The uniform symbolic topology property for diagonally F-regular algebras☆
Introduction
We are concerned with the following question: when does a finite-dimensional Noetherian ring R satisfy for all prime ideals and for some h independent of ? Here, the expression denotes the m-th symbolic power of . 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 is a prime ideal such that the -adic and symbolic topologies are equivalent, then they are in fact linearly equivalent, meaning there is a constant depending on such that for all n [29]. In particular, Swanson's result holds for every prime ideal 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 -algebra essentially of finite type, then h can be taken independently of . In fact suffices. Rings for which this number h can be taken independently of (i.e. for which there exists a uniform bound on h for all ) 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 of positive characteristic, we introduce a class of strongly F-regular algebras essentially of finite type, called diagonally F-regular -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 -algebras, as well as Segre products of polynomial rings over ,2 i.e. the affine cone over whenever is a perfect field of positive characteristic and . We also show that the class of diagonally F-regular -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: Here, denotes the test ideal of the (formal) power ; 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 ), 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 , depending on , h, and n, such that the second containment 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, holds for our choice of . Following [28], we will construct using the so-called diagonal Cartier algebras. Namely, we set where is the n-th diagonal Cartier algebra; see Definition 3.1. Then Proposition 3.4(c) demonstrates that containment (2′) holds for any reduced -algebra essentially of finite type, while (1′) holds whenever is F-regular. When this is the case for all n, we say our ring is diagonally F-regular as a -algebra. This sketches the proof of our main theorem:
Theorem A Theorem 4.1 If R is a diagonally F-regular -algebra essentially of finite type, then R has USTP with .
As we shall see, every essentially smooth -algebra is diagonally F-regular, but not conversely. Indeed, we have the following:
Theorem B Theorem 5.6 Let be a perfect field of positive characteristic, and let be integers. Then the affine cone over is diagonally F-regular.
Of course, the affine cone over 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 -algebras is closed under tensor products over : Theorem C Proposition 5.5 Let R and S be -algebras essentially of finite type, where is a field of characteristic p. If R and S are diagonally F-regular, then so is .
Finally, let K be a field of characteristic 0 and R a K-algebra. Suppose that is a finitely generated -algebra and an A-module such that descends in the sense of [14, §2]. We define R to have diagonally F-regular type if is diagonally F-regular for all maximal ideals μ in a dense open set of Spec A, 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 . Then we have for all n and all prime ideals .
Convention 1.1 Throughout this paper, all rings are defined over a field of positive characteristic p. Given a ring R, we then denote the e-th iterate of the Frobenius endomorphism by , and use the usual shorthand notation . We assume all rings are essentially of finite type over , thus Noetherian, F-finite, and so excellent. All tensor products are defined over unless explicitly stated otherwise. We also follow the convention .
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 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 over R (or Cartier R-algebra) is an -graded unitary ring4
Diagonal Cartier algebras and diagonal F-regularity
In [28], the second named author introduced the Cartier algebra consisting of -linear maps compatible with the diagonal closed embedding . 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 the n-th diagonal closed embedding given by the rule . Recall our convention that all tensor products are defined over 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 -algebra, and let be an ideal of height h. Then for all .
Proof This containment of ideals can be checked locally, and so we may assume that R is local. We can also assume that 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 -algebras are diagonally F-regular. Further, n-diagonally F-regular -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 , then is smooth and therefore regular for all n. Thus Kunz's theorem tells us that is a projective
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 -algebra . A model of R for this descent datum is an A-algebra , such that is a free A-module and ; see for instance [14] or [28, Remark 5.3]. Note that is a finite field, and in particular a perfect field of positive characteristic, for all maximal ideals . We say that R is of diagonally F-regular type if, for all choices of descent data , the set
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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.