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Excellence in Prime Characteristic Is Closely Connected to Another Common Hypothesis, That of F-finiteness

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Excellence in Prime Characteristic Is Closely Connected to Another Common Hypothesis, That of F-finiteness EXCELLENCE IN PRIME CHARACTERISTIC RANKEYA DATTA AND KAREN E. SMITH To Professor Lawrence Ein on the occasion of his sixtieth birthday. Abstract. Fix any field K of characteristic p such that [K : Kp] is finite. We discuss excellence for domains whose fraction field is K, showing for example, that R is excellent if and only if the Frobenius map is finite on R. Furthermore, we show R is excellent if and only if it admits some non- zero p−e-linear map for R (in the language of [3]), or equivalently, that the Frobenius map R → F∗R defines a solid R-algebra structure on F∗R (in the language of [11]). In particular, this means that generically F - finite, Frobenius split Noetherian domains are always excellent. We also show that non-excellent rings are abundant and easy to construct in prime characteristic, even within the world of regular local rings of dimension one inside function fields. 1. Introduction The notion of excellence for a commutative ring was introduced by Grothe- -ndieck. A Noetherian ring is excellent, essentially, if the ring satisfies a list of axioms that ensures it behaves much like a finitely generated algebra over a field; see Definition 2.1. An arbitrary Noetherian ring can be quite patho- logical, but the class of excellent rings is supposed to be the most general setting to which one can expect the deep ideas of algebraic geometry, such as resolution of singularities, to extend. Although a common hypothesis in the literature, excellent rings are not widely understood. They are often dismissed with sentences like the following arXiv:1704.03628v3 [math.AC] 18 Jan 2018 quoted from Wikipedia: “Essentially all Noetherian rings that occur naturally in algebraic geometry or number theory are excellent; in fact it is quite hard to construct examples of Noetherian rings that are not excellent” [30]. In this paper we show that on the contrary, non-excellent rings are quite easy to con- struct and are abundant, even among regular local rings of dimension one. Our setting is prime characteristic since Dedekind domains of characteristic zero are always excellent [16, Cor 8.2.40]. The examples we construct, moreover, are The first author was partially supported by a Department of Mathematics Graduate Fellowship at the University of Michigan, and the second author’s NSF grant DMS-1501625. The second author was partially supported by NSF grant DMS-1501625. 1 2 RANKEYA DATTA AND KAREN E. SMITH p generically F-finite, unlike other known examples such as Nagata’s k ⊗kp k [[t]] (whenever [k : kp] = ∞), whose completion map is purely inseparable. Excellence in prime characteristic is closely connected to another common hypothesis, that of F-finiteness. A ring of characteristic p is F -finite if the Frobenius (or p-th power) map is finite. A well-known theorem of Kunz ensures that F -finite rings of characteristic p are excellent [15, Thm 2.5]. The converse is also true under the additional hypothesis that the fraction field is F -finite. Put differently, a domain is F -finite if and only if it is excellent and generically F -finite. For example, for any domain R whose fraction field is the function field of some algebraic variety over a perfect (e.g. algebraically closed) field, R is F -finite if and only if R is excellent. Because this fact does not seem to be well-known, we show in Section 2 how this statement follows easily from known facts in the literature. In Section 3, we turn toward the issue of p−e-linear maps. For a ring R of φ prime characteristic p, a p−e-linear map is a map R → R of the underlying abelian group that satisfies φ(rpe s) = rφ(s) for all r, s ∈ S. Any splitting of the Frobenius map is such a p−1-linear map. In Section 3, we show that for Noetherian domains with F-finite fraction field, there are no non-zero p−e- linear maps at all unless R is excellent! Put differently using the language of Hochster [11], we show that a generically F -finite Noetherian domain is a solid algebra via Frobenius if and only if it is excellent. In particular, if a generically F -finite Noetherian domain is Frobenius split, it must be excellent. −e e The study of p -linear maps, or equivalently, elements of HomR(F∗ R, R), was formalized by Manuel Blickle and later used by Karl Schwede to give an alternate and more global approach to test ideals, an important topic in char- acteristic p commutative algebra. Our results show that Schwede’s approach to test ideals relies much more heavily on F -finiteness than previously under- stood. Test ideals can be viewed as “prime characteristic analogs” of multiplier ideals due to the results in [28] and [8] (see also [10] and [9]). While they have attracted great interest in birational algebraic geometry, our results in Section 4 offer a cautionary tale about the limits of this approach. In Section 4, we consider excellence in the setting of discrete valuation rings of a function field of characteristic p. Excellence in this case is equivalent to the DVR being divisorial, a topic explored in [5]. We show here how this makes it is easy to write down explicit examples of non-excellent discrete valuation rings. Moreover, a simple countability argument shows that among domains whose fraction field is, say the function field of P2, non-excellent regular local rings of dimension one are far more abundant than the excellent ones. EXCELLENCE IN PRIME CHARACTERISTIC 3 This paper is largely expository, drawing heavily on the work in [5] (and the corrections in [6]) where most of the harder proofs of the results discussed here appear. However, we are emphasizing a different aspect of the subject than in that paper, drawing conclusions not explicit there. Acknowledgments: We are honored to help celebrate the birthday of Lawrence Ein, who has been a tremendous inspiration and support for the second au- thor, both mathematically and personally. We thank the referee for their careful comments, and, in particular, for suggesting a generalization of an earlier version of Proposition 3.7. 2. Excellence An arbitrary Noetherian ring can exhibit pathological behavior. For in- stance, the integral closure of a Noetherian domain in a finite extension of its fraction field can fail to be Noetherian, and Noetherian rings can have saturated chains of prime ideals of different lengths. Excellent rings were in- troduced by Grothendieck in [7, expos´eIV] to rule out such pathologies. We review Grothendieck’s definition and some other relevant properties of excel- lent rings. Another good source is [18, Chapter 32]. Definition 2.1. [7, D´ef 7.8.2] A Noetherian ring A is excellent if it satisfies the following properties: (1) A is universally catenary. This means that every finitely generated A-algebra has the property that for any two prime ideals p ⊆ q, all saturated chains of prime ideals from p to q have the same length.1 (2) All formal fibers of A are geometrically regular. This means that for every p ∈ Spec(A), the fibers of the natural map Spec(Acp) → Spec(Ap) induced by completion along p are geometrically regular in the sense that for each x ∈ Spec(A ), the ring A ⊗ K is regular for any finite p cp Ap field extension K of the residue field κ(x). (3) For every finitely generated A algebra B, the regular locus of Spec(B) is open; that is, the set {q ∈ Spec(B): Bq is a regular local ring} is open in Spec(B). The class of excellent rings is closed under homomorphic image, localization, and finite-type algebra extensions. Since every field is excellent, it follows that nearly every ring one is likely to encounter in classical algebraic geometry is excellent. Likewise, because the ring of integers is excellent and all complete 1The first example of a non-catenary Noetherian ring was given by Nagata in [20]. 4 RANKEYA DATTA AND KAREN E. SMITH local rings are excellent, familiar rings likely to arise in number theory are excellent as well. Among the many properties of excellent rings, the following, sometimes called the Japanese or N2 property, will be important for us later. Proposition 2.2. [7, expos´eIV, 7.8.3 (vi)]. Let A be a Noetherian excellent domain. The integral closure of A in any finite extension of its fraction field is finite as an A-module. We construct discrete valuation rings of characteristic p which fail to be Japanese in Subsection 4.1. 2.1. Excellence in prime characteristic. Fix a commutative ring R of F prime characteristic p. The Frobenius map is the ring homomorphism R → R sending each element to its p-th power. It is convenient to denote the target copy of R by F∗R. Thus the notation F∗R denotes the ring R but viewed as an R-module via the Frobenius map: an element r ∈ R acts on x ∈ F∗R to pro- p e duce r x. Similarly, iterating the Frobenius map, F∗ R denotes the R-algebra e F e pe defined by the e-th iterate of Frobenius R −→ F∗ R sending r 7→ r . Definition 2.3. The ring R is F-finite if the Frobenius map is finite; that is, R is F-finite if F∗R is a finitely generated R-module. F-finite rings are ubiquitous. For example, it is easy to check that every perfect field is F-finite, and that a finitely generated algebra over an F-finite ring is F-finite.
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