Elsevier

Journal of Algebra

Volume 548, 15 April 2020, Pages 25-52
Journal of Algebra

The uniform symbolic topology property for diagonally F-regular algebras

https://doi.org/10.1016/j.jalgebra.2019.11.017Get rights and content

Abstract

Let k be a field of positive characteristic. Building on the work of the second named author, we define a new class of k-algebras, called diagonally F-regular algebras, for which the so-called Uniform Symbolic Topology Property (USTP) holds effectively. We show that this class contains all essentially smooth k-algebras. We also show that this class contains certain singular algebras, such as the affine cone over Pkr×Pks, when k is perfect. By reduction to positive characteristic, it follows that USTP holds effectively for the affine cone over PCr×PCs and more generally for complex varieties of diagonal F-regular type.

Introduction

We are concerned with the following question: when does a finite-dimensional Noetherian ring R satisfyp(hn)pnnN, for all prime ideals pR and for some h independent of p? Here, the expression p(m) denotes the m-th symbolic power of p. We invite the reader to glimpse at [6] for an excellent survey on this beautiful but tough problem.

This story starts, perhaps, with the work of I. Swanson. Swanson established that if R is a Noetherian ring and pR is a prime ideal such that the p-adic and symbolic topologies are equivalent, then they are in fact linearly equivalent, meaning there is a constant hN depending on p such that p(hn)pn for all n [29]. In particular, Swanson's result holds for every prime ideal p when R is a normal domain essentially of finite type over a field.

Later, Ein–Lazarsfeld–Smith demonstrated in their seminal work [8] that if R is a regular C-algebra essentially of finite type, then h can be taken independently of p. In fact h=dimR suffices. Rings for which this number h can be taken independently of p (i.e. for which there exists a uniform bound on h for all p) are said to have the Uniform Symbolic Topology Property, or USTP for short. Ein–Lazarsfeld–Smith's result is now known to hold for any finite-dimensional regular ring: their result was extended to regular rings of equal characteristic by M. Hochster and C. Huneke [13] and to regular rings of mixed characteristic by L. Ma and K. Schwede [19].

Since then, it has been of great interest to know which non-regular rings have USTP. For instance, Huneke–Katz–Validashti showed that, under suitable hypotheses, rings with isolated singularities have USTP, although without an effective bound on h [16]. R. Walker showed that 2-dimensional rational singularities have USTP and obtained an effective bound for h [30].

In this paper, we continue the above efforts in the strongly F-regular setting. Strong F-regularity is a weakening of regularity defined for rings of positive characteristic. Strong F-regularity is well-studied by positive characteristic commutative algebraists; see [15], [25], [27]. Given a field k of positive characteristic, we introduce a class of strongly F-regular k algebras essentially of finite type, called diagonally F-regular k-algebras, that are engineered to have USTP; in particular, (1.0.1) holds for these rings with h equal to dimension. We prove that this class includes all essentially smooth1 k-algebras, as well as Segre products of polynomial rings over k,2 i.e. the affine cone over Pkr×Pks whenever k is a perfect field of positive characteristic and r,s1. We also show that the class of diagonally F-regular k-algebras contains some non-isolated singularities.

To motivate our approach, we summarize the method introduced in [8], following the presentation of K. Schwede and K. Tucker in their survey, [25, §6.3]; see also [27] by K. Smith and W. Zhang.3 We do this with the aim of pointing out exactly where this argument breaks down for non-regular rings. In positive characteristic, the crux of Ein–Lazarsfeld–Smith's argument is the following chain of containments:p(hn)(1)τ(p(hn))=τ((p(hn))n/n)(2)τ((p(hn))1/n)n(3)pn Here, τ(at) denotes the test ideal of the (formal) power at; see Section 2 for details. Containment (1) holds in any strongly F-regular ring. Containment (2) holds by the subadditivity theorem for test ideals—this theorem requires the ambient ring R to be regular. Containment (3) holds quite generally (for h=dimR), as we shall discuss in the proof of Theorem 4.1.

So, in order to apply this technique to the non-regular case, we must deal with containment (2). Our approach here is simple: we will find an ideal t, depending on p, h, and n, such that the second containmentt(2)τ((p(hn))1/n)n is guaranteed to hold. Then the problem of deciding whether a particular F-regular ring satisfies USTP is reduced to deciding whether the first containment,p(nh)(1)t, holds for our choice of t. Following [28], we will construct t using the so-called diagonal Cartier algebras. Namely, we sett=τ(D(n);p(hn)), where D(n) is the n-th diagonal Cartier algebra; see Definition 3.1. Then Proposition 3.4(c) demonstrates that containment (2′) holds for any reduced k-algebra essentially of finite type, while (1′) holds whenever D(n) is F-regular. When this is the case for all n, we say our ring is diagonally F-regular as a k-algebra. This sketches the proof of our main theorem:

Theorem A Theorem 4.1

If R is a diagonally F-regular k-algebra essentially of finite type, then R has USTP with h=dimR.

As we shall see, every essentially smooth k-algebra is diagonally F-regular, but not conversely. Indeed, we have the following:

Theorem B Theorem 5.6

Let k be a perfect field of positive characteristic, and let r,s1 be integers. Then the affine cone over Pkr×Pks is diagonally F-regular.

Of course, the affine cone over Pkr×Pks is an isolated singularity, and so USTP is known to hold for this ring by [16]. Nonetheless, our result has the virtue of being effective in the sense that we determine the number h explicitly, and show h is as small as we might expect it to be. We also observe that the class of diagonally F-regular F-finite k-algebras is closed under tensor products over k:

Theorem C Proposition 5.5

Let R and S be k-algebras essentially of finite type, where k is a field of characteristic p. If R and S are diagonally F-regular, then so is RkS.

This implies, remarkably, that the class of diagonally F-regular singularities includes some non-isolated singularities. To our knowledge, this gives a new class of examples where USTP is known to hold. We note that R. Walker obtains orthogonal results to Theorem 4.1 and Theorem 5.6 using complementary techniques; see [31], [32] for precise statements.

Finally, let K be a field of characteristic 0 and R a K-algebra. Suppose that AK is a finitely generated Z-algebra and RAR an A-module such that ARA descends KR in the sense of [14, §2]. We define R to have diagonally F-regular type if RAA/μ is diagonally F-regular for all maximal ideals μ in a dense open set of SpecA, for all choices of A. By standard reduction-mod-p techniques, we get

Theorem D Theorem 6.1

Let K be a field of characteristic 0 and let R be a K-algebra essentially of finite type and of diagonally F-regular type. Let d=dimR. Then we have p(nd)pn for all n and all prime ideals pR.

Thus we see that the affine cone over Pkr×Pks has USTP even if char k=0.

Convention 1.1

Throughout this paper, all rings are defined over a field k of positive characteristic p. Given a ring R, we then denote the e-th iterate of the Frobenius endomorphism by Fe:RR, and use the usual shorthand notation q:=pe. We assume all rings are essentially of finite type over k, thus Noetherian, F-finite, and so excellent. All tensor products are defined over k unless explicitly stated otherwise. We also follow the convention N={0,1,2,...}.

The authors started working on this project during Craig Huneke's 65th Birthday Conference in the University of Michigan in the summer of 2016 after attending Daniel Katz's talk on the subject. We are especially grateful to Craig Huneke who originally inspired us to study this problem for strongly F-regular singularities. We are greatly thankful to Eloísa Grifo, Linquan Ma, and Ilya Smirnov for many valuable conversations. We are especially thankful to Axel Stäbler, Kevin Tucker, and Robert Walker for reading through a preliminary draft and providing valuable feedback. We are particularly thankful to Axel Stäbler for pointing out that we do not need k to be perfect in Proposition 3.4. Finally, we are deeply thankful to our advisor Karl Schwede for his generous support and constant encouragement throughout this project.

Section snippets

Preliminaries

The central objects in this paper are Cartier algebras, their test ideals, and the notion of (strong) F-regularity of a Cartier module. We briefly summarize these here, following the formalism of M. Blickle and A. Stäbler [2], [4]. It is worth mentioning that for the most part we will only be using Cartier algebras and test ideals in the generality introduced by K. Schwede in [24].

Definition 2.1 Cartier algebras

Let R be a ring. A Cartier algebra C over R (or Cartier R-algebra) is an N-graded eNCe unitary ring4

Diagonal Cartier algebras and diagonal F-regularity

In [28], the second named author introduced the Cartier algebra consisting of pe-linear maps compatible with the diagonal closed embedding Δ2:RRR. Here, we generalize this construction to higher diagonals and verify these have the required basic properties, including an analogous subadditivity formula.

For this, we consider Δn:RnR the n-th diagonal closed embedding given by the rule r1rnr1rn. Recall our convention that all tensor products are defined over k unless otherwise explicitly

USTP for diagonally F-regular singularities

In this section, we prove our main result, namely that USTP is satisfied by locally diagonally F-regular rings with h equal to the dimension. We do this by making our discussion in the introduction rigorous. For this we establish:

Theorem 4.1

Let R be a diagonally F-regular k-algebra, and let pSpecR be an ideal of height h. Then p(hn)pn for all nN.

Proof

This containment of ideals can be checked locally, and so we may assume that R is local. We can also assume that p is not the maximal ideal of R, because in

On the class of diagonally F-regular rings

Here is a simple observation about the class of diagonally F-regular rings.

Proposition 5.1

Essentially smooth k-algebras are diagonally F-regular. Further, n-diagonally F-regular k-algebras are strongly F-regular, in particular normal and Cohen–Macaulay.

Proof

The second statement is obvious, whereas the former is a consequence of Kunz's theorem [18] just as in [28, §7]. Indeed, if R is smooth over k, then Rn is smooth and therefore regular for all n. Thus Kunz's theorem tells us that FeRn is a projective Rn

USTP for KLT complex singularities of diagonal F-regular type

Let R be a ring of equicharacteristic 0. A descent datum is a finitely generated Z-algebra AK. A model of R for this descent datum is an A-algebra RAR, such that RA is a free A-module and RAAK=R; see for instance [14] or [28, Remark 5.3]. Note that A/μ is a finite field, and in particular a perfect field of positive characteristic, for all maximal ideals μA. We say that R is of diagonally F-regular type if, for all choices of descent data AK, the set{μMaxSpecA|RAAA/μ is diagonally F

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  • The first named author was supported in part by the NSF FRG Grant DMS #1265261/1501115. The second named author was supported by the NSF grants DMS #1246989/#1252860/#1265261.

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