of d_{1} × d_{1} minors of the Jacobian of R with d_{2} × d_{2} minors of the Jacobian of S. Therefore J ⊇ J_{1}J_{2}, and hence ht J ≥ min(ht J_{1}, ht J_{2}) > n, since R and S are (R_{n}). Therefore R ⊗_{k} S is smooth in codimension n.

Corollary 2.18. Let R and S be two normal domains, which are finitely generated k-algebras, where k is an algebraically closed field. Then R ⊗_{k} S is also a normal domain.

Proof. Serre’s normality criterion says that a ring being normal is equivalent to it satisfying (S_{2}) and (R_{1}), and so by Proposition 2.17 R⊗_{k} S is normal. If K and K′ are the quotient fields of R and S respectively, Theorem III.15.40 of [ZS1] proves that K ⊗_{k} K′ is a domain when k is algebraically closed, and since R ⊗_{k} S is a subring of K ⊗_{k} K′, it follows that R ⊗_{k} S is a domain.

Below, we adopt the following notation. If R⊗_{k} S is the tensor product of two finitely generated k-algebras R and S over the field k, and if x ∈ R and y ∈ S, by x ⊗_{k} y we mean the product of x ⊗_{k} 1 and 1 ⊗_{k} y, where x ⊗_{k} 1 and 1 ⊗_{k} y are the images of x and y in R ⊗_{k} S under the inclusions R ⟶ R ⊗_{k} S and S ⟶ R ⊗_{k} S, respectively (see [ZS1]).

Lemma 2.19. Let R = k[x_{1}^{1}, …, x_{m_{1}}^{1}, …, x_{1}^{n}, …, x_{m_{n}}^{n}], I and J be as in Theorem 2.14. Let M be a monomial of R that belongs to (I_{1}, …, I_{n})^{∼}. Fix an index set i_{1}, …, i_{n}, such that 1 ≤ i_{v} ≤ s_{v}, for 1 ≤ v ≤ n. Suppose that for some fixed β, 1 ≤ β ≤ n, M ∈ (f_{i_{β}}^{β})R[It]′_{(f_{i_{1}}^{1}...f_{in}^{n}t)}. Then for

any index set j_{1}, …, j_{n}, 1 ≤ j_{v} ≤ s_{v}, 1 ≤ v ≤ n, such that j_{β} = i_{β} (so that f_{j_{β}}^{β} = f_{i_{β}}^{β}), we have

M ∈ (f_{j_{β}}^{β})R[It]′_{(f_{j_{1}}^{1}...f_{jn}^{n}t)}.

Proof. We know that M ∈ (I_{1}, …, I_{n})^{∼}, so for all index sets j_{1}, …, j_{n} one has

M ∈ (JR[It]′_{(f_{j_{1}}^{1}...f_{jn}^{n}t)})^{∗} = (f_{j_{1}}^{1}, …, f_{j_{n}}^{n})R[It]′_{(f_{j_{1}}^{1}...f_{jn}^{n}t)},

as we proved earlier in this section.

Since the f_{j_{1}}^{1}, …, f_{j_{n}}^{n} are monomials in distinct sets of variables, Corollary 2.18 implies that

R[It]′_{(f_{j_{1}}^{1}...f_{jn}^{n}t)} = S_{1}[I_{1}t]′_{(f_{j_{1}}^{1}t)} ⊗_{k} … ⊗_{k} S_{n}[I_{n}t]′_{(f_{jn}^{n}t)},  (4)

where for each v, 1 ≤ v ≤ n, S_{v} = k[x_{1}^{v}, …, x_{m_{v}}^{v}].

So, for the index set i_{1}, …, i_{n}, following the structure in 4, we can write M = M_{1} ⊗ … ⊗ M_{n}, where each M_{v} is a monomial in S_{v} for 1 ≤ v ≤ n, and M_{β} ∈ (f_{i_{β}}^{β})S_{β}[I_{β}t]′_{(f_{i_{β}}^{β} t)}.

Let j_{1}, …, j_{n}, 1 ≤ j_{v} ≤ s_{v} be any set of indices such that j_{β} = i_{β}, that is, f_{j_{β}}^{β} = f_{i_{β}}^{β}. Then

since M_{β} ∈ (f_{i_{β}}^{β})S_{β}[I_{β}t]′_{(f_{i_{β}}^{β} t)}, we still have M ∈ (f_{j_{β}}^{β})R[It]′_{(f_{j_{1}}^{1}...f_{jn}^{n}t)}.

Proof of Theorem 2.14. We want to show that (I_{1}, …, I_{n})^{∼} = \overline{I_{1}} + … + \overline{I_{n}}. Clearly \overline{I_{1}} + … + \overline{I_{n}} ⊂ (I_{1}, …, I_{n})^{∼}. We need to show that the other inclusion holds. From Proposition 2.15 we know that since I_{1}, …, I_{n} are monomial ideals, (I_{1}, …, I_{n})^{∼} is a monomial ideal.

So we pick a monomial M of R = k[x_{1}^{1}, …, x_{m_{1}}^{1}, …, x_{1}^{n}, …, x_{m_{n}}^{n}] such that M ∈ (I_{1}, …, I_{n})^{∼}. Our goal is to show that for some α, 1 ≤ α ≤ n, M ∈ I_{α}R[It]′.

Fix an index set i_{1}, …, i_{n}, and let f = f_{i_{1}}^{1} … f_{i_{n}}^{n}. Then from Equation 3 we see that

M ∈ (JR[It]′_{(ft)})^{∗} = (f_{i_{1}}^{1}, …, f_{i_{n}}^{n})R[It]′_{(ft)}.

From [EGA] Lemma 2.1.6, it follows that ProjR[It]′ = ProjR[\overline{I^{h}}t] for some h ≥ 1, and we obtain

R[It]′_{(ft)} ≃ R[\overline{I^{h}}t]_{(f^{h}t)}.

Since I is a monomial ideal, \overline{I^{h}} is a monomial ideal (see [E] Chapter 4), and we can write \overline{I^{h}} = (H_{1}, …, H_{r}), where H_{1}, …, H_{r} are monomials. So we have

M ∈ (f_{i_{1}}^{1}, …, f_{i_{n}}^{n})k[x_{1}^{1}, …, x_{m_{1}}^{1}, ….x_{1}^{n}, …, x_{m_{n}}^{n}, H_{1}t, …, H_{r}t]_{(f^{h}t)},

which is isomorphic to

k[x_{1}^{1}, …, x_{m_{1}}^{1}, …, x_{1}^{n}, …, x_{m_{n}}^{n}, \frac{H_{1}}{f^{h}} , …, \frac{H_{r}}{f^{h}}],

and hence from Lemma 2.16 it follows that for some β, 1 ≤ β ≤ n, M ∈ (f_{i_{β}}^{β})R[\overline{I^{h}}t]_{(f^{h}t)} which

implies that M ∈ (f_{i_{β}}^{β})R[It]′_{(ft)}.

We would like to prove that this choice is consistent for all the affine sets, that is, there is some β such that M ∈ (f_{i_{β}}^{β})R[It]′_{(f_{i_{1}}^{1}...f_{in}^{n}t)} for all choices of index sets i_{1}, …, i_{n}.

Suppose that for each α = 1, ..., n − 1, there is some index γ_{α}, 1 ≤ γ_{α} ≤ s_{α}, and some index set i_{1}, …, i_{n} with i_{α} = γ_{α}, for which M ∉ (f_{i_{α}}^{α})R[It]′_{(f_{i_{1}}^{1}...f_{in}^{n}t)}. Then by Lemma 2.19, for all j, 1 ≤ j ≤ s_{n}, if one picks the index set γ_{1}, …, γ_{n−1}, j,

M ∉ (f_{γ_{α}}^{α})R[It]′_{(f_{γ_{1}}^{1}...f_{γ_{n−1}}^{n−1}f_{j}^{n}t)}

for 1 ≤ α ≤ n − 1. Therefore for all possible j,

M ∈ (f_{j}^{n})R[It]′_{(f_{γ_{1}}^{1}...f_{γ_{n−1}}^{n−1}f_{j}^{n}t)}.

Applying Lemma 2.19 again, one gets that M ∈ (f_{i_{n}}^{n})R[It]′_{(f_{i_{1}}^{1}...f_{in}^{n}t)}, for all index sets i_{1}, …, i_{n}.

We have now proved that M ∈ (I_{1} + … + I_{n})R[It]′, implies that for some β, 1 ≤ β ≤ n, M ∈ I_{β}R[It]′. Therefore

M ∈ R ∩ I_{β}R[It]′ = \overline{I_{β}}

by Proposition 2.20.

Proposition 2.20. Let R be a Noetherian domain, and let X be a normal scheme, with a proper birational map π : X ⟶ SpecR. Suppose that I is an ideal of R such that IO_{X} is an invertible sheaf of ideals on X. Then IO_{X} ∩ R = \overline{I}.

Proof. See Proposition 6.2 of [L1] and the remark following it for the proof.

3 Applications to T I closure

In this section we apply the notion of blowup closure to study properties of T I closure and to answer the questions mentioned in Section 1.4. It turns out (Corollary 3.5) that the T I closure of a set of ideals is the contraction of the blowup closure of certain extensions of those ideals in an extension of the original ring. In fact, the T I closure of a set of ideals can be described as the contraction of only one of the affine patches used to calculate the blowup closure mentioned above. To facilitate the arguments, we call this particular patch the multiple closure.

3.1 Definition and Basic Facts

Definition 3.1. Let R be a Noetherian ring of positive characteristic p, and let I_{1}, …, I_{n} be ideals in R. We define x in R to be in the multiple closure of I_{1}, …, I_{n}, denoted by (I_{1}, …, I_{n})^{∼^{∗}}, if and only if the image of x is in:

((w_{1}, …, w_{n})R[w_{1}, …, w_{n}, \frac{I_{1}}{w_{1}} , …, \frac{I_{n}}{w_{n}}])^{∗} ,

where w_{1}, …, w_{n} are indeterminates.

Recall that the ring

S = R[w_{1}, …, w_{n}, \frac{I_{1}}{w_{1}} , …, \frac{I_{n}}{w_{n}}]

above is obtained as follows: We take a ring extension R[w_{1}, …, w_{n}] of R, and we consider the Rees ring R[w_{1}, …, w_{n}][I′t] ring of the product I′ of the ideals I_{1} + (w_{1}), …, I_{n} + (w_{n}). We then localize this Rees ring at the element w_{1} … w_{n}t, and take the zeroth graded piece of the localized ring to obtain S. The ideal (w_{1}, …, w_{n}) in S is just the sum of the ideals I_{1} + (w_{1}), …, I_{n} + (w_{n}) extended to S. So the multiple closure of I_{1}, …, I_{n} in the ring R is one of the affine patches to be considered to compute the blowup closure of I_{1} +(w_{1}), …, I_{n} +(w_{n}) in the ring R[w_{1}, …, w_{n}] (see Theorem 2.3). It turns out that the multiple closure of I_{1}, …, I_{n} is in fact the blowup closure of I_{1} + (w_{1}), …, I_{n} + (w_{n}) in R[w_{1}, …, w_{n}], contracted back to R (Corollary 3.5). Multiple closure therefore enjoys all the basic properties of blowup closure.

We verify that multiple closure can be tested modulo minimal primes, and hence one can reduce most arguments to the case of domains.

Proposition 3.2 (multiple closure can be tested modulo minimal primes). Suppose that R is a Noetherian ring of positive characteristic p, and let I_{1}, …, I_{n} be ideals in R. An element x of R is in (I_{1}, …, I_{n})^{∼^{∗}} if and only if for all minimal primes p of R, the image \overline{x} of x in

R/p is in (I_{1}R/p, …, I_{n}R/p)^{∼^{∗}}.

Proof. The proof follows from the description of minimal primes of localized Rees rings, as in Corollary 2.7.

Let I = (I_{1} + (w_{1})) … (I_{n} + (w_{n})). Then by Proposition 2.6 and Corollary 2.7, a minimal prime p′ of

S = R[w_{1}, …, w_{n}, \frac{I_{1}}{w_{1}} , …, \frac{I_{n}}{w_{n}}] ≃ R[w_{1}, …, w_{n}, It]_{(w_{1}...w_{n}t)}

corresponds to a minimal prime p^{†} of R[w_{1}, …, w_{n}] that does not contain w_{1} … w_{n}, which in turn corresponds to a minimal prime p of R. By the same results, we have the following isomorphisms

S/p′ ≃ ((R[w_{1}, …, w_{n}]/p^{†})[I(R[w_{1}, …, w_{n}]/p^{†})t])_{(w_{1}...w_{n}t)}

which is isomorphic to

((R/p)[w_{1}, …, w_{n}] [(I(R/p)[w_{1}, …, w_{n}])t])_{(w_{1}...w_{n}t)}

which is isomorphic to

(R/p) [w_{1}, …, w_{n}, \frac{I_{1}(R/p)}{w_{1}}, …, \frac{I_{n}(R/p)}{w_{n}}] .

Now take x ∈ (I_{1}, …, I_{n})^{∼^{∗}}. By Definition 3.1, this is equivalent to

x ∈ ((w_{1}, …, w_{n})S)^{∗} ∩ R.

From the isomorphisms above, equivalently for all p ∈ M(R)

\overline{x} ∈ ((w_{1}, …, w_{n})S/p^{†})^{∗} ∩ R/p,

or, equivalently

\overline{x} ∈ ((w_{1}, …, w_{n})(R/p)[w_{1}, …, w_{n}, \frac{I_{1}(R/p)}{w_{1}}, …, \frac{I_{n}(R/p)}{w_{n}}])^{∗} ∩ R/p

By Definition 3.1 this is equivalent to \overline{x} ∈ (I_{1}R/p, …, I_{n}R/p)^{∼^{∗}}.

Theorem 3.3. Let R be a Noetherian ring of prime characteristic p such that either R is essentially of finite type over an excellent local ring, or R_{red} is F -finite. Let I_{1}, …, I_{n} be ideals of R. Then

(I_{1}, …, I_{n})^{∼^{∗}} = (I_{1}, …, I_{n})\underline{∗} .

Proof. Since T I closure and multiple closure can both be tested modulo minimal primes ([ho2] 1.4 and 3.2 above), we can assume that R is a domain. Suppose that S = R[w_{1}, …, w_{n}, \frac{I_{1}}{w_{1}} , …, \frac{I_{n}}{w_{n}}], where w_{1}, …, w_{n} are indeterminates. Pick z ∈ (I_{1}, …, I_{n})\underline{∗}. Then there is a nonzero c in R such that cz^{q} ∈ I_{1}^{q} + … + I_{n}^{q} for all large q = p^{e}. Since we have the inclusion R ⊂ S, it follows that for all large q

cz^{q} ∈ (I_{1}^{q} + … + I_{n}^{q})S ⊆ (w_{1}^{q}, …, w_{n}^{q})S = (w_{1}, …, w_{n})^{[q]}S .

Hence

z ∈ ((w_{1}, …, w_{n})S)^{∗} ∩ R = (I_{1}, …, I_{n})^{∼^{∗}}.

To show the other inclusion, we choose an element c in R such that R_{c} is regular. Then

S_{cw_{1}...w_{n}} = R_{c}[w_{1}, …, w_{n}, \frac{1}{w_{1}} , …, \frac{1}{w_{n}}]

is regular, and therefore d = cw_{1} … w_{n} has a power c′ that is a test element for the ring S (Theorem 1.10). By multiplying c′ with appropriate powers of elements in \frac{I_{1}}{w_{1}} , …, \frac{I_{n}}{w_{n}}, we may assume that c′ ∈ R.

Now take z ∈ (I_{1}, …, I_{n})^{∼^{∗}}. Then

c′z^{q} ∈ (w_{1}^{q}, …, w_{n}^{q})S

for all q = p^{e}.

So for a given q, we can find C_{1}, …, C_{n} in S, such that

c′z^{q} = C_{1}w_{1}^{q} + … + C_{n}w_{n}^{q}.

By taking common denominators, we can find a positive integer N, which we can take to be larger than q, such that C_{i} = \frac{A_{i}}{(w_{1}...w_{n})^{N}} for every i = 1, …, n, where A_{i} is a polynomial in R[w_{1}, …, w_{n}]. So we get

c′z^{q}(w_{1} … w_{n})^{N} = A_{1}w_{1}^{q} + … + A_{n}w_{n}^{q}.

Since R[w_{1}, …, w_{n}] is a free module over R generated by the monomials in w_{1}, …, w_{n}, and c′z^{q} ∈ R, we can without loss of generality take each A_{i} to be is a monomial of the form B_{i}w_{1}^{N} … w_{i−1}^{N}w_{i}^{N−q}w_{i+1}^{N} … w_{n}^{N}, where B_{i} ∈ R, for all i = 1, …, n. So we can write c′z^{q} as

\frac{A_{1}}{(w_{1} … w_{n})^{N}} w_{1}^{q} + … + \frac{A_{n}}{(w_{1} … w_{n})^{N}} w_{n}^{q} = \frac{B_{1}}{w_{1}^{q}} w_{1}^{q} + … + \frac{B_{n}}{w_{n}^{q}} w_{n}^{q}

which implies that B_{i} ∈ I_{i}^{q} for all i = 1, …, n. So

c′z^{q} ∈ I_{1}^{q} + … + I_{n}^{q}.

This holds for all q, hence z ∈ (I_{1}, …, I_{n})\underline{∗}.

This equality translates the T I closure of a set of ideals in R into the tight closure of an ideal in an extension ring of R. In particular, most properties of tight closure can now be extended to T I closure.

3.2 Basic Properties of T I Closure Via Multiple Closure

Theorem 3.4 (persistence of T I closure). Let R be a Noetherian ring of prime characteristic p that is either essentially of finite type over an excellent local ring or R_{red} is F -finite. Let I_{1}, …, I_{n} be ideals of R. Suppose R ⟶ S is a homomorphism of rings. Then T I closure persists under this map:

(I_{1}, …, I_{n})\underline{∗} S ⊆ (I_{1}S, …, I_{n}S)\underline{∗} .

Proof. The properties mentioned above for R are preserved when we pass to the ring R^{†} = R[w_{1}, …, w_{n}, \frac{I_{1}}{w_{1}} , …, \frac{I_{n}}{w_{n}}], since this is just a finitely generated algebra over R. Moreover, we have the obvious induced map

R^{†} ⟶ S^{†} = S[w_{1}, …, w_{n}, \frac{I_{1}S}{w_{1}} , …, \frac{I_{n}S}{w_{n}}]

under which tight closure persists. Therefore, by Theorem 1.11,

(w_{1}, …, w_{n})^{∗}S^{†} ⊆ ((w_{1}, …, w_{n})S^{†})^{∗}

which implies that (I_{1}, …, I_{n})\underline{∗} S ⊆ (I_{1}S, …, I_{n}S)\underline{∗} .

An interesting corollary is that multiple closure (or T I closure) is indeed a blowup closure in a larger ring.

Corollary 3.5. Let R be a Noetherian ring of prime characteristic p such that either R is essentially of finite type over an excellent local ring, or R_{red} is F -finite. Let I_{1}, …, I_{n} be ideals of R. If S = R[w_{1}, …, w_{n}], and for i = 1, …, n, I′_{i} = I_{i} + (w_{i}) is an ideal of S, then

(I_{1}, …, I_{n})\underline{∗} = (I_{1}, …, I_{n})^{∼^{∗}} = ((I′_{1}, …, I′_{n})S)^{∼} ∩ R.

Proof. From Theorem 3.4, [Ho2] Proposition 1.4, and Theorem 2.9 part (a) we have

(I_{1}, …, I_{n})\underline{∗}S ⊆ (I_{1}S, …, I_{n}S)\underline{∗} ⊆ (I′_{1}, …, I′_{n})\underline{∗} ⊆ (I′_{1}, …, I′_{n})^{∼},

and so

(I_{1}, …, I_{n})\underline{∗} ⊆ (I′_{1}, …, I′_{n})^{∼} ∩ R.

On the other hand, let I′ and J′ denote the product and sum of I′_{1}, …, I′_{n}, respectively, and let G′ be a fixed set of generators for I′ such that w_{1} … w_{n} ∈ G′. By Theorem 2.3,

(I′_{1}, …, I′_{n})^{∼} = ⋂_{f∈G′} (J′S[I′t]_{(ft)})^{∗} ∩ S.

From the previous paragraph, for every f ∈ G′, (I_{1}, …, I_{n})\underline{∗} ⊆ (J′S[I′t]_{(ft)})^{∗} ∩ R. On the other hand, by definition of multiple closure, we know that if f = w_{1} … w_{n}, then (I_{1}, …, I_{n})\underline{∗} = (J′S[I′t]_{(ft)})^{∗} ∩ R. So we have

(I_{1}, …, I_{n})\underline{∗} = ((I′_{1}, …, I′_{n})S)^{∼} ∩ R.

One can see that tight closure commuting with localization and T I closure commuting with localization are equivalent properties.

Theorem 3.6. Let R denote the class of all Noetherian rings R, such that R is either essentially of finite type over an excellent local ring or R_{red} is F -Finite. Then TI closure commutes with localization for all rings in R if and only if tight closure commutes with localization for all rings in R.

Proof. It is clear that if T I closure commutes with localization for any ring, then so will tight closure, since the tight closure of an ideal is equal to the T I closure of a set of principal ideals ([Ho2] proposition 1.4).

Now suppose that tight closure commutes with localization, and take a set of ideals I_{1}, …, I_{n} in a ring R described above. Let U be a multiplicative set in R. Then

U^{−1}(I_{1}, …, I_{n})\underline{∗} = U^{−1} [((w_{1}, …, w_{n})R[w_{1}, …, w_{n}, \frac{I_{1}}{w_{1}} , …, \frac{I_{n}}{w_{n}}])^{∗} ∩ R]
= ((w_{1}, …, w_{n})U^{−1}R[w_{1}, …, w_{n}, \frac{I_{1}}{w_{1}} , …, \frac{I_{n}}{w_{n}}])^{∗} ∩ U^{−1}R
= (U^{−1}I_{1}, …, U^{−1}I_{n})\underline{∗}.

In Section 3.4 we strengthen the statement of Theorem 3.6 using the notion of test exponents.

Theorem 3.7. Let R be a Noetherian ring of prime characteristic p such that either R is essentially of finite type over an excellent local ring, or R_{red} is F -Finite. Let I_{1}, …, I_{n} be ideals of R. Suppose that c ∈ R^{o} and z ∈ R are such that cz^{q} ∈ I_{1}^{q} + … + I_{n}^{q} for infinitely many powers q of p. Then z ∈ (I_{1}, …, I_{n})\underline{∗}.

Proof. Let

S = R[w_{1}, …, w_{n}, \frac{I_{1}}{w_{1}} , …, \frac{I_{n}}{w_{n}}].

Since c ∈ R^{o}, it immediately follows that c ∈ S^{o}; see Corollary 2.7. Also, cz^{q} ∈ I_{1}^{q} + … + I_{n}^{q} implies that cz^{q} ∈ (w_{1}, …, w_{n})^{[q]} in S for infinitely many q. Therefore

z ∈ ((w_{1}, …, w_{n})R[w_{1}, …, w_{n}, \frac{I_{1}}{w_{1}} , …, \frac{I_{n}}{w_{n}}])^{∗} ∩ R = (I_{1}, …, I_{n})\underline{∗}.

3.3 Test Elements for T I Closure

Theorem 3.3 allows us to develop a theory of test elements for T I closure (see Section 1.2). Test elements are useful since they help us decide whether a given element of a ring is in the T I closure of a given set of ideals in that ring. Test elements do not exist for integral closure (see Example 1.22), and since the T I closure of one ideal is the integral closure of that ideal (see Section 1.3), we expect the test elements for T I closure to depend on the ideals. We therefore first specify what we mean by test elements for T I closure.

Definition 3.8. Let R be a Noetherian ring of prime characteristic p. Let I_{1}, …, I_{n} be ideals in R. We say that c ∈ R^{o} is a TI closure test element for I_{1}, …, I_{n}, or in short, a test element for I_{1}, …, I_{n}, if for every z ∈ (I_{1}, …, I_{n})\underline{∗}, cz^{q} ∈ I_{1}^{q} + … + I_{n}^{q}, for all nonnegative powers q of p. We call the ideal generated by the test elements for I_{1}, …, I_{n} the test ideal for I_{1}, …, I_{n}, and denote it by τ(I_{1}, …, I_{n}).

From the proof of Theorem 3.3 it immediately follows that:

Corollary 3.9. Let R be a Noetherian ring of prime characteristic p such that either R is essentially of finite type over an excellent local ring, or R_{red} is F -finite. Let I_{1}, …, I_{n} be ideals of R, and let S = R[w_{1}, …, w_{n}, \frac{I_{1}}{w_{1}} , …, \frac{I_{n}}{w_{n}}]. Then

τ(S) ∩ R ⊆ τ(I_{1}, …, I_{n}).

Here, τ(S) is the usual tight closure test ideal for S. We also obtain locally stable T I closure test elements for a set of ideals; these are T I closure test elements of a set of ideals I_{1}, …, I_{n} that remain test elements for these ideals after we localize the ring at any multiplicative set.

Theorem 3.10. Let R be a Noetherian ring of prime characteristic p such that either R is essentially of finite type over an excellent local ring, or R_{red} is F -finite. Let I_{1}, …, I_{n} be ideals of R. Let c be an element of R^{o} such that R_{c} is regular. Then for some positive integer N and all choices a_{1} ∈ I_{1}, …, a_{n} ∈ I_{n} , (ca_{1} … a_{n})^{N} is a locally stable T I closure test element for the ideals I_{1}, …, I_{n}.

Proof. Let U be a multiplicative set in R. Suppose z ∈ (U^{−1}I_{1}, …, U^{−1}I_{n})\underline{∗}. Therefore

z ∈ ((w_{1}, …, w_{n})U^{−1}R[w_{1}, …, w_{n}, \frac{U^{−1}I_{1}}{w_{1}} , …, \frac{U^{−1}I_{n}}{w_{n}}])^{∗} = ((w_{1}, …, w_{n})U^{−1}S)^{∗},

where

S = R[w_{1}, …, w_{n}, \frac{I_{1}}{w_{1}} , …, \frac{I_{n}}{w_{n}}],

and U denotes the multiplicative set U in R, as well as the image of U in S.

Since R_{c} is regular, so is S_{cw_{1}...w_{n}}, and so cw_{1} … w_{n} to some positive power N is a locally stable test element for S (see Theorem 1.10). This means that (cw_{1} … w_{n})^{N} is a test element for U^{−1}S, and therefore (ca_{1} … a_{n})^{N} = (cw_{1} … w_{n})^{N}(\frac{a_{1}}{w_{1}} … \frac{a_{n}}{w_{n}})^{N} is a test element for U^{−1}S for any a_{i} ∈ I_{i}.

For any such test element d = (ca_{1} … a_{n})^{N} we have

dz^{q} ∈ (w_{1}, …, w_{n})^{[q]}U^{−1}S

for all powers q of p, and following the proof of Theorem 3.3, we get

dz^{q} ∈ U^{−1}I_{1}^{q} + … + U^{−1}I_{n}^{q}

for all nonnegative powers q of p.

This implies that d is a T I closure test element for U^{−1}I_{1}, …, U^{−1}I_{n}, and since this holds for all U, we conclude that d is a locally stable T I closure test element for I_{1}, …, I_{n} in R.

For ideals in a finitely generated algebra over a field of characteristic p, we are able to compute explicit T I closure test elements. Using a theorem of Lipman and Sathaye, Hochster and Huneke described specific tight closure test elements for such rings.

Theorem 3.11 (Corollary 1.5.5 [HH2]). Let k be a field of characteristic p and let R be a ddimensional geometrically reduced domain over k (meaning that \overline{k} ⊗_{k} R is reduced) that is finitely generated as a k-algebra. Let R = k[u_{1}, …, u_{m}]/(g_{1}, …, g_{s}) be a presentation of R as a homomorphic image of a polynomial ring. Then the (m − d) × (m − d) minors of the Jacobian matrix (δg_{i}/δu_{j}) are contained in the test ideal of R, and remain so after localization and completion. Thus, any element of the Jacobian ideal generated by all these minors that is in R^{o} is a completely stable test element.

Theorem 3.12. Let k be a field of characteristic p and let R be a d-dimensional geometrically reduced domain over k that is finitely generated as a k-algebra. Let R = k[u_{1}, …, u_{m}]/(g_{1}, …, g_{s}) be a presentation of R as a homomorphic image of a polynomial ring. Take ideals I_{1}, …, I_{n} of R where for each i = 1, …, n, I_{i} is minimally generated by the elements f_{1}^{i}, …, f_{m_{i}}^{i} of R. Then

I_{1}^{m_{1}−1} … I_{n}^{m_{n}−1}J_{m−d} ⊆ τ(I_{1}, …, I_{n}) ,

where J_{m−d} is the ideal generated by the (m − d) × (m − d) minors of the Jacobian matrix J (R) = (\frac{δg_{i}}{δu_{j}}).

( \begin{matrix}\frac{δg_{1}}{δu_{1}} & ⋯ & \frac{δg_{1}}{δu_{m}} \\ ⋮ & ... & ⋮ & & & & & & & 0 \\ \frac{δg_{s}}{δu_{1}} & ⋯ & \frac{δg_{s}}{δu_{m}} \\ & & & x_{1}^{1} & 0 & ⋯ & 0 & w_{1} & ⋯ & 0 & 0 & ⋯ & 0 & ⋯ & 0 & ⋯ & 0 \\ & & & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ & & & x_{m_{1}}^{1} & 0 & ⋯ & 0 & 0 & ⋯ & w_{1} & 0 & ⋯ & 0 & ⋯ & 0 & ⋯ & 0 \\ & & & 0 & x_{1}^{2} & ⋯ & 0 & 0 & ⋯ & 0 & w_{2} & ⋯ & 0 & ⋯ & 0 & ⋯ & 0 \\ & & & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ & A & & 0 & x_{m_{2}}^{2} & ⋯ & 0 & 0 & ⋯ & 0 & 0 & ⋯ & w_{2} & ⋯ & 0 & ⋯ & 0 \\ & & & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋮ \\ & & & 0 & 0 & ⋯ & x_{1}^{n} & 0 & ⋯ & 0 & 0 & ⋯ & 0 & ⋯ & w_{n} & ⋯ & 0 \\ & & & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... & ⋮ \\ & & & 0 & 0 & ⋯ & x_{m_{n}}^{n} & 0 & ⋯ & 0 & 0 & ⋯ & 0 & ⋯ & 0 & ⋯ & w_{n} \end{matrix} )

Figure 1: Part of the Jacobian matrix

Proof. Let S be the ring

S = R[w_{1}, …, w_{n}, \frac{I_{1}}{w_{1}} , …, \frac{I_{n}}{w_{n}}].

Then there is a surjective map to S from the polynomial ring

H = k[u_{1}, …, u_{m}, w_{1}, …, w_{n}, x_{1}^{1}, …, x_{m_{1}}^{1}, …, x_{1}^{n}, …, x_{m_{n}}^{n}]

mapping x_{i}^{j} to \frac{f_{i}^{j}}{w_{j}}. If for 1 ≤ i ≤ n and 1 ≤ j ≤ m_{i}, F_{j}^{i} is an element of the polynomial ring k[u_{1}, …, u_{m}] whose image in R is f_{j}^{i}, the kernel ℘ of this map is generated by

g_{1}, …, g_{s}, w_{1}x_{1}^{1} − F_{1}^{1}, …, w_{1}x_{m_{1}}^{1} − F_{m_{1}}^{1}, …, w_{n}x_{1}^{n} − F_{1}^{n}, …, w_{n}x_{m_{n}}^{n} − F_{m_{n}}^{n}

and possibly other polynomials. So S is isomorphic under this map to H/℘. A part of the Jacobian matrix for this presentation of S will then look like the matrix shown in Figure 1.

Now dim R = d, and so dim S = d+n. If m′ = m_{1}+…+m_{n}, then m+n+m′−(n+d) = m+m′−d, and so we take the ideal I of the (m + m′ − d) × (m + m′ − d) minors of this matrix. We note that I is contained in the ideal generated by the (m + m′ − d) × (m + m′ − d) minors of the Jacobian matrix of S, and so from Theorem 3.11 we can conclude that I is in the (tight closure) test ideal for S. Corollary 3.9 then implies that I ∩ R ⊆ τ(I_{1}, …, I_{n}).

To find elements of I that are in R, we make the following partition of the matrix:

Step 1. We take the m′ × m′ minor of the lower right corner of the matrix by taking the first n columns, and for each i, i = 1, …, n, removing one of the columns (say the s_{i}th column) involving a w_{i}. The outcome is x_{s_{1}}^{1} … x_{s_{n}}^{n}w_{1}^{m_{1}−1} … w_{n}^{m_{n}−1}.

Step 2. The upper left corner of this matrix is just J (R). By taking the (m − d) × (m − d) minors of the upper left corner we obtain J_{m−d}.

So

x_{s_{1}}^{1} … x_{s_{n}}^{n}w_{1}^{m_{1}−1} … w_{n}^{m_{n}−1}J_{m−d} ⊆ I.

It follows that:

(\frac{I_{1}}{w_{1}} )^{m_{1}−2} … (\frac{I_{n}}{w_{n}})^{m_{n}−2}x_{s_{1}}^{1} … x_{s_{n}}^{n}w_{1}^{m_{1}−1} … w_{n}^{m_{n}−1}J_{m−d} ⊆ I,

and so

F_{s_{1}}^{1} … F_{s_{n}}^{n}I_{1}^{m_{1}−2} … I_{n}^{m_{n}−2}J_{m−d} ⊆ I,  (5)

since for each i, x_{s_{i}}^{i}w_{i} = F_{s_{i}}^{i} ∈ I_{i}.

By repeating Step 1, and for each i = 1, …, n allowing s_{i} to vary between 1 and m_{i}, we see that the inclusion in (5) will hold for 1 ≤ s_{i} ≤ m_{i} and i = 1, …, n. It follows that

I_{1} … I_{n}I_{1}^{m_{1}−2} … I_{n}^{m_{n}−2}J_{m−d} = I_{1}^{m_{1}−1} … I_{n}^{m_{n}−1}J_{m−d} ⊆ I.

In practice, k being of characteristic zero or perfect and R being reduced ensures that R is geometrically reduced (see tensor products over fields in [ZS1]).

Example 3.13. Let R = k[x, y, z] be a polynomial ring over a perfect field k of characteristic p. Let I = (f, g) and J = (h) be ideals of R. Then every element of I is a T I closure test element for the ideals I and J.

3.4 Test Exponents and the Localization of T I Closure

A recent advance in tight closure theory is the development of test exponents. Hochster and Huneke show in [HH4] that the existence of test exponents for tight closure is roughly equivalent to tight closure commuting with localization. This result suggests that the study of test exponents could provide a breakthrough in the localization problem for tight closure.

The situation is similar for T I closure: below we define T I closure test exponents, and establish that T I closure commuting with localization is related to the existence of test exponents, although the result we get for T I closure is somewhat weaker than the corresponding result for tight closure in [HH4]. Nevertheless, Theorem 3.17 below strengthens our previous statement (Theorem 3.6) on the equivalence of tight closure commuting with localization with T I closure commuting with localization.

We begin by stating some relevant facts from tight closure theory.

Definition 3.14 ([HH4]). Let R be a reduced Noetherian ring of positive prime characteristic p. Let c be a fixed test element for R, and let I be an ideal of R. Then q = p^{e} is called a test exponent for c and I, if whenever cu^{Q} ∈ I^{[Q]} for some u ∈ R and Q ≥ q, then u ∈ I^{∗}.

Thus the existence of test exponents reduces the process of checking whether an element u of the ring is in the tight closure of an ideal I, to just checking if cu^{Q} ∈ I^{[Q]} for some large Q, rather than all large Q. It is easy to see that the existence of a test exponent for an ideal forces that ideal to commute with localization (see Proposition 2.3 of [HH4]). The converse of this statement is however difficult to prove. Here, we produce a parallel definition for T I closure test exponents. It will easily follow that the existence of such an exponent for a set of ideals will force the T I closure of those ideals to commute with localization. For the converse, we exploit the fact that T I closure can be described in terms of tight closure and apply results of [HH4] to achieve the desired statement.

Definition 3.15. Let R be a Noetherian ring of prime characteristic p, and let I_{1}, …, I_{n} be a set of ideals in R. Let c be a fixed test element for I_{1}, …, I_{n} in R. Then q = p^{e} is called a test exponent for c, I_{1}, …, I_{n}, if whenever cu^{Q} ∈ I_{1}^{Q} + … + I_{n}^{Q} for some u ∈ R and Q ≥ q, then u ∈ (I_{1}, …, I_{n})\underline{∗}.

Theorem 3.16. Let R be a Noetherian ring of prime characteristic p, and let I_{1}, …, I_{n} be ideals of R. Suppose that c is a locally stable test element for I_{1}, …, I_{n}, and suppose that c, I_{1}, …, I_{n} have a test exponent q. Then T I closure commutes with localization for I_{1}, …, I_{n}.

Proof. Let U be a multiplicative set in R, and let z/1 ∈ (U^{−1}I_{1}, …, U^{−1}I_{n})\underline{∗}. We have

cz^{q}/1 ∈ U^{−1}I_{1}^{q} + … + U^{−1}I_{n}^{q},

and so for some u ∈ U, ucz^{q} ∈ I_{1}^{q} + … + I_{n}^{q}, hence c(uz)^{q} ∈ I_{1}^{q} + … + I_{n}^{q}. Since q is a test exponent, this implies that uz ∈ (I_{1}, …, I_{n})\underline{∗}, and so

z/1 ∈ U^{−1}(I_{1}, …, I_{n})\underline{∗}.

Theorem 3.17. Let R be a Noetherian ring of prime characteristic p that is either essentially of finite type over an excellent local ring or R_{red} is F -Finite. Let I_{1}, …, I_{n} be ideals of R. Suppose that the tight closure of the ideal (w_{1}, …, w_{n}) in the ring S = R[w_{1}, …, w_{n}, I_{1}/w_{1}, …, I_{n}/w_{n}] commutes with localization at all primes ideals in Ass (S/(w_{1}, …, w_{n})^{∗}). Let c be a locally stable test element for I_{1}, …, I_{n} in R, which is also a locally stable test element for the ring S (such test elements exist; see the statement and proof of Theorem 3.10). Then c, I_{1}, …, I_{n} have a test exponent.

Proof. Since tight closure of the ideal (w_{1}, …, w_{n}) commutes with localization at all primes in Ass (S/(w_{1}, …, w_{n})^{∗}), Theorem 2.4 of [HH4] implies that c, (w_{1}, …, w_{n}) have a test exponent q.

Now suppose cu^{Q} ∈ I_{1}^{Q} + … + I_{n}^{Q} for some u ∈ R and Q ≥ q. Then cu^{Q} belongs to the ideal (w_{1}, …, w_{n})^{[Q]} in S, which implies that u ∈ (w_{1}, …, w_{n})^{∗}. Hence

u ∈ (w_{1}, …, w_{n})^{∗} ∩ R = (I_{1}, …, I_{n})\underline{∗},

which proves that q is a test exponent for c, I_{1}, …, I_{n}.

We are now able to strengthen one direction of the statement of Theorem 3.6.

Corollary 3.18. If R, S, I_{1}, …, I_{n} are as in Theorem 3.17, then the T I closure of I_{1}, …, I_{n} commutes with localization at any multiplicative set in R, if the tight closure of (w_{1}, …, w_{n}) in S commutes with localization at all primes that belong to Ass (S/(w_{1}, …, w_{n})^{∗}).

3.5 T I Closure in Equal Characteristic Zero

The translation of T I closure into tight closure enables us to extend T I closure to rings of characteristic zero in an effective way. The definition of T I closure in characteristic zero was introduced in [Ho2], but due to the difficulty of working with the positive characteristic definition, several questions remained unanswered (see Section 1.3). In this section, we address some of those questions for affine algebras over fields of characteristic zero. We show that T I closure in equal characteristic zero has similar features to tight closure in equal characteristic zero. We also introduce universal test elements for T I closure based on universal test elements for tight closure.

We begin by recalling the definition of T I closure in characteristic zero:

Definition 3.19. Let R be a finitely generated algebra over a field K of characteristic zero, and let I = {I_{1}, …, I_{n}} be a set of ideals of R. We say that an element x of R is in the tight integral closure of I, denoted by I\underline{∗}, if there exist descent data (D, R_{D}, I_{1,D}, …, I_{n,D}), such that for every maximal ideal m of D, if k = D/m, then x_{k} ∈ I_{k}\underline{∗}, where I_{k} denotes the set of ideals \{I_{1,k}, …, I_{n,k}\}, with I_{t,k} = k ⊗_{k} I_{t,D}.

See Definition 1.13 for the definition of descent data. Similar to the positive characteristic situation, we can describe T I closure in equal characteristic zero in terms of tight closure.

Theorem 3.20. Let R be an affine algebra over a field K of characteristic zero, which can be presented as

R = K[u_{1}, …, u_{m}]/(g_{1}, …, g_{s})

where u_{1}, …, u_{m} are indeterminates and g_{1}, …, g_{s} are polynomials in K[u_{1}, …, u_{m}]. Suppose I_{1}, …, I_{n} are ideals of R, and x ∈ R. Then x ∈ (I_{1}, …, I_{n})\underline{∗} if and only if

x ∈ ((w_{1}, …, w_{n})R[w_{1}, …, w_{n}, \frac{I_{1}}{w_{1}} , …, \frac{I_{n}}{w_{n}}])^{∗}

where w_{1}, …, w_{n} are indeterminates.

Proof. The main point of the proof is that one can construct descent data that work for the ideals I_{1}, …, I_{n} in R as well as for the ideal (w_{1}, …, w_{n}) in

S = R[w_{1}, …, w_{n}, \frac{I_{1}}{w_{1}} , …, \frac{I_{n}}{w_{n}}].

Suppose that for each i, I_{i} = (f_{1}^{i}, …, f_{s_{i}}^{i}) in R and for every i and j, 1 ≤ i ≤ n and 1 ≤ j ≤ s_{i}, F_{j}^{i} is a polynomial in K[u_{1}, …, u_{m}] whose image in R is f_{j}^{i}. Pick u ∈ K[u_{1}, …, u_{m}] whose image in R is x.

We can represent S as a polynomial ring

K[u_{1}, …, u_{m}, w_{1}, …, w_{n}, α_{1}^{1}, …, α_{s_{1}}^{1}, …, α_{1}^{n}, …, α_{s_{n}}^{n}]

modulo the ideal J which is generated by g_{1}, …, g_{s}, polynomials of the form w_{i}α_{j}^{i} −F_{j}^{i} for 1 ≤ i ≤ n and 1 ≤ j ≤ s_{i}, and possibly other polynomials (see the proof of Theorem 3.12 for a more detailed description of this isomorphism).

We can now construct descent data D for (w_{1}, …, w_{n}) and x in S by adjoining the coefficients of all the generators of J in K and the coefficients of u in K to the ring of integers Z. We then replace D by a localization at a single element to assure that it satisfies the properties of descent data (Lemma of Generic Freeness, [HR]). This D will also work as descent data for I_{1}, …, I_{n} and x in R, since it is an enlargement of some basic descent data that one would construct, and once some descent data work, every enlargement of them work as well (see Section 1.2).

With this construction, we have that x ∈ (I_{1}, …, I_{n})\underline{∗} if and only if for every maximal ideal m of D, setting k = D/m

x_{k} ∈ (I_{1,k}, …, I_{n,k})\underline{∗}

in R_{k}. Since k is a finite and therefore perfect field, R_{k} will be F -finite, and so equivalently

x_{k} ∈ ((w_{1}, …, w_{n})R_{k}[w_{1}, …, w_{n}, \frac{I_{1,k}}{w_{1}} , …, \frac{I_{n,k}}{w_{n}}])^{∗}.

This is equivalent to x ∈ ((w_{1}, …, w_{n})S_{k})^{∗}, and since this holds for all residue class fields k of D, we conclude that equivalently x ∈ ((w_{1}, …, w_{n})S)^{∗}.

The description of T I closure as tight closure in characteristic zero yields the following:

Theorem 3.21 (independence of choice of descent). Let K be a field of characteristic zero, R a finitely generated K-algebra, I_{1}, …, I_{n} ideals of R and u ∈ R. Let (D, I_{1,D}, …, I_{n,D}) be descent data for R, I_{1}, …, I_{n} and u. If u ∈ (I_{1}, …, I_{n})\underline{∗}, then for almost all maximal ideals m of D (i.e. for m in a dense open subset of MaxSpecD), if k = D/m, then u_{k} ∈ (I_{1,k}, …, I_{n,k})\underline{∗} in R_{k}.

Proof. By joining finitely many elements of K to D and localizing at an element, we obtain descent data (A, S_{A}, γ_{A}) for

S = R[w_{1}, …, w_{n}, \frac{I_{1}}{w_{1}} , …, \frac{I_{n}}{w_{n}}],

γ = (w_{1}, …, w_{n}) and u. We can localize both A and D at an element of D so that A will be free and hence faithfully flat over D ([HR]).

Let m ∈ MaxSpecD and let k = D/m. Take m′ ∈ MaxSpecA that lies over m, and let k′ = A/m′. Since tight closure is independent of the choice of descent, u_{k′} ∈ γ_{k′}^{∗} in S_{k′}. On the other hand S_{k′} = k′ ⊗_{k} S_{k} (where S_{k} = R_{k}[w_{1}, …, w_{n}, \frac{I_{1,k}}{w_{1}} , …, \frac{I_{n,k}}{w_{n}}]). The map k ⟶ k′ is a finite separable extension of fields, since both k and k′ are finite. It follows now from [HH3] Theorem 7.29a^{o} that u_{k} ∈ γ_{k}^{∗} in S_{k}, which implies that u_{k} ∈ (I_{1,k}, …, I_{n,k})\underline{∗} in R_{k}.

We now introduce the notion of universal test element for T I closure in the case where R is an affine domain over a field of characteristic zero.

Definition 3.22. Let R be an affine algebra over a field of characteristic zero that is a domain and let I_{1}, …, I_{n} be ideals of R. Let (D, R_{D}, I_{1,D}, …, I_{n,D}) be descent data. Then an element c_{D} ∈ R_{D}^{o} is called a universal test element for I_{1,D}, …, I_{n,D} if for every u ∈ (I_{1}, …, I_{n})\underline{∗}, and almost all m ∈ MaxSpecD, if k = D/m, then

c_{k}u_{k}^{q} ∈ I_{1,k}^{q} + … + I_{n,k}^{q},

for all positive powers q of p, where p is the characteristic of k.

Similar to the situation in the positive characteristic case, we can explicitly calculate universal test elements for the T I closure of a set of ideals. We first state the analogous theorem for tight closure.

Theorem 3.23 ([HH2] 2.4.10). Let A ⊇ Z be a domain finitely generated over Z with fraction field F, and let R_{A} be a finitely generated A-algebra. Suppose that R_{F} is an absolute domain of dimension d, that is, \overline{F} ⊗_{F} R_{F} (where \overline{F} is the algebraic closure of the field F) is a domain. Let

R_{A} = A[u_{1}, …, u_{m}]/(g_{1}, …, g_{s}).

Then every element of the ideal generated by the (m − d) × (m − d) minors of the Jacobian matrix (δg_{i}/δu_{j}) is a universal test element of R_{A} over A.

Similar to the positive characteristic situation, we can deduce

Theorem 3.24. Let K be a field of characteristic zero and let R be an equidimensional finitely generated reduced algebra over K. Take ideals I_{1}, …, I_{n} of R where for each i = 1, …, n, I_{i} is minimally generated by the elements f_{1}^{i}, …, f_{m_{i}}^{i} of R. Let (D, R_{D}, I_{1,D}, …, I_{n,D}) be descent data such that if F is the fraction field of D, then R_{F} is an absolute domain of dimension d. Suppose that R_{D} is presented as

R_{D} = D[u_{1}, …, u_{m}]/(g_{1}, …, g_{s}).

Then every nonzero element of

I_{1,D}^{m_{1}−1} … I_{n,D}^{m_{n}−1}J_{m−d}

is a universal test element for I_{1,D}, …, I_{n,D}, where J_{m−d} is the ideal generated by the (m − d) × (m − d) minors of the Jacobian matrix (δg_{i}/δu_{j}).

Proof. Let u ∈ (I_{1}, …, I_{n})\underline{∗}. Then for almost all maximal ideals m of R_{D}, if k = D/m, u_{k} ∈ (I_{1,k}, …, I_{n,k})\underline{∗}. Equivalently, by Theorem 3.20, u_{k} ∈ ((w_{1}, …, w_{n})S_{k})^{∗}, where

S_{D} = R_{D}[w_{1}, …, w_{n}, \frac{I_{1}}{w_{1}} , …, \frac{I_{n}}{w_{n}}].

If c is a universal test element for S_{D} over S, then c_{k} is a test element for S_{k} for almost all m ∈ MaxSpecD, k = D/m (see [HH2] or Theorem 1.15). If c happens to be in R_{D}, it will follow that c_{k} is a T I closure test element for the ideals I_{1,k}, …, I_{n,k} for almost all m ∈ MaxSpecD, k = D/m, and so c will be a universal test element for I_{1,D}, …, I_{n,D}.

As in Theorem 3.12 we use the D-algebra structure of S_{D} and construct part of its Jacobian matrix, and take appropriate minors of the Jacobian to generate universal test elements for I_{1,D}, …, I_{n,D} in R_{D}.

With the presentation of R_{D} as a finitely generated D-algebra given above, S_{D} will be isomorphic to the polynomial ring

D[u_{1}, …, u_{m}, w_{1}, …, w_{n}, x_{1}^{1}, …, x_{m_{1}}^{1}, …, x_{1}^{n}, …, x_{m_{n}}^{n}]

modulo the ideal generated by

g_{1}, …, g_{s}, w_{1}x_{1}^{1} − F_{1}^{1}, …, w_{1}x_{m_{1}}^{1} − F_{m_{1}}^{1}, …, w_{n}x_{1}^{n} − F_{1}^{n}, …, w_{n}x_{m_{n}}^{n} − F_{m_{n}}^{n}

and possibly other polynomials, where F_{j}^{i} is an element of the polynomial ring D[u_{1}, …, u_{m}] whose image in R_{D} is f_{j}^{i}, 1 ≤ i ≤ n, 1 ≤ j ≤ m_{i}. A part of the Jacobian matrix of S_{D} will then look like the matrix shown in Figure 1 on page 25.

We are given that dim R_{F} = d, and so dim S_{F} = d + n. If m′ = m_{1} + … + m_{n}, then m + n + m′ − (n + d) = m + m′ − d, and so to obtain universal test elements via Theorem 3.23, we are interested in the ideal generated by the (m + m′ − d) × (m + m′ − d) minors of this matrix, which is contained in the ideal generated by the (m + m′ − d) × (m + m′ − d) minors of the Jacobian matrix of S_{D}.

Following the exact same steps as in the proof of Theorem 3.12, we can see that this ideal will contain the ideal

H = I_{1,D}^{m_{1}−1} … I_{n,D}^{m_{n}−1}J_{m−d}.

Since H ⊆ R_{D}, every element of H gives a universal test element for I_{1,D}, …, I_{n,D}.

References

[B] Bourbaki, N. Elements of Mathematics, Commutative Algebra, Springer Verlag, 1989.

[BH] Bruns, W. and Herzog, J. Cohen-Macaulay Rings, vol. 39, Cambridge studies in advanced mathematics, revised edition, 1998.

[E] Eisenbud, D. Commutative Algebra with a View Towards Algebraic Geometry, Graduate Texts in Mathematics, No. 150, Springer Verlag, New York 1995.

[EGA] A. Grothendieck. Èlèments de Gèomètrie Algèbrique II. I.H.E.S Publ. math. 4 (1965).

[F] Faridi, S. Closure operations on ideals, Thesis, University of Michigan, 2000.

[Ha] Hara, H. A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), no. 5, 981-996.

[Har] Harris, J. Algebraic geometry. A first course, Graduate Texts in Mathematics, No. 133. Springer-Verlag, New York, 1992.

[HH1] Hochster, M. and Huneke, C. Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc. vol. 3 (1990), 31-116.

[HH2] Hochster, M. and Huneke, C. Tight closure in equal characteristic zero, preprint.

[HH3] Hochster, M. and Huneke, C. F -regularity, test elements, and smooth base change, Trans. Amer. Math. Soc. 346 (1994), 1-62.

[HH4] Hochster, M. and Huneke, C. Localization and test exponents for tight closure, Michigan Math. J. 48 (2000), 305–329.

[HH5] Hochster, M. and Huneke, C. Tight closure and strong F -regularity, Mm. Soc. Math. France (N.S.) No. 38, (1989), 119–133.

[Ho1] Hochster, M. The notion of tight closure in equal characteristic zero, Appendix in Tight closure and its applications by C. Huneke, CBMS Regional Conference Series in Mathematics, 88.

[Ho2] Hochster, M. The tight integral closure of a set of ideals, J. Algebra 230 (2000), no. 1, 184–203.

[Ho3] Hochster, M. Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes, Ann. of Math. (2) 96 (1972), 318–337.

[HR] Hochster, M. and Roberts, J.L. Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Advances in Math. 13 (1974), 115-175.

[Hu] Huneke, C. Tight closure and its applications, CBMS Regional Conference Series in Mathematics, 88.

[L1] Lipman, J. Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. No. 36 1969 195–279.

[L2] Lipman, J. Relative Lipschitz-saturation, Amer. J. Math. 97 (1975), no. 3, 791–813.

[M] Matsumura, H. Commutative ring theory, second edition, Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge-New York, 1989.

[Sm1] Smith, K.E. F-Rational rings have rational singularities, American Journal, 119, 159-180 (1997).

[Sm2] Smith, K.E. Tight Closure commutes with localization in binomial rings, Proc. Amer. Math. Soc. 129 (2001), no. 3, 667–669.

[V] Vasconcelos, W. The Arithmetic of Blowup Algebras, Cambridge University Press, 1994.

[Va] Valla, G. Certain graded algebras are always Cohen-Macaulay, J. Algebra 42(1976), 537-548.

[ZS1] Zariski, O. and Samuel, P. Commutative Algebra, Vol. 1, GTM 28, Springer Verlag, New York, 1975.

[ZS2] Zariski, O. and Samuel, P. Commutative Algebra, Vol. 2, Van Nostrand, Princeton, 1960.