THE STRUCTURE OF F-PURE RINGS

IAN M. ABERBACH AND FLORIAN ENESCU

Dedicated to Professor Melvin Hochster on the occasion of his sixtieth birthday

Abstract. For a reduced F -finite ring R of characteristic p > 0 and q = p^{e} one can write R^{1/q} = R^{a_{q}} ⊕ M_{q}, where M_{q} has no free direct summands over R. We investigate the structure of F -finite, F -pure rings R by studying how the numbers a_{q} grow with respect to q. This growth is quantified by the splitting dimension and the splitting ratios of R which we study in detail. We also prove the existence of a special prime ideal P(R) of R, called the splitting prime, that has the property that R/P(R) is strongly F -regular. We show that this ideal captures significant information with regard to the F -purity of R.

1. Introduction

Let (R, 𝔪, k) be a reduced, local ring of positive characteristic p > 0 and Krull dimension d. Throughout this paper, q = p^{e} denotes a power of the characteristic, R^{1/q} is the ring of qth roots of elements in R, and R will usually be assumed to be F -finite, i.e. R^{1/p} is module finite over R. This implies that R^{1/q} is module finite over R for all q. Write R^{1/q} = R^{a_{q}} ⊕M_{q}, which is a direct sum decomposition of R^{1/q} over R such that M_{q} has no free direct summands. We would like to investigate the size of the numbers a_{q}, as q grows to infinity, by studying the splitting dimension of the ring R (for short the s-dimension of R, as it was called in [2]). (For precise definitions, we refer the reader to Section 2). The rings for which some a_{q} > 0 (equivalently, all a_{q} > 0) are called F -pure. The purpose of this paper is to demonstrate that there is much more structure to be discerned in such rings, and at least part of this structure is captured by the s-dimension and the splitting ratios (defined below).

The behavior of the sequence {a_{q}}_{q} captures subtle information about the structure of the ring R with regard to the natural action of Frobenius on it. This behavior is intimately connected to the notions of F -purity and strong F -regularity. In fact, our work starts with and extends ideas originating in papers of Huneke and Leuschke ([12]), and Aberbach and Leuschke ([2]) where the relation between the strong F -regularity and the F -signature of Gorenstein rings ([12]) and Cohen-Macaulay rings ([2]) are analyzed. One can think of the concept of strong F -regularity as a very strong form of F -purity, as the definitions included below show. Our work shows that there is a spectrum of F -pure behavior, ranging from “merely F -pure,” in the case that sdim(R) = 0 (i.e., essentially a_{q} = 1 for all q), all the way up to strong F -regularity (i.e., sdim(R) = dim(R)). Moreover, the central notions of the paper, the splitting dimension and the Frobenius splitting ratios, will quantify this behavior in a clear fashion.

2000 Mathematics Subject Classification: 13A35.

The first author was partially funded by the National Security Agency.

An important outcome of our analysis is the discovery, in any F -pure ring R, of a special prime ideal P(R), which we will call the splitting prime of R, whose dimension (i.e. dim(R/P)) is an upper bound for the splitting dimension of R. A significant feature of this prime ideal P is that it defines a strongly F -regular quotient R/P. In the case of a Gorenstein ring R this ideal is the only F -stable prime of R (the study of the F -stable primes of ring R was initiated in ([4]).

Let (R, 𝔪, k) be an F -finite F -pure ring. For simplicity in making an initial statement, assume that the residue field of R is perfect. We let k be the largest integer such that lim inf_{q} \frac{a_{q}}{q^{k}} > 0 and call this number the s-dimension of R, sdim(R). We can then prove

Theorem 1.1. There exists a prime ideal P = P(R) such that R/P is strongly F -regular and sdim(R) ≤ dim(R/P). The prime P is 0 if and only if R is strongly F -regular.

The proof of this theorem is contained in Theorem 3.3 and Theorem 4.8.

The first part of the paper will introduce the relevant concepts of the paper, such as the splitting dimension and the Frobenius splitting ratios. Section 2 of the paper contains some results about the splitting dimension, the splitting prime ideal P, and the lower and upper Frobenius splitting ratios of a ring R. We also raise the question of whether or not sdim(R) = P(R) and provide some evidence in support of an affirmative answer. The study of these concepts continues in Section 3 where we investigate the case of reduced rings that are images of regular local rings and state our main Theorem (Theorem 4.8). The last section of the paper extends some of the earlier results to the class of R-modules.

We would like to conclude our introduction by outlining some of the main facts about F -purity and strong F -regularity. The notion of F -purity was introduced in the work of Hochster and Roberts ([11]) and that of strong F -regularity in a paper by Hochster and Huneke ([8]). Both concepts have been studied by many other authors; the main facts about them, which are now considered standard, will be listed below without any proofs or references.

Let F : R → R be the Frobenius homomorphism F (r) = r^{p}. We denote by F^{e} the eth iteration of F, that is F^{e}(r) = r^{q}, F^{e} : R → R. One can regard R as an R-algebra via the homomorphism F^{e}. Although as an abelian group it equals R, it has a different scalar multiplication. We will denote this new algebra by R^{(e)}.

Definition 1.2. R is F -finite if R^{(1)} is module finite over R, or, equivalently (in the case that R is reduced), R^{1/p} is module finite over R. R is called F -pure if the Frobenius homomorphism is a pure map, i.e, F ⊗_{R} M is injective for every R-module M.

If R is F -finite, then R^{1/q} is module finite over R, for every q. Moreover, any quotient and localization of an F -finite ring is F -finite. Any finitely generated algebra over a perfect field is F -finite. An F -finite ring is excellent.

If R is F -finite, then R is F -pure if and only if the inclusion R ↪ R^{1/p} splits over R. This is equivalent to the assertion that R ↪ R^{1/q} splits, for every q. The reader should note that R^{1/q} ≃ R^{(e)} as R-algebras, by sending r^{1/q} to r.

Let R^{o} be the complement of the minimal primes of the ring R.

Definition 1.3. A reduced Noetherian F -finite ring R is strongly F -regular if for every c ∈ R^{o} there exists q such that the R-linear map R → R^{1/q} that sends 1 to c^{1/q} splits over R, or equivalently Rc^{1/q} ⊂ R^{1/q} splits over R.

The notion is related to that of F -purity by the fact that, in the definition above, if a splitting exists for a choice of c ∈ R^{o} and q, then the inclusion R ↪ R^{1/q′} (sending 1 ↦ 1) also splits for every q′. Also, if a splitting of Rc^{1/q} ⊂ R^{1/q} exists, then a splitting of Rc^{1/q′} ⊂ R^{1/q′} exists too for all q′ ≥ q.

The notion of strong F -regularity localizes well, and all ideals are tightly closed in strongly F -regular rings. Regular rings are strongly F -regular and strongly F -regular rings are Cohen-Macaulay and normal. To show that a reduced Noetherian and F -finite ring is strongly F -regular it is necessary and sufficient to show that, for some c ∈ R^{o} such that R_{c} is strongly F -regular, there exists q such that Rc^{1/q} ⊂ R^{1/q} splits over R.

Let E_{R}(k) be the injective hull of the residue field of R. Then R is strongly F -regular if and only if 0_{E_{R}}^{∗} = 0. The ideal Ann_{R}(0_{E_{R}}^{∗}) is called the CS-test ideal of R and is denoted by τ̃(R). Note that τ̃(R) = R if and only if R is strongly F -regular.

Definition 1.4. The ring (R, 𝔪) is approximately Gorenstein if R has a sequence of 𝔪primary irreducible ideals {I_{t}}_{t} cofinal with the powers of 𝔪. See [7].

By taking a subsequence, we may assume that I_{t} ⊃ I_{t+1}. For each t, let u_{t} be an element of R which represents a socle element modulo I_{t}. Then there is, for each t, a homomorphism R/I_{t} ↪ R/I_{t+1} such that u_{t} + I_{t} ↦ u_{t+1} + I_{t+1}. The direct limit of the system will be the injective hull E = E_{R}(R/𝔪) and each u_{t} will map to the socle element of E, which we will denote by u. Hochster has shown that every excellent, reduced local ring is approximately Gorenstein ([7]). In particular, every F -finite reduced ring is approximately Gorenstein.

2. Notation and terminology

Let (R, 𝔪, k) be a reduced, local, F -finite ring of positive characteristic p > 0 and Krull dimension d. As in the introduction, let

R^{1/q} = R^{a_{q}} ⊕ M_{q}

be a direct sum decomposition of R^{1/q} such that M_{q} has no free direct summands. If R is complete, such a decomposition is unique up to isomorphism, but it is always the case that the values of the a_{q} are independent of the decomposition. As mentioned in the introduction, we investigate here the size of the numbers a_{q} with respect to q.

For a local ring (R, 𝔪, k), we set α(R) = log_{p}[k : k^{p}]. For I an ideal of R, let I^{[q]} = (i^{q} : i ∈ I)R. It is easy to see that, for an 𝔪-primary ideal I of R, λ_{R}(R^{1/q}/IR^{1/q}) = λ(R/I^{[q]})q^{α(R)}. If P is a prime ideal of R, then α(P) will simply denote α(R_{P}). It is often convenient to understand many of the statements in this paper first in the case that k is perfect, i.e., α(R) = 0. However, in order to use induction and have the concepts localize well we must keep track of the exponent α(R).

Before we proceed, we would like to remind the reader of the following fact ([13]).

Lemma 2.1. Let R be an F -finite Noetherian ring of characteristic p. Then for any prime ideals P ⊂ Q of R,

α(P) = α(Q) + ht(Q/P).

We would like to first define the notion of F -signature as it appears in [2] and [12].

Definition 2.2. Let R be an F -finite local ring. The F -signature of R is

s(R) = lim_{q→∞} a_{q}/q^{d+α(R)},

if it exists.

The following result, due to Aberbach and Leuschke, holds:

Theorem 2.3. Let (R, 𝔪, k) be a reduced Noetherian ring of positive characteristic p. Then lim inf_{q→∞} a_{q}/q^{d+α(R)} > 0 if and only if lim sup_{q→∞} a_{q}/q^{d+α(R)} > 0 if and only if R is strongly F -regular.

Moreover, Huneke and Leuschke proved that if R is local, reduced and Gorenstein, then the F -signature exists. Yao has recently extended this result to rings that are Gorenstein on their punctured spectrum.

We are now able to define the central concepts to be investigated in this paper.

Definition 2.4. Let R be an F -finite local ring. The s-dimension of R, sdim(R), is the largest integer k such that

lim inf_{q→∞} \frac{a_{q}}{q^{k+α(R)}}

is not zero. If R is not F -pure then sdim(R) = −∞. The lower Frobenius splitting ratio of R, r_{F}^{−}(R), equals the value of the limit introduced above, that is r_{F}^{−}(R) = lim inf_{q→∞} \frac{a_{q}}{q^{sdim(R)+α(R)}} . Similarly, the upper Frobenius splitting ratio of R, r_{F}^{+}(R), equals lim sup_{q→∞} \frac{a_{q}}{q^{sdim(R)+α(R)}} . Whenever r_{F}^{−}(R) = r_{F}^{+}(R), we call this number the Frobenius splitting ratio of R, r_{F} (R).

Whenever sdim(R) = dim(R) (equivalently, R is strongly F -regular), we recover the notion of F -signature of R, denoted by s(R), and, we will use this terminology whenever we are in such a special case. In particular, whenever sdim(R) = dim(R), we will be speaking of the lower and upper signature of R, denoted accordingly by s^{−}(R) and s^{+}(R), instead of its lower and, respectively, upper Frobenius splitting ratio.

We show below that R is F -pure if and only if sdim(R) ≥ 0.

Remark 2.5. In [14] the notion of minimal Hilbert-Kunz multiplicity is defined, denoted m_{HK}(R). Theorem 2.3 shows that this number is non-zero if and only if R is strongly F - regular. In our terminology, m_{HK}(R) = 0 if sdim(R) < dim(R) and m_{HK}(R) = s^{−}(R) otherwise.

The following result is classical and very helpful in understanding the number of free direct summands of R^{1/q}. We will quote it here in the form stated in [10].

Theorem 2.6. Let (R, 𝔪, k) be a local ring and F be a free finitely generated R-module. Let M be a finitely generated R-module and ϕ : F → M an R-linear map. Then ϕ splits over R if and only if ϕ ⊗ E_{R}(k) is an injective map.

If M is a finite R-module that has no nonzero free direct summands, then for every m ∈ M, the map R → M that sends 1 to m does not split. By tensoring with E_{R}(k) and using the above Theorem, we conclude that the map E_{R}(k) → E_{R}(k) ⊗ M that sends e to e ⊗ m is not injective. Denote by u the socle generator for E_{R}(k). The following holds:

Proposition 2.7. Let (R, 𝔪, k) be a local ring, let M be a finite R-module, and let u be the socle generator of E_{R}(k). Then M has no nonzero free direct summands if and only if for every m ∈ M, u ⊗ m = 0 in E_{R}(k) ⊗ M.

Proof. For each m ∈ M, define f_{m} : R → M by f_{m}(r) = rm. Then f_{m}does not split over R if and only if f_{m} ⊗ E_{R}(k) is not injective if and only if ker(f_{m} ⊗ E_{R}(k)) is nonzero (Theorem 2.6). This holds if and only if ker(f_{m} ⊗ E_{R}(k)) intersects the socle of E_{R}(k) non trivially. But, Soc(E_{R}(k)) = Ru. So, f_{m} ⊗E_{R}(k) is not injective if and only if u⊗m = 0 □

The following Corollary will prove to be of much use in what follows. Yao has also noted this result in his recent work [15].

Corollary 2.8. Let (R, 𝔪, k) be an F -finite reduced local ring of characteristic p. Let R^{1/q} = R^{a_{q}} ⊕ M_{q} be a direct sum decomposition of R^{1/q} over R, where M_{q} has no R-free direct summands. Define

J_{q} := {r ∈ R^{1/q} : r ⊗ u = 0 in R^{1/q} ⊗ E_{R}(k)}.

Then, a_{q} = λ_{R}(R^{1/q}/J_{q}).

Proof. R^{1/q} = R^{a_{q}} ⊕ M_{q}; so, by tensoring with Ru ⊂ E_{R}(k), we get

R^{1/q} ⊗ R · u = (R^{a_{q}} ⊗ R · u) ⊕ (M_{q} ⊗ R · u).

Map this further to R^{1/q} ⊗ E_{R}(k) and apply Proposition 2.7 to M_{q}. We have that Im(R^{1/q} ⊗ R · u → R^{1/q} ⊗ E_{R}(k)) = Im(g : R^{a_{q}} ⊗ Ru → R^{1/q} ⊗ E_{R}(k)), where g(a ⊗ ru) = a ⊗ ru. But g factors through R^{a_{q}} ⊗ E_{R}(k) → R^{1/q} ⊗ E_{R}(k) which is an injective map (again by Theorem 2.6), so g is also injective. But λ(Ru) = 1, and hence λ(R^{a_{q}} ⊗ Ru) = a_{q}. In conclusion,

λ(Im(R^{1/q} ⊗ R · u → R^{1/q} ⊗ E_{R}(k))) = a_{q}.

There is a natural R-linear map R^{1/q} → R^{1/q} ⊗ E_{R}(k) that sends r ∈ R^{1/q} to r ⊗ u. The kernel of this map equals J_{q} and the theorem is proved. □

3. Splittings, s-dimension and strong F -regularity

Let F^{e} : R → R be the eth iteration of the Frobenius map. We can regard R as a new (right) R-algebra, under this map, which we denote by R^{(e)}. Corollary 2.8 leads us to define the following sequence of ideals.

Definition 3.1. Let (R, 𝔪, k) be an F -finite local ring. We define

A_{e} = A_{e}(R) = {r ∈ R : r ⊗ u = 0 ∈ R^{(e)} ⊗_{R} E_{R}(k)}.

If R is not reduced then the map R → R^{(e)} cannot be injective for e > 0, and hence by Theorem 2.6, A_{e} = R for all e ≥ 1. So assume that R is reduced (hence approximately Gorenstein). Then clearly, A_{e} = (J_{q})^{[q]} (note that R^{1/q} ≃ R^{(e)} by r^{1/q} ⟼ r). In fact, A_{e} is an ideal both of R and R^{(e)}. Using the notation in Definition 1.4, we see that A_{e} = ∪_{t}(I_{t}^{[q]} : u_{t}^{q}). We claim that the sequence {A_{e}}_{e} is non-increasing. If R is not F -pure then A_{e} = R for all e. Otherwise, let r ∈ A_{e}. For t ≫ 0, ru_{t}^{q} ∈ I_{t}^{[q]} where q = p^{e}, hence r^{p}u_{t}^{q} ∈ I_{t}^{[q]}. Taking pth roots and applying the splitting yields, ru_{t}^{q/p} ∈ I_{t}^{[q/p]}, showing that A_{e} ⊆ A_{e−1}.

Definition 3.2. Let (R, 𝔪, k) be an F -finite reduced local ring. Let

P(R) = {r ∈ R : r ⊗ u = 0 ∈ R^{(e)} ⊗_{R} E_{R}(k) for all e ≫ 0} = ∩_{e}A_{e}.

If the ring is understood we will sometimes refer to this ideal simply as P.

As we will show in Theorem 3.3 below, the ideal P(R) is prime.

Since u is the socle generator for E_{R}(k), we have r^{q} ⊗ u = 1 ⊗ ru, which is 0 if r ∈ 𝔪, so m^{[q]} ⊂ A_{e} for every q. On the other hand, if u ∉ 0_{E_{R}}^{∗}, then P = 0 (see [1], Proposition 2.4). If R is a domain, then P is nonzero only if R is not strongly F -regular, since 0_{E_{R}}^{∗} = 0 in strongly F -regular rings (see also Corollary 3.4). The reader is invited to note that there is a similarity between the ideal P(R) introduced here and the concept of F -stable primes of R introduced in [4]. In fact, when R is Gorenstein, then R admits a unique F -stable prime, which, indeed, is P. This similarity is underlined by our next Theorem that asserts that P(R) is in fact a prime ideal of R. We will refer to P(R) as the splitting prime for R.

Before we state the result, we need to introduce some notation. For every c ∈ R, one can define an R-linear map ϕ_{c,e} : R → R^{1/q}, ϕ_{c,e}(1) = c^{1/q}, where q = p^{e}.

Theorem 3.3. Let (R, 𝔪, k) be an F -finite reduced local ring of characteristic p. Then, A_{e} = {c ∈ R : ϕ_{c,q} does not split over R} and P(R) is a prime ideal or the unit ideal.

Proof. By Theorem 2.6 we know that ϕ_{c,e} splits if and only ϕ_{c,e} ⊗ E_{R}(k) is injective.

Take c ∈ A_{e}. Then c^{1/q} belongs to J_{q}. Hence c^{1/q} ⊗ u = 0, and therefore ϕ_{c,e} does not split. Now, assume that c ∈ R is an element such that the map ϕ_{c,q} does not split. Then again u ∈ Ker(ϕ_{c,e} ⊗ E_{R}(k)) which means that c^{1/q} ∈ J_{q}. So, c ∈ A_{e}. This shows the first assertion of the Theorem.

For the second assertion assume that P(R) is not the unit ideal and let us write

P = {c ∈ R : ϕ_{c,e} does not split for any e ≫ 0}.

If P is not prime, then there exist a and b not in P such that ab ∈ P. Since a and b are not in P we can find e_{1} and e_{2} large enough such that the R-linear maps ϕ_{a,e_{1}} and ϕ_{b,e_{2}} split. Let θ_{1} : R^{1/q_{1}} → R an R-linear that maps a^{1/q_{1}} to 1. This map exists because ϕ_{a,e_{1}} splits over R. Similarly, we can find θ_{2} : R^{1/q_{1}q_{2}} → R^{1/q_{1}}, such that θ_{2}(b^{1/q_{1}q_{2}}) = 1 and θ_{2} is R^{1/q_{1}}-linear (we can do this by taking the q_{1}-root of a splitting for ϕ_{b,e_{2}}).

Let θ = θ_{1}θ_{2}. It is easy to check that θ is R-linear and θ((a^{q_{2}}b)^{1/q_{1}q_{2}}) = 1. So, for e_{0} = e_{1} + e_{2}, ϕ_{a^{q_{2}}b,e_{0}} splits.

We know that ab ∈ P, and this implies that a^{q_{2}}b ∈ P. So, if we fix e_{2}, this means that ϕ_{a^{q_{2}}b,e} does not split for any e ≫ 0 large enough. This is a contradiction, since we have just shown the existence of a splitting for values q of the form q = q_{0} = p^{e_{0}} = p^{e_{1}+e_{2}}, where q_{1} = p^{e_{1}} can be arbitrarily large.

In conclusion, P is a prime ideal. □

Corollary 3.4. Let (R, 𝔪, k) be an F -finite reduced local ring of characteristic p. Then, R is F -pure if and only if P(R) is a proper ideal if and only if sdim(R) ≥ 0. In this case,

sdim(R) ≤ dim(R/P).

The ring R is strongly F -regular if and only if P = 0. Also, dim(R/P) = 0 if and only if P = 𝔪 if and only if sdim(R) = 0, in which case r_{F} (R) = 1.

Proof. R is F -pure if and only if some a_{q} > 0 if and only if A_{e} is proper, and since the sequence {A_{e}}_{e} is non-increasing this occurs if and only if P(R) ≠ R. By Theorem 2.8, a_{q} = λ(R^{1/q}/J_{q}) and hence a_{q} = λ(R/A_{e}) · q^{α(R)} with α(R) = log_{p}[k : k^{p}] . From this we can see that R is F -pure if and only if sdim(R) ≥ 0.

We next show that sdim(R) ≤ dim(R/P). Since 𝔪^{[q]} ⊂ A_{e}, for every q = p^{e}, we have that 𝔪^{[q]} + P ⊂ A_{e}. Then a_{q}/q^{k+α(R)} = λ(R/A_{q})/q^{k} ≤ λ(R/𝔪^{[q]} + P)/q^{k}. If k > dim(R/P), then by the theory of Hilbert-Kunz functions, lim_{q→∞} λ(R/𝔪^{[q]} + P)/q^{k} = 0. So, sdim(R) ≤ dim(R/P).

The ideal P = 0 if and only if for all c ≠ 0 there exists a q such that ϕ_{c,q} splits; this is precisely the definition of a strongly F -regular ring.

Since P is prime, dim(R/P) = 0 if and only if P = 𝔪. Clearly P ≠ R if and only if R is F -pure if and only if A_{e} ≠ R for all e. So, P = 𝔪 if and only if R is F -pure and A_{e} = 𝔪 for all e. In conclusion, a_{q}/q^{α(R)} = λ(R/A_{e}) = λ(R/𝔪) = 1. Hence, in particular, r_{F} (R) = 1. □

Remark 3.5. We note the following facts:

1) The proof of the above Corollary shows that, in fact, when R is F -pure and P = 𝔪 (an important such case is that of an isolated non-strongly F -regular point), then a_{q} = q^{α(R)}.

2) Assume that R is strongly F -regular. A result of Aberbach ([1]) shows that there exists e_{o} such that A_{e} ⊂ 𝔪^{[q/q_{o}]} for every e ≥ e_{o}. This shows that a_{q}/q^{d+α(R)} ≥ λ(R/𝔪^{[q/q_{o}]})/q^{d}; since e_{HK}(𝔪) > 0, we conclude that sdim(R) = d and r_{F} (R) ≥ e_{HK}(𝔪)/q_{o}^{d}. This result was first established in [2] and our remark is along the lines of the argument given there.

3) Also, let us take c ∈ τ̃(R) = Ann_{R}(0_{E_{R}}^{∗}), a CS-test element. The ring R_{c} is strongly F -regular. If c ∉ A_{e}, for some e, then R is strongly F -regular (as explained in the introduction). So, if R is not strongly F -regular, then τ̃(R) ⊆ P.

The splitting prime of R does not localize well in general. The most obvious example is when P(R) = 𝔪. Then for all Q ∈ Spec(R), R_{Q} is F -pure, so P(R_{Q}) ≠ R_{Q} = P(R)R_{Q}. However, when Q ⊇ P(R) we do get a nice localization result.

Proposition 3.6. Let (R, 𝔪, k) be a local ring of characteristic p. Then for every Q ∈ Spec(R) such that P ⊂ Q we have

P(R_{Q}) = P(R)R_{Q}.

Proof. We first note that (R^{1/q})_{Q} = (R_{Q})^{1/q} = R^{1/q} ⊗ R_{Q}. Moreover, if ϕ_{c,e} splits, then ϕ_{c,e} ⊗ R_{Q} splits, too.

Take c/s ∈ P(R_{Q}). Then c/1 belongs to P(R_{Q}). We would like to show that c ∈ P(R). Assume the contrary; then, ϕ_{c,e} : R → R^{1/q} does split for e ≫ 0. By tensoring with R_{Q} we get a splitting over R_{Q}, which is a contradiction.

Now, take c ∈ P(R). We want to show that c/1 belongs to P(R_{Q}). If not, for large q, the map R_{Q} → R_{Q}^{1/q} that sends 1 to c^{1/q}/1 splits. So, for each such q = p^{e}, there is an element d_{q} ∉ Q and a map R^{1/q} → R that sends c^{1/q} to d_{q}. But, d_{q} is not in P(R), since P(R) ⊂ Q. So, for q′ ≫ 0, there is an R-linear map R^{1/q′} → R that sends d_{q}^{1/q′} to 1. Combining these last two assertions, we conclude that c is not in P(R) and this is a contradiction.

□

Proposition 3.7. Let R be an F -pure local ring. Then Q ⊂ P, for every Q ∈ Min(R).

Proof. Let c ∈ R. For every q, the map ϕ_{c,q} is not injective if c is a zero-divisor on R. Since R is reduced this means that c belongs to some minimal prime Q of R. So, if c ∈ Q, with Q a minimal prime, then ϕ_{c,q} cannot split over R for any q. The assertion of the Proposition follows.

□

We would like to show that the splitting prime of the completion is extended. In order to do this we need the next Lemma.

Lemma 3.8. Let (R, 𝔪) → (S, 𝔫) be a faithfully flat map of F -finite local reduced rings such that the closed fiber S/𝔪S is regular of dimension n. Then A_{e}(S) = A_{e}(R)S + 𝔫^{[q]}. If sdim R = 0, then sdim S = n, P(S) = 𝔪S, and r_{F}^{−}(S) = r_{F}^{+}(S) = 1.

Proof. Both rings are approximately Gorenstein, and the closed fiber S/𝔪S is regular of dimension n. Let z_{1}, ..., z_{n} be a sequence of elements in S that define a regular sequence on S/𝔪S. Then for any q = p^{e}, z_{1}^{q}, ..., z_{n}^{q} form a regular sequence on S/𝔪S and, in fact, R → S/(z_{1}^{q}, ..., z_{n}^{q})S if a faithfully flat map.

If {I_{t}}_{t} is a sequence of irreducible 𝔪-primary ideals of R, cofinal with the powers of 𝔪, then {J_{t}}_{t} = {I_{t}S + (z_{1}^{t}, ..., z_{n}^{t})}_{t} is a sequence of irreducible 𝔫-primary ideals of S, cofinal with the powers of 𝔫. Let u_{t} be elements of R which are socle elements modulo I_{t}. Then the images of the u_{t} ·(z_{1} ⋯ z_{n})^{t−1} in S are socle elements modulo J_{t}. Moreover, R/I_{t}^{[q]} → S/I_{t}^{[q]}S is flat with regular closed fiber of dimension n. Since R/I_{t}^{[q]} has depth 0, we see that S/I_{t}^{[q]}S has depth n. It follows that S/I_{t}^{[q]}S is Cohen-Macaulay. In particular, this shows that z_{1}, ..., z_{n} form a regular sequence on S/I_{t}^{[q]}S, because their images in S/I_{t}^{[q]}S form a system of parameters.

Let us compute A_{e}(S) = ∪_{t}((I_{t}^{[q]}S + (z_{1}^{tq}, ⋯, z_{n}^{tq})) : u_{t}^{q} · (z_{1} ⋯ z_{n})^{(t−1)q}). If a ∈ S such that

a · u_{t}^{q} · (z_{1} ⋯ z_{n})^{(t−1)q} ∈ I_{t}^{[q]}S + (z_{1}^{tq}, ⋯, z_{n}^{tq})

then, by using that z_{1}, ..., z_{n} form a regular sequence on S/I_{t}^{[q]}S, we have that

au_{t}^{q} ∈ I_{t}^{[q]}S + (z_{1}^{q}, ..., z_{n}^{q}).

This is equivalent to a ∈ (I_{t}^{[q]}S + (z_{1}^{q}, ..., z_{n}^{q})) :_{S} u_{t}^{q}).

By flatness of S/(z_{1}^{q}, ⋯, z_{n}^{q}) over R, (I_{t}^{[q]}S + (z_{1}^{q}, ..., z_{n}^{q})) :_{S} u_{t}^{q} = (I_{t}^{[q]} :_{R} u_{t}^{q})S + (z_{1}^{q}, ..., z_{n}^{q}). In conclusion, A_{e}(S) = ∪_{t}((I_{t}^{[q]} :_{R} u_{t}^{q})S+(z_{1}^{q}, ..., z_{n}^{q})) and hence A_{e}(S) = A_{e}(R)S+(z_{1}^{q}, ..., z_{n}^{q}). So, if sdim(R) = 0, then A_{e}(S) = 𝔪^{[q]}S + (z_{1}^{q}, ..., z_{n}^{q}). Hence, a_{q} = λ_{S}(S/A_{e}(S)) = λ_{S}(S/(𝔪S + (z_{1}^{q}, ..., z_{n}^{q})) which shows that sdim(S) = n. □

Proposition 3.9. Let (R, 𝔪) and (S, 𝔫) be local, reduced, F -finite rings of characteristic p and let f : (R, 𝔪) → (S, 𝔫) be a flat local map with regular fibers. Then A_{e}(S) = A_{e}(R)S+𝔫^{[q]} for every q = p^{e}, and P(S) = P(R)S. In particular, if R is local, reduced and F -finite, then A_{e}(R)R̂ = A_{e}(R̂) and P(R)R̂ = P(R̂).

Proof. The assertion that A_{e}(S) = A_{e}(R) + 𝔫^{[q]} has been shown in Lemma 3.8.

To see that P(S) = P(R)S we first observe that P(S) = ∩_{e}A_{e}(S) is a prime ideal of S lying over P(R).

We first show that any prime of S that is minimal over P(R)S is in fact contained in P(S). Let Q ≠ P(S) be a prime ideal of S that is minimal over P(R)S. Since R → S is flat then Q lies over P. Thus the map R_{P(R)} → S_{Q} is faithfully flat and P(R)S_{Q} = QS_{Q}. By Lemma 3.8, sdim(S_{Q}) = 0. In particular, A_{e}(S_{Q}) = QS_{Q} for all e. However, if we take c ∈ Q − P(S), then for q ≫ 0, c^{1/q}S ⊆ S^{1/q} splits over S. This remains true after localizing, showing that A_{e}(S_{Q}) ≠ QS_{Q}, a contradiction.

In conclusion, Q ⊆ P(S).

Note that P(R) is prime in R, and since R → S has regular fibers, then P(R)S is a radical ideal in S.

Now, we would like to argue that P(S) itself is a minimal prime of P(R)S. The homomorphism R_{P(R)} → S_{P(S)} is flat because P(S) lies over P(R). Its closed fiber is regular and sdim(R_{P(R)}) = 0. If the dimension of its closed fiber is positive, then, by Lemma 3.8 below, sdim(S_{P(S)}) > 0. However, this is impossible. In conclusion, the closed fiber of R_{P(R)} → S_{P(S)} is 0-dimensional, so P(S) is minimal over P(R)S.

Now, since P(R)S is radical and its only minimal prime is P(S), we obtain the equality P(R)S = P(S).

The last assertion about the completion follows from the fact that, under our hypotheses, R is excellent, so all the fibers of the completion homomorphism are smooth, hence regular. □

Proposition 3.9 enables us to reduce the study of the s-dimension and the Frobenius splitting ratio of reduced excellent rings to the case of complete local rings. These rings are images of regular local rings. Following in the paths of Fedder, Glassbrenner, and Cowden-Vassilev we can analyze the finer structure of F -pure rings in this way.

4. Images of regular rings

In this section, we continue our analysis in the case of local rings that are images of regular local rings. The more general case of reduced F-finite rings can be reduced to this class of rings by Proposition 3.9.

Let us write R = S/I where S is regular and local and π : S → R the natural projection. Assume that R is reduced, i.e., I is a radical ideal. There are two results that describe under what conditions such rings are F -pure or strongly F -regular. The first one is due to Fedder (Theorem 1.12 in [5]). It does not require R to be F -finite.

Theorem 4.1. Let (S, 𝔪, k) be a regular local ring of characteristic p and let R = S/I. Then R is F -pure if and only if (I^{[p]} : I) ⊄ 𝔪^{[p]}.

The idea behind this criterion for F -purity has been used by Glassbrenner to give a similar criterion for strong F -regularity (Theorem 2.3 in [6]). We will state it here in the form we need it later.

Theorem 4.2. Let (S, 𝔪, k) be a local regular ring and let R = S/I and c ∈ S. Then, for every q = p^{e}, the map ϕ_{π(c),e} splits if and only if c ∉ 𝔪^{[q]} :_{S} (I^{[q]} : I). Moreover, R is strongly F -regular if and only if I = ∩_{q}(𝔪^{[q]} :_{S} (I^{[q]} : I)).

Using the notation introduced previously, we see that A_{e} = (𝔪^{[q]} :_{S} (I^{[q]} : I))/I. So, a_{q} = q^{α(R)} λ_{S}(S/(𝔪^{[q]} :_{S} (I^{[q]} : I))). Note that S/𝔪^{[q]} is zero dimensional and Gorenstein. The quotient S/(𝔪^{[q]} :_{S} (I^{[q]} : I)) is an S/𝔪^{[q]}-module and its Matlis dual is \frac{(I^{[q]}:I)+𝔪^{[q]}}{𝔪^{[q]}}. Hence, a_{q} = q^{α(R)} λ_{S}(\frac{(I^{[q]}:I)+𝔪^{[q]}}{𝔪^{[q]}}).

Although Theorem 4.2 was stated in [6] under the assumption that S is local, regular and F -finite, one can note that the F -finiteness hypothesis can be removed along the lines of the argument used by Fedder in proving his criterion for F -purity (where we interpret “strongly F -regular” to mean that 0_{E}^{∗} = 0). The point is that one can make a flat base change, by enlarging the residue field to its perfect closure, to get to the case where S is F -finite. Strong F -regularity commutes with this base change (see, for example, Theorem 3.6 [1]).

Remark 4.3 (for proofs, see [5], 1.4 and 1.5). For every q = p^{e}, Hom_{S}(S^{1/q}, S) has an S^{1/q}- structure given by s^{1/q} · ϕ(t^{1/q}) := ϕ((st)^{1/q}), for every s, t ∈ S and ϕ ∈ Hom_{S}(S^{1/q}, S). Moreover, Hom_{S}(S^{1/q}, S) ≃ S^{1/q}. Let T be a generator for Hom_{S}(S^{1/q}, S) and s^{1/q} ∈ S. Then s^{1/q}T defines an element in Hom_{R}(R^{1/q}, R) if and only if s^{1/q} ∈ (IS^{1/q} : I^{1/q}) if and only if s ∈ (I^{[q]} : I).

The following result is similar to Theorem 3.1 in [3], in which Cowden-Vassilev showed that if R is F -pure then so is R/τ(R), where τ(R) is the test ideal of R. We give two proofs: one is similar to the proof of Theorem 3.1 given in [3], while the other one uses some ideas originating in Fedder’s work, [5].

Proposition 4.4. Let (S, 𝔪, k) be an F -finite local regular ring and let R = S/I be reduced and F -pure. Set Q to be the full preimage of P(R) in S. Then for all q, (I^{[q]} : I) ⊂ (Q^{[q]} : Q). In particular, if R is F -finite then sdim(R/P) ≥ sdim(R).

Proof. Proof 1. R is excellent and reduced, so it is approximately Gorenstein. Let {I_{t}} be a collection of irreducible 𝔪-primary ideals in R cofinal with the powers of 𝔪, and denote by {J_{t}} their full preimages in S. Let u_{t} be an element of S mapping to the socle in S/J_{t}. Then we may describe Q as Q = ∩_{q}(∪_{t}(J_{t}^{[q]} + I) :_{S} u_{t}^{q}). Let w ∈ (I^{[q]} : I) and v ∈ Q.

We want to show that vw ∈ Q^{[q]} = (∩_{q′} ∪_{t} (J_{t}^{[q′]} + I) : u_{t}^{q′})^{[q]}, which by flatness of the Frobenius endomorphism over S is ∩_{q′} ∪_{t} ((J_{t}^{[qq′]} + I^{[q]}) : u_{t}^{qq′}). For all q′ there is a t such that vu_{t}^{qq′} ∈ J_{t}^{[qq′]} + I, hence vwu_{t}^{qq′} ∈ J_{t}^{[qq′]} + I^{[q]}. This shows that vw ∈ Q^{[q]}, as desired.

Proof 2. Let T be as in Remark 4.3. In the light of this Remark we need to show that if ψ := s^{1/q}T ∈ Hom_{R}(R^{1/q}, R), then (sQ)^{1/q} ∈ QS^{1/q}. By the same Remark, this is equivalent to the assertion that ψ induces an R-linear map R^{1/q}/Q^{1/q} → R/Q.

We can define an R^{1/q′}-linear map by ψ_{q′} : R^{1/qq′} → R^{1/q′}, ψ_{q′}(a) = ψ(a^{q′})^{1/q′}. This map is, in particular, R-linear.

Take c ∈ Q; then, c^{1/qq′} ⊗ u = 0 in R^{1/qq′} ⊗ E_{R}(k) where u is the socle generator in E_{R}(k). Clearly, ψ_{q′}(c^{1/qq′}) ⊗ u = 0. That is, ψ(c^{1/q})^{1/q′} ⊗ u = 0. So, s^{1/q}T (c^{1/q}) = ψ(c^{1/q}) ∈ Q.

Hence, s^{1/q}T takes Q^{1/q} into Q. So, s^{1/q}T defines an element in Hom_{R}(R^{1/q}/Q^{1/q}, R/Q), and hence s ∈ (Q^{[q]} : Q).

The last statement now follows, after noting that α(R) = α(R/P), since

a_{q}(R) = q^{α(R)} λ_{S} (\frac{I^{[q]} : I + 𝔪^{[q]}}{𝔪^{[q]}} ) ≤ q^{α(R/P)} λ_{S} (\frac{Q^{[q]} : Q + 𝔪^{[q]}}{𝔪^{[q]}} ) = a_{q}(R/P).

□

Remark 4.5. This result improves Theorem 3.4, since the s-dimension of a ring is bounded above by its dimension.

Corollary 4.6. Let R be a characteristic p local F -pure ring that is a homomorphic image of a regular local ring (S, 𝔪, k), that is R = S/I. Then sdim(R_{P}) = 0.

Proof. Denote by k(P) the residue field of the localization of R at P. Apply Theorem 4.4 to R_{P}. We get that 0 = sdim(k(P)) ≥ sdim(R_{P}) and we are done. □

Theorem 4.7. Let R = S/I be a characteristic p local F -pure, F -finite ring that is a homomorphic image of a regular local ring S. Then R/P is strongly F -regular.

Proof. If sdim R = 0, i.e., P = 𝔪_{R} this is clear. Thus we may assume that sdim R > 0. If R/P is not strongly F -regular we may localize a prime minimal among the set {P |(R/P)_{P} is not strongly F -regular}. After relabeling this ring R, we note that the sdimension is still positive. Thus we can assume that (R/P)_{Q} is strongly F -regular for all Q, different from 𝔪, containing P. We know that R/P is F -pure by Proposition 4.4.

If R/P is not strongly F -regular, then P(R/P) equals 𝔪_{R} since 𝔪_{R} = τ̃(R) ⊆ P(R). But then sdim(R/P) = 0, a contradiction. □

Now we are in a position to describe the s-dimension of a local ring of positive characteristic. We also give some information on the splitting ratios. We expect that the upper and lower splitting ratios always coincide. At present, we can only prove this if sdim(R) ≤ 1, however, we can give an upper bound on the ratio r_{F}^{+}(R)/r_{R}^{−}(F), if sdim(R) = dim(R/P).

Theorem 4.8. Let R be an F -pure, F -finite local ring of positive characteristic. Then

i) R/P is strongly F -regular, sdim(R) ≤ dim(R/P). Moreover, if sdim(R) = dim(R/P), then r_{F}^{−}(R) ≤ s^{−}(R/P) ≤ 1, r_{F}^{+}(R) ≤ s^{+}(R/P) ≤ 1.

ii) If sdim(R) = dim(R/P), then r_{F}^{+}(R) ≤ r_{F}^{−}(R) · e_{HK}(R/P). This shows that, in this case, r_{F}^{+}(R) > 0 if and only if r_{F}^{−}(R) > 0.

iii) If dim(R/P) = 1, then sdim(R) = 1 and R/P is a DVR.

iv) If dim(R/P) ≤ 1 then r_{F}^{+}(R) = r_{F}^{−}(R).

v) dim(R/P) = dim(R) if and only if R is strongly F -regular. In this case, s^{−}(R) ≤ s^{+}(R)e_{HK}(R).

vi) depth R ≥ sdim R.

Proof. One can pass to the completion of R by Proposition 3.9, and hence assume that R = S/I with S regular local.

For part i), let us note that since R is a homomorphic image of a regular ring, R/P is strongly F -regular by Theorem 4.7.

The inequalities stated in i) follow easily from the proof of Proposition 4.4.

To prove ii), let j = sdim(R) and choose two sequences of indices q and q′ such that λ(R/A_{e})/q^{j} approaches r_{F}^{−} and λ(R/A_{e+e′})/(qq′)^{j} approaches r_{F}^{+}. Let us note that P +A_{e′}^{[q]} ⊂ A_{e+e′}.

Claim:

λ(R/A_{e+e′}) ≤ λ(R/P + A_{e}^{[q′]}) ≤ λ(R/A_{e}) · λ(R/P + 𝔪^{[q′]}).

Indeed, the left side of the inequality is immediate since P +A_{e′}^{[q]} ⊂ A_{e+e′}. Set k = λ(R/A_{e}) and write a composition series A_{e} = I_{o} ⊂ ⋯ ⊂ I_{i} ⊂ ⋯ ⊂ I_{k} = R. Then (P + A_{e}^{[q′]}) ⊂ ⋯ ⊂ (P + I_{i}) ⊂ ⋯ ⊂ (P + I_{k}) = R. The successive quotients of this filtration are homomorphic images of R/P + 𝔪^{[q]}. Hence, λ(R/A_{e+e′}) ≤ λ(R/P + A_{e}^{[q′]}) ≤ k · λ(R/P + 𝔪^{[q]}).

By dividing on both sides by (qq′)^{j}, with j = sdim(R) = dim(R/P), and letting q, q′ approach infinity we obtain the inequality stated in ii).

For iii), note that a one-dimensional strongly F -regular ring R is a DVR. Part iv) follows at once from i), ii) and iii) by noting that the Hilbert-Kunz multiplicity of a local regular ring equals one.

For v), dim(R/P) = dim(R) implies that P is a minimal prime of R. Since P contains all the minimal primes of R, it follows that P is, in fact, the only minimal prime of R. But R is reduced, so P = 0. Hence, R is strongly F -regular.

That depth R ≥ sdim R follows from [16], Lemma 2.2.

□

Question 4.9. Is is true that, for a local reduced and F -finite ring (R, 𝔪, k) , sdim(R) = dim(R/P)?

Our results so far seem to indicate an affirmative answer to this question, since we have the equality sdim(R) = dim(R/P) if dim(R/P) ∈ {0, 1, dim(R)}.

Proposition 4.10. Let R = S/I be a Stanley-Reisner ring, then ∑ Q = P where the sum runs over all the minimal primes Q of R.

Proof. By the definition of Stanley-Reisner rings, I is a square-free monomial ideal. In particular, I can be written as an intersection of some prime ideals in S, each of them generated by a string of indeterminates.

Let us assume that the union of all the minimal primes of R lifted to S involves all the indeterminates of S. But then ∑ Q = 𝔪 ⊂ P ⊂ 𝔪 and we are done.

In the general case, let us assume that x_{1}, ..., x_{k} are all the indeterminates that do not appear in any of the minimal primes of R. Then one can write R = A[x_{1}, ..., x_{k}], with A a Stanley-Reisner ring that satisfies the condition on the minimal primes of the previous paragraph. Hence 𝔪_{A} = P(A). It is easy to show that P(R) = P(A)R. But then, P(R) = 𝔪_{A}R and this equals the sum of all the minimal primes Q of R as it can be easily checked. □

5. The module case

In what follows, we would like to extend some of our considerations to the case of Rmodules. Let M be a finitely generated R-module. For every e, we can define a new R-module on the Z-module M with the R-multiplication given as follows: r · m = r^{p^{e}}m, for every r ∈ R and m ∈ M. Denote this new module by M^{(e)} and, as before, let us denote q = p^{e}. The reader should note that M^{(e)} = M as abelian groups.

Denote by a_{q} the maximal rank of a free direct summand of M. Hence, M^{(e)} = R^{a_{q}} ⊕ M_{q} with M_{q} an R-module with no free direct summands. Also, for every m ∈ M, denote ϕ_{m,e} : R → M^{(e)} the map that takes r to r^{q}m for every r ∈ R. This is clearly a map of R-modules.

Definition 5.1. Let u be the socle generator for E_{R}(k). For every positive integer e, one can define

A_{e}(M) = {m ∈ M^{(e)} : m ⊗ u = 0}

in M^{(e)} ⊗ E_{R}(k). Here M^{(e)} is the R-module just introduced above. It is easy to see that A_{e}(M) is a submodule of M = M^{(e)} over R with either of the two module structures that can be considered. Let us also define, P(M) := ∩_{e≫0}A_{e}(M) seen as an R-submodule of M.

Note that the above Definition naturally extends the Definition 3.1.

Also, 𝔪^{[q]}M +P(M) ⊂ A_{e}(M) for every e: if r ∈ 𝔪 and m ∈ M, then r^{q}m⊗u = m⊗ru = m ⊗ 0 = 0.

The following Theorem is similar to the Theorem 2.8.

Theorem 5.2. With the notations introduced above, one has that

λ_{R}(M/A_{e}(M)) · q^{α(R)} = a_{q}.

Proof. Regard A_{e}(M) as an R-submodule of M. Then (M/A_{e}(M))^{(e)} = M^{(e)}/A_{e}(M).

The proof is similar to that of Theorem 2.8 and one can show that a_{q} = λ_{R}(M^{(e)}/A_{e}(M)). But, for every R-module N one has that λ_{R}(N^{(e)}) = λ_{R}(N)·q^{α(R)}. Hence, a_{q} = λ_{R}(M/A_{e}(M))· q^{α(R)}.

□

Theorem 5.3. Let M be a finitely generated R-module and r ∈ R and m ∈ M. Then rm ∈ P(M) if and only if r ∈ P(R) or m ∈ P(M).

Proof. The proof is similar to the that of Theorem 3.3.

Let r ∈ P(R) and m ∈ M. We need to show that rm ∈ P(M). Consider the map R^{(e)} → M^{(e)} that sends r′ to r′m (seen as an element of M under the usual R-module structure) for every r′ ∈ R^{(e)}. Tensor this map with E_{R}(k) and let u be the socle element of this module. For every e ≫ 0, r ⊗ u = 0. This maps further to zero, so rm ⊗ u = 0 for e ≫ 0. Hence, rm ∈ P(M).

For the converse implication, assume that rm ∈ P(M) and that r ∉ P(R) and m ∉ P(M). Since r ∉ P(R), we have that for e ≫ 0 the map ϕ_{r,e} : R → R^{(e)} splits over R. Similarly,

for m ∉ P(M), we have that ϕ_{m,e′} : R → M^{(e′)} splits over R.

Fix e′ ≫ 0 so that ϕ_{m,e′} splits over R. Let q′ = p^{e′}. We will show that r^{q′}m ∉ P(M) which contradicts the fact that rm ∈ P(M). This will prove the converse implication.

Take e ≫ 0, so that ϕ_{r,e} splits. Define the map ψ_{m,e′} : R^{(e)} → M^{(e+e′)} by ψ_{m,e′}(r′) = r^{′q′}m. This is essentially the map ϕ_{m,e′} “lifted” to R^{(e)}. It is an R-linear map and it splits over R. The composition of the two maps ϕ_{r,e} and ψ_{m,e′} defines an R-linear map R → M^{(e+e′)} that sends 1 → r^{q′}m. This map splits over R for all e ≫ 0 (or, equivalently, e + e′ ≫ 0) and hence r^{q′}m ∉ P(M). □

We can define the s-dimension and the Frobenius splitting ratio of an R-module.

Definition 5.4. The s-dimension of M, sdim(M), is the largest integer k such that

lim inf_{q→∞} \frac{a_{q}}{q^{k+α(R)}}

is not zero. The lower Frobenius splitting ratio of M, r_{F} (M), equals the value of the limit introduced above, that is r_{F}^{−}(M) = lim inf_{q→∞} \frac{a_{q}}{q^{sdim(M)+α(R)}} . Similarly, the upper Frobenius splitting ratio of M, r_{F}^{+}(M), equals lim sup_{q→∞} \frac{a_{q}}{q^{sdim(M)+α(R)}} . Whenever r_{F}^{−}(M) = r_{F}^{+}(M), we call this number the Frobenius splitting ratio of M.

The following result is an extension of the case when M = R.

Theorem 5.5. Let M be an R-module. Then sdim(M) ≤ dim(M/P(M)). If R is strongly F -regular and M is an torsion free R-module, then sdim(M) = dim(R) = dim(M).

Proof. From the definition of M^{(e)} one can see that (Ann_{R}(M^{(e)}))^{q} ⊂ Ann_{R}(M), so dim(M^{(e)}) = dim(M). Write M^{(e)} = R^{a_{q}} ⊕ M_{q}. If a_{q} ≠ 0, for some q, then dim(M^{(e)}) = dim(R), because R injects into M^{(e)}.

We have seen that a_{q} = λ_{R}(M/A_{e}(M)), and 𝔪^{[q]}M + P(M) ⊂ A_{e}(M). So, a_{q} ≥ λ(M/𝔪^{[q]}M + P(M)). Since, e_{HK}(𝔪, M/P(M)) > 0, the first part of the Theorem follows.

If R is strongly F -regular, then P(R) = 0. If M is torsion-free over R, then M injects into R^{n} for some n. Hence, if m⊗u = 0 in M^{(e)} ⊗E_{R}(k) for all e ≫ 0, then the image of m in R^{n}, say (r_{1}, ..., r_{n}), belongs to P(R) component-wise. However, P(R) = 0, and so P(M) = 0. So, a_{q} ≤ λ(M/𝔪^{[q]}M), and since e_{HK}(𝔪, M) > 0 we see that sdim(M) = dim(M) = dim(R).

□

Acknowledgments Much of the work for this paper has been done while the authors attended the Commutative Algebra program at the Mathematical Sciences Research Institute, Berkeley, CA, Fall of 2002. The authors would like to thank MSRI for financial support and excellent working conditions as well as the organizers of the program for making this possible.

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Department of Mathematics, University of Missouri, Columbia, MO 65211, USA

E-mail address: aberbach@math.missouri.edu

Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA and The Institute of Mathematics of the Romanian Academy, Bucharest, Romania

E-mail address: enescu@math.utah.edu