Excellence, F-singularities, and solidity
aa r X i v : . [ m a t h . A C ] J u l EXCELLENCE, F -SINGULARITIES, AND SOLIDITY RANKEYA DATTA AND TAKUMI MURAYAMAWITH AN APPENDIX BY KAREN E. SMITH
Abstract. An R -algebra S is R -solid if there exists a nonzero R -linear map S → R . In charac-teristic p , the study of F -singularities such as Frobenius splittings implicitly rely on the R -solidityof R /p . Following recent results of the first two authors on the Frobenius non-splitting of cer-tain excellent F -pure rings, in this paper we use the notion of solidity to systematically study thenotion of excellence, with an emphasis on F -singularities. We show that for rings R essentiallyof finite type over complete local rings of characteristic p , reducedness implies the R -solidity of R /p , F -purity implies Frobenius splitting, and F -pure regularity implies split F -regularity. Wedemonstrate that Henselizations and completions are not solid, providing obstructions for the R -solidity of R /p for arbitrary excellent rings. This also has negative consequences for the solidityof big Cohen–Macaulay algebras, an important example of which are absolute integral closures ofexcellent local rings in prime characteristic. We establish a close relationship between the solidityof absolute integral closures and the notion of Japanese rings. Analyzing the Japanese property re-veals that Dedekind domains R for which R /p is R -solid are excellent, despite our recent examplesof excellent Euclidean domains with no nonzero p − -linear maps. Additionally, we show that whileperfect closures are often solid in algebro-geometric situations, there exist locally excellent domainswith solid perfect closures whose absolute integral closures are not solid. In an appendix, Karen E.Smith uses the solidity of absolute integral closures to characterize the test ideal for a large classof Gorenstein domains of prime characteristic.2010 Mathematics Subject Classification.
Primary 13F40; Secondary 13B22, 13A35, 14G22, 13H10.
Key words and phrases.
Excellent ring, Japanese ring, solid algebra, F -singularities, F -solidity, big Cohen–Macaulay algebra, Tate algebra, absolute integral closure, perfect closure, test ideal.The first author was supported by an AMS-Simons Travel Grant, administered by the American MathematicalSociety with support from the Simons Foundation.The second author was supported by the National Science Foundation under Grant No. DMS-1902616.The third author was supported by the National Science Foundation under Grant No. DMS-1801697. Contents
1. Introduction 22. Notation and preliminaries 63. F -purity, Frobenius splitting, and variants of strong F -regularity 103.1. F -singularities for rings of finite type over complete local rings 103.2. On F -pure regularity of regular rings 113.3. Excellent F -pure regular rings are not always split F -regular 143.4. Split F -regularity of discrete valuation rings 194. F -purity vs. Frobenius splitting: a contrast in permanence properties 204.1. Relative Frobenius and the Radu–Andr´e theorem 214.2. Ascent under regular maps and faithfully flat descent 215. F -solidity and non- F -finite excellent rings 245.1. Solid modules and algebras 245.2. F -solidity 255.3. F -solidity for rings of finite type over complete local rings 265.4. Excellent rings are not always F -solid 286. Some obstructions to solidity 296.1. Henselizations and completions are not solid 296.2. Non-solidity of some big Cohen–Macaulay algebras 327. Solidity of absolute integral and perfect closures – a deeper analysis 337.1. N-1 and Japanese rings 347.2. Solidity of absolute integral closures and Japanese rings 357.3. Solidity of filtered colimits and Japanese rings of characteristic zero 367.4. Solidity and excellence in Krull dimension one 387.5. A meta construction of Hochster and solidity of perfect closures 418. Some open questions 45 Appendix by Karen E. Smith
Appendix A. Solidity of absolute integral closures and the test ideal 48A.1. Tight Closure and Test Ideals 49A.2. The Proof of Theorem A.0.2 50A.3. Graded variants of absolute integral closures 52References 541.
Introduction
In the 1950s, Nagata constructed various examples of Noetherian rings that behave badly undertaking completions and integral closures (see [Nag75, App. A1]). In order to avoid such pathologies,Grothendieck and Dieudonn´e introduced excellent rings, which form a large class of rings for whichdeep theorems of algebraic geometry such as resolutions of singularities are conjectured to hold[EGAIV , D´ef. 7.8.2 and Rem. 7.9.6]. In prime characteristic p >
0, Smith and the first authorcharacterized excellent domains R in terms of the existence of nonzero R -linear maps F ∗ R → R when the fraction field K of R satisfies [ K : K p ] < ∞ , that is, when K is F -finite [DS18]. On theother hand, the first two authors recently constructed examples of excellent regular domains forwhich there exist no nonzero R -linear maps F ∗ R → R when one relaxes F -finiteness of the fractionfield [DM(b)]. Here, F ∗ R denotes the ring R with the R -algebra structure given by restrictingscalars via the Frobenius ( p -th power) map F : R → F ∗ R . We will refer to R -linear maps of theform F ∗ R → R as p − -linear maps . XCELLENCE, F -SINGULARITIES, AND SOLIDITY 3 Our goal in this paper is to systematically study how the excellence of a ring R interacts with theexistence of nonzero p − -linear maps in prime characteristic, and more generally, with the existenceof nonzero R -linear maps S → R for certain R -algebras S in arbitrary characteristic. Hochsterutilized such R -algebras to propose a closure operation called solid closure as a characteristicindependent substitute for tight closure [Hoc94]. Following Hochster [Hoc94, Def. 1.1], an R -algebra S is R -solid if there exists a nonzero R -linear map S → R . For example, for a ring R ofprime characteristic p >
0, the existence of a nonzero p − -linear map such as a Frobenius splittingis precisely the assertion that F ∗ R is a solid R -algebra. Thus, the notion of solid algebras providesa convenient framework to investigate questions related to the non-triviality of the dual space ofcertain algebras.We focus first on the case of prime characteristic p >
0. In this setting, the theory of F -singularities studies how the properties of the Frobenius map F : R → F ∗ R are related to thesingularities of a Noetherian ring R of characteristic p . This theory began with the work of Kunz[Kun69; Kun76], and was subsequently developed in parallel in the theories of F -purity and tightclosure in commutative algebra (see, for example, [HR76; HH90]) and in the theory of Frobeniussplittings in algebraic geometry (see, for example, [MR85]). One differentiating aspect betweenthese approaches is that Hochster and Roberts focused on the purity of the Frobenius map, whileMehta and Ramanthan focused on the splitting of the Frobenius map. Their approaches wereknown to coincide in the setting of Noetherian F -finite rings and for complete local rings, but thisdoes not seem to have been known in general, even for rings of finite type over an arbitrary field ofpositive characteristic.Our first main result says that for the purposes of classical algebraic geometry, the approachesof Hochster–Roberts and of Mehta–Ramanathan coincide. Theorem 1.1 (see Theorem 3.1.1) . An F -pure (resp. strongly F -regular) ring essentially of finitetype over a complete local ring of prime characteristic p > is Frobenius split (resp. split F -regular). In particular, F -purity and Frobenius splitting coincide for rings essentially of finite type over fieldsof positive characteristic that are not necessarily F -finite. Here, the notion of strong F -regularitythat we use is one defined in terms of tight closure, following Hochster, while split F -regularityis defined in terms of splittings of maps R → F e ∗ R sending 1 to certain elements c ∈ F e ∗ R . Bothvariants (see Definition 2.4.1) coincide with the usual notion of strong F -regularity in the settingof Noetherian F -finite rings, and Theorem 1.1 shows that the notions coincide for a large class ofnon- F -finite Noetherian rings as well.Despite the equivalence of the related notions of singularity proved in Theorem 1.1, the firsttwo authors recently constructed examples which exhibit that F -purity is distinct from Frobeniussplitting for excellent rings in general [DM(b), Thm. A]. Using these same examples arising fromrigid analytic geometry, we show in the present paper that while F -purity behaves well undercompletions for excellent rings (and more generally for faithfully flat maps with sufficiently nicefibers), Frobenius splittings do not in general. Proposition 1.2 (see Propositions 4.2.1 and 4.2.3) . Let ϕ : R → S be a map of Noetherian ringsof prime characteristic p > . ( i ) If ϕ is flat and has geometrically regular fibers, then F -purity ascends from R to S . ( ii ) If ϕ is faithfully flat (more generally, is pure), then F -purity descends from S to R .However, there exist examples of excellent regular Henselian local rings R and S for which Frobeniussplitting fails to ascend under regular maps, and fails to descend under faithfully flat maps. Note that properties ( i ) and ( ii ) for F -purity are well-known (see [Has10, Prop. 2.4.4] and [HR76,Prop. 5.13], respectively). The new contents in Proposition 1.2 are the examples which illustrate RANKEYA DATTA AND TAKUMI MURAYAMA that F -purity is a better behaved notion of singularity than Frobenius splitting with respect topermanence properties for excellent rings that are not essentially of finite type over complete localrings. At the same time, although Frobenius splittings do not ascend under arbitrary flat mapswith geometrically regular fibers, we show that the property behaves well under the closely relatednotion of smooth maps even in a non- F -finite setting (see Proposition 4.2.1( ii )). This indicatesthat as a notion of singularity, Frobenius splitting behaves best under additional hypotheses suchas being of finite type.Just as Frobenius splitting behaves well only under additional hypotheses, we demonstrate thatthe two most natural variants of the usual notion of strong F -regularity, namely split F -regularity and F -pure regularity , do not coincide for arbitrary excellent rings (see Theorem 3.3.4 and Corol-lary 3.3.5). More specifically, we prove that Tate algebras and convergent power series rings overcomplete non-Archimedean fields of prime characteristic are always F -pure regular, while [DM(b)]provides examples of non-Archimedean fields over which such rings are not always split F -regular.This is despite the fact that Tate algebras and convergent power series rings are the rigid analyticanalogues of polynomial rings, their localizations, and the completions thereof, all three of whichare split F -regular by Theorem 1.1.An interesting aspect of the authors’ examples in [DM(b)] is that not only are the excellentregular rings R therein not split F -regular, but the rings R are not even F -solid , that is, R doesnot admit any nonzero R -linear maps F ∗ R → R . Given the connection between F -solidity andexcellence established in [DS18], we show that a large class of excellent Noetherian domains arenevertheless F -solid, without assuming that their fraction fields are F -finite. Theorem 1.3 (see Theorem 5.3.1 and Proposition 5.4.1) . A reduced ring which is essentially offinite type over a complete local ring of prime characteristic p > is F -solid. Nevertheless, foreach integer n > , there exist excellent Henselian regular local rings of characteristic p > andKrull dimension n that are not F -solid. Theorem 1.3 should be viewed as a natural extension of Theorem 1.1 because a Frobenius splittingis a special instance of a nonzero p − -linear map. To better explain the difference in behaviorbetween rings of finite type over complete local rings and more general excellent rings such as Tatealgebras from the point of view of F -solidity, we prove some obstructions for descending F -solidityfrom the complete to the non-complete cases in the following result. Theorem 1.4 (see Theorems 6.1.1 and 6.1.3) . Let R be a Noetherian domain of arbitrary charac-teristic. ( i ) If R is a non-Henselian local ring whose Henselization R h is a domain, then R h is not asolid R -algebra. ( ii ) If I is an ideal of R and R is not I -adically complete, then the I -adic completion b R I is nota solid R -algebra. In particular, Theorem 1.4 illustrates that there is no general method to reduce the question of F -solidity of rings essentially of finite type over excellent local rings to that of F -solidity of ringsthat are essentially of finite type over a complete local ring. This provides another perspective onthe non- F -solid excellent local examples obtained in [DM(b)].Theorem 1.4 has consequences for Hochster and Huneke’s theory of big Cohen–Macaulay algebras.In particular, while big Cohen–Macaulay algebras for complete local rings are always solid, if( R, m ) is a mixed characteristic Noetherian domain that is not m -adically complete, then Theorem1.4 implies that all known constructions of big Cohen–Macaulay R -algebras are non- R -solid (seePropositions 6.2.1 and 6.2.3). XCELLENCE, F -SINGULARITIES, AND SOLIDITY 5 One of the principal examples of a big Cohen–Macaulay algebra in prime characteristic is theabsolute integral closure of an excellent local domain. Recall that if R is a domain, then the absolute integral closure R + of R is the integral closure of R in an algebraic closure of its fractionfield. Since big Cohen–Macaulay algebras in prime and mixed characteristics are often constructedas extensions of R + , it is natural to ask whether R + itself is a solid R -algebra. The following resultshows that the R -solidity of R + is closely connected with an important property that is implied byexcellence, namely that of being Japanese (see Definition 7.1.1). Theorem 1.5 (see Theorem 7.2.1 and Corollary 7.3.2) . Let R be a Noetherian domain of arbitrarycharacteristic. ( i ) If the absolute integral closure R + is a solid R -algebra, then R is a Japanese ring. Theconverse holds if R contains the rational numbers Q . ( ii ) Suppose R has prime characteristic. If the perfect closure R perf is a solid R -algebra, ormore generally, if R is F -solid, then R is N-1 if and only if R is Japanese. The perfect closure R perf of R is the universal perfect ring with respect to ring homomorphismsfrom R to a perfect ring (see (2.0.0.1)), while the N-1 condition is a generalization of the Japaneseproperty (see Definition 7.1.1). The N-1 and Japanese properties are well-known to be equivalentwhen the fraction field K of R has characteristic zero due to the absence of inseparability phenomena(see [Mat80, (31.B), Cor. 1]). Thus, Theorem 1.5 can be interpreted as saying that in primecharacteristic, F -solidity is an antidote to inseparability.One of the more surprising aspects of the constructions in [DM(b)] are the examples of excellentEuclidean domains which illustrate the failure of F -solidity even for prime characteristic excellentrings of Krull dimension one. However, by carefully analyzing the N-1 and Japanese properties,we show in this paper that for most rings of Krull dimension one, the R -solidity of R + or of F ∗ R in fact implies that R is excellent. Our result does not impose any F -finiteness restrictions on thefraction field of R , which was the case in [DS18]. Theorem 1.6 (see Theorem 7.4.1) . Let R be a Noetherian N-1 domain of Krull dimension one.Suppose R + is a solid R -algebra (in which case one does not need R to be N-1), or that R is an F -solid domain of prime characteristic p > . Then, R is excellent. In particular, a Frobenius split(or F -solid) Dedekind domain is always excellent. Furthermore, Theorem 1.6 yields a characterization of excellence for DVRs of prime characteristicin terms of F -solidity. This extends [DS18, Thm. 4.1] to a larger class of DVRs, and provides analmost complete classification for which DVRs are F -solid. Corollary 1.7 (see Corollary 7.4.3) . Let ( R, m ) be a DVR of prime characteristic p > . Considerthe following statements: ( i ) R is F -solid. ( ii ) R is Frobenius split. ( iii ) R is split F -regular. ( iv ) R is excellent.Then, ( i ) , ( ii ) , and ( iii ) are equivalent, and always imply ( iv ) . All four assertions are equivalent if R is essentially of finite type over a complete local ring, or the fraction field K of R is such that K /p is countably generated over K . However, there exist excellent Henselian DVRs that are not F -solid. A natural question that arises from Theorem 1.6 and Corollary 1.7 is whether there exist non-excellent domains that are F -solid. We end the main body of the paper by addressing this questionusing a meta-construction of Hochster. Hochster’s construction (see Theorem 7.5.1) provides an RANKEYA DATTA AND TAKUMI MURAYAMA abundance of examples of biequidimensional locally excellent Noetherian domains that are Frobe-nius split (see Corollary 7.5.3), and consequently F -solid, but not excellent. In fact, the constructionof Corollary 7.5.3, when specialized to Krull dimension one, provides examples of locally excellentNoetherian domains R of prime characteristic whose perfect closures R perf are solid, but whoseabsolute integral closures R + are not (see Example 7.5.4). This illustrates that solidity of R perf is more common than solidity of R + , further evidenced by the fact that most algebro-geometricrings of prime characteristic have solid perfect closures (see Proposition 7.5.7). On the other hand,surprisingly little seems to be known about the solidity of absolute integral closures of Noetherianrings that are not complete local, even for polynomial rings over fields of positive characteristic (seeQuestion 8.4). We list this and other related open questions in § § Theorem 1.8 (see Theorem A.0.2) . If R is a complete Noetherian Gorenstein local domain ofprime characteristic p > , then the image of the evaluation at map Hom R ( R + , R ) eval@1 −−−−→ R is the test ideal τ ( R ) of R . Smith obtains a similar characterization of the test ideal τ ( R ) when R is an N -graded Gorensteinring of finite type over R , when R is a field of prime characteristic (see Theorem A.3.2). For thisgraded characterization, Smith replaces the absolute integral closure R + by its graded analogues R +GR and R +gr (see Definition A.3.1) and Hom R by its graded analogue * Hom R in the statementof Theorem 1.8. Acknowledgments.
We would first like to thank Karen E. Smith for contributing the appendix.This paper is a continuation of work the first author began with Karen, so we also thank herfor all of her insights and for helpful conversations regarding absolute integral closures. The firstauthor would also like to thank Karl Schwede from whom he first learned of some of the questionsaddressed in this paper at the 2015 AMS Math Research Communities in commutative algebra.Both authors are very grateful to Kevin Tucker for numerous insightful conversations on the topicof solidity, especially on solidity of completions, and for his interest in non- F -finite rings. We arealso grateful to Melvin Hochster for answering questions about solidity, absolute integral closures,and for sharing with us some of the history behind Lemma 2.3.3. In addition, we would like tothank Linquan Ma for discussions on the solidity of absolute integral closures, and Bhargav Bhattfor conversations about relatively perfect maps. Finally, we are grateful to Eric Canton, NaokiEndo, Shubhodip Mondal, Alapan Mukhopadhyay, Mircea Mustat¸˘a, and Matthew Stevenson forhelpful conversations. 2. Notation and preliminaries
All rings will be commutative with identity. Given a domain R , the integral closure of R in itsfraction field (that is, the normalization of R ) will often be denoted as R N . Most rings in this paperwill be Noetherian. On the other hand, if R is a Noetherian domain, we will consider its absoluteintegral closure R + , which is the integral closure of R in an algebraic closure of its fraction field.Similarly, if R is a Noetherian domain of prime characteristic, we will consider the perfect closure R perf := colim e ∈ Z > (cid:16) R F −→ R F −→ · · · (cid:17) , (2.0.0.1)where F : R → R is the Frobenius map (see Definition 2.1.1). Both rings R + and R perf are notNoetherian unless R is a field. XCELLENCE, F -SINGULARITIES, AND SOLIDITY 7 A discrete valuation ring (abbreviated DVR) is a Noetherian regular local ring of Krull dimensionone, or equivalently, a Noetherian normal ring with a unique nonzero maximal ideal that is principal.2.1. Frobenius.
We fix our notation for the Frobenius map.
Definition 2.1.1.
Let R be a ring of prime characteristic p >
0. The
Frobenius map is the ringhomomorphism F R : R F R ∗ R.r r p We often consider the target copy of R as an R -algebra via F R , in which case we denote it by F R ∗ R .In other words, F R ∗ R is the same underlying ring as R , but for r ∈ R and x ∈ F R ∗ R , the R -algebrastructure is given by r · x = r p x. The e -th iterate of the Frobenius map is denoted F eR ∗ : R → F eR ∗ R . If R is a domain, then F eR ∗ R will occasionally be identified with the ring R /p e of p e -th roots of elements of R . Under thisidentification, the map F eR corresponds to the inclusion R ֒ → R /p e . An R -linear map of the form F eR ∗ R → R will often be called a p − e -linear map . We say R is F -finite if the Frobenius map isfinite (equivalently, of finite type).We will sometimes drop the subscript R from F eR ∗ R and just write F e ∗ R instead when R is clearfrom context or is too cumbersome to write as a subscript. We hope this does not cause anyconfusion.2.2. Excellent rings.
A focus of this paper will be to better understand various notions of singu-larities defined via the Frobenius map in the setting of excellent rings. These are a class of rings,introduced by Grothendieck and Dieudonn´e and defined below, to which deep results in algebraicgeometry such as resolution of singularities are expected to extend.
Definition 2.2.1.
Let R be a Noetherian ring. We say that R is excellent if it satisfies the followingaxioms:(1) R is universally catenary , that is, for every finite type R -algebra S , if p ( q are two primeideals of S , then all maximal chains of primes ideals from p to q have the same length.(2) R is J-2 , that is, for every finite type R -algebra S , the regular locus in Spec( S ) is open.(3) R is a G -ring , that is, for every prime ideal p of R , the p R p -adic completion map R p → c R p has geometrically regular fibers.We say that R is quasi-excellent if it satisfies axioms (2) and (3), but not necessarily (1). We saythat R is locally excellent if for all prime (equivalently, maximal) ideals p of R , the local ring R p isexcellent. Example 2.2.2. (1) Local G -rings are quasi-excellent by [Mat80, Thm. 76 and footnote on p. 252].(2) If R is an excellent ring, then any ring which is essentially of finite type over R is alsoexcellent. In particular, since a complete local Noetherian ring is excellent, any algebrathat is essentially of finite type over a complete local ring is also excellent. The sameapplies for quasi-excellent rings. See [Mat80, (34.A) Def.] or [EGAIV , Prop. 5.6.1( b ), Rem.5.6.3( i ), Thms. 6.12.4 and 7.4.4( ii )].(3) A Noetherian F -finite ring of prime characteristic is excellent by [Kun76, Thm. 2.5].(4) Examples of locally excellent rings that are not excellent are abundant. See Theorem 7.5.1for a construction due to Hochster that provides many such examples. There are alsoexamples of locally F -finite rings of prime characteristic that are not F -finite. See [DI],which uses a construction of Nagata [Nag59, § RANKEYA DATTA AND TAKUMI MURAYAMA
Pure maps.
We will study the notion of a pure map of modules that we introduce next.
Definition 2.3.1.
Let R be a ring. A map of R modules ϕ : M → N is pure if for all R -modules P , the induced map id P ⊗ ϕ : P ⊗ R M → P ⊗ R N is injective. Remark 2.3.2. (1) A pure map of modules is automatically injective (by taking P = R ).(2) Since any R -module P is a filtered colimit of its finitely generated submodules, and sincetensor products commute with filtered colimits, it follows that to check that an R -linearmap ϕ : M → N is pure, it suffices to check that for all finitely generated R -modules P , themap id P ⊗ ϕ is injective.(3) Any map of R -modules M → N that admits an R -linear left inverse is pure.(4) Any faithfully flat ring homomorphism R → S is pure as a map of R -modules [BouCA, Ch.I, §
3, n o
5, Prop. 9].We will use the following result about pure maps of complete local rings. The “in particular”statement was communicated to Hochster by Auslander, although it may be older.
Lemma 2.3.3 (cf. [HH90, Cor. 6.24]) . Let ( R, m , κ ) be a Noetherian local ring, and let f : R → M be a map of R -modules. Then, the following are equivalent: ( i ) f is pure. ( ii ) There exists an R -linear map g : M → b R such that g ◦ f is the canonical map R → b R .In particular, if R is complete local, then every pure map f : R → M splits as a map of R -modules.Proof. We first prove ( ii ) ⇒ ( i ). The canonical map R → b R is faithfully flat, hence pure. Therefore,assuming ( ii ), the composition g ◦ f is a pure map of R -modules, hence so is f .We now show show ( i ) ⇒ ( ii ). Denoting by E R ( κ ) the injective hull of the residue field κ , we claimwe have the following commutative diagram with exact rows, where the vertical homomorphismsare isomorphisms:Hom R (cid:0) E R ( κ ) ⊗ R M, E R ( κ ) (cid:1) Hom R (cid:0) E R ( κ ) ⊗ R R, E R ( κ ) (cid:1) R (cid:0) M, Hom R (cid:0) E R ( κ ) , E R ( κ ) (cid:1)(cid:1) Hom R (cid:0) R, Hom R (cid:0) E R ( κ ) , E R ( κ ) (cid:1)(cid:1) R ( M, b R ) Hom R ( R, b R ) 0 (id ER ( κ ) ⊗ f ) ∗ ∼ ∼ f ∗ f ∗ ∼ ∼ The top horizontal arrow is the Matlis dual of id E R ( κ ) ⊗ f , and is surjective by the purity of f . Themiddle row is obtained from the top row by tensor-hom adjunction, and the last row is obtainedusing the isomorphism Hom R ( E R ( κ ) , E R ( κ )) ≃ b R from [Mat89, Thm. 18.6( iv )]. Since the lastrow is exact by the commutativity of the diagram, there exists a map g ∈ Hom R ( M, b R ) such that f ∗ ( g ) = g ◦ f is the completion homomorphism R → b R .Finally, the “in particular” statement follows since the map g in ( ii ) is a splitting for f when R = b R . (cid:3) Remark 2.3.4.
Lemma 2.3.3 implies that a pure map R → M from a Noetherian local ring R issplit precisely when among the non-empty set of b R -linear splittings of the induced map b R → M ⊗ R b R ,there exists a splitting f : M ⊗ R b R → b R such that the image of the composition M → M ⊗ R b R f −→ b R lands inside the canonical image of R in b R . XCELLENCE, F -SINGULARITIES, AND SOLIDITY 9 F -singularities. We now recall various notions of F -singularities that will be studied subse-quently in this paper. Definition 2.4.1.
Let R be a Noetherian ring of prime characteristic p >
0, and R ◦ denote theset of elements of R not in any minimal prime. For every c ∈ R and every integer e >
0, we denoteby λ ec the composition R F e −→ F eR ∗ R F eR ∗ ( −· c ) −−−−−→ F eR ∗ R. In other words, λ ec is the unique R -linear map R → F eR ∗ R that maps 1 to c . If c ∈ R , then following[DS16, Def. 6.1.1], we say that R is F -pure along c if λ ec is pure for some e >
0, and that R is Frobenius split along c if λ ec splits as an R -module homomorphism for some e >
0. We then saythat( a ) R is split F -regular if R is Frobenius split along every c ∈ R ◦ [HH94(a), Def. 5.1];( b ) R is F -pure regular if R is F -pure along every c ∈ R ◦ [HH94(a), Rem. 5.3];( c ) R is strongly F -regular if for every R -module M and every submodule N of M , N is tightlyclosed in M [Has10, Def. 3.3];( d ) R is Frobenius split if R is Frobenius split along 1 ∈ R [MR85, Def. 1]; and( e ) R is F -pure if R is F -pure along 1 ∈ R [HR76, p. 121].The definition in ( c ) is due to Hochster. See [HH90, Def. 8.2] for the definition of tight closure formodules.Note that ( a ) is the usual definition of strong F -regularity in the F -finite setting. The terminologyin ( a ) and ( b ) is from [DS16, Defs. 6.6.1 and 6.1.1]. F -pure regular rings are called very strongly F -regular in [Has10, Def. 3.4]. Remark 2.4.2.
We have the following implications between the classes of singularities above. F -split regular F -pure regularstrongly F -regular regularFrobenius split F -pure split maps are pureDefinition [Has10, Lem. 3.8] local[Has10, Lem. 3.6] F -finite[Has10, Lem. 3.9] [Has10, Cor. 3.7] [DS16, Thm. 6.2.1]split maps are pure F -finite[HR76, Cor. 5.3] For the implication “regular ⇒ strongly F -regular,” the reference [DS16, Thm. 6.2.1] proves thatregular local rings are F -pure regular. This shows that “regular ⇒ strongly F -regular” in general,since a ring R of prime characteristic is strongly F -regular if and only if every localization at aprime ideal is F -pure regular [Has10, Lem. 3.6]. We expect that all excellent regular rings are F -pure regular (see Theorem 3.2.1 and Question 8.1).2.5. The gamma construction.
We give a brief account of the gamma construction of Hochsterand Huneke [HH94(a)].
Construction 2.5.1 [HH94(a), (6.7) and (6.11)] . Let ( R, m , κ ) be a Noetherian complete localring of prime characteristic p >
0. By the Cohen structure theorem, we may identify the residuefield κ with a coefficient field κ ⊆ R , and by Zorn’s lemma, there exists a p -basis Λ for κ , which isa subset Λ ⊆ κ such that κ = κ p (Λ), and such that for every finite subset Σ ⊆ Λ with s elements,we have [ κ p (Σ) : κ p ] = p s . See [Mat89, p. 202].Now for every cofinite subset Γ ⊆ Λ and for every integer e ≥
0, we consider the subfield κ Γ e = κ (cid:2) λ /p e (cid:3) λ ∈ Γ ⊆ κ perf0 RANKEYA DATTA AND TAKUMI MURAYAMA of a perfect closure κ perf of κ . These subfields of κ perf form an ascending chain, and we set R Γ := lim −→ e (cid:16) κ Γ e J R K (cid:17) , where κ Γ e J R K is the completion of κ Γ e ⊗ k R at the extended ideal m · ( κ Γ e ⊗ κ R ). Finally, if S is an R -algebra essentially of finite type, we set S Γ := S ⊗ R R Γ .The gamma construction satisfies the following properties that will be important for us. Lemma 2.5.2.
Fix notation as in Construction 2.5.1. ( i ) The rings R Γ and S Γ are Noetherian and F -finite. ( ii ) If S is reduced, then there exists a cofinite subset Γ ⊆ Λ such that S Γ is reduced for everycofinite subset Γ ⊆ Γ .Proof. ( i ) follows from [HH94(a), (6.11)] and the fact that algebras essentially of finite type over F -finite rings are F -finite. ( ii ) is [HH94(a), Lem. 6.13( a )]. (cid:3) F -purity, Frobenius splitting, and variants of strong F -regularity In [DM(b)], the authors constructed examples of excellent rings of prime characteristic that are F -pure but not Frobenius split, thereby answering the long-standing open question of whetherexcellent F -pure rings are always Frobenius split. The origins of this question can be traced toHochster and Roberts’s observation that F -purity and Frobenius splitting coincide for Noetherian F -finite rings, and to Auslander’s lemma (Lemma 2.3.3) which shows that the same result holdsfor complete local rings. Our goal in this section is to show that, despite the examples of [DM(b)], F -purity and Frobenius splitting coincide for most rings that arise in classical algebraic geometry(Theorem 3.1.1), even in a non- F -finite setting. In Theorem 3.1.1, we will also prove an analogousresult for the variants of strong F -regularity defined in Definition 2.4.1. This in turn will allow usto establish that most excellent regular rings arising in arithmetic and geometry are F -pure regular(Theorem 3.2.1). At the same time, we will show that the constructions considered in [DM(b)]are always F -pure regular, but not necessarily split F -regular (Theorem 3.3.4), giving the firstexamples of excellent regular local rings (even DVRs) where these two notions of singularity donot coincide. We will then end this section with a characterization of split F -regularity for DVRs(Proposition 3.4.1).Our explorations of the various notions of F -singularities in this section and Section 4 willmotivate the discussion of solidity of algebra extensions, which is a topic that we will formallyintroduce in Section 5 and study in considerable depth thereafter.3.1. F -singularities for rings of finite type over complete local rings. We show that forrings to which the gamma construction applies, the notions of F -purity and Frobenius splittingcoincide, as do the variants of strong F -regularity defined in Definition 2.4.1. Theorem 3.1.1.
Let S be a ring essentially of finite type over a Noetherian complete local ring ( R, m , κ ) of prime characteristic p > . If S is strongly F -regular (resp. F -pure), then S is split F -regular (resp. Frobenius split). Consequently, the notions ( a ) , ( b ) , and ( c ) (resp. ( d ) and ( e ) ) inDefinition 2.4.1 are equivalent for such S . This extends [Fed83, Lem. 1.2], which proves the case when S = R , that is, when S is completelocal. Proof.
By the gamma construction 2.5.1 (see also Lemma 2.5.2) and [Mur, Thm. 3.4], there exists afaithfully flat ring extension
R ֒ → R Γ such that S Γ := S ⊗ R R Γ is strongly F -regular (resp. F -pure) XCELLENCE, F -SINGULARITIES, AND SOLIDITY 11 and F -finite. By F -finiteness, the ring S Γ is split F -regular by [Has10, Lem. 3.9] (resp. F -split by[HR76, Cor. 5.3]). Now consider the commutative diagram R S F eS ∗ S F eS ∗ SR Γ S Γ F eS Γ ∗ S Γ F eS Γ ∗ S Γ F eS F eS ∗ ( −· c ) F eS Γ F eS Γ ∗ ( −· ( c ⊗ for every c ∈ S ◦ and every integer e >
0, where the left square is cocartesian by definition of S Γ .Note that if c ∈ S ◦ , then c ⊗ ∈ ( S Γ ) ◦ , since S → S Γ satisfies Going Down by faithful flatness[Mat89, Thm. 9.5], and so, minimal primes contract to minimal primes.Since the inclusion R ֒ → R Γ is faithfully flat, it is pure, and hence splits as an R -modulehomomorphism by Lemma 2.3.3. By base change, this implies the inclusion S ֒ → S Γ splits as an S -module homomorphism. For both split F -regularity and F -splitting of S , it then suffices to notethat if F eS Γ ∗ (cid:0) − · ( c ⊗ (cid:1) ◦ F eS Γ : S Γ −→ F eS Γ ∗ S Γ splits for some c ∈ S ◦ and for some e >
0, then restricting this splitting to F eS ∗ S and composingwith a splitting of S ֒ → S Γ gives a splitting of F eS ∗ ( − · c ) ◦ F eS by the commutativity of the diagramabove. Thus, S is split F -regular if it is strongly F -regular. The proof that S is Frobenius split ifit is F -pure follows by taking c = 1 and e = 1 in the aformentioned argument. (cid:3) As a corollary, we answer a question raised by Karl Schwede which inspired this paper.
Corollary 3.1.2.
Let k be an arbitrary field of prime characteristic p > . Let X be a normal k -variety and let D be a prime divisor on X . Then, the divisorial valuation ring O X,D is Frobeniussplit.Proof.
The local ring O X,D is essentially of finite type over k and is F -pure (since it is regular).Therefore O X,D is Frobenius split by Theorem 3.1.1. (cid:3)
Remark 3.1.3.
The equivalence of F -purity and Frobenius splitting for a Noetherian ring whichis essentially of finite type over a complete local ring of prime characteristic is in some sense thebest result one can hope for. Indeed, using fundamental constructions from rigid analytic geometry,the authors have shown recently that this equivalence fails even for excellent Henselian DVRs[DM(b), Cor. C]. We will have more to say about the constructions from [DM(b)] in Subsection3.3.3.2. On F -pure regularity of regular rings. In this subsection, we further explore the connec-tions between the various variants of strong F -regularity (see Definition 2.4.1), which we showedare all equivalent for rings essentially of finite type over a complete local ring in Theorem 3.1.1.Somewhat surprisingly, these connections are not completely understood even for the class of reg-ular rings. Regular rings are always strongly F -regular (Remark 2.4.2). However, even a regular local ring may not be split F -regular [DS18, Cor. 4.4], although such a local ring is always F -pureregular [DS16, Thm. 6.2.1]. The main result of this subsection is that most prime characteristicregular rings arising in arithmetic and geometric settings are F -pure regular even in the non-localsetting. Theorem 3.2.1.
Let ( R, m , κ ) be a Noetherian local G -ring of prime characteristic p > . If S isa regular ring that is essentially of finite type over R , then S is F -pure regular.Proof. By assumption, the canonical map R → b R is a regular homomorphism. Since S is essentiallyof finite type over R , the ring b R ⊗ R S is also Noetherian and the map S −→ b R ⊗ R S is regular [EGAIV , Prop. 6.8.3( iii )] and faithfully flat by base change. Thus, S b R := b R ⊗ R S isregular because S is regular [Mat89, Thm. 23.7( ii )]. The regular ring S b R is strongly F -regular byRemark 2.4.2, and hence F -pure regular by Theorem 3.1.1. Thus, S is F -pure regular by faithfullyflat descent of F -pure regularity [Has10, Lem. 3.14] (see also [DS16, Prop. 6.1.3( d )]). (cid:3) Remark 3.2.2. (1) Although there exist local G -rings that are not excellent, any regular ring that is essentiallyof finite type over a local G -ring is automatically excellent. Indeed, regular rings are Cohen–Macaulay, and hence universally catenary [Mat80, Thm. 33]. Moreover, since local G -rings are quasi-excellent (Example 2.2.2(1)) and since quasi-excellence is preserved underessentially of finite type maps (Example 2.2.2(2)), it follows that a regular ring that isessentially of finite type over a local G -ring is quasi-excellent and universally catenary,that is, such a ring is excellent. Thus, a large class of excellent regular rings of primecharacteristic are F -pure regular by Theorem 3.2.1, which suggests that the same statementholds for all excellent regular rings of prime characteristic (see Question 8.1). At the sametime, one should note that F -pure regularity of regular rings often holds even without theexcellence hypothesis. For example, any regular local ring is always F -pure regular (Remark2.4.2).(2) Quasi-excellence of rings essentially of finite type over local G -rings can be used to showthat if S is essentially of finite type over a local G -ring R of prime characteristic p > F -regular, then S is F -pure regular. This generalizes Theorem 3.2.1, and gives ananalogue of a similar result for F -rationality due to V´elez [V´el95, Thm. 3.8]. The only pointwhere the proof differs from the proof of Theorem 3.2.1 is that one has to show that thering S b R is strongly F -regular, and to then use Theorem 3.1.1 to conclude that S b R is F -pureregular. To show strong F -regularity of S b R , one can use the following lemma essentially dueto Hashimoto which establishes permanence properties of strong F -regularity and F -pureregularity under regular maps: Lemma . Let A → B be a regular map of Noetherian ringsof prime characteristic p > . Assume that A is F -pure regular (resp. strongly F -regular)and quasi-excellent, and that B is essentially of finite type over a Noetherian local G -ring(resp. is a G -ring). Then, B is F -pure regular (resp. strongly F -regular). One applies Lemma 3.2.2.1 to A = R and B = S in our situation and notation. Note thatHashimoto states Lemma 3.2.2.1 under the stronger assumption that A is excellent. Weneed A to be quasi-excellent since the ring S in the proof of Theorem 3.2.1 is quasi-excellent(see (1)), but not a priori excellent. Of course, once one proves that S b R is strongly F -regular,it then follows that S b R is Cohen–Macaulay by [HH94(a), Thm. 3.4( c )] and [Kaw02, Cor.1.2], and consequently, so is S by faithfully flat descent. This then implies S is excellent asin (1).The proof of Lemma 3.2.2.1 proceeds as in [Has10], with the following changes: • [Has10, Lem. 3.27] only uses the assumption that R is a G -ring in Hashimoto’s notation. • In the F -pure regular case, [Has10, Lem. 3.28] does not use the assumption that A isuniversally catenary. • In the strongly F -regular case, [Has10, Lem. 3.28] reduces to the F -pure regular caseby localization. Indeed if B is a G -ring, then as in the proof of [Has10, Lem. 3.28], forevery maximal ideal m of B , if n = A ∩ m , then B m is a local G -ring, A n is strongly F -regular hence also F -pure regular, and A n → B m is a regular map. Then B m is A flat homomorphism of Noetherian rings with geometrically regular fibers is called a regular map.
XCELLENCE, F -SINGULARITIES, AND SOLIDITY 13 F -pure regular by Lemma 3.2.2.1 for the F -pure regular case, and so, B is strongly F -regular since strong F -regularity is a local property [Has10, Lem. 3.6].We expect, though cannot prove, that an arbitrary excellent regular ring of prime characteristicis F -pure regular (see Question 8.1). However, using flatness of Frobenius for regular rings, we cangive a purely ideal-theoretic criterion for when a regular ring is F -pure regular. Proposition 3.2.3.
Let R be a ring of prime characteristic p > . Consider the following state-ments. ( i ) R is F -pure regular. ( ii ) For every nonzerodivisor c ∈ R , there exists e ∈ Z > such that for all maximal ideals m , wehave c / ∈ m [ p e ] . ( iii ) For every countable collection of maximal ideals { m e } e ∈ Z > , we have T e m [ p e ] e = 0 .We then have the following implications: ( i ) ( ii ) ( iii ) . R regular R domain Proof.
We first show ( i ) ⇒ ( ii ). Let c ∈ R be a nonzerodivisor and let e ∈ Z > such that the map λ ec : R −→ F eR ∗ R mapping 1 to c is pure. In particular, for every maximal ideal m of R , the map( R/ m ) ⊗ R λ ec : R/ m −→ F eR ∗ ( R/ m [ p e ] )is injective. But this latter map sends 1 + m to c + m [ p e ] , and so, c / ∈ m [ p e ] .We next show ( ii ) ⇒ ( i ) when R is regular. Suppose there exists e ∈ Z > satisfying ( ii ). Wewill show that the map λ ec : R −→ F eR ∗ R mapping 1 to c is pure. We first show that for every proper ideal I ⊆ R , the map( R/I ) ⊗ R λ ec : R/I −→ F eR ∗ ( R/I [ p e ] ) r + I r p e c + I [ p e ] is injective. We have r p e c + I [ p e ] = 0 = ⇒ r p e c ∈ I [ p e ] = ⇒ c ∈ ( I [ p e ] : r p e ) = ( I : r ) [ p e ] , where the last equality follows by flatness of the Frobenius map on R since R is regular [Kun69, Thm.2.1]. Since c / ∈ m [ p e ] for any maximal ideal m of R , it follows that ( I : r ) is not contained in anymaximal ideal of R . Thus, ( I : r ) = R , which shows r ∈ I . Consequently, ( R/I ) ⊗ R λ ec is injective,proving the claim.The injectivity of ( R/I ) ⊗ R λ ec for every proper ideal I ⊆ R implies that λ ec is pure by [Hoc77](see Remark 3.2.4) using the notion of approximately Gorenstein rings, but we give a direct proofas follows. To show that λ ec is a pure map of R -modules, it suffices to show that for every finitelygenerated R -module M , the map µ M : M −→ M ⊗ R F eR ∗ Rm m ⊗ c is injective. Recall that every finitely generated module over a Noetherian ring has a finite filtration M = M n ) M n − ) · · · ) M ) M = 0 by cyclic modules. We now proceed by induction on the length n of this filtration. If n = 1, then M is a nonzero cyclic module and µ M is injective by the claim. Otherwise, consider the short exactsequence 0 −→ M n − −→ M −→ MM n − −→ . Here,
M/M n − is cyclic, and M n − is a module with a filtration of length n −
1. By the flatness of F eR ∗ R [Kun69, Thm. 2.1], we get a commutative diagram of short exact sequences0 M n − M M/M n − M n − ⊗ R F eR ∗ R M ⊗ R F eR ∗ R ( M/M n − ) ⊗ R F eR ∗ R µ Mn − µ M µ M/Mn − The map µ M n − is injective by the induction hypothesis, and the map µ M/M n − is injective by ourclaim since M/M n − is cyclic. Then µ M must also be injective.Finally, the negation of ( ii ) implies the existence of a collection of maximal ideals { m e } e ∈ Z > such that T e m [ p e ] e = 0, and hence ( iii ) ⇒ ( ii ). The converse ( ii ) ⇒ ( iii ) holds when R is a domain,since a nonzerodivisor on R is precisely a nonzero element of R . (cid:3) Remark 3.2.4.
One can also use Hochster’s notion of approximately Gorenstein rings [Hoc77] togive another proof of ( ii ) ⇒ ( i ). Indeed, since R is regular it is Gorenstein and hence approximatelyGorenstein (see [Hoc77, Def. 1.3 and Cor. 2.2( b )]). Thus, the purity of ( R/I ) ⊗ R λ ec for every properideal I ⊆ R implies that λ ec is pure by [Hoc77, Thm. 2.6( iv )]. However, our argument is simpler. Corollary 3.2.5.
Let R be a regular ring of prime characteristic p > . Then R is F -pure regularif R is semi-local or R is a domain such that every nonzero element is contained in only finitelymany maximal ideals (for example, a Dedekind domain).Proof. Suppose R is semi-local. Let m , m , . . . , m n be the maximal ideals of R . If c ∈ R is anonzerodivisor, then c is a nonzerodivisor in each R m i . By Krull’s intersection theorem for localrings, there exists e ≫ c / ∈ m [ p e ] i R m i . Then c / ∈ m ei for each i , so R is F -pure regularby Proposition 3.2.3( ii ). Similarly, if R is a domain where every nonzero element is contained inonly finitely many maximal ideals, R satisfies Proposition 3.2.3( ii ) by an application of Krull’sintersection theorem for domains. (cid:3) Example 3.2.6.
As an application of Corollary 3.2.5, Nagata’s example of an infinite-dimensionalNoetherian regular domain [Nag75, App. A1, Ex. 1] is F -pure regular, since every nonzero elementof this ring is only contained in finitely many maximal ideals by construction.3.3. Excellent F -pure regular rings are not always split F -regular. Let R be a Noetherianring of prime characteristic p >
0. While the notions of F -pure regularity and split F -regularitycoincide if R is F -finite by [HR76, Cor. 5.2] or if R is essentially of finite type over a complete localring by Theorem 3.1.1, the two notions are not equivalent in general, even when R is a regular localring. The first such examples were obtained in [DS16, Ex. 4.5.1], where the first author and Smithconstructed DVRs in the function field of P F p that are not Frobenius split. However, the rings in[DS16, Ex. 4.5.1] are not excellent, and a general expectation since Hochster and Huneke’s work ontight closure [HH90; HH94(a)] is that the various notions of F -singularities are well-behaved onlyin the setting of excellent rings. The goal of this subsection is to use the constructions of [DM(b)]to give examples of excellent regular (local) rings for which F -pure regularity and split F -regularitydo not coincide. These are the first examples which illustrate that the two notions do not coincidefor excellent rings.Before we can state our main result, we need to introduce the rigid analytic analogue of apolynomial ring over a field and its local variant. XCELLENCE, F -SINGULARITIES, AND SOLIDITY 15 Definition 3.3.1.
Let ( k, | · | ) be a complete non-Archimedean field. For every positive integer n >
0, the
Tate algebra in n indeterminates over k is the k -subalgebra T n ( k ) := k { X , X , . . . , X n } := ( X ν ∈ Z n ≥ a ν X ν : a ν ∈ k and | a ν | → ν + ν + · · · + ν n → ∞ ) of the formal power series k J X , X , . . . , X n K in n indeterminates over k . Similarly, for every positiveinteger n >
0, the convergent power series ring in n indeterminates over k is the k -subalgebra K n ( k ) := k h X , X , . . . , X n i := X ν ∈ Z n ≥ a ν X ν : a ν ∈ k and there exists r , r , . . . , r n , M ∈ R > such that | a ν | r ν r ν · · · r ν n n ≤ M for all ν ∈ Z n ≥ of k J X , X , . . . , X n K .For the notion of a non-Archimedean field, which we will not define in this paper, we refer thereader to [Bos14, § Proposition 3.3.2.
Let ( k, | · | ) be a complete non-Archimedean field. Then for any integer n > ,we have the following: ( i ) T n ( k ) and K n ( k ) are Noetherian and regular of dimension n . ( ii ) T n ( k ) and K n ( k ) are excellent. ( iii ) K n ( k ) is local and Henselian.Indication of proof. For precise references for all these properties and additional ones, we refer thereader to [DM(b), Thms. 2.7 and 4.3]. (cid:3)
We also prove the following preliminary result.
Lemma 3.3.3.
Let ( k, | · | ) be a complete non-Archimedean field of characteristic p > such that [ k /p : k ] < ∞ . Then, T n ( k ) and K n ( k ) are F -finite for every integer n > . Although the statement for T n ( k ) is contained in the proof of [GV74, Thm. 12], we give an elemen-tary proof. Proof.
Fix a basis { a , a , . . . , a m } of F k ∗ k over k . Since k is complete, and F k ∗ k is an algebraicextension of k via the Frobenius map F k , there exists a unique extension of the norm | · | on k toa norm on F k ∗ k [Bos14, App. A, Thm. 3]. Denoting the norm on F k ∗ also by | · | , uniqueness and[Bos14, App. A, Thm. 1] then gives us that for x = x a + x a + · · · + x m a m ∈ F k ∗ k , | x | = max ≤ i ≤ m (cid:8) | x i | (cid:9) . (3.3.3.1)Now consider the formal power series ring A := k J X , X , . . . , X n K . Since k is F -finite, F A ∗ A is afree A -module with basis given by B := (cid:8) a i X β X β · · · X β n n : 1 ≤ i ≤ m, ≤ β j ≤ p − (cid:9) . We next claim that B is also a basis of F T n ∗ T n over T n . Indeed, let f = P ν ∈ Z n ≥ b ν X ν be anelement of F T n ∗ T n . Expressing f , considered as an element of A , in terms of the basis B , to get thecoefficient of a i X β X β · · · X β n n , we first collect all the terms in the expansion of f for which thepower of X i is congruent to β i modulo p , for all 1 ≤ i ≤ n . This gives us a power series of the form X ν ∈ Z n ≥ b pν + β X pν + β , where β := ( β , β , . . . , β n ). Note that we still have | b pν + β | → ν + ν + · · · + ν n → ∞ . Expressingeach b pν + β in terms of the basis { a , a , . . . , a m } of F k ∗ k over k , we find that the coefficients b i,pν + β of a i also satisfy | b i,pν + β | → ν + ν + · · · + ν n → ∞ by (3 . . . a i X β X β · · · X β n n in the expansion of f with respect to the basis B , which is precisely X ν ∈ Z n ≥ b i,pν + β X pν + β , (3.3.3.2)is also an element of T n by the discussion above. This implies that B is also a free T n -basis of F T n ∗ T n , and so, T n is F -finite.Finally, we show that B is also a free basis of F K n ∗ K n over K n . To see this, note that for f = X ν ∈ Z n ≥ b ν X ν ∈ F K n ∗ K n considered as an element of F ∗ ( k J X , X , . . . , X n K ), the coefficient of a i X β X β · · · X β n n is againgiven by (3.3.3.2), where b i,pν + β is the coefficient of a i when b pν + β is expressed in terms ofthe basis { a , a , . . . , a m } of F k ∗ k over k . Now choose r , r , . . . , r n , M ∈ R > such that for all | b ν | r ν r ν · · · r ν n n ≤ M for all ν ∈ Z n ≥ . Then | b i,pν + β | r pν + β r pν + β · · · r pν n + β n n ≤ | b pν + β | r pν + β r pν + β · · · r pν n + β n n ≤ M for all ν ∈ Z n ≥ , where the first inequality again follows by (3.3.3.1). Thus, the same choice of r , r , . . . , r n , M shows that P ν ∈ Z n ≥ b i,pν + β X pν + β is also an element of K n . Thus B is also a free K n -basis of F K n ∗ K n . (cid:3) With these preliminaries, we can now state and prove the main result of this subsection.
Theorem 3.3.4.
Let ( k, | · | ) be a complete non-Archimedean field of characteristic p > . Thenwe have the following: ( i ) For every integer n > , T n ( k ) and K n ( k ) are F -pure regular. ( ii ) There exists a complete non-Archimedean field ( k, | · | ) of characteristic p > such that forevery integer n > , T n ( k ) and K n ( k ) are not Frobenius split, and consequently, not split F -regular.Proof. We first prove ( i ). Note that since K n ( k ) is local by Proposition 3.3.2, it is F -pure regular byCorollary 3.2.5. To show F -pure regularity of T n ( k ), we first assume that k satisfies [ k /p : k ] < ∞ ,that is, k is F -finite. Since T n ( k ) is F -finite by Lemma 3.3.3 and is regular by Proposition 3.3.2, itfollows that T n is split F -regular, and hence F -pure regular, when k is F -finite.Now suppose ( k, | · | ) is an arbitrary complete non-Archimedean field of characteristic p >
0. Let ℓ be the completion of the algebraic closure of k , where the completion is taken with respect to themetric induced by the unique norm on the algebraic closure of k that extends the norm on k . Thenby Krasner’s lemma [Bos14, App. A, Lem. 6], ℓ is an algebraically closed non-Archimedean field,hence in particular, F -finite. Therefore T n ( ℓ ) is F -pure regular by the previous paragraph. Sincethe canonical inclusion T n ( k ) ֒ −→ T n ( ℓ )is faithfully flat by [Ber93, Lem. 2.1.2] and [Bos14, App. B, Prop. 5], and since F -pure regularitydescends under faithfully flat maps [Has10, Lem. 3.14] (see also [DS16, Prop. 6.1.3( d )]), it followsthat T n ( k ) is F -pure regular. This completes the proof of ( i ).( ii ) follows from [DM(b), Thm. A] and [DM(b), Rem. 5.5] by working over a complete non-Archimedean field ( k, | · | ) of characteristic p > k admits no nonzero continuous linearfunctionals k /p → k . An explicit example of such a field is constructed based on ideas of Gabberin [DM(b), Thm. 5.2]. (cid:3) As a consequence, we obtain examples of excellent Henselian F -pure regular local rings that arenot split F -regular. XCELLENCE, F -SINGULARITIES, AND SOLIDITY 17 Corollary 3.3.5.
There exists an excellent Henselian DVR of prime characteristic p > that isnot split F -regular.Proof. Choose a non-Archimedean field ( k, | · | ) as in Theorem 3.3.4( ii ). Then the convergent powerseries ring K ( k ) is an excellent Henselian discrete valuation ring by Proposition 3.3.2 that is noteven Frobenius split by Theorem 3.3.4( ii ), hence also not split F -regular. (cid:3) Remark 3.3.6.
As far as we are aware, it is not known if Hochster’s notion of strong F -regularitydefined via tight closure (Definition 2.4.1( c )) coincides with F -pure regularity for excellent Noe-therian rings (see Question 8.1). Note that if a counterexample exists, then it necessarily has to benon-local by [Has10, Lem. 3.8] (see also [DS16, Prop. 6.3.2]). We expect the two notions to not beequivalent in general even for regular rings, because we expect there to exist regular rings that donot satisfy the ideal-theoretic characterization of F -pure regularity given in Proposition 3.2.3. Notethat any regular ring is always strongly F -regular by [Has10, Lem. 3.6] because strong F -regularityis a local property and regularity localizes.In Lemma 3.3.3, we showed that if ( k, | · | ) is an F -finite non-Archimedean field, then T n ( k ) is F -finite, and hence, split F -regular for each integer n >
0. However, using some non-Archimedeanfunctional analysis, one can also show that Tate algebras are often split F -regular, even if k is not F -finite. To do so, we will use the following: Lemma 3.3.7.
Let ( k, | · | ) be a complete non-Archimedean field of characteristic p > . Let k perf be a perfect closure of k and let ℓ be the completion of k perf with respect to the unique norm on k perf that extends the norm on k . Then, ℓ is perfect.Proof. Let a ∈ ℓ be a nonzero element, and choose a sequence ( b n ) n of elements in k perf such that b n → a as n → ∞ . Such a sequence exists because k perf is dense in ℓ . Since k perf is perfect, thesequence ( b /pn ) n also consists of elements in k perf . Moreover, ( b /pn ) n is a Cauchy sequence because( b n ) n is Cauchy. Let a ′ = lim n →∞ b /pn be the limit of ( b /pn ) n in ℓ . Then ( a ′ ) p is the limit of ( b n ) n ,and so, ( a ′ ) p = a because sequences have unique limits in a metric space. Thus, any element of ℓ has a p -th root in ℓ , that is, ℓ is perfect. (cid:3) We now show that Tate algebras are often split F -regular. Proposition 3.3.8.
Let ( k, | · | ) be a complete non-Archimedean field of characteristic p > . Let k perf be a perfect closure of k and let ℓ be the completion of k perf with respect to the unique normon k perf that extends the norm on k . Then for every n > , T n ( k ) and K n ( k ) are split F -regular inthe following cases: ( i ) ( k, | · | ) is spherically complete. ( ii ) k /p has a dense k -subspace V which has a countable k -basis. ( iii ) | k × | is not discrete, and the norm on k perf is polar with respect to the norm on k . For the definitions of spherically complete non-Archimedean fields and polar norms, we refer thereader to [DM(b), Def. 2.13] and [DM(b), Def. 2.12] respectively.
Proof.
We first claim that if ( ii ) holds, then ℓ has a dense k -subspace which is countably generatedover k . Since k perf is dense in ℓ , to prove the claim it suffices to show that k perf has a dense k -subspace which is countably generated over k . Let S be a countable generating set for the dense k -subspace V of k /p . We will inductively show that for all e > k /p e has a dense k -subspacethat is countably generated over k . The case e = 1 holds by the assumption of ( ii ). For e >
1, bythe inductive hypothesis, let T e − be a countable subset of k /p e − that generates a dense k -linearsubspace. Define S /p e − := (cid:8) x p/p e : x ∈ S (cid:9) ⊆ k /p e . Then note that the k /p e − -linear subspace V e − of k /p e generated by S /p e − is dense in k /p e because the k -linear space V generated by S is dense in k /p . Now consider the countable set S e := (cid:8) xy ∈ k /p e : x ∈ T e − , y ∈ S /p e − (cid:9) . To prove the induction statement, it suffices to show that the k -linear subspace k · { S e } of k /p e generated by S e is dense in k /p e . For any z ∈ k /p e and any real number ǫ >
0, let B ǫ ( z ) ⊆ k /p e be the ball of radius ǫ centered at z . Since V e − is dense in k /p e , there exists y ∈ V e − ∩ B ǫ ( z ).Choose y , y , . . . , y n ∈ S /p e − and a , a , . . . , a n ∈ k /p e − such that y = a y + a y + · · · + a n y n . We may assume each y i = 0. Since k · { T e − } is dense in k /p e − , for all 1 ≤ i ≤ n , choose x i ∈ k · { T e − } such that | a i − x i | < ǫ/ | y i | . Then x y + x y + · · · + x n y n ∈ k · { S e } , and | z − x y + x y + · · · + x n y n |≤ max (cid:8) | z − y | , | y − x y + x y + · · · + x n y n | (cid:9) = max (cid:8) | z − y | , | ( a − x ) y + ( a − x ) y + · · · + ( a n − x n ) y n | (cid:9) ≤ max (cid:8) | z − y | , | ( a − x ) y | , | ( a − x ) y | , . . . , | ( a − x ) y | (cid:9) ≤ ǫ. Here the first and penultimate inequalities follow by the non-Archimedean triangle inequality. Thus, x y + x y + . . . + x n y n ∈ B ǫ ( z ), and since z and ǫ were chosen arbitrarily, this implies k h S e i isdense in k /p e , as desired. Finally, because k perf = S e> k /p e , it follows that k perf has a dense k -subspace that is countably generated over k by [PGS10, Thm. 4.2.13( iv )], proving the claim.We can now show that T n ( k ) and K n ( k ) are split F -regular. The ring T n ( ℓ ) is split F -regularby Lemma 3.3.3 because ℓ is F -finite by Lemma 3.3.7. Therefore, it suffices to show that for k satisfying any of the three conditions of this Proposition, the inclusion T n ( k ) ֒ → T n ( ℓ ) splits. Thisis because a direct summand of a split F -regular ring is split F -regular. However, if k satisfies anyof the conditions ( i ), ( ii ), or ( iii ), then variants of the Hahn-Banach theorem for normed spacesover R and C also hold for normed spaces over k ; see for example [DM(b), Thm. 2.15]. When k satisfies ( ii ), in order to apply [DM(b), Thm. 2.15] one needs the observation made in the aboveclaim that ℓ has a countably generated dense k -subspace. When k satisfies ( iii ), then in order toapply [DM(b), Thm. 2.15] one needs that the norm on ℓ is polar with respect to the norm on k .But this follows by the hypothesis of ( iii ) and [PGS10, Thm. 4.4.16( ii )] because the norm on k perf is polar and k perf is a dense k -subspace of ℓ . The upshot is that the identity map id k : k → k liftsto a continuous k -linear functional φ : ℓ −→ k when k satisfies ( i ), ( ii ), or ( iii ). By continuity, φ maps sequences in ℓ whose norms converge to 0to sequences in k whose norms converge to 0 [DM(b), Lem. 2.11]. Therefore, the induced map e φ : T n ( ℓ ) −→ T n ( k ) X ν ∈ Z n ≥ a ν X ν X ν ∈ Z n ≥ φ ( a ν ) X ν is a T n ( k )-linear splitting of T n ( k ) ֒ → T n ( ℓ ), as desired.Similarly, K n ( ℓ ) is split F -regular because ℓ is F -finite by Lemma 3.3.3. Now, for φ : ℓ → k asabove, we get an induced map φ ′ : K n ( ℓ ) −→ K n ( k ) X ν ∈ Z n ≥ a ν X ν X ν ∈ Z n ≥ φ ( a ν ) X ν . XCELLENCE, F -SINGULARITIES, AND SOLIDITY 19 The reason why φ ( a ν ) X ν is an element K n ( k ) is because by continuity of φ , there exists a realnumber B > a ∈ ℓ , | φ ( a ) | ≤ B | a | [DM(b), Lem. 2.11]. Hence if r , . . . , r n , M ∈ R > are such that | a ν | r ν . . . r ν n n ≤ M for all ν = ( ν , . . . , ν n ) ∈ Z ≥ , then | φ ( a ν ) | r ν . . . r ν n n ≤ BM for all ν ∈ Z n ≥ , which shows that P ν φ ( a ν ) X ν ∈ K n ( k ) by Definition 3.3.1. Thus, K n ( k ) is a directsummand of the split F -regular ring K n ( ℓ ), and consequently, also split F -regular. (cid:3) Remark 3.3.9.
Let ( k, | · | ) be a complete non-Archimedean field of characteristic p >
0. In[DM(b), Thms. 3.1 and 4.4], the authors show that a necessary and sufficient condition for T n ( k )and K n ( k ) to be Frobenius split is for there to exist a nonzero continuous k -linear functional k /p → k . However, we do not know if the existence of such a functional is also sufficient for split F -regularity of T n ( k ) and K n ( k ). As far as we can ascertain, it is not clear if the existence of anonzero continuous functional k /p → k implies the existence of a nonzero continuous functional ℓ → k , where ℓ = d k perf , as in the proof of Proposition 3.3.8. The main obstruction seems to be thatif ( M, | · | ) is a field equipped with a non-Archimedean valuation that is not complete, then theremay not be any nonzero continuous functionals c M → M . Indeed, this fails even when the valuegroup | M × | ≃ Z . For such discrete valuations, if M ◦ is the corresponding DVR, then c M is thefraction field of the M ◦◦ -adic completion d M ◦ of M ◦ . Here M ◦◦ is the principal maximal ideal of M ◦ .However, we show in Theorem 6.1.3 that there are no nonzero M ◦ -linear maps d M ◦ → M ◦ . Thisimplies that are no nonzero continuous M -linear maps c M → M . Indeed, assume for contradictionthat φ : c M → M is a nonzero continuous M -linear map. We may assume without loss of generalitythat φ (1) = 0. By [DM(b), Lem. 2.11], choose B ∈ R > such that for all x ∈ c M , we have | φ ( x ) | ≤ B | x | . Since the value group of M is nontrivial, choose a ∈ M × such that B < | a | . Then,the composition c M −· a −−→ c M φ −→ M is a nonzero continuous M -linear map (because c M −· a −−→ c M is an isomorphism), which induces a M ◦ -linear map of the corresponding valuation rings d M ◦ → M ◦ since for all x ∈ d M ◦ , | φ ( xa − ) | = | a − | · | φ ( x ) | ≤ | a − | · B | x | < | x | ≤ . Note that the induced map d M ◦ → M ◦ is nonzero because φ (1) = 0. But this contradicts Theorem6.1.3, as observed above. This remark motivates the interesting question of whether Frobeniussplitting of a regular ring is sufficient to imply split F -regularity (see Question 8.3), which to thebest of our knowledge is not known. However, in the next subsection we show that all Frobeniussplit DVRs are split F -regular (Proposition 3.4.1).3.4. Split F -regularity of discrete valuation rings. We have seen that even DVRs of primecharacteristic exhibit varied behvaior from the point of view of F -singularities despite being thesimplest examples of regular local rings. In particular, while DVRs of prime characteristic arealways F-pure regular, they are not always split F -regular, or even Frobenius split. Our goal in thissubsection will be to show that Frobenius splitting, or more generally, the existence of a nonzero p − -linear map is the only obstruction to a DVR of characteristic p being split F -regular. Thisremoves generic F -finiteness assumptions from a result of the first author and Smith [DS16, Cor.6.6.3]. Proposition 3.4.1.
Let ( R, m , κ ) be a DVR of prime characteristic p > . Then, the following areequivalent: ( i ) R has nonzero p − -linear map. ( ii ) R is Frobenius split. ( iii ) R is split F -regular.Proof. For the proof of ( i ) ⇒ ( ii ) we will use that fact that R is a principal ideal domain (PID).Let ϕ : F R ∗ R → R be a nonzero R -linear map. Since R is a PID, im( ϕ ) = aR , for some nonzero a ∈ R . Restricting the codomain of ϕ to im( ϕ ) = aR , and then using the fact that aR ≃ R as R -modules gives us an R -linear surjection e ϕ : F R ∗ R −→−→ R. Choose x ∈ F R ∗ R such that e ϕ ( x ) = 1. Then the composition F R ∗ R F R ∗ ( −· x ) −−−−−−→ F R ∗ R e ϕ −→ R is a Frobenius splitting of R . We have ( iii ) ⇒ ( i ) because any split F -regular ring is Frobeniussplit, and a Frobenius splitting is a nonzero p − -linear map.It remains to show ( ii ) ⇒ ( iii ). Let π be a generator of the maximal ideal m . It suffices to showthat for any integer n ≥
0, the ring R is Frobenius split along π n . Indeed, any x ∈ R is of the form x = uπ n for some unit u ∈ R × and n ≥
0. If R is Frobenius split along π n , then choose e > ϕ : F eR ∗ R → R such that ϕ ( π n ) = 1. Then the composition F eR ∗ F eR ∗ ( −· u − ) −−−−−−−→ F eR ∗ R ϕ −→ R maps x
1. Now to show that R is Frobenius split along π n , choose e ≫ p e − n > n ,and let ϕ e : F eR ∗ R → R be a splitting of F eR : R → F eR ∗ R (such a splitting exists for any e > R is Frobenius split). Then consider the composition φ : F eR ∗ R F eR ∗ ( −· π pe − n ) −−−−−−−−−→ F eR ∗ R ϕ e −→ R. We have φ ( π n ) = ϕ e ( π p e ) = πϕ e (1) = π . Thus, im( φ ), which is an ideal of R , is either πR or R . If im( φ ) = πR , then restricting the codomain of φ to πR and composing with the canonicalisomorphism πR ≃ R (that sends π to 1) shows that R is Frobenius split along π n . On the otherhand, if im( φ ) = R , this means that ϕ e ( π p e − n R ) = R . Choose b ∈ R such that ϕ e ( bπ p e − n ) = 1.Since p e − n > n by our choice of e , we see that π n divides bπ p e − n . Thus, let bπ p e − n = aπ n , forsome a ∈ R . The composition F eR ∗ R F eR ∗ ( −· a ) −−−−−−→ F eR ∗ R ϕ e −→ R maps π n ϕ e ( aπ n ) = ϕ e ( bπ p e − n ) = 1, which again shows that R is Frobenius split along π n . Thiscompletes the proof of ( ii ) ⇒ ( iii ). (cid:3) Remark 3.4.2. If R is generically F -finite, then [DS16, Cor. 6.6.3] shows that the equivalentconditions of Proposition 3.4.1 are further equivalent to R being excellent. Since there are excellentHenselian DVRs that are not split F -regular, we cannot hope to get rid of the generic F -finitenessassumption from [DS16, Cor. 6.6.3] to show that all excellent DVRs are split F -regular. However,we will show in Theorem 7.4.1 that any split F -regular DVR will be excellent, without any genericrestrictions on such a ring.4. F -purity vs. Frobenius splitting: a contrast in permanence properties In this section we continue exploring the closely related notions of F -purity and Frobenius split-ting from the perspective of permanence properties. Even though these two notions of singularitycoincide for most rings arising in algebro-geometric applications by Theorem 3.1.1, we will now il-lustrate some ways in which they differ. The general slogan is that F -purity is a more stable notionof singularity than Frobenius splitting. As evidence of the better stability properties of F -purity,we will show in this section that while F -purity descends under faithfully flat maps and ascendsunder regular maps, the same does not hold for Frobenius splitting in general.Recall that given a field k , a Noetherian k -algebra R is geometrically regular over k if, for everyfinite field extension L/k , the ring L ⊗ k R is regular. A map of rings R → S is regular if it is flatand has geometrically regular fibers. Implicit in this definition is the assertion that all the fibers of XCELLENCE, F -SINGULARITIES, AND SOLIDITY 21 R → S are Noetherian even if R and S are not. A map is (essentially) smooth if it is regular and(essentially) of finite type.Most notions of singularities such as reduced, normal, Cohen–Macaulay, Gorenstein ascend underregular maps, that is, if R satisfies a certain notion of singularity then so does S for a regular map R → S . We will use a characterization of regular maps in prime characteristic due to Radu andAndr´e to show that while Frobenius splitting ascends under essentially smooth maps, it does notascend under regular maps in general. However, F -purity always ascends under regular maps,providing evidence for the assertion that F -purity is a better behaved notion of singularity.4.1. Relative Frobenius and the Radu–Andr´e theorem.
The result of Radu and Andr´e thatwe will employ is a relative version of Kunz’s celebrated theorem which characterizes regularity ofa Noetherian ring in terms of flatness of the (absolute) Frobenius map [Kun69, Thm. 2.1]. To stateRadu and Andr´e’s result we need to introduce the relative Frobenius map [SGA5, Exp. XV, Def. 3],which is also known as the
Radu–Andr´e homomorphism in the commutative algebraic literature.
Definition 4.1.1.
Let ϕ : R → S be a homomorphism of rings of prime characteristic p >
0. Forevery integer e >
0, consider the cocartesian diagram
R F eR ∗ RS S ⊗ R F eR ∗ R F eS ∗ S F eR ϕ F eR ∗ ϕϕ ⊗ R F eR ∗ R id S ⊗ R F eR F eS F eS/R in the category of rings. The e -th Radu–Andr´e ring is the ring S ⊗ R F eR ∗ R , and the e -th relativeFrobenius homomorphism associated to ϕ is the ring homomorphism F eS/R : S ⊗ R F eR ∗ R F eS ∗ Ss ⊗ r s p e ϕ ( r )If e = 1, we denote F S/R by F S/R . We also sometimes denote F eS/R by F eϕ .Radu and Andr´e’s result is the following: Theorem 4.1.2 [Rad92, Thm. 4; And93, Thm. 1] . A homomorphism ϕ : R → S of Noetherianrings of prime characteristic p > is regular if and only if F S/R is faithfully flat.
One then recovers [Kun69, Thm. 2.1] upon applying Theorem 4.1.2 to the homomorphism F p → R , for a Noetherian ring R of prime characteristic p > Ascent under regular maps and faithfully flat descent.
Theorem 4.1.2 has the followingconsequence for Frobenius splitting and F -purity: Proposition 4.2.1.
Let ϕ : R → S be a regular map of Noetherian rings of prime characteristic p > . We have the following: ( i ) If R is F -pure, then S is F -pure. ( ii ) If ϕ is essentially smooth and R is Frobenius split, then S is Frobenius split. ( iii ) Suppose R and S are arbitrary rings (not necessarily Noetherian). If F S/R is an isomor-phism and R is Frobenius split, then S is Frobenius split. ( iv ) If S is a filtered colimit of ´etale R -algebras and R is Frobenius split, then S is Frobeniussplit. Thus, the (strict) Henselization of a Frobenius split local ring is Frobenius split. ( v ) There exists a regular map ϕ such that R and S are excellent local, R is Frobenius split, but S is not Frobenius split.Proof. Since ϕ is a regular map, F S/R is faithfully flat by Theorem 4.1.2. It is well-known thatfaithfully flat maps are pure [BouCA, Ch. I, §
3, n o
5, Prop. 9], and so, F S/R is a pure map. Withthis preliminary observation, we can now begin the proof of the Proposition.( i ) Since purity is preserved under base change, F -purity of R implies that the map ϕ ⊗ R F R ∗ R is pure. Consequently, the composition F S = F S/R ◦ ( ϕ ⊗ R F R )is also pure, that is, S is F -pure. See also [Has10, Prop. 2.4.4].( ii ) If ϕ is essentially of finite type, then F S/R is a finite map. Indeed, it is well-known thatthe relative Frobenius of a finite type map is always finite. Now, if S is a localization of a finitetype R -algebra B , then it is straightforward to check that F S/R is a localization of the finite map F B/R , and consequently also finite. Thus, if ϕ is essentially smooth, then F S/R is a finite map ofNoetherian rings which is also pure. It follows by [HR76, Cor. 5.2] that F S/R splits. Moreover,since R is Frobenius split, the map id S ⊗ R F R splits by base change. Therefore the composition F S = F S/R ◦ (id S ⊗ R F R ) also splits, that is, S is Frobenius split.( iii ) If the relative Frobenius is an isomorphism, then R SF R ∗ R F S ∗ S F R F S is a pushout square and the assertion follows.( iv ) If R → A is an ´etale map of rings of characteristic p >
0, then the relative Frobenius F R/A is always an isomorphism [SGA5, Exp. XV, Prop. 1( c )]. Thus, if S is a filtered colimit of ´etale R -algebras, then F S/R is a filtered colimit of relative Frobenii that are all isomorphisms. Consequently, F S/R is also an isomorphism. Then, Frobenius splitting is preserved under filtered colimits of ´etalemaps by what we just proved. Frobenius splitting is preserved by (strict) Henselizations becausethe (strict) Henselization of a local ring is a filtered colimit of ´etale maps.( v ) Choose a non-Archimedean field ( k, | · | ) of prime characteristic p > K ( k ) is not Frobenius split; see [DM(b), Rem. 5.5]. Then K ( k ) is anexcellent Henselian DVR (Proposition 3.3.2), the unique ring homomorphism F p → K ( k ) is regular[EGAIV , Prop. 6.7.7], and F p is Frobenius split while K ( k ) is not. (cid:3) Remark 4.2.2. (1) For Proposition 4.2.1( v ), one can construct examples of R and S even in the function fieldof P F p if one relaxes the requirement for S to be excellent. Indeed, there exist discretevaluation rings in the function field of P F p that are not excellent [DS18, Cor. 4.4], andconsequently also not Frobenius split [DS18, Thm. 4.1]. Take such a discrete valuation ring S . Then the map F p → S is regular, and F p is Frobenius split, but S itself is not.(2) Proposition 4.2.1( i ) holds more generally for any ring homomorphism ϕ : R → S of Noe-therian rings for which the relative Frobenius F S/R is a pure map [Has10, Prop. 2.4.4]. Suchring homomorphisms are called F -pure homomorphisms by Hashimoto [Has10, (2.3)] andshould be thought of as a relative version of the notion of F -purity of a ring [Has10, Prop.2.4.3].(3) The key assumption of Proposition 4.2.1( iii ) is that the relative Frobenius of R → S isan isomorphism. Such maps are usually called relatively perfect in the literature, becausethey are a relative version of perfect rings. Relative perfection plays an important rolein the proofs of various invariance properties in the theory of F -singularities under ´etale XCELLENCE, F -SINGULARITIES, AND SOLIDITY 23 base change. Thus, it is natural to wonder how close the notion of relative perfection isto that of being ´etale. If R → S is relatively perfect, then R [ S p ] = S , which immediatelyimplies Ω S/R = 0 (since d ( s p ) = pd ( s ) = 0). That is, a relatively perfect map is formallyunramified. Moreover, the set { } is a p -basis of S over R in the sense of [EGAIV , Ch. 0,D´ef. 21.1.9]. Then injectivity of F S/R implies that R → S is formally smooth by [EGAIV ,Ch. 0, Thm. 21.2.7]. In other words, a relatively perfect map is formally ´etale. In fact,a flat relatively perfect map R → S satisfies the stronger property that the cotangentcomplex L S/R is acyclic [Bha19, Prop. 3.12] (formally ´etale is equivalent to acyclicity of thecotangent complex in degrees 0 and −
1; see [Stacks, Tags 0D11, 08RB, 06E5]). Nevertheless,a relatively perfect map need not be the filtered colimit of ´etale maps. For example, F S/R is always an isomorphism if R and S are perfect. However, if S is not a flat R -algebra, then S cannot be a filtered colimit of ´etale (hence flat) R -algebras. Failure of flatness is not theonly obstruction that prevents a relatively perfect map from being a filtered colimit of ´etalemaps. Indeed, if R → S is an injective map of domains that is a filtered colimit of ´etalemaps, then the induced map on fraction fields has to be algebraic. Thus, F p ֒ → F p [ T ] perf is a flat relatively perfect map that cannot be a filtered colimit of ´etale maps.(4) Relative perfection also shows that if S is a filtered colimit of ´etale R -algebras and R is F -finite, then S is also F -finite. Thus, (strict) Henselizations of F -finite rings are F -finite.Finally, while it is well-known that F -purity descends under faithfully flat maps, we show thatFrobenius splitting does not. Proposition 4.2.3.
Let ϕ : R → S be a faithfully flat map of rings of prime characteristic p > .If S is F -pure, then so is R . However, Frobenius splitting does not descend under faithfully flatmaps even when R and S are both excellent, Henselian and regular.Proof. The proof of the first statement is in [HR76, Prop. 5.13], but we reprove it here for conve-nience. Note that ϕ is pure since it is faithfully flat (Remark 2.3.2(4)). Therefore the composition F S ◦ ϕ is pure as a map of R -modules if S is F -pure. However, F S ◦ ϕ = ϕ ◦ F R , and so, F R is pure, that is, R is F -pure. This shows that F -purity satisfies faithfully flat descent.For the second statement, choose a non-Archimedean field ( k, | ·| ) such that the convergent powerseries ring K ( k ) is not Frobenius split as in [DM(b), Rem. 5.5]. Take R := K ( k ), and considerthe completion S := b R = k J X K of R with respect to the maximal ideal ( X ). Then, the canonicalcompletion map R → S is regular (since K ( k ) is excellent) and faithfully flat, and R is notFrobenius split even though S is. Thus, Frobenius splitting does not satisfy faithfully flat descenteven for faithfully flat maps between excellent Henselian regular rings of prime characteristic. (cid:3) Remark 4.2.4.
While the Henselization of a Frobenius split local ring ( R, m ) of prime characteristicis always Frobenius split, as far as we are aware the converse is not known except in special cases.For one such special case, suppose R is a generically F -finite Noetherian normal local domain suchthat R h is Frobenius split. Then R h is also a Noetherian normal local domain [Mat89, Cor. to Thm.23.9]. Let K be the fraction field of R and K h be the fraction field of R h . Since the field extension K ⊆ K h is separable algebraic, the map K → K h is relatively perfect. The diagram K K h F K ∗ K F K h ∗ K hF K F Kh We thank Alapan Mukhopadhyay for this observation. is therefore cocartesian, and hence K h is F -finite by base change (see also [Has15, Lem. 2(7)]).Now, because R h is Frobenius split and generically F -finite, it follows that R h is also F -finite by[DS18, Thm. 3.2]. This implies that R is also F -finite. Indeed, since R → R h is relatively perfect,it follows by faithfully flat descent of module finiteness that if F R h ∗ R h = F R ∗ R ⊗ R R h is a finite R h -module, then F R ∗ R is a finite R -module (see also [Has15, Cor. 22]). Thus, R is F -finite andalso F -pure by Proposition 4.2.3. But an F -finite and F -pure ring is Frobenius split by [HR76, Cor.5.2].If ( R, m ) is a non-Henselian DVR such that R h is Frobenius split, then the only obstruction toFrobenius splitting of R is for the latter to possess a nonzero p − -linear map by Theorem 3.4.1.Note that if Hom R ( R h , R ) is nontrivial, then there would exist a nonzero R -linear map φ : R h → R such that φ (1) = 0. The composition F R ∗ R ֒ −→ F R h ∗ R h F -splitting −−−−−−→ R h φ −→ R would consequently give a nonzero p − -linear map of R , allowing us to conclude that R is Frobeniussplit. However, we will show soon that Hom R ( R h , R ) is trivial for any non-Henselian Noetherianlocal domain R for which R h is a domain (Theorem 6.1.1). Therefore, the question of whetherFrobenius splitting descends over Henselizations seems nontrivial even for DVRs.5. F -solidity and non- F -finite excellent rings An underlying theme of the previous sections was an exploration of when nonzero p − e -linearmaps exist for excellent rings of prime characteristic. The existence of a nonzero p − e -linear map isa special instance of a more general phenomenon studied by Hochster in [Hoc94], with the goal offormulating a characteristic independent closure operation that satisfies properties similar to tightclosure. Hochster called this notion a solid module, and our goal in this section, and the rest of thepaper, is to revisit his notion of solidity from the point of view of the excellence condition. However,our focus is not solid closure, for which we refer the reader to [Hoc94].5.1. Solid modules and algebras.
We first define solid modules and algebras.
Definition 5.1.1 [Hoc94, Def. 1.1] . Let R be a ring. An R -module M is solid if there exists anonzero R -linear map from M → R . An R -algebra S is solid if it is solid as an R -module. Remark 5.1.2. (1) If R is a ring of prime characteristic, then F R ∗ R is a solid R -algebra precisely when R admits a nonzero p − -linear map.(2) A simple, but surprisingly useful, observation is that an R -algebra S is solid if and only ifthere exists an R -linear map ϕ : S → R such that ϕ (1) = 0. Indeed, the existence of such amap implies R -solidity of S by definition. Conversely, if S is a solid R -algebra, then chooseany R -linear map φ : S → R such that φ ( s ) = 0, for some s ∈ S . Then the composition S −· s −−→ S φ −→ R (5.1.2.1)is a map that sends 1 to φ ( s ) = 0. Moreover, the same argument shows that the ideal P φ im( φ ) of R , as φ ranges over all elements of Hom R ( S, R ), is the same as the ideal of I S/R := { φ (1) : φ ∈ Hom R ( S, R ) } . Indeed, the fact that I S/R is an ideal follows fromthe observation that I S/R is the image of the R -linear map Hom R ( S, R ) → R given byevaluation at 1. Furthermore, the equality P φ im( φ ) = I S/R follows because { φ ( s ) : φ ∈ Hom R ( S, R ) , s ∈ S } is a generating set for P φ im( φ ), and each φ ( s ) can be rewritten as theimage of 1 under the R -linear map S → R from (5.1.2.1). XCELLENCE, F -SINGULARITIES, AND SOLIDITY 25 (3) If T is a solid R -algebra and the map R → T factors through an R -algebra S , then S is alsoa solid R -algebra. Indeed, choose an R -linear map ϕ : T → R such that ϕ (1) = 0. Then,the composition S → T ϕ → R is a nonzero R -linear map. Example 5.1.3.
Suppose
R ֒ → S is a finite extension of rings (not necessarily Noetherian) suchthat R is reduced and has finitely many minimal primes. Then, S is a solid R -algebra. Indeed, if K is the total ring of fractions of R and L := K ⊗ R S , then one can choose a splitting L → K andrestrict it to S , giving a nonzero R -linear map S → K . Note such a splitting exists because K isa finite direct product of fields by [Stacks, Tag 02LX] and the fact that the nonzerodivisors of areduced ring are precisely the elements contained in minimal primes. Since S is finitely generatedas an R -module, the image of S → K is contained in the R -submodule of K generated by 1 /f , forsome nonzerodivisor f ∈ R . Then, the composition S −→ K −· f −−→ K is an R -linear map that sends 1 to f = 0, and whose image lands inside R . Thus, S is a solid R -algebra.It is natural to ask how the notion of solidity behaves under compositions. While we do notknow the answer to this question in general, the following special case will be useful in the sequel. Lemma 5.1.4.
Let R → S be an extension of domains such that Frac( S ) is an algebraic extensionof Frac( R ) . If S is a solid R -algebra and T is a solid S -algebra, then T is a solid R -algebra.Proof. Since T is S -solid, choose an S -linear map ϕ : T → S such that b := ϕ (1) = 0. By ourhypothesis on the extension R → S , it follows by [DS18, Prop. 3.7] that the canonical map S −→ Hom R (cid:0) Hom R ( S, R ) , R (cid:1) is injective. Since b is a nonzero element of S , this means that we can find some φ ∈ Hom R ( S, R )such that φ ( b ) = 0. Then the composition T ϕ −→ S φ −→ R is a nonzero R -linear map because φ ◦ ϕ (1) = φ ( b ) = 0. In other words, T is a solid R -algebra. (cid:3) Remark 5.1.5.
Analyzing the proof of [DS18, Prop. 3.7], one can show that the injectivity of thecanonical map S → Hom R (Hom R ( S, R ) , R ) holds as long as S is R -solid and the extension R → S has the property that for any nonzero ideal J of S , the contraction J ∩ R is also a nonzero idealof R . Thus, solidity is preserved under composition as long as R → S is an extension of domainsthat satisfies this ideal contraction property.5.2. F -solidity. We now define and study a special case of a solid algebra in prime characteristic,which we call F -solidity. Definition 5.2.1.
Let R be a ring of prime characteristic p >
0. We say R is F -solid if F R ∗ R is asolid R -algebra.In other words, R is F -solid precisely when it admits a nonzero p − -linear map. We make a fewpreliminary observations about F -solidity. Remark 5.2.2.
Let R be a ring of prime characteristic p > R is Frobenius split, then R is F -solid. This does not need R to be Noetherian.(2) If R is a reduced F -finite Noetherian ring, then R is F -solid. Indeed, let K be thetotal quotient ring of R . Then, K is a finite direct product of F -finite fields. Thus,Hom K ( F K ∗ K, K ) = 0. But since F R ∗ R is a finitely presented R -module and K is a flat R -module, we have Hom K ( F K ∗ K, K ) = Hom R ( F R ∗ R, R ) ⊗ R K. This implies that Hom R ( F R ∗ R, R ) = 0, or equivalently, that R is F -solid.(3) If R is a domain, then R is F -solid if and only if for all (equivalently, for some) e > F eR ∗ R is a solid R -module. This is well-known, but we provide a brief justification. Note thatsince F eR ∗ R is always an F R ∗ R -algebra for any e > R -solidity of F eR ∗ R implies R -solidityof F R ∗ R by Remark 5.1.2(3). Conversely, suppose F R ∗ R is R -solid and let φ : F R ∗ R → R bea map such that c := φ (1) = 0. Assume by induction that there exists a nonzero R -linearmap ϕ : F eR ∗ R → R such that ϕ (1) = 0. Then, the composition F e +1 R ∗ R F eR ∗ φ −−−→ F eR ∗ R ϕ −→ R maps c ( p e − p to cϕ (1) = 0, showing that F e +1 R ∗ R is R -solid and completing the proof byinduction. Note that we use R is a domain in the step where we conclude cϕ (1) = 0 fromthe fact that both c and ϕ (1) are nonzero.(4) Suppose R → S is a finite extension of Noetherian domains. Then, R is F -solid if and onlyif S is F -solid. For the forward implication, since R is Noetherian, by [Hoc94, Cor. 2.3]it suffices to show that F S ∗ S is a solid R -algebra. We will show that F eS ∗ S is R -solid forany e >
0. Consider the composition R → F eR ∗ R → F eS ∗ S , where the first map is the e -thiterate of the Frobenius on R and the second map is just restriction of scalars of the finiteextension R → S . Since F eS ∗ S is a finite F eR ∗ R -algebra, we know that F eS ∗ S is a solid F eR ∗ R -algebra by Example 5.1.3. Moreover, R → F eR ∗ R is generically integral and F eR ∗ R is a solid R -algebra by (3) since R is F -solid. Therefore by Lemma 5.1.4 we conclude that F eS ∗ S is asolid R -algebra, proving the forward implication. Conversely, suppose S is F -solid. Since R → S is a finite extension, S is R -solid by Example 5.1.3. Then F S ∗ S is a solid R -algebraby Lemma 5.1.4. Since R −→ S F S −→ F S ∗ S factors through F R ∗ R , it follows that F R ∗ R is a solid R -algebra (Remark 5.1.2(3)), that is, R is F -solid.(5) Let R be a Noetherian domain. Since polynomial algebras or power series rings over fieldsare F -solid (they are even Frobenius split), it follows by part (3) of this Remark that R is F -solid either when R is of finite type over a field (via Noether normalization), or when R is a complete local domain (by Cohen’s structure theorem). If R is finite type extensionof an F -solid domain A , then similar reasoning shows that there exists a nonzero element a ∈ A such that R a is F -solid by [Stacks, Tag 07NA]. Here, we use the fact that F -solidityof A is preserved under localization, and also that a polynomial ring over an F -solid ring is F -solid. However, it is unclear whether a ring R essentially of finite type over a completelocal domain is F -solid using this line of reasoning. We will show that such rings are indeed F -solid via the gamma construction 2.5.1 (see Theorem 5.3.1 below).(6) If R is an F -solid domain, then any localization of R is also F -solid. This is a simpleapplication of the fact that for a multiplicative set S ⊆ R , we have S − R ⊗ R F R ∗ R ≃ F S − R ∗ S − R . However, F -solidity is not a local property, in the sense that if R is a ring ofprime characteristic whose local rings are all F -solid, it is not necessarily the case that R is F -solid. This follows from the failure of F -finiteness being a local property (see Example2.2.2(4)). Indeed, choose a Noetherian domain R of prime characteristic p > R p is F -finite for all prime ideals p , but R itself is not as in [DI]. Then the fraction field of R is F -finite, and so, if R was F -solid, then [DS18, Thm. 3.2] would imply that R is F -finite,a contradiction.5.3. F -solidity for rings of finite type over complete local rings. In Remark 5.2.2(5) weobserved that rings to which one can apply a Noether normalization type result (such as finite typealgebras over fields or complete local rings) are often F -solid in prime characteristic. However, thisnormalization technique does not yield F -solidity for algebras that are essentially of finite type over XCELLENCE, F -SINGULARITIES, AND SOLIDITY 27 a complete local ring, which, as we saw in Section 3, often behave like Noetherian F -finite rings.Drawing inspiration from the F -solidity of reduced F -finite Noetherian rings, the main result ofthis section shows that F -solidity often holds even in a non- F -finite setting. Theorem 5.3.1.
Let S be a reduced ring which is essentially of finite type over a complete localring R of prime characteristic p > . Then, S is F -solid. The proof of this result proceeds by reducing to the F -finite case via an argument similar to theone in Theorem 3.1.1, and by using a descent result on F -solidity which we establish first. Lemma 5.3.2.
Let ϕ : R → S be an injective homomorphism of rings of prime characteristic p > ,and suppose S is a solid R -algebra. ( i ) If R and S are domains, ϕ is injective, and the induced map Frac( R ) → Frac( S ) is algebraic,then R is F -solid. ( ii ) If ϕ is purely inseparable, S is reduced, and there exists an R -linear map g : S → R suchthat g (1) is a nonzerodivisor, then R is F -solid.Proof. Throughout this proof, let φ : F S ∗ S → S be an S -linear map such that φ (1) =: c = 0 . ( i ) Since Frac( R ) ֒ → Frac( S ) is algebraic, clearing denominators from an equation of algebraicdependence of c over Frac( R ), it follows that there exists a nonzero s ∈ S such that sc ∈ R . Let ℓ s : S → S denote left multiplication by s . Then ℓ s ◦ φ maps 1 to sc ∈ R r { } . Since S is a solid R -algebra, choose an R -linear map f : S → R such that f (1) = 0. Then the composition F R ∗ R F ∗ ϕ −−→ F S ∗ S ℓ s ◦ φ −−−→ S f −→ R maps 1 to scf (1), and the latter is a nonzero element of R because the elements sc and f (1) arenonzero, and R is a domain. This shows R is F -solid.( ii ) We may assume without loss of generality that ϕ is an inclusion. Since ϕ is purely inseparable,choose e > c p e ∈ R . Note that c p e = 0 because S is reduced and c is a nonzero elementby choice. Let ℓ c pe − denote left multiplication by c p e − . The composition F R ∗ R ֒ −→ F S ∗ S ℓ cpe − ◦ φ −−−−−→ S g −→ R maps 1 to c p e g (1), which is a nonzero element of R because g (1) is a nonzerodivisor on R byhypothesis. This completes the proof. (cid:3) We can now prove Theorem 5.3.1.
Proof of Theorem 5.3.1.
Suppose S is essentially of finite type over the complete Noetherian lo-cal ring ( R, m ). Again, using the gamma construction 2.5.1, there exists a faithfully flat purelyinseparable ring extension R ֒ → R Γ such that S Γ := S ⊗ R R Γ is F -finite and reduced (Lemma2.5.2( ii )). In particular, S Γ is then F -solid by Remark 5.2.2(2). Since R is complete, the pureextension R ֒ → R Γ splits (Lemma 2.3.3), hence so does the purely inseparable extension S ֒ → S Γ .By F -solidity of S Γ and an application of Lemma 5.3.2( ii ), it follows that S is also F -solid. (cid:3) Remark 5.3.3.
In order to establish F -solidity of domains that are essentially of finite type overany field of prime characteristic, one can avoid the gamma construction approach of Theorem 5.3.1.Namely, suppose k is a field of characteristic p > R is a domain that is essentially of finitetype over k . Let T be a finite type k -algebra and S ⊂ T a multiplicative set such that R = S − T .Note that T can be chosen to be a domain. Indeed, let P be a prime ideal of T disjoint from S such that S − P = (0) (such a P exists because R is a domain). Since P is finitely generated, thereexists some f ∈ S such that P T f = (0). Upon replacing T by T f , we can assume T is a domain.Then T is F -solid by Remark 5.2.2(5), and consequently, R is F -solid by Remark 5.2.2(6). This argument provides an alternate proof of Frobenius splitting of divisorial valuation rings (Corollary3.1.2). Namely, if R is a divisorial valuation ring, then R is essentially of finite type over a fieldand hence F -solid. Then R is Frobenius split by Proposition 3.4.1.5.4. Excellent rings are not always F -solid. A natural question is whether Theorem 5.3.1 canbe generalized to reduced excellent rings, or at least, to reduced excellent rings that are of finitetype over an an excellent local ring. This turns out to be false, as shown in the next result. Theseare the first excellent examples of prime characteristic domains that are not F -solid. Proposition 5.4.1.
There exists a complete non-Archimedean field ( k, | · | ) of characteristic p > such that for each integer n > , we have the following: ( i ) The Tate algebra T n ( k ) is an excellent regular ring of Krull dimension n that is not F -solid. ( ii ) Any k -affinoid domain of Krull dimension n is not F -solid. ( iii ) The convergent power series ring K n ( k ) is an excellent Henselian regular local ring of Krulldimension n that is not F -solid. Here by a k -affinoid domain we mean an integral domain that is the quotient of some Tatealgebra. These are the rigid analytic analogues of coordinate rings of affine varieties. Proof. ( i ) follows from [DM(b), Thm. A] and ( iii ) follows from [DM(b), Rem. 5.5] (see also [DM(b),Thm. 4.4]).For ( ii ), choose a complete non-Archimedean field ( k, | · | ) for which ( i ) holds, and suppose that A is an k -affinoid domain of Krull dimension n >
0. Then by the rigid analytic analogue of Noethernormalization [Bos14, Cor. 2.2/11], there exists a module-finite ring extension T n ( k ) ֒ → A . Since T n ( k ) is not F -solid by ( i ), it follows that A is not F -solid by Remark 5.2.2(4). (cid:3) Remark 5.4.2. (1) Examples of non- F -solid regular local rings can be constructed even in the function field of P F p . But any such ring will not be excellent; see [DS18, § k be as in Proposition 5.4.1. Then the completion of the local ring K ( k ) at its maximalideal ( X ) is the power series ring k J X K , which is F -solid. This illustrates that F -soliditydoes not descend over faithfully flat maps. The same example also shows that F -soliditydoes not ascend over regular maps, since F p → K ( k ) is regular, F p is F -solid and K ( k )is not.Despite the examples obtained in Proposition 5.4.1, affinoid domains of prime characteristic areoften F -solid. Proposition 5.4.3.
Let ( k, | · | ) be a complete non-Archimedean field that satisfies any of thefollowing additional properties: ( i ) ( k, | · | ) is spherically complete. ( ii ) k /p has a dense k -subspace V which has a countable k -basis, hence in particular if [ k /p : k ] < ∞ . ( iii ) | k × | is not discrete, and the norm on k /p is polar with respect to the norm on k .Then any k -affinoid domain is F -solid.Proof. Let A be a k -affinoid domain. If A has Krull dimension zero, then A = k and there is nothingto prove since any field of prime characteristic is F -solid. If A has Krull dimension n >
0, then A is a module-finite extension of T n ( k ) as in the proof of Proposition 5.4.1( ii ). When k satisfies anyof the conditions of this Proposition, then T n ( k ) is Frobenius split by [DM(b), Cor. D], hence also F -solid. Then A is F -solid by Remark 5.2.2(4). (cid:3) XCELLENCE, F -SINGULARITIES, AND SOLIDITY 29 Remark 5.4.4.
The observant reader will notice that the hypothesis of Proposition 5.4.3( iii ) isweaker than the hypothesis of Proposition 3.3.8( iii ). This is because to show F -solidity of k -affinoid domains one only needs Tate algebras to be F -solid, which is a weaker requirement thansplit F -regularity. 6. Some obstructions to solidity
In the previous section, we showed that a reduced ring R which is essentially of finite type over acomplete local Noetherian ring of prime characteristic p > F -solid (Theorem 5.3.1). The proofproceeds by constructing an F -finite solid Noetherian R -algebra R Γ , and then using non-trivial p − e -linear maps on R Γ and its solidity as an R -algebra to produce non-trivial p − e -linear maps on R .At the same time, we also exhibited examples of excellent Henselian regular local rings that are not F -solid (Proposition 5.4.1). The goal of this section is to highlight some obstructions to F -solidityof excellent local rings which will more systematically explain why the gamma construction haslittle hope of producing examples of F -solid excellent rings beyond those that are essentially offinite type over a complete local ring.6.1. Henselizations and completions are not solid.
Suppose S is a reduced ring which isessentially of finite type over an excellent local ring ( R, m ) of prime characteristic p > p − e -linear maps on the change of base ring S b R := b R ⊗ R S can help produce nonzero p − e -linear maps on R . As we now demonstrate, the maindifficulty with this approach is that S b R is rarely a solid S -algebra. Thus, descent type argumentsin the spirit of Lemma 5.3.2, crucial in the proof of Theorem 5.3.1, will almost never work.The following simple observation will be useful: if R → S → T are ring maps such that T isa solid R -algebra, then S is also a solid R -algebra (Remark 5.1.2(3)). Said differently, if S is nota solid R -algebra, then T is not a solid R -algebra. We will use this observation to show that thecompletion of a local ring is rarely solid. In fact, we first prove the following stronger result forNoetherian local rings that are not Henselian. Theorem 6.1.1.
Let ( R, m ) be a Noetherian local ring of arbitrary characteristic such that theHenselization R h is a domain (for example, if R is normal). Assume R is not Henselian. Then,we have the following: ( i ) If ϕ : R → S is an essentially ´etale extension of domains, then S is a solid R -algebra if andonly if ϕ is finite ´etale. ( ii ) The Henselization R h is not a solid R -algebra. ( iii ) If ( R, m ) → ( S, n ) is a local ring homomorphism such that S is Henselian (for example, if S = R sh or S = b R ), then S is not solid R -algebra. The main ingredient in the proof of Theorem 6.1.1 is the following observation of Smith and thefirst author:
Proposition 6.1.2 [DS18, Prop. 3.7] . Let ϕ : R → S be an injective homomorphism of Noetheriandomains which is generically finite. If S is a solid R -algebra, then ϕ is a finite map.Proof of Theorem 6.1.1. ( iii ) follows from ( ii ) by Remark 5.1.2(3), since the local map R → S factors through R h by the universal property of Henselization.For ( ii ), choose an essentially ´etale local homomorphism φ : ( R, m ) → ( S, n ) such that the inducedmap on residue fields is an isomorphism and φ is not an isomorphism. Note that such an S existsbecause R is not Henselian. We claim that S is not a finite R -algebra. Indeed, otherwise S is afree R -module of finite rank because φ is flat and R is Noetherian local. The free rank of S mustthen equal 1 because κ ( n ) = S ⊗ R κ ( m ) and the extension κ ( m ) ֒ → κ ( n ) is trivial. But this means φ is an isomorphism, contrary to its choice. Thus S is not a finite R -algebra. Since R h is also the Henselization of S , it follows that S is also a domain. Then S is not a solid R -algebra by ( i ), andso, R h also cannot be a solid R -algebra. This proves ( ii ).Thus, it remains to show ( i ). If ϕ : R → S is an essentially ´etale extension of domains, thenthe generic fiber is a finite separable extension of fields. If S is a solid R -algebra, then S must bemodule-finite over R by Proposition 6.1.2. Conversely, if S is a finite R -algebra, then S is R -solidby Example 5.1.3 regardless of whether ϕ is ´etale. (cid:3) If ( R, m ) is a Henselian Noetherian local domain that is not complete, then Theorem 6.1.1 givesno information on whether b R is a solid R -algebra. The goal in the remainder of this subsection is togive a more direct proof of the following result which, as a special case, implies that the completionof a Noetherian local domain, Henselian or not, is never solid. Theorem 6.1.3.
Let R be a Noetherian ring (not necessarily local) and let I be a nonzero idealof R such that R is I -adically separated. Denote by b R I the I -adic completion of R . Consider anelement ϕ ∈ Hom R ( b R I , R ) . Then, we have the following: ( i ) ϕ = 0 if and only if ϕ (1) = 0 . ( ii ) If R is not I -adically complete and ϕ = 0 , then ϕ (1) is a zerodivisor on R . ( iii ) If R is not I -adically complete, the canonical map i : R → b R I never splits. ( iv ) If R is a domain that is not I -adically complete, then b R I is not a solid R -algebra. For the proof of Theorem 6.1.3, we first make a few preliminary observations about I -adiccompletions. Recall that for an ideal I of a ring R , we say that an R -module M is I -adicallyseparated if T n ∈ N I n M = (0). Said differently, a module M is I -adically separated if the I -adictopology on M is Hausdorff. Lemma 6.1.4.
Let I be an ideal of a Noetherian ring R . The canonical map i : R −→ b R I satisfies the following properties: ( i ) For all integers n > , we have b R I = i ( R ) + I n b R I . ( ii ) The map i is flat, and i is faithfully flat if and only if I is contained in the Jacobson radicalof R . ( iii ) The map i : R → b R is finite if and only if i is surjective. In particular, if R is I -adicallyseparated but not I -adically complete, then i is not a finite map. ( iv ) If M is an I -adically separated R -module, then the map i ∗ : Hom R ( b R I , M ) −→ Hom R ( R, M ) induced by pre-composition with i is injective. ( v ) For every ϕ ∈ Hom R ( b R I , R ) , we have ϕ (1) b R I ⊆ i ( R ) .Proof of Lemma 6.1.4. ( i ) follows from the isomorphisms R/I n ≃ b R I /I n b R I for all integers n > ii ) see [Mat89, Thms. 8.8 and 8.14].( iii ) If i is surjective, then it is a finite map. Conversely, assume i is finite. Tensoring the exactsequence R i −→ b R I −→ coker( i ) −→ R/I , we get an exact sequence
R/I −→ b R I /I b R I −→ coker( i ) /I coker( i ) −→ . As the induced map
R/I → b R I /I b R I is an isomorphism, it follows thatcoker( i ) = I coker( i ) . XCELLENCE, F -SINGULARITIES, AND SOLIDITY 31 Since i is finite, coker( i ) is a finitely generated R -module, and consequently, coker( i ) is annihilatedby an element of the form 1 + a for some a ∈ I by Nakayama’s lemma [Mat89, Thm. 2.2]. Thismeans that the ideal of b R I generated by 1 + a is contained in i ( R ). However, 1 + a is a unit in b R I by [BouCA, Ch. III, §
2, n o
13, Lem. 3]. Therefore b R I = (1 + a ) b R I ⊆ i ( R ) ⊆ b R I , which implies that the map i : R → b R I is surjective. For the second assertion of ( iii ), if A is I -adically separated, then i is injective. If i is also finite, then i is surjective by what we just proved,which would imply i is an isomorphism. But this is impossible because A is not I -adically completeby hypothesis.( iv ) Let ϕ : b R I → M be an R -linear map such that ϕ ◦ i = 0. It suffices to show that ϕ = 0. Let x ∈ b R . By ( i ), for every integer n >
0, there exists a n ∈ R and x n ∈ I n b R I such that x = i ( a n ) + x n . Then, ϕ ( x ) = ϕ (cid:0) i ( a n ) (cid:1) + ϕ ( x n ) = ϕ ( x n ) , and ϕ ( x n ) ∈ I n M by R -linearity. This shows that ϕ ( x ) ∈ \ n ∈ Z > I n M = (0) , where the last equality follows because M is I -adically separated. Thus, ϕ = 0.( v ) Consider the R -linear map i ◦ ϕ : b R I −→ b R I , and let a := ϕ (1) ∈ R . We also have the R -linear map ℓ a : b R I → b R I given by left multiplication by a . Now for any x ∈ R , we have ℓ a (cid:0) i ( x ) (cid:1) = ai ( x ) = i ( ax ) = i (cid:0) ϕ (1) x (cid:1) = i ◦ ϕ (cid:0) i ( x ) (cid:1) , that is, i ∗ ( ℓ a ) = i ∗ ( i ◦ ϕ ) , where i ∗ : Hom R ( b R I , b R I ) → Hom R ( R, b R I ) is the map induced by pre-composition with i . Since b R I is an I -adically separated R -module, it follows from ( iv ) that i ∗ is injective. Thus ℓ a = i ◦ ϕ , andso, ϕ (1) b R I = ℓ a ( b R I ) = i ◦ ϕ ( b R I ) ⊆ i ( R ) . (cid:3) Remark 6.1.5. (1) If I is an idempotent ideal of a Noetherian ring R , then b R I = R/I and the completion map R → b R I is just the quotient map. In particular, the completion map is finite. Hence thesecond assertion of Lemma 6.1.4( iii ) fails if R is not I -adically separated.(2) The proofs of parts ( i ) , ( iii ) , ( iv ), and ( v ) of Lemma 6.1.4 work even when R is not Noe-therian as long as I is a finitely generated ideal of R . We need I to be finitely generatedin order for the isomorphisms R/I n ≃ b R I /I n b R I to hold (see [Stacks, Tag 05GG] for a niceproof of this due to Poonen).We now prove Theorem 6.1.3 as follows. Proof of Theorem 6.1.3. ( i ) follows by Lemma 6.1.4( iv ) applied to M = R because the pullbackmap i ∗ : Hom R ( b R I , R ) −→ Hom R ( R, R )is injective. Note that here we need R to be I -adically separated, and the latter follows by thehypotheses of the Theorem. For ( ii ), if ϕ = 0, then by ( i ), we have ϕ (1) = 0. Assume that ϕ (1) is a nonzerodivisor on R .Then, by flatness of i : R → b R I (Lemma 6.1.4( ii )), we see that ϕ (1) is also a nonzerodivisor on b R I .In particular, ϕ (1) b R I ≃ b R I . At the same time, Lemma 6.1.4( v ) shows that ϕ (1) b R I ⊆ i ( R ) , and so ϕ (1) b R I is a finitely generated R -module since it is a submodule of the finitely generated R -module i ( R ) and R is Noetherian. Thus, b R I is a finitely generated R -module, which is impossibleby Lemma 6.1.4( iii ) and the hypotheses of ( ii ).( iii ) follows from ( ii ) because a splitting of R → b R I maps 1 to 1, and 1 is a nonzerodivisor.Similarly, ( iv ) also follows from ( ii ). Indeed, if R is a domain and b R I is R -solid, then there existsan R -linear map φ : b R I → R such that φ (1) = 0 (Remark 5.1.2). But this is impossible because ( ii )implies φ (1) must be a zerodivisor and a domain has no nonzero zerodivisors. (cid:3) Remark 6.1.6.
Let R be a product of two rings R × R and let I be the idempotent idealgenerated by (1 , b R I = R and the associated map R → b R I is just projection onto R . Inthis case b R I is a solid R -algebra because one has the R -linear inclusion b R I = R → R that sends a to (0 , a ). Thus, Theorem 6.1.3( iv ) fails for arbitrary Noetherian rings R .6.2. Non-solidity of some big Cohen–Macaulay algebras.
Apart from providing obstructionsto F -solidity of rings essentially of finite type over an excellent local ring, the results of the previoussubsection have implications for the non-solidity of certain algebra extensions that are at the heartof the recent solution of Hochster’s direct summand conjecture in mixed characteritic [And18].Throughout this subsection, ( R, m ) will denote a Noetherian local ring of dimension d . Recallthat an R -algebra B is a big Cohen–Macaulay R -algebra if there exists a system of parameters x , x , . . . , x d ∈ m which form a regular sequence on B . The adjective “big” is meant to indicatethat the R -algebra B need not be finite over R (in practice, B is not even Noetherian). We willcall a big Cohen–Macaulay R -algebra a BCM R -algebra for brevity. We say B is a balanced BCM R -algebra if every system of parameters of R is a regular sequence on B .The existence of (balanced) BCM R -algebras has far-reaching consequences for the homologicalconjectures [HH95] and has been recently used by Ma and Schwede to develop a singularity theoryin mixed characteristic that draws inspiration from various notions of singularities in equal char-acteristic [MS] (see also [MSTWW]). Hochster and Huneke constructed BCM R -algebras when R has equal characteristic using their characterization of the absolute integral closure R + as a BCM R -algebra for an excellent local domain R of prime characteristic p > R -algebraswere constructed by Andr´e in mixed characteristic in the same paper where he settled the directsummand conjecture [And18], following which Shimomoto showed that one can even construct aBCM R -algebra that is simultaneously an R + -algebra [Shi18]. Andr´e later showed that a (moreuseful) weakly functorial version of BCM R -algebras also exists in mixed characteristic [And20](see also [HM18]).One of the pleasing properties of BCM R -algebras is that they are often R -solid, a well-knownfact summarized in the following result. Proposition 6.2.1.
Let ( R, m , κ ) be a complete local domain of dimension d of arbitrary charac-teristic. Then, every BCM R -algebra is R -solid.Indication of proof. Let B be a BCM R -algebra. Since there exists a system of parameters of R that is a regular sequence on B , we see that H d m ( B ) = 0. It now follows that B is a solid R -algebrausing [Hoc94, Cor. 2.4], where the argument proceeds by reducing to the case where R is regularvia the analogue of Noether normalization for complete local rings and then using Matlis dualityin a way similar to how it is used in Lemma 2.3.3. (cid:3) XCELLENCE, F -SINGULARITIES, AND SOLIDITY 33 Remark 6.2.2. (1) Proposition 6.2.1 provides a different perspective on F -solidity of a complete local domain( R, m ) in prime characteristic. Since R is excellent, [HH93, Thm. 5.15] shows that R + is abig Cohen–Macaulay R -algebra. Thus, R + is a solid R -algebra by Proposition 6.2.1, andsince R → R + factors via F R ∗ R , we see that F R ∗ R is also a solid R -algebra by Remark5.1.2.(2) Now that we know the direct summand theorem, one can give a simple proof of the fact that R + is always a solid R -algebra for a complete local Noetherian domain ( R, m ), even when R + is not known to be BCM (for example, in equal characteristic 0 and mixed characteristic).Note that we do not even need that there exist BCM R -algebras that contain R + . UsingCohen’s theorem, choose a module finite extension A ֒ → R where A is a power series ringover a field or a mixed characteristic complete DVR. Then one can check that A + = R + . Themap A → A + is pure because A is splinter (here we use the direct summand theorem), andso, A → A + splits by Lemma 2.3.3 because A is complete. In other words, the composition A ֒ → R → R + = A + is A -solid. Then R + is R -solid by [Hoc94, Cor. 2.3] because A ֒ → R is a finite extension of Noetherian domains.We now show that we lose solidity of BCM R -algebras if we drop the hypothesis that R iscomplete. There exist excellent regular rings that behave like polynomial or power series rings overfields for which R + is not R -solid. In fact, examples exist even for a class of excellent local ringsthat are closest in behavior to complete local rings, namely those that are Henselian. As far as weare aware, these examples are the first of their kind. Proposition 6.2.3.
Let R be a Noetherian ring. We have the following: ( i ) For each integer n > , there exist excellent regular rings R of prime characteristic p > and Krull dimension n for which R + is not a solid R -algebra. Moreover, R can be chosento be local and Henselian. ( ii ) Let ( R, m , κ ) be a Noetherian local domain of arbitrary characteristic that is not m -adicallycomplete. For any BCM R -algebra B , the m -adic completion b B m is a balanced BCM R -algebra that is not R -solid.Proof. ( i ) For each n >
0, there are Tate algebras T n ( k ) and convergent power series rings K n ( k )that are not F -solid [DM(b), Thm. A and Rem. 5.5] for some appropriately chosen non-Archimedeanfield ( k, |·| ) of characteristic p >
0. Consequently, the absolute integral closures R + of such excellentregular rings R cannot be R -solid. Note that the convergent power series rings K n ( k ) are evenHenselian, and both T n ( k ) and K n ( k ) have Krull dimension n (see Proposition 3.3.2).( ii ) The fact that b B m is a balanced BCM R -algebra follows from [BH98, Cor. 8.5.3]. By con-struction, there is a factorization R → b R m → b B m . Thus, if b B m is R -solid, then b R m is also a solid R -algebra, which is impossible by Theorem 6.1.3( iv ). (cid:3) Remark 6.2.4.
Let ( R, m ) be a Noetherian local domain of mixed characteristic that is not com-plete. As far as we are aware, all constructions of balanced BCM R -algebras in the literatureproceed by first passing to b R m and then constructing a balanced BCM b R m -algebra that is automat-ically also a balanced BCM R -algebra. See [And18, n o o R -algebras can be R -solid because they contain b R m .7. Solidity of absolute integral and perfect closures – a deeper analysis
In the previous subsection, one of our goals was to show that R + fails to be a solid R -algebra fornice excellent local domains (Proposition 6.2.3), even though R + is always a solid R -algebra when R is a complete local domain as a consequence of the direct summand theorem (Remark 6.2.2(2)).In this section, we will examine the restrictions that solidity of R + imposes on the ring R . For example, we will see that solidity of absolute integral closures often implies excellence in primecharacteristic (Theorem 7.2.1( iii )), whereas most domains of equal characteristic zero have solidabsolute integral closures because of the existence of splittings arising from trace maps (Proposition7.3.1( v )). In prime characteristic, solidity of absolute integral closures implies solidity of perfectclosures of a domain. Hence we will also spend some effort understanding when the perfect closureof a Noetherian domain of prime characteristic is solid. We begin our investigation by exhibitingclose connections between solidity of absolute integral and perfect closures and the well-studiednotions of N-1 and Japanese (aka N-2) rings.7.1. N-1 and Japanese rings.
N-1 and Japanese rings, defined below, are rings for which nor-malizations satisfy the familiar finiteness properties of finite type algebras over a field and rings ofintegers of number fields.
Definition 7.1.1 [Mat80, (31.A), Defs.] . Let R be an arbitrary integral domain with fraction field K . Then, we have the following notions: • R is N-1 if the integral closure of R in K is a finite R -algebra. • R is Japanese or N-2 if the integral closure of R in every finite field extension of K is afinite R -algebra.A ring R , not necessarily a domain, is Nagata if R is Noetherian and for every prime ideal p of R ,the quotient ring R/ p is Japanese. Remark 7.1.2.
We will use the terminology “Japanese ring,” which is due to Grothendieck andDieudonn´e [EGAIV , Ch. 0, Def. 23.1.1]. By a theorem of Nagata [EGAIV , Thm. 7.7.2], a Noe-therian ring is Nagata if and only if it is universally Japanese in the sense of Grothendieck andDieudonn´e [EGAIV , Ch. 0, Def. 23.1.1]. Note that (quasi-)excellent rings are Nagata by theZariski–Nagata theorem (see [EGAIV , Cor. 7.7.3]).We first record some basic properties of the notions introduced in Definition 7.1.1 for the reader’sconvenience, with brief indications for why the properties are true. Most, if not all, of the propertiesappear in [Mat80, §
31] or [EGAIV , Ch. 0, § Lemma 7.1.3.
Let R be a Noetherian domain with fraction field K . We have the following: ( i ) If every finite extension domain of R is N-1, then R is Japanese. ( ii ) Suppose that for every finite purely inseparable field extension K ⊆ L , the integral closureof R in L is a module-finite R -algebra. Then R is Japanese. ( iii ) When K has characteristic zero, then R is N-1 if and only if R is Japanese. ( iv ) If R is N-1, then its normal locus in Spec( R ) is open. See [Mat80, (31.F), Lem. 4] for a partial converse to ( iv ). Proof. ( i ) Let K ⊆ L be a finite field extension. We have to show that the integral closure S of R in L is a finite R -algebra. Choose a basis { l , l , . . . , l n } of L/K such that the l i are integral over R . Let S ′ be the finite R -algebra R [ l , l , . . . , l n ]. Note that the fraction field of S ′ is L . Since S ′ is N-1 by hypothesis, it follows that the integral closure of S ′ in L is a finite S ′ , hence also, a finite R -algebra. But the integral closure of S ′ in L is S .( ii ) We have to show that if K ⊆ M is a finite field extension, then the integral closure of R in M is module-finite over R . Since a submodule of a finitely generated module over a Noetherianring is finitely generated, we may enlarge M to assume K ֒ → M is normal. The normality of M then implies that there is a factorization K ֒ → L ֒ → M such that L/K is purely inseparable and
M/L is separable [Stacks, Tag 030M]. The proof then follows by the hypothesis of the Lemma and
XCELLENCE, F -SINGULARITIES, AND SOLIDITY 35 the fact that integral closures of Noetherian domains in finite separable extensions of their fractionfields are always module-finite (see [Mat80, (31.B), Prop.]).( iii ) It is clear that a Japanese ring is N-1. Conversely, an N-1 ring whose fraction field hascharacteristic zero is Japanese by ( ii ).( iv ) Let R N be the normalization of R in K . Since R is N-1, R ֒ → R N is a finite map, andhence the cokernel Q is also a finitely generated R -module. The support of Q as an R -module istherefore closed, and it consists of those points p ∈ Spec( R ) for which R p is not normal by the factthat taking integral closures commutes with localization [BouCA, Ch. V, §
1, n o
5, Prop. 16]. Thenormal locus in Spec( R ) is the complement of the support of Q , and is therefore open. (cid:3) A simple sufficient condition for a ring of prime characteristic to be N-1 is the following:
Lemma 7.1.4 (cf. [EG, Proof of Cor. 2.2]) . Let ( R, m ) be a Noetherian local ring of prime charac-teristic p > . If R is F -injective (for example, if R is F -pure), then R is N-1.Proof. If R is F -injective, then b R is F -injective as well by [Fed83, Rem. on p. 473]. Since F -injectiverings are reduced [QS17, Lem. 3.11], this means that R is analytically unramified, and hence R isN-1 by [Mat80, (31.E)]. (cid:3) Solidity of absolute integral closures and Japanese rings.
Our main result of thissubsection relates the solidity of absolute integral closures of Noetherian domains to the Japanesecondition. The notion of F -solidity also allows us to prove a prime characteristic analogue ofLemma 7.1.3( iii ). Theorem 7.2.1.
Let R be a Noetherian domain of arbitrary characteristic and let R + be theabsolute integral closure of R . Then, we have the following: ( i ) If R + is a solid R -algebra, then R is a Japanese ring. ( ii ) Suppose R has prime characteristic. If the perfect closure R perf of R is a solid R -algebra,or more generally, if R is F -solid, then R is N-1 if and only if R is Japanese. ( iii ) If R has prime characteristic, is generically F -finite, and the condition in ( i ) holds, then R is excellent.Proof. Throughout this proof, let K denote the fraction field of R .( i ) Let L be a finite extension of K . Let S be the integral closure of R in L . We may assumewithout loss of generality that L is a subfield of Frac( R + ) and S is a subring of R + (since S is anintegral extension of R ). Therefore R -solidity of R + implies that S is a solid R -algebra (Remark5.1.2). Since R is Noetherian and the extension R ֒ → S is generically finite (the fraction field of S is L ), it follows by Proposition 6.1.2 that S is a finite R -algebra.( ii ) The backward implication is clear because a Japanese ring is always N-1. To show theforward implication, by Lemma 7.1.3( ii ) it suffices to show that if L is a finite purely inseparablefield extension of K , then the integral closure of R in L is a finite R -algebra. If R perf is a solid R -algebra, then F R ∗ R is also a solid R -algebra because it embeds in R perf (Remark 5.1.2). Therefore,it suffices to show that every F -solid Noetherian N-1 domain is Japanese. Let R N be the integralclosure of R in K . Note that for every integer e > F eR N ∗ R N is the integral closure of R in F eK ∗ K .Since R is N-1, R ֒ → R N is a finite extension of domains. Therefore, F eR N ∗ R N is a solid R N -algebrafor every e > R is F -solid. Then for every integer e > F eR N ∗ R N is a solid R -algebra by Lemma 5.1.4 because R N is a solid R -algebra by module-finiteness(see Example 5.1.3). Returning to our finite inseparable extension L/K , there exists e ≫ K -algebra embedding of L in F e K ∗ K . The integral closure S of R in L is then a subring of F e R N ∗ R N .Then S is a solid R -algebra because F e R N ∗ R N is a solid R -algebra. By Proposition 6.1.2 we againconclude that S is module-finite over R because R ֒ → S is generically finite.We will use ( i ) to prove ( iii ) directly, although ( i ) implies that R is F -solid (after embedding F R ∗ R in R + ), which then implies ( iii ) by [DS18, Thm. 3.2]. Note that by ( i ), R is a Japanese ring. Since R p is isomorphic to R , it follows that R p is also Japanese. Using the assumption that R is a generically F -finite, the extension K p ֒ → K is finite. Thus, the integral closure S of R p in K is module finite over R p . But R is contained in S , so that R is also module finite over R p byNoetherianity. Thus, R is F -finite, and consequently, also excellent by [Kun76, Thm. 2.5]. (cid:3) Remark 7.2.2. (1) Lemma 7.1.3( iii ) indicates that the difference between N-1 and the Japanese property arisesbecause of inseparability. Thus, Theorem 7.2.1( ii ) shows that F -solidity can be viewed asan antidote for the bad behavior of inseparable extensions in prime characteristic.(2) It is natural to ask to what extent the R -solidity of R + differs from the R -solidity of R perf for a ring R of prime characteristic. We will see in Example 7.5.4 that there exist one-dimensional locally excellent (but not excellent) Noetherian domains R that are not N-1(hence also not Japanese), but such that R perf is a solid R -algebra. Thus, R -solidity of R perf ,or more generally, F -solidity of R , is weaker than R -solidity of R + by Theorem 7.2.1( i ).7.3. Solidity of filtered colimits and Japanese rings of characteristic zero.
Let R bedomain. Both R + and the perfect closure R perf (if R has positive characteristic) can be expressedas filtered colimits of module-finite R -subalgebras, each of which are R -solid by Example 5.1.3.Thus, it is natural to wonder how the notion of solidity behaves under filtered colimits of rings.This topic is pursued in this subsection. Our analysis (Proposition 7.3.1) shows, for example, that R perf is always R -solid for a Frobenius split ring. We also find a natural class of excellent DVRsof prime characteristic that are Frobenius split, but that are not necessarily F -finite or complete.A partial converse of Theorem 7.2.1( i ) is obtained for normal domains that contain the rationalnumbers. Although this latter observation is known to experts, the conclusion we draw using it(Corollary 7.3.2) gives a new characterization of Japanese Noetherian rings in equal characteristiczero. Proposition 7.3.1.
Let R be a ring, not necessarily Noetherian. ( i ) Let { R i } i ∈ I be a collection of R -algebras indexed by a filtered poset. Suppose there existsan index i ∈ I such that for all i ≥ i there exists an R -linear map ϕ i : R i → R . Assumethe maps ϕ i are compatible with the transition maps φ ij : R i → R j in the obvious sense. If ϕ i is nonzero map, then colim i R i is a solid R -algebra. In particular, if ϕ i is a splittingof R → R i , then R is a direct summand of colim i R i . ( ii ) If R has prime characteristic, R is Frobenius split if and only if R → R perf splits. Inparticular, R perf is a solid R -algebra when R is Frobenius split. ( iii ) Suppose R is a Japanese Dedekind domain with fraction field K . Let L be an algebraicextension of K that is countably generated over K . If R L is the integral closure of R in L ,then R is a direct summand of R L . In particular, R L is a solid R -algebra. ( iv ) If R is a Japanese DVR (or more generally, a Japanese Dedekind domain) of prime char-acteristic such that F K ∗ K is countably generated over the fraction field K of R , then R isFrobenius split. ( v ) Suppose R is a domain (not necessarily Noetherian) containing the rational numbers Q .Then, R is normal if and only if the natural map R → R + splits.Proof. For ( i ), the existence of an R -linear map Ψ : colim i R i → R follows by the universal propertyof colimits and the compatibility condition on the ϕ i ’s. Since the composition R i −→ colim i ∈ I R i Ψ −→ R This result was observed by Karl Schwede and the first author during the 2015 Math Research Communities incommutative algebra at Snowbird, Utah.
XCELLENCE, F -SINGULARITIES, AND SOLIDITY 37 coincides with ϕ i , it follows that if the latter is nonzero, then Ψ must be as well. Consequently,colim i R i is a solid R -algebra if ϕ i = 0. If ϕ i is additionally a splitting of R → R i , then ϕ i maps1 to 1. Then, Ψ also maps 1 to 1, completing the proof of ( i ).For ( ii ), we apply ( i ) to the filtered system of R -algebras { F eR ∗ R } e ∈ Z > where the transition maps F eR ∗ R → F e +1 R ∗ R are just the p -th power Frobenius map. If R is Frobenius split, let ϕ : F R ∗ R → R be a splitting of the Frobenius map. Then for each e ≥
1, define ϕ e : F eR ∗ R → R to be thecomposition ϕ e : F eR ∗ R F e − R ∗ ϕ −−−−−→ F e − R ∗ R F e − R ∗ ϕ −−−−−→ F e − R ∗ R −→ · · · −→ F R ∗ R ϕ −→ R. Said more simply, ϕ e as a map of sets from R → R is just the composition of ϕ with itself e -times.Then, using the fact that ϕ is a left inverse of the Frobenius map, it is easy to verify that thecollection { ϕ e } is compatible with the transition maps of the filtered system { F eR ∗ R } e ∈ Z > . Nowtaking colimit and using ( i ), we get that R → R perf splits. Conversely, if R → R perf splits, thenthe factorization R F −→ F R ∗ R → R perf shows that R → F R ∗ R splits as well, that is, R is Frobeniussplit.( iv ) follows readily from ( iii ) by taking L to be F K ∗ K and using the observation that since R is normal, the integral closure of R in F K ∗ K is precisely F R ∗ R . Thus, we now prove ( iii ). Fix acountable set of generators { ℓ n : n ∈ Z > } of L over K . For each n ∈ Z ≥ , let K n := K ( ℓ , ℓ , . . . , ℓ n ) = K [ ℓ , ℓ , . . . , ℓ n ] . Moreover, let R n be the integral closure of R in K n . Note that K = K and R = R because R isnormal. Furthermore, R n +1 is the integral closure of R n in K n +1 , hence the collection of rings { R n : n ∈ Z ≥ } is filtered with inclusions as the transition maps. Since L is the union of the subfields K n ,it follows that R L is the colimit of the rings R n , which are themselves Dedekind domains. As K n is a finite extension of K , it follows by the Japanese property that R n is a module-finite R -algebra,for all n ≥
0. Thus, R n is also module-finite over R n − , for all n ≥
1. Now using the fact that R n is a regular ring, fix an R n − -linear splitting s n : R n −→ R n − of the inclusion R n − ֒ → R n for all n ≥ ϕ n := s ◦ s ◦ · · · ◦ s n . Thus, for n ≥
1, the map ϕ n : R n → R is an R -linear splitting of the inclusion R ֒ → R n chosen ina way so that the splittings are compatible with the transition maps of the filtered system { R n } .Therefore, the induced map R L → R is a splitting of R ֒ → R L by ( i ).( v ) We first show that if R is normal, then R → R + splits. Write R + as a filtered colimit ofits module-finite R -subalgebras R i with the transition maps given by inclusions. Choose R i = R and ϕ i : R i → R to be the identity map on R . Given any finite subalgebra R → R i of R + , define ϕ i : R i → R to be the restriction of the normalized trace map1[ K i : K ] Tr K i /K : K i −→ K to R i , where K = Frac( R ) and K i = Frac( R i ). The restricted normalized trace maps R i into R .Indeed, if a ∈ R i , then the minimal polynomial f a ( x ) of a over K has coefficients in R by normalityof R . Since Tr K i /K ( a ) is an integer multiple of a coefficient of f a ( x ) [Stacks, Tag 0BIH], and since R contains Q , it follows that for all a ∈ R i ,1[ K i : K ] Tr K i /K ( a ) ∈ R. One can check that ϕ i is a splitting of R → R i . We now verify that the maps ϕ i are compatiblewith the transition inclusions R i ֒ → R j . So let a ∈ R i . Then, ϕ j ( a ) := 1[ K j : K ] Tr K j /K ( a )= 1[ K j : K i ][ K i : K ] Tr K i /K (cid:0) Tr K j /K i ( a ) (cid:1) = 1[ K j : K i ][ K i : K ] Tr K i /K (cid:0) [ K j : K i ] a (cid:1) = 1[ K i : K ] Tr K i /K ( a ) =: ϕ i ( a ) . Here, the second equality follows by the behavior of trace maps under a tower of field extensions,and the third equality follows because a is already an element of K i . Thus, the ϕ i give a compatiblesystem of splittings R i → R , and so, by ( i ) we get a splitting of R + → R .Conversely suppose R ֒ → R + splits. Let R N be the normalization of R . Using the factorization R ֒ → R N ֒ → R + we see that R ֒ → R N also splits. As R ֒ → R N is generically an isomorphism, thisimplies R ֒ → R N must be an isomorphism as well. Indeed, coker( R ֒ → R N ) is R -torsion free as itcan identified as a submodule of the R -torsion free module R N , and so, coker( R ֒ → R N ) = 0 sinceit is generically trivial. (cid:3) As a consequence of Proposition 7.3.1 and Theorem 7.2.1, we obtain the following characterizationof the N-1 property for Noetherian domains of equal characteristic zero. Note that this corollaryalso gives a new proof of the fact that the N-1 condition coincides with the Japanese property inequal characteristic zero (cf. the proof of Lemma 7.1.3( iii )).
Corollary 7.3.2.
Let R be a Noetherian domain containing the rational numbers Q . Then, thefollowing are equivalent. ( i ) R + is a solid R -algebra. ( ii ) R is Japanese. ( iii ) R is N-1.Proof. We have ( i ) ⇒ ( ii ) by Theorem 7.2.1( i ). Note this is the only place where we will use that R is Noetherian. The implication ( ii ) ⇒ ( iii ) follows by the definitions of N-1 and Japanese rings(Definition 7.1.1).Thus, it remains to show that ( iii ) ⇒ ( i ). Let R N be the normalization of R . Then by assumption, R N is a module-finite R -algebra, hence R -solid by Example 5.1.3. Moreover, it is clear that R + =( R N ) + . That ( R N ) + is a solid R N -algebra follows by Proposition 7.3.1( v ) because R N is normal.Then, Lemma 5.1.4 implies that R + = ( R N ) + is also R -solid. (cid:3) Solidity and excellence in Krull dimension one.
Theorem 7.2.1 and Proposition 7.3.1raise the question of whether there exist non-excellent domains R of prime characteristic p > R + is R -solid, or more generally, if R is F -solid. In this subsection our main resultshows that N-1 F -solid Noetherian domains of prime characteristic p > F -solidDedekind domains of prime characteristic are excellent, although the converse fails even for excellentHenselian discrete valuation rings by Proposition 5.4.1. The novel aspect of Theorem 7.4.1 is thatunlike [DS18], no generic F -finiteness assumptions are made. Note that it is well-known that aDedekind domain whose field of fractions has characteristic zero is excellent (we prove it below forthe reader’s convenience), making the question of excellence for Dedekind domains only interestingin prime characteristic. XCELLENCE, F -SINGULARITIES, AND SOLIDITY 39 Theorem 7.4.1.
Let R be a Noetherian N-1 domain of Krull dimension one. Suppose R + is asolid R -algebra (in which case one does not need R to be N-1), or that R is an F -solid domainof prime characteristic p > . Then, R is excellent. In particular, a Frobenius split (or F -solid)Dedekind domain is always excellent.Proof. We directly verify all the axioms for R to be excellent to illustrate to the reader that theseaxioms are quite checkable for low-dimensional rings.We will focus on verifying the axioms when R has prime characteristic p >
0, and mention howto adapt the proof when the fraction field K of R has characteristic zero. We have to check thefollowing three axioms that characterize an excellent ring (see Definition 2.2.1).(1) R is universally catenary: This is immediate because R is a one-dimensional domain, henceCohen–Macaulay. And it is well-known that Cohen–Macaulay rings are universally catenary [Mat80,Thm. 33].(2) If S is a finite type R -algebra, the regular locus of S is open: By [Mat80, Thm. 73], it sufficesto show that for every p ∈ Spec( R ) and for every finite field extension K ′ of the residue field κ ( p )at p , there exists a finite R -algebra R ′ with fraction field K ′ , such that R ′ contains R/ p and suchthat the regular locus in Spec( R ′ ) contains a nonempty open set. Since R is one-dimensional, p iseither a maximal ideal or the zero ideal. If p is maximal, then one can just take R ′ to equal K ′ . If p = (0), then κ ( p ) is the fraction field K of R . By F -solidity and the fact that A is N-1, Theorem7.2.1( ii ) implies that R is a Japanese ring. Then one can take R ′ to be the integral closure of R in K ′ . Note R ′ is a one-dimensional normal domain (hence regular) that is module-finite over R withfraction field K ′ .The argument for property (2) only uses that R has Krull dimension one and is Japanese. Notethat if K has characteristic zero, then R is automatically Japanese when R + is a solid R -algebraby Theorem 7.2.1( i ).(3) The formal fibers of the local rings of R are geometrically regular: Let p ∈ Spec( R ). We haveto check that the map R p −→ c R p from R p to its p R p -adic completion has geometrically regular fibers. If p = (0), the completionmap is an isomorphism and there is nothing to verify. Suppose p is a nonzero maximal ideal of R . Then R p is one-dimensional local domain. Thus, R p has two formal fibers, the closed fiber andthe generic fiber. The closed formal fiber is just an isomorphism of the residue fields of R p and c R p , hence it is always geometrically regular. Let K be the fraction field of R p (which is also thefraction field of R ), and consider the generic formal fiber( c R p ) K := K ⊗ R p c R p . Observe that ( c R p ) K is a zero-dimensional Noetherian ring. Thus, to show that K ֒ → ( c R p ) K is geo-metrically regular, it suffices to show that ( b R p ) K is a geometrically reduced K -algebra because thenotions “geometrically regular” and “geometrically reduced” agree for zero-dimensional Noether-ian algebras over a field. Indeed, a geometrically regular Noetherian algebra over a field is alwaysgeometrically reduced because regular rings are reduced. Conversely, if R is a zero-dimensionalNoetherian geometrically reduced k -algebra, then for every finite field extension k ′ of k , k ′ ⊗ k R isalso zero-dimensional, Noetherian and reduced. However, we know that a reduced Noetherian ringof dimension zero always decomposes as a finite direct product of fields by the Chinese RemainderTheorem. Thus, k ′ ⊗ k R is regular, which implies that R is geometrically regular over k .To show that the formal fibers of R p are geometrically reduced, we will again use that R isJapanese by F -solidity and the N-1 property (Theorem 7.2.1( ii )). Since taking integral closurescommutes with localization, R p is also a Japanese ring. Now because R p has only two prime ideals(it is a one-dimensional local domain), this implies that R p is in fact a Nagata ring by Definition7.1.1. But a Noetherian local ring is a Nagata ring if and only if its formal fibers are geometrically reduced by a result of Zariski and Nagata [EGAIV , Thm. 7.6.4]. This proves that all formalfibers of all local rings of R are geometrically regular, completing the proof that an F -solid N-1Noetherian domain of positive characteristic is excellent.Again we only used that R is Japanese in the verification of property (3). This shows that one-dimensional Japanese domains whose fraction fields are of characteristic zero are always excellent.In particular, Dedekind domains whose fraction fields are of characteristic zero are excellent becausesuch rings are N-1, and hence Japanese by Lemma 7.1.3( iii ). (cid:3) Said differently, Theorem 7.4.1 shows that non-excellent Dedekind domains of prime characteris-tic p > p − e -linear maps. In particular, Nagata’s non-excellent discrete valuationring k ⊗ k p k p J t K where [ k : k p ] = ∞ (see [Nag75, App. A1, Ex. 3]), has no nonzero p − e -linear maps. Remark 7.4.2. (1) Combined with Lemma 7.1.4, Theorem 7.4.1 shows that if a Noetherian domain of Krulldimension one is F -pure (or more generally, F -injective) and F -solid, then it is locallyexcellent without any N-1 hypotheses. This is because both F -solidity and F -injectivityare preserved under localization (see [DM(a), Prop. 3.3] for the latter fact). Thus, such aring is locally N-1 and locally F-solid, and hence, locally excellent. Note that F-solidity iscrucial and just F-injectivity (or even F-purity) alone does not guarantee that a Noetheriandomain of Krull dimension 1 is locally excellent. For example, there exist non-excellentDVRs in the function field of P F p by [DS18, § F -solid. In the next subsection we will construct a one-dimensional locally excellentNoetherian domain R for which R perf is R -solid but R is not N-1, hence also not excellent(Example 7.5.4).(3) Theorem 7.4.1 has the interesting consequence that if R is a normal Noetherian domain ofprime characteristic such that R + is R -solid, or more generally, if R is F -solid, then forevery height one prime p of R , the localization R p is excellent. Indeed, since the formationof absolute integral closures commutes with localization, and since all rings involved aredomains, one can check that if S ⊂ R is a multiplicative set, then ( S − R ) + is also a solid R -algebra (resp. S − R is F -solid) by just localizing a nonzero R -linear map from R + (resp. F R ∗ R ) to R . Thus, Theorem 7.4.1 applied to the localization R p implies R p is excellent. Inother words, an F -solid normal (or, more generally, N-1) Noetherian domain R of primecharacteristic is “excellent in codimension 1.”Theorem 7.4.1, the results of this paper, and those of [DM(b)] (see also [DS18]) allow us to obtaina classification of F -solid DVRs, which we now collect and summarize for the reader’s convenience. Corollary 7.4.3.
Let ( R, m ) be a DVR of prime characteristic p > . Consider the followingstatements: ( i ) R is F -solid. ( ii ) R is Frobenius split. ( iii ) R is split F -regular. ( iv ) R is excellent.Then, ( i ) , ( ii ) , and ( iii ) are equivalent, and always imply ( iv ) . All four assertions are equivalent if R is essentially of finite type over a complete local ring, or the fraction field K of R is such that K /p is countably generated over K . However, there exist excellent Henselian DVRs that are not F -solid.Proof. The equivalence of ( i ), ( ii ), and ( iii ) follow from Proposition 3.4.1, while each of theseequivalent statements implies ( iv ) by Theorem 7.4.1. XCELLENCE, F -SINGULARITIES, AND SOLIDITY 41 That all four statements are equivalent when R is essentially of finite type over a completelocal ring, or if the fraction field K of R has the property that K /p is countably generated over K follows from Theorem 3.1.1 and Proposition 7.3.1( iv ). Finally, the existence of non- F -solidexcellent Henselian discrete valuation rings follows from Proposition 5.4.1, which in turn dependson the results of [DM(b)]. (cid:3) A meta construction of Hochster and solidity of perfect closures.
In the previousSubsection we proved that Noetherian N-1 domains of Krull dimension one and prime characteristic p > F -solid are also excellent (Theorem 7.4.1). Our goal in this subsection will be to usethe following meta construction due to Hochster to give examples of non-excellent one-dimensionaldomains that are F -solid, and even Frobenius split. In other words, we will show that the N-1assumption cannot be weakened in Theorem 7.4.1. Moreover, we will also obtain examples ofNoetherian domains of prime characteristic whose perfect closures are solid, but whose absoluteintegral closures are not (Example 7.5.4). In a sense Hochster’s construction, and consequently ourexamples, are as nice as possible because the rings will be locally excellent. Theorem 7.5.1 [Hoc73, Props. 1 and 2] . Let P be a property of Noetherian local rings. Let k bea field, and let ( R, m ) be a local ring essentially of finite type over k such that ( i ) R is geometrically integral; ( ii ) R/ m = k ; and ( iii ) for every field extension L ⊇ k , the ring ( L ⊗ k R ) m ( L ⊗ k R ) fails to satisfy P .Moreover, suppose every field extension L ⊇ k satisfies P . For all n ∈ Z > , let R n be a copy of R with maximal ideal m n = m . Let R ′ := N n ∈ Z > R n , where the infinite tensor product is taken over k . Then, each m n R ′ is a prime ideal of R ′ . Moreover, if S = R ′ r ( S n m n R ′ ) , then the ring T := S − R ′ is a Noetherian domain whose locus of primes that satisfy P is not open in Spec( T ) . Furthermore,the map n m n T induces a one-to-one correspondence between Z > and the maximal ideals of T ,and T m n T ≃ ( L n ⊗ k R n ) m n ( L n ⊗ k R n ) , where L n is the fraction field of the domain N m = n R m . Thus, each local ring of T is essentiallyof finite type over some appropriate field extension of k . In particular, all local rings of T areexcellent, that is, T is locally excellent. Remark 7.5.2.
By definition, the infinite tensor product R ′ = N n ∈ Z > R n is the filtered colimitcolim j ∈ Z > (cid:18) j O n =1 R ⊗ k R ⊗ k · · · ⊗ k R j (cid:19) of finite tensor products. Also, implicit in Theorem 7.5.1 is the assertion that for each n ∈ Z > ,the ideal m n ( L n ⊗ k R n ) is a prime ideal of L n ⊗ k R n (Hochster calls this property absolutely prime in his paper). This follows from the fact that since R n / m n is isomorphic to the base field k , forever field extension L of k , we have L ⊗ k R n / m n ≃ L. Thus, m n ( L n ⊗ k R n ) is in fact a maximal ideal of L n ⊗ k R n for all n ∈ Z > . Moreover, as aconsequence of Hochster’s construction, it follows that the dimension of each local ring T m n T equalsthe dimension of R . Indeed, the extension R n ֒ → L n ⊗ k R n is faithfully flat, and induces a faithfullyflat local extension R n = ( R n ) m n ֒ → ( L n ⊗ k R n ) m n ( L n ⊗ k R n ) . But the closed fiber of this localextension is a field (the maximal ideal of R n expands to the maximal ideal of ( L n ⊗ k R n ) m n ( L n ⊗ k R n ) ),hence is zero-dimensional. Therefore,dim( R ) = dim( R n ) = dim (cid:0) ( L n ⊗ k R n ) m n ( L n ⊗ k R n ) (cid:1) = dim( T m n T ) , where the first equality follows because R n is just a copy of R , the second equality follows by faithfulflatness of the local extension and [Mat89, Thm. 15.1( ii )], and the third equality follows by theisomorphism in Hochster’s result. In other words, each maximal ideal of T has the same height,that is, T is equicodimensional and dim( T ) = dim( R ). In particular, T has finite Krull dimension.We can now construct many non-excellent Frobenius split (hence F -solid) rings. Corollary 7.5.3.
Let ( R, m ) be a Noetherian local F p -algebra such that (1) R is essentially of finite type over F p , (2) R is geometrically integral over F p , (3) R/ m = F p , (4) R is Frobenius split, and (5) R is not regular.Then, the ring T from Theorem 7.5.1 constructed using any R satisfying properties (1) – (5) is alocally excellent Frobenius split ring whose regular locus is not open. In particular, T perf is a solid T -algebra and T is not excellent.Proof. First, we give an explicit example of a ring R satisfying properties (1)–(5). Let k be a field,and consider the homogeneous coordinate ring k [ x, y, z ]( y − xz )of a conic in P k . Note that this ring is an integral domain for any k because it can be identifiedwith the Veronese subring k [ x , xy, y ] of k [ x, y ]. Moreover, for any field k of prime characteristic p >
0, this ring is split F -regular because k [ x , xy, y ] is a direct summand of the split F -regularring k [ x, y ]. In particular, k [ x, y, z ] / ( y − xz ) is Frobenius split for any k , and consequently, so areall its local rings. Let R = (cid:18) F p [ x, y, z ]( y − xz ) (cid:19) ( x,y,z ) . Then by the above discussion, it follows that R satisfies properties (1)–(4). Furthermore, R is notregular since it is the local ring at the cone point. Therefore R satisfies all five properties in thestatement of the corollary. From here on, we do not need the specifics of what R actually is, butonly that it has the aforementioned five properties.Let P be the property of a Noetherian local ring being regular, and fix any F p -algebra R thatsatisfies (1)–(5). Since R is not regular and since regularity descends under faithfully flat maps[Mat89, Thm. 23.7( i )], it follows that R also satisfies properties ( i )-( iii ) of Theorem 7.5.1. Construct T as in Theorem 7.5.1 using a countable number of copies of R . By construction, T is locallyexcellent and the regular locus of T is not open (we know this locus is nonempty because T isgenerically regular). We will now show that T is Frobenius split. For this it suffices to show thatthe ring R ′ := O n ∈ Z > R n := colim j ∈ Z > (cid:0) R ⊗ F p R ⊗ F p · · · ⊗ F p R j (cid:1) . is Frobenius split, because T is a localization of R ′ and Frobenius splittings are preserved underlocalization. Note each R n , being a copy of R , is Frobenius split by hypothesis. Fix a Frobeniussplitting ϕ : F R ∗ R → R of R , and denote by ϕ n : F R n ∗ R n → R n the corresponding splitting of the n -th copy of R . Then, for each j , we get an ( R ⊗ F p · · · ⊗ F p R j )-linear map ϕ ⊗ · · · ⊗ ϕ j : F R ∗ R ⊗ F p · · · ⊗ F p F R j ∗ R j −→ R ⊗ F p · · · ⊗ F p R j that maps 1 to 1. But note that since the Frobenius map of F p is just the identity, F R ∗ R ⊗ F p · · · ⊗ F p F R j ∗ R j = F ∗ ( R ⊗ F p · · · ⊗ F p R j ) . XCELLENCE, F -SINGULARITIES, AND SOLIDITY 43 Thus, for each j , ϕ ⊗· · ·⊗ ϕ j is a Frobenius splitting of R ⊗ F p · · ·⊗ F p R j . Moreover, these Frobeniussplittings are compatible with the transition maps of the filtered system { R ⊗ F p · · · ⊗ F p R j } j inthe sense that for every pair of indices ℓ > j , we have a commutative diagram F ∗ ( R ⊗ F p · · · ⊗ F p R j ) F ∗ ( R ⊗ F p · · · ⊗ F p R ℓ ) R ⊗ F p · · · ⊗ F p R j R ⊗ F p · · · ⊗ F p R ℓ . ϕ ⊗···⊗ ϕ j ϕ ⊗···⊗ ϕ ℓ where the horizontal maps come from the transition map R ⊗ F p · · · ⊗ F p R j → R ⊗ F p · · · ⊗ F p R ℓ that sends x ⊗ · · · ⊗ x j to x ⊗ · · · ⊗ x j ⊗ ⊗ · · · ⊗
1. Taking colimits now gives a Frobenius splittingof the ring R ′ , which is what we wanted. Note that if T is Frobenius split, then T perf is a solid T -algebra by Proposition 7.3.1( ii ). Hence the last assertion of the Corollary follows. (cid:3) Using Corollary 7.5.3 we will now show that there exists a locally excellent Noetherian domain R of Krull dimension one which is Frobenius split (in which case R perf is a solid R -algebra), but forwhich R + is not a solid R -algebra. In particular, this example will also show that F -solidity alonecannot imply the Japanese property without the N-1 hypothesis in Theorem 7.2.1( ii ), and that,similarly, F -solidity alone cannot imply excellence in Krull dimension one (see Theorem 7.4.1). Example 7.5.4.
Let R be the local ring at the origin of the node over F p , that is, R = (cid:18) F p [ x, y ]( y − x − x ) (cid:19) ( x,y ) . Since the coordinate ring of the node is a domain over any field, it follows that R is geometricallyintegral over F p . We know that R is not regular (hence not normal since it is one-dimensional).However, an application of Fedder’s criterion [Fed83, Thm. 1.12] shows that R is Frobenius split.By Corollary 7.5.3, the ring T constructed using R is locally excellent, Frobenius split, but notexcellent. In particular, T perf is a solid T -algebra. That T has Krull dimension one follows fromRemark 7.5.2. We claim that T + is not a solid T -algebra. Indeed, if it is, then Theorem 7.2.1( i ) willimply that T is Japanese, and hence N-1. However, this is impossible by Theorem 7.4.1 becausean F -solid N-1 Noetherian domain of Krull dimension one is always excellent, whereas T is not. Remark 7.5.5.
Hochster’s construction provides a wealth of examples of locally excellent domainsthat are not excellent (Theorem 7.5.1). Said differently, excellence in not a local property. Example7.5.4 shows that the properties of being N-1 and Japanese are also not local. At the same time, theN-1 property is often local by [Mat80, (31.G), Lem. 4].Example 7.5.4 illustrates that R perf can be a solid R -algebra even when R + is not. We now showsolidity of perfect closures for most rings that arise in algebro-geometric applications. To establishthis, we exploit the following observation of Hochster and Huneke that is related to the notion oftest elements in tight closure theory. Lemma 7.5.6 [HH90, Lem. 6.5] . Let ϕ : A → R be a module-finite and generically smooth (equiv-alently, generically geometrically reduced) map, where A is a normal domain and R is torsion-freeas an A -module. Let r , r , . . . , r d ∈ R be a vector space basis for K ′ = K ⊗ A R over the fractionfield K of A , and let c := det (cid:0) Tr K ′ /K ( r i r j ) (cid:1) . Then, c is a nonzero element of A such that cR perf ⊆ A perf [ R ] . In the above lemma, A perf [ R ] is the image of the unique map A perf ⊗ A R → R perf obtained bythe universal property of the tensor product that makes the diagram A A perf
R R perf ϕ ϕ perf commute. Moreover, since ϕ is a finite map, having a smooth generic fiber is the same as having ageometrically reduced generic fiber, which is in turn the same as having an ´etale generic fiber.Using Lemma 7.5.6, we can show solidity of the perfect closure in many geometric situations. Proposition 7.5.7.
Let ϕ : A → R be a module-finite extension of domains of prime characteristic p > such that A is normal and Noetherian. If A perf is a solid A -algebra (for example, if A isFrobenius split), then R perf is a solid R -algebra.Proof. Since A → R is a module-finite extension of Noetherian domains, to show that R perf is asolid R -algebra, it suffices to show that R perf is a solid A -algebra by [Hoc94, Cor. 2.3]. Considerthe chain of maps A −→ A perf ϕ perf −−−→ R perf . Since A → A perf induces an algebraic extension of fraction fields, by Lemma 5.1.4 it further sufficesto show that R perf is a solid A perf -algebra. Note that A perf → R perf need not be a finite map eventhough A → R is. Hence more work has to be done to establish A perf -solidity of R perf . For every e >
0, let A /p e [ R ] denote the image of the relative Frobenius map F eR/A : A /p e ⊗ A R −→ R /p e . We claim that the kernel of F eR/A is precisely the nilradical of A /p e ⊗ A R . Indeed, F eR/A induces ahomeomorphism on spectra, which means that ker( F eR/A ) is contained in the nilradical of A /p e ⊗ A R .On the other hand, the nilradical of A /p e ⊗ A R must be contained in ker( F eR/A ) because R is reduced.The upshot of this argument is that for each e > A /p e ⊗ A R ) red ≃ A /p e [ R ] . Note also that the module-finiteness of ϕ : A → R implies that for each e >
0, the map A /p e → A /p e [ R ] is a module-finite extension. By [DS19, Prop. 2.4.2.1], we may choose e ≫ A /p e −→ ( A /p e ⊗ A R ) red ≃ A /p e [ R ]has geometrically reduced generic fiber. Module-finiteness implies that having a geometricallyreduced generic fiber is the same as having a smooth generic fiber (see the proof of (3) in Theorem7.4.1). Moreover, A /p e [ R ] is a torsion-free module over the normal domain A /p e . Hence, byLemma 7.5.6 applied to the finite extension A /p e ֒ → A /p e [ R ], there exists a nonzero c ∈ A /p e such that c (cid:0) A /p e [ R ] perf (cid:1) ⊆ A /p e perf (cid:2) A /p e [ R ] (cid:3) = A /p e perf [ R ] . But A /p e [ R ] perf = R perf and A /p e perf = A perf , which means that left multiplication by c inducesa nonzero map R perf → R perf whose image lands inside A perf [ R ]. Restricting the codomain ofthis map shows that R perf is a solid A perf [ R ]-algebra. On the other hand, A perf ֒ → A perf [ R ] is amodule-finite extension of domains because it is the composition of the module-finite maps A perf id ⊗ ϕ −−−→ A perf ⊗ A R −→−→ A perf [ R ] . Then, A perf [ R ] is A perf -solid by Example 5.1.3. Thus, R perf is a solid A perf -algebra upon applyingLemma 5.1.4 to the composition A perf ֒ → A perf [ R ] ֒ → R perf . (cid:3) XCELLENCE, F -SINGULARITIES, AND SOLIDITY 45 Upon taking a Noether normalization one can now conclude that the perfect closure of anyalgebra that is of finite type over a prime characteristic field is solid over the algebra. In fact weobtain the following more general consequence.
Corollary 7.5.8.
Let A → R be a module-finite extension of domains such that A is regular andessentially of finite type over a complete local ring of prime characteristic p > . Then, R perf is asolid R -algebra.Proof. Since regular rings are F -pure, A is Frobenius split by Theorem 3.1.1, and hence A perf is A -solid by Proposition 7.3.1( ii ). Since A is also normal, we are done by Proposition 7.5.7. (cid:3) Finally, we observe that the analogue of Proposition 7.5.7 for the solidity of the absolute integralclosure has a significantly simpler proof.
Proposition 7.5.9.
Let A → R be a module-finite extension of Noetherian domains. If A + is asolid A -algebra, then R + is a solid R -algebra.Proof. The module-finite extension A → R induces an an isomorphism A + ∼ → R + . Thus, R + is asolid A -algebra, and so, R + is a solid R -algebra by [Hoc94, Cor. 2.3]. (cid:3) Remark 7.5.10. If R is a domain that is also a finite type algebra over a field k , then Proposition7.5.9 shows that R -solidity of R + reduces to the question of the solidity of the absolute integral clo-sure of a polynomial ring over k upon taking a suitable Noether normalization. However, somewhatsurprisingly, the question of whether k [ x , x , . . . , x n ] + is a solid k [ x , x , . . . , x n ]-algebra seems tobe completely open unless unless k has characteristic zero, in which case solidity follows by Proposi-tion 7.3.1( v ). See Question 8.4 and Proposition 8.5 for some further details regarding this. In fact,the authors do not know of a single non-excellent example of a Noetherian domain A of positivecharacteristic for which A + is a solid A -algebra (Question 8.10).8. Some open questions
For the reader’s convenience, we collect a few questions that to the best of our knowledge areopen and related to the results of this paper and [DM(b)]. We will summarize what is known aboutthese questions.We begin with some questions on the variants of the usual notion of strong F -regularity. Whilesplit F -regularity and F -pure regularity do not coincide even for excellent Henselian regular localrings (Theorem 3.3.4), we showed that every regular ring essentially of finite type over an excellentlocal ring is F -pure regular (Theorem 3.2.1). This raises the following question. Question 8.1.
Is every excellent regular ring of prime characteristic F -pure regular?We expect (though cannot show) that there are regular rings that do not satisfy the ideal-theoreticcharacterization of F -pure regularity from Proposition 3.2.3. However, we do not know if any suchexamples can be excellent. Theorem 3.2.1 and Remark 3.2.2 show that the characterization of F -pure regularity in Proposition 3.2.3 is satisfied for regular or even strongly F -regular rings ofprime characteristic that are essentially of finite type over an excellent local ring. This suggests thatwhile Question 8.1 may be true for excellent regular rings, there may be counterexamples for non-excellent regular rings. On the other hand, any example of a regular ring that is not F -pure regular,excellent or not, would be interesting in its own right because it will show that Hochster’s tightclosure notion of strong F -regularity (Definition 2.4.1( c )) does not coincide with F -pure regularityfor regular rings. In particular, this would indicate that Definition 2.4.1( c ) is the most appropriatevariant of the usual notion of strong F -regularity in a non- F -finite setting.In [DM(b)], we obtained examples of excellent F -pure rings that are not Frobenius split. However,the rings that we construct satisfy the stronger property that they do not admit any nonzero p − -linear maps. Thus the following question arises. Question 8.2.
Suppose R is a Noetherian F -pure ring of prime characteristic p > p − -linear map. Is R Frobenius split?We showed in Proposition 3.4.1 that Question 8.2 has an affirmative answer when R is a DVR, andin [DS18] it was shown that the question has an affirmative answer when R is a generically F -finitedomain. However, to the best of our knowledge Question 8.2 is open even for excellent local ringsof Krull dimension one that are not regular, and for regular local rings of Krull dimension greaterthan one. We expect any progress on Question 8.2 to also shed light on the following question. Question 8.3.
Suppose R is a regular local ring that is Frobenius split. Is R split F -regular?This question also seems to be open when R has Krull dimension greater than one, while the caseof Krull dimension one follows by Proposition 3.4.1. A likely first step is to try and see if theconvergent power series rings K n ( k ) are split F -regular whenever k is a complete non-Archimedeanfield that admits nonzero continuous functionals k /p → k and n >
1. Note that we know that K n ( k ) is Frobenius split for any such k by [DM(b), Thm. 4.4], while split F -regularity is known ina few cases; see Proposition 3.3.8 and Remark 3.3.9.Switching gears, we now highlight some open questions on the solidity of absolute integral closures.The next question is particularly surprising. Question 8.4.
Let k be a field of prime characteristic p > k = F p ). Is theabsolute integral closure k [ x ] + a solid k [ x ] algebra?Note that the question has an affirmative answer if we replace the polynomial ring over k by thepower series ring k J x K by Lemma 2.3.3, because the map k J x K → k J x K + is pure since k J x K is regular,and hence a splinter. We would now like to show that if k is F -finite, then solidity of k [ x ] + impliesthat the map k [ x ] → k [ x ] + splits. Proposition 8.5.
Let R be any domain of prime characteristic p > . Then, we have the following: ( i ) Let S be any R -algebra. Then, I S/R := im (cid:0)
Hom R ( S, R ) eval@1 −−−−→ R (cid:1) is a uniformly F -compatible ideal of R . ( ii ) Suppose R is Noetherian, generically F -finite, and split F -regular. If S is a solid R -algebra,then R → S splits. ( iii ) If R is Noetherian, generically F -finite, split F -regular, and R + is a solid R -algebra, then R → R + splits. Recall that an ideal I of a ring R of prime characteristic is uniformly F -compatible if, for all e ∈ Z > and for every p − e -linear map ϕ , we have ϕ ( F eR ∗ I ) ⊆ I . See [Sch10, Def. 3.1]. Proof.
The basic idea of the proof of ( i ) comes from the proof of [MS, Thm. 5]. If R is not F -solid,then I S/R is uniformly F -compatible because the only p − e -linear map on R is the trivial one for any e > R is F -solid. Consider aninteger e >
0, and a p − -linear map φ ∈ Hom R ( F eR ∗ R, R ). Note that we have a natural map η φ : Hom F eR ∗ R ( F eS ∗ S, F eR ∗ R ) −→ Hom R ( S, R )given by η φ ( ψ ) := φ ◦ ψ ◦ F eS , where F eS is the e -th iterate of the Frobenius on S . We then get acommutative square Hom F eR ∗ R ( F eS ∗ S, F eR ∗ R ) Hom R ( S, R ) F R ∗ R R η φ eval@1 eval@1 φ XCELLENCE, F -SINGULARITIES, AND SOLIDITY 47 The image of the left vertical evaluation map is just F eR ∗ I S/R . By the commutativity of the diagramand the definition of I S/R , it now follows that φ ( F eR ∗ I S/R ) ⊆ I S/R . Since φ was an arbitrary p − e -linear map of R for an arbitrary e >
0, this proves ( i ).( iii ) clearly follows from ( ii ), and hence it remains to show ( ii ). If R is Noetherian, generically F -finite and split F -regular, then R is F -solid, and hence, F -finite by [DS18, Thm. 3.2]. Then, R has a smallest nonzero uniformly F -compatible ideal with respect to inclusion, namely the big ornon-finitistic test ideal τ [Sch10, Thm. 6.3]. However, F -finiteness and split F -regularity impliesthat τ = R . Now if S is a solid R -algebra, then the ideal I S/R from ( i ) is nonzero by Remark 5.1.2.Since I S/R is a uniformly F -compatible ideal by ( i ) , it follows that R = τ ⊆ I S/R by the minimalityof τ . Thus, 1 ∈ I S/R , which is equivalent to the splitting of R → S . (cid:3) Remark 8.6.
A result of Hochster and Huneke shows that every weakly F -regular Noetheriandomain R is a splinter, in the sense that every module-finite extension of domains R ֒ → S splits[HH94(b), Thm. 5.17]. Note that S is a solid R -algebra for any such finite extension by Example5.1.3. Thus, it appears that the property of being a Noetherian F -finite split F -regular domainis quite a bit stronger than that of being a splinter by Proposition 8.5( ii ) because this propertyimplies splitting of any solid algebra, module-finite or not. Proposition 8.5 and Lemma 2.3.3 alsohave the interesting consequence that when ( R, m ) is F -finite, split F -regular, and complete local,then purity of an R -algebra S is equivalent to the R -solidity of S .Since conjecturally Noetherian F -finite splinters are split F -regular, the following question thenarises. Question 8.7.
Let R be a Noetherian F -finite splinter. If S is a solid R -algebra (not necessarilymodule-finite), then does R → S split?A negative answer to Question 8.7 will imply that Noetherian F -finite splinters are not split F -regular in general, although both notions are known to coincide for Q -Gorenstein rings [Sin99, Thm.1.1]. On the other hand, a positive answer will likely shed light on the equivalence of weak andsplit F -regularity for Noetherian F -finite rings, which is one of the outstanding open problems inprime characteristic commutative algebra.The uniform F -compatibility of the images of evaluation at 1 maps raises the question of char-acterizing the images of such maps, at least for some special extensions of rings. Question 8.8.
Suppose R is a Noetherian excellent domain of prime characteristic p > R is F -finite or complete local). If R + is a solid R -algebra, then what is the image ofevaluation at 1 map Hom R ( R + , R ) eval@1 −−−−→ R ?Note that the solidity assumption is important in Question 8.8 because Proposition 6.2.3 showsthat there are excellent Henselian regular local rings R of prime characteristic for which R + isnot a solid R -algebra. Using the notation of Proposition 8.5, suppose I R + /R is the image of thisevaluation at 1 map. If I R + /R = R , then R is a splinter because the map R ֒ → R + splits, andso, must also be pure. However, we do not know if the converse holds, even in the case where R is F -finite. In Appendix A, Karen E. Smith will provide an answer to Question 8.8 when R isGorenstein and either complete local or N -graded and finitely generated over R = k a field, byreinterpreting results from her fundamental thesis (see Theorems A.0.2 and A.3.2).One can replace R + by R perf in Question 8.8 and ask what the image I R perf /R of the evaluation at1 map Hom R ( R perf , R ) → R is. In this case, regardless of whether R is excellent, I R perf /R = R if andonly if R is Frobenius split. This equivalence follows by Proposition 7.3.1( ii ). Now suppose R is notnecessarily Frobenius split. Then I R perf /R = R . Let p be a prime ideal of R such that I R perf /R * p (no such prime ideal will exist if R perf is not a R -solid). Choose an element a ∈ I R perf /R such that a / ∈ p . Then there exists an R -linear map φ : R perf → R that sends 1 to a . Localizing at p and usingthe fact that perfect closures commute with localization, we then get a map φ p : ( R p ) perf → R p thatmaps 1 to a unit in R p because a / ∈ p . Then the composition( R p ) perf φ p −→ R p −· a − −−−−→ R p sends 1 to 1. Consequently, R p is Frobenius split. Thus, the closed subset Z of Spec( R ) defined by I R perf /R contains the non-Frobenius split locus of Spec( R ). However, since R p ⊗ R Hom R ( R perf , R ) = Hom R p (cid:0) ( R p ) perf , R p (cid:1) in general, it is not clear to the authors if Z coincides with the non-Frobenius split locus of Spec( R ).Thus, we raise the following question: Question 8.9.
Let R be a Noetherian domain of prime characteristic p >
0. Suppose R perf is asolid R -algebra. Then, does the image of the evaluation at 1 mapHom( R perf , R ) eval@1 −−−−→ R define the non-Frobenius split locus of R ?We showed that a Noetherian domain R with solid R + is Japanese. However, we do not knowany examples in prime characteristic of Noetherian domains with solid absolute integral closurewhen such domains are not complete local. This raises the final question in our summary of openquestions. Question 8.10.
Is there a non-excellent Noetherian domain of prime characteristic p > R + is a solid R -algebra?Examples of non-excellent Noetherian domains R with solid R + abound when R contains Q becausesolidity of R + is then equivalent to R being Japanese by Corollary 7.3.2. In prime characteristic, R + can be replaced by R perf and then the analogue of Question 8.10 for R perf has an affirmativeanswer by Corollary 7.5.3. Appendix A. Solidity of absolute integral closures and the test ideal
KAREN E. SMITH
The purpose of this appendix is to address the following question raised in Section 8:
Question A.0.1 (see Question 8.8) . Suppose R is a complete local domain of prime characteristic p >
0. What is the image of the “evaluation at 1” mapHom R ( R + , R ) −→ R ?Here, R + denotes the absolute integral closure of R – that is, the integral closure of R in analgebraic closure of its fraction field.We prove the following satisfying answer, at least in the Gorenstein case: Theorem A.0.2.
Let ( R, m ) be a complete local Noetherian Gorenstein domain of prime charac-teristic p > . Then the image of the evaluation at map Hom R ( R + , R ) eval@1 −−−−→ R is the test ideal τ ( R ) of R . XCELLENCE, F -SINGULARITIES, AND SOLIDITY 49 A graded version also holds; see Theorem A.3.2.The image of the evaluation map in Question A.0.1 was shown to be a uniformly F -compatibleideal in general in Proposition 8.5 of the main paper. Uniformly F -compatible ideals, introducedby Schwede as a prime characteristic analog of log canonical centers (see [Sch10, Def. 3.1]), are ageneralization of the F -ideals introduced in [Smi94] as annihilators of Frobenius-stable submodulesof the top local cohomology module of a Noetherian local ring ( R, m ) and later generalized in[Smi95(a)] and [LS01]. In the Gorenstein case, there is a unique smallest nonzero F -ideal (oruniformly F -compatible ideal), which was shown in [Smi94] to be the famous test ideal τ ( A ) ofHochster and Huneke’s tight closure theory. This was a key step in the proof that the tight closureof a parameter ideal in a local excellent ring is equal to its plus closure [Smi94]. Our proof ofTheorem A.0.2 invokes the methods of [Smi94]. Remark A.0.3.
Theorem A.0.2, in particular, implies that R + is a solid R -algebra (see Definition5.1.1), in light of Hochster and Huneke’s result that a complete local domain always admits a testelement [HH90, § Tight Closure and Test Ideals.
Tight closure, introduced by Hochster and Huneke, isa closure operation on submodules of a fixed ambient module M over an excellent ring of primecharacteristic p >
0. One property is that, for modules over a regular ring, all submodules aretightly closed. Another property is that local domains for which all ideals are tightly closed areCohen-Macaulay [HH90, Rem. 7.1]. Using these two properties, Hochster and Huneke gave a simpleconceptual proof that direct summands of regular rings are Cohen-Macaulay in prime characteristic,and many other important results in commutative algebra. The basic theory is developed in [HH90].
Definition A.1.1.
Let R be a Noetherian domain of prime characteristic p >
0. Let M be any R -module, and let N ⊂ M be a submodule. The tight closure of N in M , denoted N ∗ M , is thecollection of elements m ∈ M for which there exists a nonzero element c ∈ R such that c ⊗ m liesin the image of the canonical map F e ∗ R ⊗ R N → F e ∗ R ⊗ R M (A.1.1.1)for all e ≫ M is R , so we arecomputing tight closures of ideals , and the other is when M is arbitrary but N is zero.To verify a module is tightly closed using the definition above, we would need to consider maps ofthe form (A.1.1.1) above for all nonzero c . Fortunately, it turns out that so-called test elements exist that can be used to check an element is in the tight closure by checking relations (A.1.1.1) for just one “test element” c . Definition A.1.2.
Let R be a Noetherian domain of prime characteristic p >
0. A nonzero element c ∈ R is a test element if for all ideals I of R , cI ∗ ⊆ I (equivalently, for all a ∈ I ∗ , ca p e ∈ I [ p e ] for all e > c is a test element if and onlyif c annihilates the tight closure of zero in every finitely generated module M . Definition A.1.3.
The test ideal of R , denoted τ ( R ), is the ideal of R generated by all testelements.A deep theorem of Hochster and Huneke ensures that the test ideal τ ( R ) is nonzero when ( R, m )is a complete local domain (and quite a bit more generally); see [HH94(a), § M , the finitistic tight closure of zero in M is 0 ∗ fg M := [ M ′ ∗ M ′ , where M ′ ranges over all finitely generated submodules of M . Note that 0 ∗ fg M ⊆ ∗ M , and that theseare equal if M is finitely generated.The test ideal can be characterized as follows: Proposition A.1.4 [HH90, Prop. (8.23)(d)] . Let ( R, m ) be a Noetherian local domain of primecharacteristic p > . Let E denote an injective hull of its residue field. Then τ ( R ) = Ann R (0 ∗ fg E ) . A.2.
The Proof of Theorem A.0.2.
We will deduce Theorem A.0.2 from the following result:
Theorem A.2.1 [Smi94, Thm. 5.6.1.] . Let ( R, m ) be an excellent local domain of prime character-istic p > and dimension d . Then the kernel of the canonical map H d m ( R ) → H d m ( R + ) is precisely ∗ fg H d m ( R ) . Remark A.2.2.
A key point in the proof of Theorem A.2.1 above is that0 ∗ fg H d m ( R ) = 0 ∗ H d m ( R ) in the Gorenstein case. An open question predicts that 0 ∗ E = 0 ∗ fg E in general for the injective hull ofthe residue field E of a local ring [Smi93, § F -regularity; see [Smi93, Prop. 7.1.2], [LS01, Prop. 2.9].In general, since 0 ∗ fg E ⊆ ∗ E , this conjecture is equivalent to saying that the corresponding inclusionof annihilators Ann R (0 ∗ E ) ⊆ Ann R (0 ∗ fg E ) is an equality by [Smi94, Lem. 3.1(v)]. Theorem A.0.2 will follow by applying Matlis duality to Theorem A.2.1.A.2.3.
Matlis Duality.
Let ( R, m ) be a complete Noetherian local ring of dimension d , and let E be an injective hull of its residue field. Matlis duality is the exact contravariant functor on R -modules sending each M to M ∨ := Hom R ( M, R ). This functor takes Noetherian modules toArtinian modules, and vice versa, and is involutive when restricted to either class of modules: thatis ( M ∨ ) ∨ ∼ = M if M satisfies either the ACC or DCC condition on submodules.The canonical module of ( R, m ) in this context is defined as any R -module ω R that is a Matlisdual to the local cohomology module H d m ( R ). Since H d m ( R ) is Artinian, ω R is Noetherian. The ring R is a canonical module of itself when R is Gorenstein – a sister fact to the fact that E can betaken to be H d m ( R ). See [BH98; HK71] for basics on Matlis duality and the canonical module.Matlis duality gives us a perfect pairing ω R × H d m ( R ) → E which can be concretely understood by viewing w ∈ ω R as inducing the R -linear map “evaluationmap” ϕ w : H d m ( R ) → E The ideal Ann R (0 ∗ E ) of R is called the non-finitistic test ideal or the big test ideal , although the latter term isa bit confusing since it is a priori smaller than the usual (finitistic) test ideal Ann R (0 ∗ fg E ). The name “big” comesfrom the fact that the elements of Ann R (0 ∗ E ) can be used as test elements even for non-finitely generated moduleswhereas the elements of the usual test ideal Ann R (0 ∗ fg E ) works ( a priori ) only for tests where the ambient module isNoetherian. XCELLENCE, F -SINGULARITIES, AND SOLIDITY 51 given that, by definition, ω R = Hom R ( H dm ( R ) , E ). In particular, for an R -submodule M of H d m ( R ),we can define the annihilator of M in ω R , denoted Ann ω R M , { w ∈ ω R | ϕ w ( m ) = 0 for all m ∈ M } ⊂ ω R where ϕ w : H d m ( R ) → E is as defined above. See [Smi93, § Lemma A.2.4.
Let ( R, m ) be a Noetherian, S , complete local ring of dimension d , and let E bean injective hull of its residue field. If M is a submodule of H d m ( R ) , then M ∨ ∼ = ω R / Ann ω R M .Proof. See [Smi93, Prop. 2.4.1.(i)]. (cid:3)
Lemma A.2.5. ( R, m ) be a Noetherian complete local ring of dimension d , and let E be an injectivehull of its residue field. For any R -module map φ : R → S , the Matlis dual of the induced map H dm ( R ) φ ∗ −→ H dm ( S ) can be identified with the map Hom R ( S, ω R ) → ω R Ψ Ψ( φ (1)) . Proof.
This is really a much more general fact, following from the adjointness of tensor and Hom.First note that H d m ( R ) φ ∗ −→ H d m ( S ) can be identified with the map R ⊗ R H d m ( R ) φ ⊗ id −−−→ S ⊗ R H d m ( R ) . So applying the Matlis dual functor Hom R ( − , E ) we haveHom R ( S ⊗ R H d m ( R ) , E ) → Hom R ( R ⊗ R H d m ( R ) , E ) sending Φ Φ ◦ ( φ ⊗ id ) , which, under adjunction, becomes the mapHom R (cid:0) S, Hom R ( H d m ( R ) , E ) (cid:1) → Hom R (cid:0) R, Hom R ( H d m ( R ) , E ) (cid:1) sending Ψ Ψ ◦ φ. Now using the identification of Hom R ( H d m ( R ) , E ) with ω R , we have a mapHom R ( S, ω R ) → Hom R ( R, ω R ) sending Ψ Ψ ◦ φ, and since the latter is identified with ω R via the map f f (1), this becomesHom R ( S, ω R ) → ω R Ψ Ψ( φ (1)) , that is, the “evaluation at φ (1)” map. (cid:3) Proof of Theorem A.0.2.
Let ( R, m ) be a complete local Gorenstein domain of dimension d . Wefirst recall that because ( R, m ) is Gorenstein, the local cohomology module H d m ( R ) is an injectivehull of the residue field of R . Thus the test ideal for R is the annihilator of the R -module 0 ∗ fg H d m ( R ) .By Theorem A.2.1, the test ideal is then also the annihilator of the kernel of the natural map H d m ( R ) → H d m ( R + ) . To relate this to the evaluation at 1 map, we apply Matlis duality to the exact sequence0 → ∗ fg H d m ( R ) → H d m ( R ) → H d m ( R + ) . (A.2.5.1)Using Lemma A.2.5 to analyze the Matlis dual of this sequence, we get an exact sequenceHom R ( R + , ω R ) eval@1 −−−−→ ω R → (0 ∗ fg H d m ( R ) ) ∨ → . On the other hand, this sequence can be rewritten asHom R ( R + , ω R ) eval@1 −−−−→ ω R → ω R / Ann ω R ∗ fg H d m ( R ) → , where we have used Lemma A.2.4 for the identification(0 ∗ fg H d m ( R ) ) ∨ ∼ = ω R / Ann ω R ∗ H d m ( R ) . Thus, by exactness, the evaluation at 1 mapHom R ( R + , ω R ) eval@1 −−−−→ ω R φ φ (1)has image Ann ω R ∗ fg H d m ( R ) , an important submodule of ω R called the parameter test module in [Smi95(a)]. In the Gorensteincase, using the fact that ω R ∼ = R and H dm ( R ) ∼ = E , we finally conclude that the image ofHom R ( R + , R ) eval@1 −−−−→ r φ φ (1)is the test ideal by Proposition A.1.4. Theorem A.0.2 now follows. (cid:3) A.3.
Graded variants of absolute integral closures.
Suppose that ( R, m ) is an N -gradeddomain, finitely generated over R = k , a field, and with unique homogeneous maximal ideal m . Inthis case, there are two graded variants of R + .Take any z ∈ R + , and let f ( x ) = x n + a x n − + · · · + a n − x + a n (A.3.0.1)be a monic polynomial over R witnessing the integrality of z over R . We say that f ( x ) is homoge-nous if each a i is homogeneous in R and there is a choice of rational number β ∈ Q such that,setting deg x = β , the polynomial f ( x ) becomes homogeneous. Equivalently, this says that thereexists a rational number β such that deg a i = iβ for all i . We say that z ∈ R + has degree β in thiscase; it is not hard to see that this is well-defined [HH93, Lemma 4.1]. Using this, Hochster andHuneke defined the following two natural subrings of R + [HH93]: Definition A.3.1.
Let R be an N -graded domain, finitely generated over R = k , where k is afield of characteristic p >
0. Then • R +GR is the subring of T + generated by all elements a ∈ T + that satisfy some homogeneous equation (A.3.0.1) of integral dependence. Note that R +GR is Q -graded. • R +gr is the subring of R + generated by all elements a ∈ R +GR of integral degree . Note that R +gr is N -graded.Both R +gr and R +GR are big Cohen-Macaulay R -algebras in prime characteristic p >
0, by [HH93,Main Thm. 5.15].We work in the category of Z -graded R -modules. In particular, for Z -graded R -modules M and N , and d ∈ Z , we say that an R -linear homomorphism f : M → N has degree d if f ( M n ) ⊂ N n + d for all n ∈ Z . The set of all homomorphisms of degree d is written Hom d ( M, N ). We are interestedin the module of graded homomorphisms * Hom R ( M, N ) = M d ∈ Z Hom d ( M, N ) , which is a Z -graded R -module. If M is finitely generated, it is not hard to see that * Hom R ( M, N )is the same as Hom R ( M, N ), but in general the former module may be strictly smaller.The following analogue of Theorem A.0.2 holds:
Theorem A.3.2.
Let R be an N -graded Gorenstein domain, finitely generated over R = k , where k is a field of characteristic p > . Then the image of the evaluation at map * Hom R ( R +gr , R ) eval@1 −−−−→ R is the test ideal τ ( R ) of R . The same holds if we replace R +gr by R +GR , and define * Hom R ( R +GR , R ) = M d ∈ Q Hom d ( R +GR , R ) . XCELLENCE, F -SINGULARITIES, AND SOLIDITY 53 Proof.
First note that for simple degree reasons, R +gr ֒ → R +GR splits over R +gr and hence over R .This implies that the maps * Hom R ( R +gr , R ) eval@1 −−−−→ R and * Hom R ( R +GR , R ) eval@1 −−−−→ R have the same image. So it suffices to prove the statement for the Z -graded R -module R +gr .Now the proof is essentially the same as in the complete case, but we work in the graded ( ∗ )category instead, using graded Matlis duality and a graded analog of Theorem A.2.1.A.3.3. Graded Matlis duality. [For details, see [BH98].] If E denotes an injective hull of the residuefield of the graded ring ( R, m ) at the unique homogeneous maximal ideal, then E is Z -graded, andwe can define a graded Matlis dual functor M M ∨ = * Hom R ( M, E )which has all the same properties we reviewed for complete local Noetherian rings, provided werestrict to graded R -modules. In particular, because H d m ( R ) is graded, its graded Matlis dual is thecanonical module ω R , which is a graded R -module. We still have a perfect pairing ω R × H d m ( R ) → E which respects the grading, so the annihilator of a graded submodule of H d m ( R ) in ω R can be defined.With this set-up, the following graded analogues of Lemmas A.2.4 and A.2.5 hold, whose proof weomit. Lemma A.3.4.
Let ( R, m ) be an S N -graded domain of dimension d , finitely generated over R = k , a field, and with unique homogeneous maximal ideal m . Let E be an injective hull of itsresidue field R/ m .(1) If M is a graded submodule of H d m ( R ) , then M ∨ ∼ = ω R / Ann ω R M .(2) For any graded map of graded R -modules φ : R → S , the graded Matlis dual of the inducedmap H dm ( R ) φ ∗ −→ H dm ( S ) can be identified with the map * Hom R ( S, ω R ) → ω R Ψ Ψ( φ (1)) . Returning to the proof of Theorem A.3.2, the main theorem of [Smi95(b)] states that I ∗ = IR +gr ∩ R = IR +GR ∩ R for homogeneous parameter ideals I in R . So using the method of [Smi94],we see also that the finitistic tight closure of zero in H d m ( R ) is graded, and is the kernel of eitherof the natural maps H d m ( R ) → H d m ( R +gr ) or H d m ( R ) → H d m ( R +GR ). In particular, we have an exactsequence of graded modules 0 → ∗ fg H d m ( R ) → H d m ( R ) → H d m ( R +gr )whose graded Matlis dual produces an exact sequence of graded R -modules * Hom R ( R +gr , ω R ) eval@1 −−−−→ ω R → ω R / Ann ω R (0 ∗ fg H d m ( R ) ) → R is Gorenstein, we conclude that the image of * Hom R ( R +gr , R ) eval@1 −−−−→ R is the test ideal τ ( R ) following the same argument as in the complete case, finishing the proof ofTheorem A.3.2. (cid:3) References [And93] M. Andr´e. “Homomorphismes r´eguliers en caract´eristique p .” C. R. Acad. Sci. Paris S´er. I Math. url : https://gallica.bnf.fr/ark:/12148/bpt6k5471009x/f645.item . mr : . 21[And18] Y. Andr´e. “La conjecture du facteur direct.” Publ. Math. Inst. Hautes ´Etudes Sci.
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E-mail address : [email protected] URL : https://rankeya.people.uic.edu/ (T. Murayama) Department of Mathematics, Princeton University, Princeton, NJ 08544-1000, USA
E-mail address : [email protected] URL : https://web.math.princeton.edu/~takumim/ (K. E. Smith) Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA
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