Generalized F-depth and graded nilpotent singularities
aa r X i v : . [ m a t h . A C ] F e b GENERALIZED F -DEPTH AND GRADED NILPOTENT SINGULARITIES KYLE MADDOX AND LANCE EDWARD MILLER
Abstract.
We study generalized F -depth and F -depth, depth-like invariants associated to the canonical Frobeniusaction on the local cohomology modules of a local ring of prime characteristic. These invariants are naturallyassociated to the singularity types generalized weakly F -nilpotent and weakly F -nilpotent, which have uniformbehavior among the Frobenius closure of all parameter ideals simultaneously. By developing natural lower boundson these invariants, we are able to provide sufficient conditions which produce broad classes of rings that have thesesingularity types, including constructions like gluing schemes along a common subscheme, Segre products of gradedrings, and Veronese subrings of graded rings. We further analyze upper bounds on Frobenius test exponents of theseconstructions in terms of the input data. Introduction
In this paper we introduce and study generalized F -depth, a depth-like invariant inspired by Lyubeznik’s F -depth.Given a local or graded ring of prime characteristic, these F -depth and generalized F -depth track the degree to whichthe canonical Frobenius action on the local cohomology modules is nilpotent under composition. Moreover, theseinvariants are naturally associated to singularity types called weakly F -nilpotent and generalized weakly F -nilpotentsingularities as outlined below. The first of these is a weak form of F -nilpotent singularities, which were studied byBlickle in [Bli01] and formally introduced by Srinivas-Takagi in [ST17]. Recently, there has been a significant growingbody of literature exploring the properties of these singularity types, see [ST17, PQ19, Quy19, Mad19, KMPS].Throughout the introduction, we loosely refer to these three classes of singularities as nilpotent singularities .It is well-known that, for a local ring ( R, m ) of dimension d , the Frobenius action on H d m ( R ) is never nilpotent. Infact, observations in early work of Smith [Smi97] already showed the elements of H d m ( R ) nilpotent under Frobeniusmust be contained in the proper submodule 0 ∗ H d m ( R ) , the tight closure of 0 in H d m ( R ). As the Frobenius action onthe lower local cohomology modules is possibly nilpotent, a local ring R is called weakly F -nilpotent if the canonicalFrobenius action on each lower local cohomology module is nilpotent. A weakly F -nilpotent ring is F -nilpotent if,in addition, the Frobenius action restricted to 0 ∗ H d m ( R ) is nilpotent. It is easy to verify that F -rational singularitiesare F -nilpotent, the converse holding if the ring in question is also F -injective. There are significant connectionsbetween F -nilpotent singularities and vanishing of Hodge-type filtrations outlined in [ST17], which is naturallyrelated to the weak ordinarity conjecture.In [Quy19], Quy shows that weakly F -nilpotent local rings have finite Frobenius test exponents . This is a uniformexponent which checks membership inside the Frobenius closure of each parameter ideal of the ring simultaneously.One cannot hope to obtain an exponent that works for all ideals, see [Bre06] and [KS06]. Generalizing [Quy19], thefirst named author introduced generalized weakly F -nilpotent rings in [Mad19] as those for which the submodule ofeach lower local cohomology module consisting of elements annihilated by a large iterate of the Frobenius actionhas finite colength. These singularities are the widest known class to admit a finite Frobenius test exponent.Our approach is based on a natural analogy between Cohen-Macaulay rings and weakly F -nilpotent rings,initiated in work of Lyubeznik [Lyu06] and highlighted by the above listed works on nilpotent singularities. Inparticular, we study similarities between ordinary depth and Lyubeznik’s F -depth to codify that weakly F -nilpotentsingularities are analogous to Cohen-Macaulay singularities. Indeed, weakly F -nilpotent singularities are Cohen-Macaulay after applying the Lyubeznik functor; thus rings enjoying these conditions share many properties incommon. Just as a Cohen-Macaulay ring is one of full depth, a weakly F -nilpotent ring is one of full F -depth. Tocomplete the picture, we introduce generalized F -depth , and recognize a generalized weakly F -nilpotent ring as oneof full generalized F -depth. Date : February 1, 2021.
Important applications come from theorems we prove about F -depth and generalized F -depth. In particular,we are able to provide ample sufficient conditions to ensure standard constructions in the local and graded settingproduce rings with nilpotent singularities. Thus far, no systematic framework for such constructions is present inthe literature. This makes the study of nilpotent singularities much less developed than other, more well-studied, F -singularity types. As these singularities are inextricably connected to deep conjectures and important conceptsin tight closure, it is of critical value to have suitable classes of examples to study in detail.In the body of the paper, stronger forms of the results explained below are provided, in technical terms, aslower bounds on either F -depth or generalized F -depth. These sharper statements establish the importance inrecasting questions about nilpotent singularities as statements about controlling these depth parameters; therebystrengthening the valuable analogy between weakly F -nilpotent and Cohen-Macaulay singularities. We present inthe introduction however, a summary of only the results pertinent to constructing rings with nilpotent singularities.To avoid technicalities, we retain for the rest of the introduction that k denotes a perfect field of prime characteristic.The first of our results concerns gluing two schemes together along a common subscheme. It is natural to expectthat if the schemes involved have nilpotent singularities, there should be simple conditions to ensure the gluing alsohas nilpotent singularities. In particular, we provide a nilpotent version of a result of Schwede, see [Sch09, Prop. 4.8],which concerns gluing F -injective, Cohen-Macaulay singularities. We obtain gluing results for weakly F -nilpotentand generalized weakly F -nilpotent singularities, as well as the following result for F -nilpotent singularities. Theorem 1. (Theorem 3.4) If ( R, m , k ) is a local ring with ideals a and a so that dim R/ ( a ∩ a ) = dim( R/ a ) =dim( R/ a ) = dim R/ ( a + a ) + 1, and R/ a and R/ a are F -nilpotent and R/ ( a + a ) is weakly F -nilpotent, then R/ a ∩ a is F -nilpotent.We then turn our attention to graded Frobenius actions and graded nilpotent singularities. In particular, weanalyze how F -depth and generalized F -depth behave under taking Segre products and Veronese subrings of gradedrings. Graded modules with Frobenius actions are well studied from the point of view of Lyubeznik’s F -modules,see [Bli01, LSW16] and references therein. The full framework of F -modules is not needed for our considerations,which simplifies the statements and applications in our approach. Furthermore, careful study of the Hartshorne-Speiser-Lyubeznik numbers are needed to obtain effective bounds on Frobenius test exponents, and this informationis lost in the context of F -modules. The novelty of working with graded rings and graded Frobenius actions is thatone obtains a natural Frobenius action on the degree 0 part of a graded module with Frobenius action. This degree0 part controls much of the story for graded nilpotence and is important for studying graded nilpotent singularities.We call a module nilpotent in degree Theorem 2. (Corollary 4.11 and Theorem 4.14) Suppose R and S are graded rings of depth at least two anddimensions d R and d S respectively, both with degree 0 part k . Set T = R S the Segre product. If R and S aregeneralized weakly F -nilpotent, so is T . If we further assume R and S are weakly F -nilpotent and H d R m R ( R ) and H d S m S ( S ) are nilpotent in degree 0, then T is weakly F -nilpotent.To further demonstrate these applications, we establish that an interesting class of well-known Segre productsare weakly F -nilpotent. The singularities of Segre products of elliptic curves and the projective line are essentiallythe simplest normal non-Cohen-Macaulay domains of dimension 3, c.f., the example at the end of Section 5 [Lyu06].Replacing the elliptic curve by a general Fermat hypersurface of degree d , we find that a certain class are not thatfar off from being Cohen-Macaulay under the analogy described above. Theorem 3. (Theorem 4.18) Let d ≥ p the characteristic of k . Set S = k [ u, v ] and consider the Fermathypersurface R = k [ x , . . . , x n ] / ( x d + · · · + x dn − − x dn ) with p > d and n ≥
2. If p ≡ − d , then the Segreproduct R S is weakly F -nilpotent.Concerning Veronese subrings, we show that the nilpotence of the local cohomology modules descends from agraded ring to all of its Veronese subrings, and in particular if R is (generalized) weakly F -nilpotent, so is S . Weutilize this to provide a nilpotent version of a theorem of Singh, [Sin00, Prop. 3.1], concerning how F -rationalsingularities behave under taking Veronese subrings. Theorem 4. (Theorem 4.20) Suppose R is a graded ring of dimension d and degree 0 part k . If R is F -nilpotenton the punctured spectrum, then for all n ≫
0, the Veronese R ( n ) is weakly F -nilpotent. Furthermore, H d m R ( R ) isnilpotent in degree 0 if and only if for all n ≫
0, the Veronese R ( n ) is F -nilpotent. ENERALIZED F -DEPTH AND GRADED NILPOTENT SINGULARITIES 3 To provide a wide class of generic examples, we apply the techniques developed to provide a nilpotent version ofthe F -rational characterization from [KSSW09, Thm. 3.1], see Theorem 4.21. In particular, this provides a methodfor easily constructing hypersurfaces over multigraded rings which have diagonal Veronese subalgebras that havethe same F -depth as the original ring.In Section 5, we address natural applications of these theorems by giving explicit bounds on Frobenius testexponents of the constructions in terms of their input data. These arise from bounds on the Harshorne-Speiser-Lyubeznik numbers of the constructions we produce earlier in the paper, see Theorems 5.5, 5.8 and Lemma 5.7for these calculations. We can then apply the main results of [Quy19] and [Mad19], adapted to the graded settingas necessary. These bounds for Frobenius test exponents or the appropriate graded analogue (see Definition 5.2)are given for gluing in Theorem 5.5 and Corollary 5.6, Segre products in Corollaries 5.10 and 5.12, and Veronesesubrings in Theorem 5.13. Acknowledgments:
We are extremely grateful to Luis N´u˜nez-Betancourt, Jack Jeffires, Paolo Mantero, andThomas Polstra for enlightening discussions related to the work here. We thank Ian Aberbach for reviewing initialdrafts of this paper. We are also thankful to Austyn Simpson for discussion and suggesting improvements to thestatement of Theorem 3.2. 2.
Preliminaries
Throughout, we assume all rings are noetherian, of prime characteristic p >
0, and F -finite, i.e. the Frobeniusmap F : R → R is finite. We denote by R ◦ the complement of the minimal primes; R ◦ = R \ S q ∈ min( R ) q . For e ≥ R -module M , denote by F e ∗ M the R -module with R -action twisted by the e -th iterated Frobenius. Thesingularities types we study here are defined in terms of Frobenius actions on local cohomology modules, however,when possible we state results purely in terms of modules with Frobenius actions. We briefly review the necessaryconcepts, but for a more elaborate treatment see [EH08]. Definition 2.1.
For R -modules M and N , an element of Hom R ( M, F e ∗ N ) is called a p e -linear map . A p -linearendomorphism of M is called a Frobenius action on M .It is convenient to describe Frobenius actions using the Frobenius skew polynomial ring. Specifically, consider R h χ i the non-commutative polynomial ring in one variable over R and let R [ F ] := R h χ i / ( { r p χ − χr | r ∈ R } ). Thecategory of left R [ F ]-modules is equivalent to the category whose objects are pairs ( M, ρ ) where ρ : M → M is aFrobenius action and morphisms ( M, ρ ) → ( M ′ , ρ ′ ) are R -module maps ϕ : M → M ′ for which ρ ′ ϕ = ϕρ . We freelyswitch between these perspectives and establish when necessary that R -linear maps between left R [ F ]-modulesare also left R [ F ]-module maps. That is to say, we refer to a left R [ F ]-module by a pair ( M, ρ M ) where ρ M is aFrobenius action on M and we ruthlessly suppress the subscript when M is clear. Remark 2.2.
We will tacitly use that left modules over a non-commutative ring still form an abelian category. Forexample, we will repeatedly use homological algebra in the category of R [ F ]-modules to obtain induced Frobeniusactions on (co)homology of complexes and connecting maps which commute with the Frobenius actions and are afortiori R -linear.The most important examples of R [ F ]-modules are arguably local cohomology modules. Specifically, for eachideal a in a ring R , the Frobenius endomorphism F : R → F ∗ R induces a natural Frobenius action on the localcohomology modules F : H i a ( R ) → F ∗ H i a ( R ). If a = ( x , . . . , x t ), then we identify H t a ( R ) = lim −→ R/ ( x j , . . . , x jt )and the induced Frobenius action on H t a ( R ) sends a class [ z + ( x j , . . . , x jt )] to [ z p + ( x jp , . . . , x jpt )]. All mentions of R [ F ]-module structures on H i a ( R ) will be assumed to be this canonical one.We will primarily focus on Frobenius actions which are nilpotent under composition. We now articulate thefollowing basic but important concepts which arise in the study of left R [ F ]-modules. Definition 2.3.
Let R be a ring and let ( M, ρ ) and ( M ′ , ρ ′ ) be R [ F ]-modules. • An R -submodule N ⊂ M is ρ -stable or ρ -compatible if ρ ( N ) ⊂ N . • The orbit closure of a ρ -stable submodule N ⊂ M is defined: N ρM = { m ∈ M | ρ e ( m ) ∈ N for some e ∈ N } , and N is ρ -closed if N ρM = N . • M is nilpotent if 0 ρM = M , i.e. for each m ∈ M there is an e ∈ N such that ρ e ( m ) = 0.It is easy to check that N ρM is a ρ -closed R -submodule of M and ρ -stable submodules of M correspond to left R [ F ]-submodules of M . KYLE MADDOX AND LANCE EDWARD MILLER
Example 2.4.
Let (
M, ρ ) be a R [ F ]-module and let I ⊂ R be an ideal. Then, ρ ( IM ) = I [ p ] ρ ( M ) ⊂ IM , so IM isa ρ -stable submodule of M .In [ST17], Srinivas and Takagi formally defined an important class of singularities ( R, m ) where the canonicalFrobenius action on H i m ( R ) is nilpotent. We will give definitions of several related classes of singularities inDefinition 2.10. When ( R, m ) is local it is also useful to study modules which are finite length away from beginnilpotent, i.e. modules M for which the orbit closure of 0 ρM is of finite colength in M . For that reason, we make thefollowing definition inspired by generalized Cohen-Macaulay rings, for which the lower local cohomology is finitelength. Definition 2.5.
Let ( R, m ) be a local ring and let ( M, ρ ) be an R [ F ]-module. Call M generalized nilpotent if M/ ρM is finite length.2.1. Tight and Frobenius closure of submodules.
The Frobenius action on the top local cohomology of aCohen-Macaulay local ring has been studied frequently for its connections to tight closure theory. K. E. Smithshowed that in many cases, the tight closure of 0 in H d m ( R ) was the largest F-stable submodule of H d m ( R ). See[Smi97, Prop. 2.5] and [EH08, Discussion 2.10] for more. We now review tight and Frobenius closures of modules. Definition 2.6.
Let N ⊂ M be R -modules. For any c ∈ R and e ∈ N , define the map µ ec : M → ( M/N ) ⊗ R F e ∗ R to be the composition of the maps below. M M/N ( M/N ) ⊗ R F e ∗ R ( M/N ) ⊗ R F e ∗ R π id ⊗ F e id ⊗ F e ∗ ( c ) The tight closure of N in M , denoted N ∗ M , is defined as N ∗ M = { m ∈ M | there exists c ∈ R ◦ such that m ∈ ker( µ ec ) for all e ≫ } . and the Frobenius closure of N in M , denoted N FM , is similarly defined as: N FM = { m ∈ M | m ∈ ker( µ e ) for all e ≫ } . By definition, we have N ⊂ N FM ⊂ N ∗ M ⊂ M . Remark 2.7.
There is an unfortunate overlap of terminology for experts familiar with tight and Frobenius closureof ideals a in rings R . The notions of Frobenius closure of ideals a F and the orbit closure of a as an R -submodulediffer. The reason is that, unlike the situations we’ve considered so far, the Frobenius closure of ideals as definedabove is not a statement about R [ F ]-modules. This notion of Frobenius closure for submodules extends the notion ofFrobenius closure of ideals, however, taking R to be an R [ F ]-module in terms of the usual Frobenius endomorphism,the orbit closure of a is √ a , instead of the much finer-grained information of the Frobenius closure a F . We onlyapply ( ) F in situations when the context is clear.When ( R, m ) is a local ring of dimension d , the map µ e on H d m ( R ) is simply the canonical Frobenius action F e on H d m ( R ). In particular, the two distinct notions (using the maps { µ e } e ∈ N versus the orbit closure) of 0 FH d m ( R ) agree,and furthermore the tight closure of zero 0 ∗ H d m ( R ) is F -stable in H d m ( R ). Remark 2.8.
The existential quantifier in Definition 2.6 can be simplified by using test elements. Recall, reducedexcellent local rings ( R, m ) admit big test elements, i.e. elements c ∈ R ◦ so that for any pair of R -modules N ⊂ M , η ∈ N ∗ M if and only if µ ec ( η ) = 0 for e ≫
0. Here big refers to lack of finite generation assumptions on the modules,and the nuance is that c is independent of the modules N and M . For proof, see [HH90, Thm. 6.17, Cor. 6.26].2.2. Singularity types.
Before we define the central singularity types of interest, we first recall the definition ofa characteristic-free singularity which generalizes the Cohen-Macaulay property.
Definition 2.9.
Let ( R, m ) be a local ring. R is generalized Cohen-Macaulay if H j m ( R ) is finite length for0 ≤ j < dim( R ).The definitions of the first two singularity types below are intended to mimic the relationship between Cohen-Macaulay, identified by vanishing lower local cohomology, and generalized Cohen-Macaulay, identified by finitelength lower cohomology, by replacing vanishing and finite length by nilpotent and generalized nilpotent. ENERALIZED F -DEPTH AND GRADED NILPOTENT SINGULARITIES 5 Definition 2.10.
Let ( R, m ) be a local ring of dimension d . • R is generalized weakly F -nilpotent if H i m ( R ) is generalized nilpotent for i < d . • R is weakly F -nilpotent if H i m ( R ) is nilpotent for i < d . • R is F -nilpotent if it is weakly F -nilpotent and 0 ∗ H d m ( R ) = 0 FH d m ( R ) .The terminology of F -nilpotence was introduced in [ST17], though the notion predates this paper, see [BB05, Def.4.1]. It is easy to see that rings which are both F -injective and F -nilpotent are F -rational, see [ST17, Prop. 2.4].In this setting, F -rational is equivalent to every parameter ideal being tightly closed in R .The class of generalized weakly F -nilpotent rings was introduced in [Mad19, Def. 3.4], where it was shownthat such rings have finite Frobenius test exponent for the class of parameter ideals. See [Mad19, Thm. 3.6] whichgeneralizes the same theorem for weakly F -nilpotent rings by Quy in [Quy19, Main Theorem (2)]. Similar resultswere already known for Cohen-Macaulay [KS06], and generalized Cohen-Macaulay rings [HKSY06].We conclude this subsection by outlining a hierarchy of the singularity classes mentioned above. F -rational Cohen-Macaulay generalized Cohen-Macaulay F -nilpotent weakly F -nilpotent generalized weakly F -nilpotent2.3. Nilpotent Frobenius actions.
We now outline some preparatory facts and definitions about nilpotent R [ F ]-modules which will be aimed at constructing rings where the local cohomology modules have a prescribed level ofnilpotence. The following remarkable finiteness theorem plays a vital role. It is stated in terms of the Hartshorne-Speizer-Lyubeznik numbers. Definition 2.11.
Let R be a ring and ( M, ρ ) be an R [ F ]-module. The Hartshorne-Speiser-Lyubeznik numberof M is defined as follows:HSL( M ) = inf { e ∈ N | ρ e ( m ) = 0 for all m ∈ ρM } ∈ N ∪ {∞} . Note if M is finitely generated, then HSL( M ) < ∞ since 0 ρM is finitely generated. If M is not finitely generated,there could be a sequence of elements { m n } n ∈ N such that the required iterate of ρ to kill m n tends to infinity.However, the following result regarding when the Hartshrone-Speiser-Lyubeznik number of an R [ F ]-module is finiteis extremely important because it may be applied to the local cohomology modules of R . See [Lyu97, Prop. 4.4],[HS77, Prop. 1.11], and [Sha06, Cor. 1.8] for proofs. Theorem 2.12 (The Hartshorne-Speiser-Lyubeznik Theorem).
Let M be an R [ F ] -module which is artinianas an R module. Then HSL( M ) < ∞ . Definition 2.13.
For a local ring ( R, m ), (re-)define:HSL( R ) = max (cid:8) HSL( H j m ( R )) | ≤ j ≤ dim R (cid:9) , which is notably finite by the proceeding theorem.This single exponent uniformly annihilates annihilates all nilpotent elements in every local cohomology moduleof R . There are many advantages to viewing 0 ρM as the kernel of the single map ρ HSL( M ) . We now need severalbasic lemmas about nilpotent modules with Frobenius actions. Lemma 2.14.
Let R be a ring of prime characteristic p > and let A B C be a short exact sequence of R [ F ] -modules. Then B is nilpotent if and only if A and C are nilpotent.Proof. Since A ⊂ B , if B is nilpotent then A is nilpotent. Further, if c + A ∈ C , then ρ eC ( c + A ) = ρ eB ( c )+ A = 0+ A forsome e , so C is also nilpotent. Now suppose A and C are nilpotent, and let b ∈ B . If b ∈ A , then ρ eB ( b ) = ρ eA ( b ) = 0for some e . Otherwise, taking b + A ∈ C , we have ρ eC ( b + A ) = ρ eB ( b ) + A = 0 + A for some e , which implies ρ eB ( b ) ∈ A . But A is nilpotent, for some e ′ ∈ N , ρ e ′ A ( ρ eB ( b )) = ρ e + e ′ B ( b ) = 0, so B is nilpotent.This shows that the full subcategory of nilpotent R [ F ]-modules forms a Serre subcategory of R [ F ]-modules. Wealso have a generalized nilpotent version of the previous lemma. KYLE MADDOX AND LANCE EDWARD MILLER
Lemma 2.15.
Let ( R, m ) be a local ring and let A B C be a short exact sequence of R [ F ] -modules. If each has finite HSL number, then B is generalized nilpotent if andonly if A and C are generalized nilpotent.Proof. Suppose B is generalized nilpotent. Since 0 ρ A A = 0 ρ B B ∩ A , we have m N A ⊂ m N B ⊂ ρ B B for some N ∈ N .Consequently, m N A ⊂ ρ A A which shows A is generalized nilpotent. Now, if ξ + A ∈ C and N ∈ N is such that m N B ⊂ ρ B B , then m N ( ξ + A ) = m N ξ + A and ρ eC ( m N ξ + A ) = ρ eB ( m N ξ ) + A . But for e = HSL( B ) we have ρ eB ( m N ξ ) = 0, so m N ( ξ + A ) ⊂ ρ C C implying that C is generalized nilpotent.Now suppose A and C are generalized nilpotent and let N = max { λ ( A/ ρ A A ) , λ ( C/ ρ C C ) } . Further, set e =max { HSL( A ) , HSL( C ) } . Now, for any ξ ∈ B , in C we have m N ( ξ + A ) ⊂ ρ C C so ρ eB ( m N ξ ) + A = A which implies ρ eB ( m N ξ ) ⊂ A . Then, in A , we have:( m N ) [ p e ] ( ρ eB ( m N ξ )) = ρ eB ( m N ξ ) ⊂ ρ A A ⊂ ρ B B , and consequently m N ξ ⊂ ρ B B . Since ξ is arbitrary and 2 N only depends on A and C , we have that B isgeneralized nilpotent as required.2.4. The Relative Frobenius Action.
When studying how F -singularities descend along hypersurfaces, it hasbeen fruitful to study the relative Frobenius map, which arises from the following commutative diagram. It has beenuseful in addressing deformation questions about several classes of F -singularities, including F -injective singularitiesin [HMS14] and F -nilpotent singularities in [PQ19]. Remark 2.16.
Let R be a ring and let I ⊂ R be an ideal. For each e ∈ N , we have a commutative diagram: R/I R/IR/I [ p e ] F e f eR π e where f eR ( y + I ) = y p e + I [ p e ] is a p e -linear map of R -modules and π e is canonical projection. This induces acommutative diagram for any j ∈ N and J ⊂ R an ideal: H jJ ( R/I ) H jJ ( R/I ) H jJ ( R/I [ p e ] ) F e f eR π e where F e is the canonical Frobenius action on H jJ ( R/I ), f eR is p e -linear, and π e is R -linear.In particular, since F e = π e ◦ f eR , if f eR is the zero map for some e ∈ N , then H jJ ( R/I ) is nilpotent as an R [ F ]-module.2.5. The F -depth of a local ring. Introduced by Lyubeznik, in [Lyu06], the F -depth of a local ring measuresthe first non-nilpotent index in the long exact sequence for local cohomology just as depth measures the firstnon-vanishing index. Definition 2.17.
Let ( R, m ) be a local ring of prime characteristic p >
0. The F -depth of R is defined asF-depth R = inf { j | H j m ( R ) is not nilpotent } . Remark 2.18.
By definition, R is weakly F -nilpotent if and only if F-depth R = dim R . We will show severalresults about natural lower bounds for F-depth R which will specialize to statements about weakly F -nilpotent ringswhen F-depth R = dim R . F -depth was used by Lyubeznik to answer a question of Grothendeick about vanishing of local cohomology. Wenow mention several basic facts about F -depth recorded throughout [Lyu06], where it was also compared with the F -depth of a scheme defined by Hartshorne and Speiser in [HS77]. Lemma 2.19.
Suppose ( R, m ) is a local ring of prime characteristic p > . (1) We have ≤ F-depth R ≤ dim R . If dim R > , then F-depth R ≥ . (2) We have
F-depth R = F-depth b R and, unlike ordinary depth, F-depth R = F-depth R/ √ . Note that H d m ( R ) is never nilpotent, see [Lyu06, Lem. 4.2]. Under mild hypotheses, H d m ( R ) cannot be generalizednilpotent either. We are grateful to T. Polstra for discussions leading to this observation. ENERALIZED F -DEPTH AND GRADED NILPOTENT SINGULARITIES 7 Lemma 2.20. If R is a locally equidimensional ring of dimension d > , then, H d m ( R ) is not generalized nilpotent.Proof. For any q ∈ Spec ( R ), set H ( q ) = H ht qq R q ( R q ) / FH ht qq R q ( R q ) and H = H ( m ). Notably H ( q ) = 0 for any q ∈ Spec ( R ). Denoting by ( ) ∨ the Matlis dual over R , [KMPS, Lem.5.1] gives that H ∨ localizes to the Matlis dual over R q of H ( q ). Thus, Supp( H ) = Supp( H ∨ ) = Spec ( R ) and so H cannot be finite length unless d = 0.We can define a similar notion to F -depth using generalized nilpotence. Definition 2.21.
Suppose ( R, m ) is a local ring of prime characteristic p >
0. We define the generalized F -depthof R , written gF-depth R , as the least index j where H j m ( R ) is not generalized nilpotent, or 0 if dim( R ) = 0.It is apparent that gF-depth R ≥ F-depth R . In fact, in light of Lemma 2.20, when R is locally equidimensional,one has depth R ≤ F-depth R ≤ gF-depth R ≤ dim R. We now extend the definition of F -depth and generalized F -depth to R [ F ]-modules. Since F ∗ commutes withlocalization, given an R [ F ]-module M , ρ M : M → F ∗ M induces a Frobenius action ρ : H jI ( M ) → F ∗ H jI ( M ). Definition 2.22.
Let ( R, m ) be a local ring and let M be a finitely generated R -module with Frobenius action ρ .We define the F -depth of M , written F-depth M , to be the least index j such that H j m ( M ) is not nilpotent underthe induced action ρ . Similarly, the generalized F -depth of M , written gF-depth M , is the least index j suchthat H j m ( M ) is not generalized nilpotent under the induced action.Since local cohomology commutes with direct sum and a direct sum is (generalized) nilpotent if and only if eachsummand is by Lemmas 2.14 and 2.15, we can use this setting to study F -depth and generalized F -depth across avariety of natural short exact sequences. First, we establish an analogue to the behavior of depth on short exactsequences. We restrict to the case of modules given by a quotient of a finitely generated free module R n by an F -stable submodule, endowed with the induced action coming from F ⊕ n . Lemma 2.23.
Suppose ( R, m ) is a local ring of prime characteristic p > and let: A B C be a short exact sequence of R [ F ] -modules where A, B, C are quotients of finitely generated free R -modules asdescribed above. Then: (1) F-depth B ≥ min { F-depth A, F-depth C } , (2) F-depth A ≥ min { F-depth B, F-depth C + 1 } , and (3) F-depth C ≥ min { F-depth B, F-depth A − } . If the sequence is split, then
F-depth B = min { F-depth A, F-depth C } . Finally, the same inequalities and equalityhold replacing F-depth with gF-depth .Proof.
We will prove (2), the others are similar. Suppose t < min { F-depth B, F-depth C + 1 } and we will show H j m ( A ) is nilpotent for j ≤ t . Since we have the long exact sequence of R [ F ]-modules: · · · H j − m ( C ) H j m ( A ) H j m ( B ) · · · δ α we know that the image of H j m ( A ) is nilpotent as j < F-depth B so since HSL( H j m ( B )) = e < ∞ , ρ eH j m ( A ) ( H j m ( A ))is inside ker( α ) = im( δ ). But, im( δ ) is a subquotient of H j − m ( C ) and j − < F-depth C and is thus nilpotent byLemma 2.14. Consequently, ρ eH j m ( A ) ( H j m ( A )) is nilpotent which implies H j m ( A ) is also nilpotent, as required.When the sequence is split, we apply Lemma 2.14. To see the final claim, repeat the above proof with Lemma 2.15instead of Lemma 2.14, noting that the modules in question have finite Hartshorne-Speiser-Lyubeznik numbers as A , B , and C are quotients of finitely generated free R -modules.3. Gluing results
We now study how F -depth and generalized F -depth behave under gluing. Specifically, suppose P is a propertyof local rings. The property P glues for schemes if, given any scheme X and two subschemes Y and Y such that X = Y ∪ Y , all local rings of Y , Y , and Y ∩ Y have property P implies all local rings of X have property P .An example of an F -singularity type which glues was given by Schwede in [Sch09, Prop. 4.8], which states thatif X is Cohen-Macaulay, dim Y = dim Y = dim X and all local rings of Y , Y , and Y ∩ Y have F -injective KYLE MADDOX AND LANCE EDWARD MILLER singularities, then X has F -injective singularities. Quy and Shimomoto also showed that the class of stably F H -finite singularities glues, see [QS17, Thm. 5.7]. In the characteristic-independent case, Dao-De Stefani-Ma proved in[DDSM, Prop. 2.8] that cohomologically full singularities glue, but in prime characteristic p >
0, cohomologicallyfull is equivalent to F -full , a singularity type introduced by Ma and Quy in [MQ18].As the conditions are local, in checking that a property P glues, one may without loss of generality assume X = Spec R where ( R, m ) is a local ring and Y and Y are defined by ideals a and a in R with a ∩ a = 0.Conditions are then imposed on the quotient rings R/ a , R/ a , and R/ ( a + a ). Throughout the remainder of thissection, we work modulo a ∩ a . Thus we fix the following setting and notation, and use it in theorem statementsbelow: fix ( R, m ) a local ring of dimension d with ideals a and a which have a ∩ a = 0 and set b = a + a . Wenotably have dim( R/ a i ) = dim( R ) in this case.If dim( R ) = 0 , R ) ≥
2. We also willtypically require conditions on dim( R/ b ); the following fact is often helpful. Lemma 3.1.
Suppose dim( R/ b ) = dim( R ) − , and that R is equidimensional. Then, H d m ( R ) ≃ H d m ( R/ a ) ⊕ H d m ( R/ a ) as R [ F ] -modules.Proof. Since R is equidimensional, height( b ) = 1 and so b contains a parameter x ∈ R . We have H d m ( R ) is an R x -module, so by hom-tensor adjointness we have:Hom R ( H d − m ( R/ b ) , H d m ( R )) ≃ Hom R x (cid:0) H d − m ( R/ b ) ⊗ R R x , H d m ( R ) (cid:1) , and since x ∈ b , this module vanishes. Thus, the connecting map H d − m ( R ) → H d m ( R ) is identically zero in the longexact sequence in local cohomology induced by the short exact sequence of R [ F ]-modules below.0 R R/ a ⊕ R/ a R/ b Theorem 3.2. If F-depth R/ b > d − and F-depth R/ a i > d − for i = 1 , , then R is weakly F -nilpotent. Inparticular, if dim R/ b ≥ dim R − and each of R/ b , R/ a , and R/ a are weakly F -nilpotent, then so is R .Similarly, if gF-depth R/ b > d − and gF-depth R/ a i > d − for each i , then, R is generalized weakly F -nilpotent.If dim R/ b ≥ dim R − and each of R/ b , R/ a , and R/ a are generalized weakly F -nilpotent, then so is R .Proof. Note that the short exact sequence:0
R R/ a ⊕ R/ a R/ b R [ F ]-modules, considering each of the quotient rings with the Frobenius map andthe induced action on the direct sum. From Lemmas 2.19 and 2.23, we have:F-depth R ≥ min { F-depth R/ a , F-depth R/ a , F-depth R/ b + 1 } > d − . This forces F-depth R = d , i.e. R is weakly F -nilpotent. For the generalized case, apply the generalized F -depthresult from the same lemma. Remark 3.3.
The dimension of R/ b cannot be lowered further and obtain a similar gluing result for weak F -nilpotence. If R , R/ a i , and R/ b are as in the theorem, but b = dim( R/ b ) < dim( R ) −
1, we have an exact sequenceof R [ F ]-modules: A b H b m ( R/ b ) H b +1 m ( R ) A b +1 β δ and notably A b and A b +1 are nilpotent, but H b m ( R/ b ) is not nilpotent since it is a top local cohomology module.Since im( β ) = ker( δ ) is nilpotent, we cannot have that im( δ ) is nilpotent by Lemma 2.14. Thus, H b +1 m ( R ) is notnilpotent and since b + 1 < dim( R ), we cannot have F-depth( R ) = dim( R ).For generalized weak F -nilpotence, the same obstruction will occur if R/ b is locally equidimeinsional, as in thiscase, H b m ( R/ b ) is not generalized nilpotent by Lemma 2.20, and similarly to the weakly F -nilpotent case, im( δ ) isnot generalized nilpotent either. So gF-depth R = b + 1 < dim( R ), and R cannot be generalized weakly F -nilpotent.We now show a similar result for F -nilpotent singularities. Theorem 3.4.
Suppose d = dim R = dim R/ a i for i = 1 , and dim R/ b ≥ d − . If R/ a and R/ a are F -nilpotentand R/ b is weakly F -nilpotent, then R is F -nilpotent. ENERALIZED F -DEPTH AND GRADED NILPOTENT SINGULARITIES 9 Proof.
By Theorem 3.2, we know R is weakly F -nilpotent, and consequently equidimensional by [PQ19, Prop. 2.8(3)].Now, if dim( R/ b ) = d −
1, we have by Lemma 3.1 that R is F -nilpotent since tight closure and Frobenius closurecommute with direct sums and R/ a and R/ a are F -nilpotent.If dim( R/ b ) = d , we consider the exact sequence in local cohomology coming from the natural short exactsequence. For convenience, we let: A d = H d m ( R/ a ) ⊕ H d m ( R/ a ) and A d − = H d − m ( R/ a ) ⊕ H d − m ( R/ a ) . We get a commutative diagram as below where the vertical maps are either the appropriate canonical Frobeniusaction or the direct sum of canonical Frobenius actions (denoted by ρ e ) in each place. · · · A d − H d − m ( R/ b ) H d m ( R ) A d · · · A d − H d − m ( R/ b ) H d m ( R ) A d βρ e δF e αF e ρ e β δ α We need to show 0 ∗ H d m ( R ) is nilpotent. Recall R is F -nilpotent if and only if R/ √ F -nilpotent by [PQ19,Prop. 2.8(2)], so without loss of generality we may assume R is reduced, whence big test elements exist – seeRemark 2.8. For an R [ F ]-module M and a test element c , one has a composition µ eM,c : M ⊗ R R id ⊗ F eR −−−−→ M ⊗ R F e ∗ R id ⊗ F e ∗ ( c ) −−−−−−→ M ⊗ R F e ∗ R and x ∈ ∗ M if and only if x ∈ ker µ eM,c for e ≫ x ∈ FM if and only if F e ( x ) = 0 for some e ∈ N . By [PQ19,Rmk. 2.6], applying this for M a top local cohomology module, we may replace id ⊗ F eR with F e , the canonicalFrobenius action on local cohomology. Thus we may consider tight closure arguments without using the maps µ eM,c and only using the Frobenius action F e . The same argument applies to the direct sum A d considering Lemma 2.14.Now, let ξ ∈ ∗ H d m ( R ) and let c ∈ R ◦ be a test element. We have for all e ≫ cF e ( ξ ) = 0, so in particular cρ e ( α ( ξ )) = 0. But 0 ∗ A d = 0 ρA d by definition of F -nilpotence and the fact that tight closure commutes with directsum, so for e = HSL( A ) we have F e ( ξ ) ∈ ker( α ) = im( δ ). But, H d − m ( R/ b ) is nilpotent, and consequently so is F e ( ξ ), thus ξ ∈ FH d m ( R ) .We can apply Remark 3.3 again to show no similar result can hold for a smaller-dimensional R/ b , since R mustbe weakly F -nilpotent to be F -nilpotent. Example 3.5.
Suppose S = k [ x, y, z, w ] ( x,y,z,w ) . Set a = ( xw − yz, yw − z , xz − y w, x z − y ) and a = ( x, z ).One has S/ a ∼ = k [ s , s t, st , t ] is weakly F -nilpotent which can be seen by deformation along x using [PQ19,Thm. 4.2]. Clearly a is Cohen-Macaulay, whence weakly F -nilpotent. Similarly, b = a + a = ( x, z, yw , y w, y )and S/ b ∼ = k [ y, w ] / ( yw , y w, y ) is easily seen to be weakly F -nilpotent. Thus by Theorem 3.2, one has R = S/ ( a ∩ a ) = S/ ( yz − xw, z − xw , xz − xyw , x z − xy w, xy − x z )is also weakly F -nilpotent.Now, notice that S/ a is regular, and so F -nilpotent. Furthermore, we show that S/ a is F -nilpotent. We cancompare R = k [ s , s t, st , t ] with the full Veronese subring T = k [ s, t ] (4) , which is F -rational. The cokernel of R → T is a one-dimensional k -vector space, and consequently 0 ∗ H m ( R ) = 0 ∗ H m ( T ) = 0. Therefore, by Theorem 3.4, S/ ( a ∩ a ) is F -nilpotent as well.By adjusting the ideal a in the example above, large families of examples can be easily generated. Example 3.6.
Set S = k [ a, b, c, d, e, f ] ( a,b,c,d,e,f ) . Set a to be the ideal defining the Segre product k [ x, y, z ] / ( x + y − z ) k [ u, v ], i.e., the kernel of the map S → k [ x, y, z, u, v ] / ( x + y − z ) mapping the algebra generators of S onto the monomials xu, yu, zu, xv, yv, zv . It will follow from Example 4.15 that this ideal defines a generalizedweakly F -nilpotent ring of dimension 3. Set a = ( ac − b , bd − c , cd − ea ). It is immediate to verify that R/ a isa Cohen-Macaulay ring of dimension 3 whence also weakly F -nilpotent. In this case, b = a + a = ( ce − bf, ae − af, cd − af, bd − af, c − af, b − ac, d + e − l , ad + be − cf , a d , a d, a )defines a two dimensional ring. This ring is also weakly F -nilpotent, which can be checked again by deformationalong d using [PQ19, Thm. 4.2]. Thus, by Theorem 3.2, the ring S/ ( a ∩ a ) is also generalized weakly F -nilpotent.These concepts and results globalize to statements about schemes in the natural way, see [Lyu06, Sec. 5]. Definition 3.7.
We say a scheme X defined over a field of prime characteristic p > generalized weakly F -nilpotent (respectively weakly F -nilpotent , F -nilpotent ) if all local rings O X,x are generalized weakly F -nilpotent (respectively weakly F -nilpotent, F -nilpotent). Corollary 3.8.
Suppose X is an equidimensional scheme which is the union of two closed subschemes X = Y ∪ Y ,which has d = dim( X ) = dim( Y i ) and dim( Y ∩ Y ) ≥ d − . If Y , Y and Y ∩ Y are F -nilpotent, then X is F -nilpotent. We are thankful to J. Jeffries, A. Simpson, and K. Tucker for observations and discussions leading to the followingremark.
Remark 3.9. If R is an F -finite local ring, then R is F -rational if and only if it is both F -injective and F -nilpotent.Given [Sch09, Prop. 4.8] and Corollary 3.8, there is an obvious corollary for gluing F -rational schemes. However,such a statement is vacuous as F -rational rings are normal domains. There can not be distinct ideals a and a of R such that dim( R ) = dim( R/ a i ) if both R/ a i and R are domains, as this forces a = a = 0. Geometrically, theaffine scheme Spec R is irreducible, so cannot be the union of closed subschemes V ( a ) and V ( a ).4. Graded Frobenius actions
Throughout this section, we are concerned with graded rings and graded local cohomology. We recall the basicfacts and set notation here. The reader is referred to [GW78a, GW78b] for a more detailed review. An N -gradedring R is one with a decomposition R = L n ∈ N [ R ] n where each [ R ] n is an abelian group called the n -degree pieceof R . We assume [ R ] is a field and R finitely generated over [ R ] , so each [ R ] n is naturally a [ R ] vector space. Forany such ring, we denote by m R = L n =0 [ R ] n its unique homogeneous maximal ideal. As this will be the primarytype of graded ring we consider, we refer to this simply as a graded ring and call R standard graded if it is generatedby [ R ] .Fixing a graded ring R , we consider Z -graded modules M = L n ∈ Z [ M ] n , and, for two such modules M and N , the Z -graded module Hom R ( M, N ) with n -graded piece given by the degree preserving maps Hom R ( M, N ( n )), where N ( n ) is the module N with shifted grading given by [ N ( n )] m = [ N ] n + m . Analogous to the ungraded situation, thederived functors of Hom R are Ext R and for homogeneous ideals a , one has H i a ( M ) = lim −→ t →∞ Ext iR ( R/ a t , M )the graded local cohomology.Suppose R is a graded ring. If M is a graded R -module, denote by a ( M ) := sup { n | [ M ] n = 0 } ∈ N ∪ {−∞} ,the a -invariant of M . When M is artinian, [ M ] n = 0 for n ≫
0, hence we may replace supremum with maximum.For each j ≥
0, denote by a j ( R ) = a ( H j m R ( R )), and by a ( R ) = max { j | a j ( R ) } , the a -invariant of R .Now let R be a graded ring of prime characteristic p > F be the Frobenius endomorphism. For any r ∈ [ R ] n , F ( r ) = r p ∈ [ R ] pn . Analogously, we say an R [ F ]-module ( M, ρ ) is a graded R [ F ] -module provided M is Z -graded and ρ ([ M ] n ) ⊂ [ M ] pn for each n ∈ Z . Similarly, a graded submodule N is ρ -stable if ( N, ρ ) is a graded R [ F ]-module. Note that [ M ] is always ρ -stable, since by hypothesis ρ ([ M ] ) ⊂ [ M ] p · = [ M ] . Definition 4.1.
Let R be a graded ring and ( M, ρ ) a graded R [ F ]-module. For n ∈ Z , call M nilpotent in degree n provided for each homogeneous m ∈ [ M ] n there is an e ∈ N such that ρ e ( m ) = 0 ∈ [ M ] p e n . Call M nilpotent if M is nilpotent in every degree.For a graded R [ F ]-module M , the orbit closure 0 ρM is graded with [0 ρM ] n = 0 ρM ∩ [ M ] n for n ∈ Z . When M isartinian, it is easy to see that M is nilpotent in degree n for n >
0. Additionally, if M is concentrated in finitedegrees, then M is nilpotent in degree n = 0. Remark 4.2.
Fix a graded ring R of dimension d . The condition that H d m R ( R ) is nilpotent in degree 0 is a necessarycondition for R to be F -nilpotent. To see this, note first that [ H d m R ( R )] ⊂ ∗ H d m R ( R ) as the module in questionis artinian and so, by definition, the Frobenius of any class is annihilated by elements of sufficiently large degree.Furthermore, when R is F -nilpotent, 0 ∗ H d m R ( R ) = 0 FH d m R ( R ) , so H d m R ( R ) is nilpotent in degree 0.In [ST17, Ex. 2.7(1)], the authors noted that if R is F -rational on the punctured spectrum, then it is generalizedCohen-Macualay and 0 ∗ H d m R ( R ) has finite length, whence only concentrated in finitely many degrees. Thus, in thiscase R is F -nilpotent if and only if H d m R ( R ) is nilpotent in degree 0. ENERALIZED F -DEPTH AND GRADED NILPOTENT SINGULARITIES 11 The nilpotence of the degree 0 part of an artinian module with a Frobenius action was studied in [LSW16]over F p [ x , . . . , x n ], where connections were made with the composition series for the canonical D -module and F -module structures, see [LSW16, Thm. 2.9]. We avoid the language of F -modules here in favor of a more elementaryapproach. Example 4.3.
Vacuously, any standard graded ring F -nilpotent on the punctured spectrum with negative a -invariant is F -nilpotent. Example 4.4.
The explicit calculation in [Bli01, Ex. 5.28] shows that the nilpotence of the Cohen-Macaulay ring R = k [ x, y, z ] / ( x + y − z ) is dependent on the characteristic of k modulo 4. If p ≡ R is F -nilpotentand unlike the previous example, we have [ H m ( R )] = 0. Conversely, if p ≡ H m ( R )] ⊂ ∗ H m ( R ) , so R is not F -nilpotent. See Lemma 4.16 for a family of similar examples.We describe a criteria on homogeneous systems of parameters to detect when the top local cohomology is nilpotentin degree 0. Such a condition should be useful in practical situations. Theorem 4.5.
Let R be a standard graded ring of dimension d and x = x , . . . , x d be a homogeneous system ofparameters with deg( x i ) = 1 for all i with x = x · · · x d . If for each t there is s so that [ R ] dt ⊂ ( x t + s ) F : x s , then H d m R ( R ) is nilpotent in degree .Proof. Represent a class η in H d m R ( R ) ∼ = lim −→ t R/ ( x t ) by a representative [ z + ( x ) t ]. Recall, in the grading for H d m R ( R ), deg[ z + ( x ) t ] = deg z − dt . For e ≥ q = p e – the natural Frobenius action has ρ e ( η ) = [ z q + ( x ) tq ].This is zero if and only if for some s , ( x s z ) q ∈ ( x ( t + s ) q ), i.e., z ∈ ( x t + s ) F : x s . Thus we have0 FH d m R ( R ) = { η = [ z + ( x ) t ] : there is s so that z ∈ ( x t + s ) F : x s } . Any such [ z + ( x ) t ] has degree 0 if and only if deg z = dt . By assumption, there is s with z ∈ [ R ] dt ⊂ ( x t + s ) F : x s whence [ z + ( x ) t ] ∈ FH d m R ( R ) .4.1. Segre Products.
We start our considerations of graded nilpotent Frobenius actions with Segre products. Inthis section, we keep the following standard setup. Fix R and S graded rings of positive dimension with R = S = k a field. The Segre product T = R S is the graded ring defined by R S = L n ≥ [ R ] n ⊗ k [ S ] n . If dim R = d R anddim S = d S , then dim T = d R + d S − M and N two Z -graded modules over R and S respectively, the Segre product M N is the Z -graded T -module with [ M N ] n = [ M ] n ⊗ k [ N ] n . When the characteristic of k is prime and ( M, ρ M ) and ( N, ρ N ) are graded R [ F ]-module and S [ F ]-modules, we consider M N as a graded T [ F ]-module under its diagonal Frobenius actiondenoted ρ M ρ N . Lemma 4.6. M N is nilpotent in degree t if and only if either M or N is nilpotent in degree t .Proof. Let ρ be the induced Frobenius action on M N . Take x y homogeneous in M t ⊗ k N t . If ρ e ( x ) = 0 in M tp e and ρ e ( y ) = 0 in N tp e for all e , then ρ e ( x ) ⊗ ρ e ( y ) = 0 for all e . Consequently, ρ e ( x y ) = 0 for all e . Furthermore,if x ∈ M t is nilpotent then ρ e ( x ) = 0 for some e , implying for any y ∈ N t we have ρ e ( x y ) = ρ e ( x ) ρ e ( y ) =0 ρ e ( y ) = 0.We note that M N may still be nilpotent even if M and N are not nilpotent, e.g. modules with complementarysupports. In fact, M N is nilpotent if and only if there is a partition of Z = A ⊔ B such that M a is nilpotent forall a ∈ A and N b is nilpotent for all b ∈ B .Our next aim is to consider how F -depth and generalized F -depth behave under Segre products. Recall that,given a reflexive graded R module M and and a reflexive graded S -module N , we have via the K¨unneth formula,[GW78a, Thm. 4.1.5], H j m T ( M N ) ∼ = H j m R ( M ) N ⊕ M H j m S ( N ) ⊕ M r + s = j +1 H r m R ( M ) H s m S ( N ) . In light of Lemma 2.14, and that the identification above is of graded T [ F ]-modules, we can study nilpotence bystudying each summand in turn. In particular, if R and S are of depth at least 2, then the K¨unneth formula canbe applied to T itself.By [GW78a, Thm. 4.4.4 (i)], the “expected” depth formula for Segre products, i.e., the equality depth( T ) =depth( R ) + depth( S ) −
1, is obstructed by the a -invariants of R and S . We introduce an invariant below, whichserves as an obstruction for a similar formula for F -depth. Definition 4.7.
Let R be a graded ring of dimension d . We define j -th base-nilpotent index of R to be: b j ( R ) = sup { t | [ H j m ( R )] t is not nilpotent } ∈ Z ∪ {−∞} . Further, we set b ( R ) = inf { j | b j ( R ) = 0 } ∈ N ∪ {∞} . Remark 4.8.
We record several observations about this sequence of invariants.(1) If H j m ( R ) is nilpotent, b j ( R ) = −∞ .(2) We must have b j ( R ) ≤
0, since H j m ( R ) is nilpotent in degree t for t > b ( R ) ≥ F-depth R , since a module which is nilpotent must be nilpotent in degree zero.(4) If R is weakly F -nilpotent, then b ( R ) ≥ dim( R ), and if R is F -nilpotent, then b ( R ) = ∞ as noted afterDefinition 4.1.(5) If f = F-depth R < gF-depth R , then H f m ( R ) must be nilpotent in non-zero degree but not nilpotent, hence b f ( R ) = 0. Example 4.9.
Returning to the calculation with the ring R = k [ x, y, z ] / ( x + y − z ) in [Bli01, Ex. 5.28], when k is of characteristic p ≡ H m ( R )] is nilpotent so b ( R ) < p ≡ H m ( R )] is notnilpotent so b ( R ) = 0. Theorem 4.10.
Suppose R and S are of depth at least two. Set T = R S and let f = F-depth R + F-depth S − .We have F-depth( T ) ≥ min { b ( R ) , b ( S ) , f } . Furthermore, if F-depth( R ) = b ( R ) and F-depth( S ) = b ( S ) , we have F-depth( T ) = f .Proof. For j < b ( R ), each H j m R ( R ) is nilpotent in degree 0. As S is only supported in non-negative degree and H j m R ( R ) is nilpotent in non-negative degree, we have by Lemma 4.6 that H j m R ( R ) S is nilpotent for j < b ( R ).However, if b ( R ) < ∞ , then, [ H b ( R ) m R ( R ) S ] = [ H b ( R ) m R ( R )] ⊗ k S = [ H b ( R ) m R ( R )] is not nilpotent in degree 0 by hypothesis. Consequently H b ( R ) m T ( T ) has a non-nilpotent summand. Similarly, if b ( S ) < ∞ , then H b ( S ) m T ( T ) is not nilpotent.For convenience, let M j = L r + s = j +1 H r m R ( R ) H s m S ( S ). Note the first index j for which M j may contain anon-nilpotent summand is when j + 1 = f R + f S by Lemma 4.6, i.e. when j = f . Thus, for j < f , M j is nilpotent.Consequently, when j < min { f, b ( R ) , b ( S ) } , H j m T ( T ) is nilpotent as required.When j = f , we have the summand H f R m R ( R ) H f S m S ( S ), which is nilpotent if and only if H f R m R ( R ) and H f S m S ( S )have a non-nilpotent degree in common. One way to guarantee this is the stated condition, that b ( R ) = f R and b ( S ) = f S . Corollary 4.11. If R and S are weakly F -nilpotent and depth at least two, then T is weakly F -nilpotent if and onlyif b ( R ) = b ( S ) = ∞ , in which case b ( T ) = ∞ as well. We now address the question of generalized weakly F -nilpotent singularities. Lemma 4.12.
Let R be a graded ring and let M be a graded R [ F ] -module. Suppose that M t is finite dimensionalover k = [ R ] for each t ∈ Z . The module M is generalized nilpotent if and only if M is nilpotent in degree t for all t = 0 . In particular, if M is concentrated in finite degree, M is generalized nilpotent.Proof. Suppose M is generalized nilpotent, so M/ ρM is finite length and the induced action on M/ ρM is injecvtive,so we must have M/ ρM = [ M/ ρM ] = [ M ] / [0 ρM ] , so M is nilpotent in degree t when t = 0.Now suppose M is nilpotent in degree t for all t = 0, then M/ ρM = [ M ] / [0 ρM ] is finite dimensional over k , so0 ρM has finite colength, i.e. M is generalized nilpotent.We can combine this with Lemma 4.6 to obtain the following. Corollary 4.13.
Let M be a graded R [ F ] -module and let N be a graded S [ F ] -module. Further assume all gradedpieces of M and N are finite-dimensional k -vector spaces. If either M or N is generalized nilpotent, then so is M N . Furthermore, if M N is generalized nilpotent, then it is nilpotent if and only if [ M ] is nilpotent or [ N ] is nilpotent. By graded duality, we can see that [ H j m ( R )] t is finite dimensional over k for each t ∈ Z , so we may apply theseresults to local cohomology modules as well. Interestingly, for generalized F -depth the “expected” formula for Segreproducts holds without additional hypotheses, unlike depth and F -depth. ENERALIZED F -DEPTH AND GRADED NILPOTENT SINGULARITIES 13 Theorem 4.14.
Suppose R and S are depth at least two. We have: gF-depth T ≥ gF-depth R + gF-depth S − . In particular, if R and S are generalized weakly F -nilpotent, so is T .Proof. Adopt the notation for M j as used in the proof of Theorem 4.10. The summands H j m R ( R ) S and R H j m S ( S )are generalized nilpotent for all j . Utilizing Corollary 4.13, we can see that the first index which might contributea summand which is not generalized nilpotent is when j + 1 = gF-depth R + gF-depth S , which shows the desiredinequality.We can use the formulas for F -depth and generalized F -depth to study Segre products of Fermat hypersurfacesand the projective line. Example 4.15.
Let R = k [ x, y, z ] / ( x + y − z ) and S = k [ u, v ]. Both are Cohen-Macaulay rings of dimension 2,with a ( R ) = 0 and a ( S ) = −
2. Set T = R S . Below is a decomposition of the local cohomology for T using theK¨unneth formula. H m T ( T ) = 0 H m T ( T ) = 0 H m T ( T ) = H m R ( R ) SH m T ( T ) = H m R ( R ) H m S ( S )Since a ( R ) = 0, H m T ( T ) = 0 and thus depth( T ) = 2, so T is not Cohen-Macaulay as dim( T ) = 3, however T isgeneralized Cohen-Macaulay hence generalized weakly F -nilpotent. One may also use Theorem 4.14 to see that T is generalized weakly F -nilpotent.When p ≡ b ( R ) = 0 so H m T ( T ) is not nilpotent and consequently F-depth T = 2. However, when p ≡ H m T ( T ) is nilpotent so F-depth T = 3 and T is weakly F -nilpotent.We give an expanded version of Example 4.15. In order to do this we give a sufficient condition for the ring R = k [ x , . . . , x n ] / ( x d + · · · + x dn − − x dn ) to be F -nilpotent, where d ≥ F -nilpotent and trivially F -nilpotent on the punctured spectrum. By Remark 4.2, we candetermine a condition for when H n m ( R ) is nilpotent in degree 0 which will ensure the ring is F -nilpotent. Lemma 4.16.
For R = k [ x , . . . , x n ] / ( x d + · · · + x dn − − x dn ) , if p > d , n ≥ , and p ≡ − d , then H n m ( R ) isnilpotent in degree . In fact, the Frobenius is zero on [ H n m ( R )] .Proof. Much as in [Bli01, Lem. 5.26], the calculation comes down to an explicit ˇCech computation. The submodule[ H n m ( R )] has basis constructed from partitions I with i = ( i , . . . , i n ) ∈ I if and only if i j ≥ P nj =0 i j = d .In particular, the set of classes η i := (cid:20) x d − inn x i ··· x in − n − (cid:21) forms the desired basis. Write ( d − i n ) p − i n = dr for integer r .Directly we see, F ( η i ) = " x ( d − i n ) pn x pi · · · x pi n − n − = z i n " x ( d − i n ) p − i n n x pi · · · x pi n − n − = z i n " x drn x pi · · · x pi n − n − = z i n " ( x d + · · · + x dn − ) r x pi · · · x pi n − n − = z i n X j + ··· + j n − = r c j ,...,j n − x dj · · · x dj n − n − x pi · · · x pi n − n − where c j ,...,j n − is a multinomial coefficient. For any summand in this expression to be non-zero, it is necessarythat dj t < pi t for all 0 ≤ t ≤ n −
1. As p ≡ − d , there is no solution for j t in an equation dj t = pi t −
1, in fact, that dj t < pi t forces dj t ≤ pi t − ( d − n − X t =0 dj t ≤ p ( n − X i =0 i t ) − n ( d −
1) = ( d − i n ) p − n ( d − . This creates a contradiction, as P n − t =0 dj t = dr = ( d − i n ) p − i n , thus ( d − i n ) p − i n ≤ ( d − i n ) p − n ( d −
1) thenforces n ( d − ≤ i n < d which is clearly false. Therefore, for some t , dj t ≥ pi t and so each term in the expressionfor F ( η i ) is zero. Remark 4.17.
It seems likely that the Frobenius is nilpotent in degree zero on H n m ( R ) for any p d .Indeed, this is asserted and easily shown for d = 5 in [Bli01, Ex. 5.28]. Establishing this fact requires a finer ˇCechcomputation than we give here, but we will not remark further on these intermediate cases.Now following the same argument in Example 4.15 with Lemma 4.16 we have the following result. Theorem 4.18.
Let p > d and n ≥ , and let R = k [ x , . . . , x n ] / ( x d + · · · + x dn − − x dn ) , S = k [ u, v ] , and T = R S . If p ≡ d , then b ( R ) = 0 and F-depth T = 2 . When p ≡ − d , then H m T ( T ) is nilpotent so F-depth T = 3 and T is weakly F -nilpotent. Veronese Subrings.
Fix R a graded ring. For n ∈ N and a graded R -module M , set M ( n ) to be the n -thVeronese submodule. In the case M = R , the Veronese R ( n ) is a direct summand of R and consequently the localcohomology of R ( n ) splits out of the local cohomology for R as an R ( n ) -module. We prove a descent result forgeneralized weakly F -nilpotent and weakly F -nilpotent singularities for Veronese subrings. Notably these resultsdo not immediately follow from graded versions of the results in [KMPS]. Theorem 4.19.
Let R be a standard graded ring of positive dimension over a field k . We have f := F-depth R ≤ F-depth R ( n ) , and if b f ( R ) = 0 , then F-depth R ( n ) = f for all n ∈ N . Furthermore, gF-depth R ≤ gF-depth R ( n ) for all n ∈ N . In particular, if R is (generalized) weakly F -nilpotent, so is R ( n ) for all n ∈ N .Proof. For any t ∈ Z we have the following commutative diagram of k -vector spaces where the horizontal mapsare isomorphisms and the vertical maps the restriction of the canonical Frobenius action on local cohomology to asingle degree, whence p -linear. h H j m R ( n ) ( R ( n ) ) i t h H j m R ( R ) i nt h H j m R ( n ) ( R ( n ) ) i pt h H j m R ( R ) i pnt ∼ F F ∼ Consequently, if H j m R ( R ) is nilpotent, there is an e ∈ N such that F e : H j m R ( R ) → H j m R ( R ) is the zero map, andso F e : [ H j m R ( R )] nt → [ H j m R ( R )] p e nt is also the zero map. This implies H j m R ( n ) ( R ( n ) ) is nilpotent. Thus, for j < f , H j m R ( n ) ( R ( n ) ) is nilpotent and consequently F-depth R ( n ) ≥ f .Now, since R is standard graded and dim k [ H j m R ( R )] t < ∞ for all t , we have that H j m R ( R ) / FH j m R ( R ) is finite lengthif and only if it is supported in finitely many degrees. Since the induced Frobenius map on the quotient is injective,it can only be supported in degree 0. Hence, H j m R ( R ) is generalized nilpotent if and only if it is nilpotent in allnonzero degrees. The same argument holds for H j m R ( n ) ( R ( n ) ) since we may regrade R ( n ) to be standard graded. Ineither grading, the underlying degree zero part [ H j m R ( n ) ( R ( n ) )] = [ H j m R ( R )] is unaffected, so that b j ( R ) = 0 if andonly if b j ( R ( n ) ) = 0 for all n ∈ N .We may then apply the argument with the commutative diagram above, restricting to t = 0 instead, so that if H j m R ( R ) is generalized nilpotent, so is H j m R ( n ) ( R ( n ) ). So for j < gF-depth R , we have H j m R ( n ) ( R ( n ) ) is generalizednilpotent, implying gF-depth R ( n ) ≥ gF-depth R . The final claim follows since dim( R ( n ) ) = dim( R ).We note that the R ( n ) -module R/R ( n ) controls the ascent of weak and generalized weak F -nilpotence along R ( n ) → R . Now apply our results so far to verify when a graded rings with controlled singularities on the puncturedspectrum have high Veronese subrings which are F -nilpotent, similar to [Sin00, Prop. 3.1]. Theorem 4.20.
Suppose R is a graded ring. If R is F -nilpotent on the punctured spectrum, then for all n ≫ , theVeronese R ( n ) is weakly F -nilpotent. Furthermore, [ H d m ( R )] ⊂ FH d m ( R ) if and only if for all n ≫ , the Veronese R ( n ) is F -nilpotent. ENERALIZED F -DEPTH AND GRADED NILPOTENT SINGULARITIES 15 Proof.
Our approach is a direct application of the ideas in [Sin00, Prop. 3.1]. Set d = dim R . Throughout theproof, we employ [GW78a, Thm. 3.1.1] which ensures for fixed n ∈ N an isomorphism H d m R ( n ) ( R ( n ) ) ∼ = H d m R ( R ) ( n ) as R ( n ) -modules. Set H ( n ) = 0 ∗ H d m R ( n ) ( R ( n ) ) / FH d m R ( n ) ( R ( n ) ) and H = H (1).Corollary 4.19 ensures that R is generalized weakly F -nilpotent, and H has finite length, and so is only supportedin finitely many graded degrees. It is clear that R ( n ) is weakly F -nilpotent for n ≫ H supported in only finitely many graded degrees is equivalent to the condition that [ H ( n )] t =0 for t = 0 and all n ≫
0. So it suffices to consider when the vector space [ H ( n )] vanishes. By the proof of [Sin00,Prop. 3.1], we see that [ H ( n )] ∼ = [ H ] = [ H d m R ( R )] / [0 FH d m R ( R ) ] which is zero by assumption.4.3. Hypersurfaces of diagonal subalgebras.
Veronese subalgebras of multigraded hypersurface rings provideinteresting examples of rings with distinguished singularities [KSSW09]. We determine a criteria for when suchrings have nilpotent-type singularities. We first set notation. For T a N -graded ring and n = ( n , n ), recall T ( n )is the multigraded ring with grading shifted by n . For ∆ = ( g, h ) ∈ N , one may consider the Veronese subring,also called the diagonal subalgebra, T ∆ := L k ∈ N [ T ] ( gk,hk ) . Any f ∈ T of degree ( d, e ) > (0 ,
0) defines a short exactsequence as below. 0 T ( − d, − e ) T T /f · f This is compatible with taking Veronese along ∆ and when T is a tensor product of polynomial rings A and B ofdimensions m and n respectively, one has a natural extension of the K¨unneth formula, [KSSW09, Lem. 2.1]. Thisallows one to give simple criteria for when ( T /f ) ∆ is Gorenstein or just Cohen-Macaulay. In particular, [KSSW09,Thm. 3.1] identifies that ( T /f ) ∆ is Cohen-Macaulay if and only if ⌊ d − mg ⌋ < eh and ⌊ e − nh ⌋ < dg . Following the proofof [KSSW09, Thm. 3.1], one can quickly show that in this case ( T /f ) ∆ is weakly F -nilpotent if and only if it isCohen-Macaulay. We seek to generalize the setting from polynomial rings to more general standard graded ringsand examine the F -depth of ( T /f ) ∆ in more detail.Fix an algebraically closed field k and ∆ = ( g, h ) ∈ N . Suppose A and B are standard graded normal domainswith [ A ] = [ B ] = k , and let T = A ⊗ k B be the tensor product. Define an N grading on T by picking generators x , · · · , x m for [ A ] over k and y , · · · , y n for [ B ] over k , and extending the grading defined by deg( x i ⊗
1) = (1 , ⊗ y i ) = (0 ,
1) in a natural way.
Theorem 4.21.
Adopt the setup in the previous paragraph. For f ∈ T of degree ( d, e ) > (0 , , we have that F-depth(
T /f ) ∆ ≥ F-depth T ∆ − . In particular, if T ∆ is weakly F -nilpotent and dim(( T /f ) ∆ ) = dim( T ∆ ) − , then ( T /f ) ∆ is weakly F -nilpotent. If T ∆ is not weakly F -nilpotent, d/g, e/h Z , and (cid:22) b dim A ( A ) + dg (cid:23) < eh and (cid:22) b dim B ( B ) + eh (cid:23) < dg , then F-depth(
T /f ) ∆ ≥ F-depth T ∆ .Proof. We use notation similar to the proof of [KSSW09, Thm. 3.1]. Set a , b , and m to be the homogeneous maximalideals of A , B , and T ∆ respectively. For any t ∈ N , we have the following commutative diagram of k -vector spaces,where the horizontal maps of graded T ∆ -modules and the vertical diagrams are p t -linear.0 T ( − d, − e ) ∆ T ∆ ( T /f ) ∆ T ( − dp t , − ep t ) ∆ T ∆ ( T /f p t ) ∆ F t | · f F t f tT ∆ · f pt The p t -linear map F t | is the restriction of F t to T ( − d, − e ) and f tT δ is the relative Frobenius map as outlined inSection 2.3. We apply local cohomology at m , and for each q we get the commutative diagram below, noting thehorizontal maps are graded T ∆ -module maps and vertical arrows are p t -linear. · · · H q m ( T ∆ ) H q m (( T /f ) ∆ ) H q +1 m ( T ( − d, − e ) ∆ ) · · ·· · · H q m ( T ∆ ) H q m (( T /f p t ) ∆ ) H q +1 m ( T ( − dp t , − ep t ) ∆ ) · · · · f F t f tT ∆ F t |· f pt In the above diagram, when H q m ( T ∆ ) is nilpotent we may take t ≥ HSL( T ∆ ) which forces F t = 0. Further, if F t | = 0 for some t ≫ f tT ∆ = 0. As noted after Remark 2.16, this willimply that the canonical Frobenius map F t on H q m (( T /f ) ∆ ) is also the zero map.When q < F-depth T ∆ − F t on H q m ( T ∆ ) and H q +1 m ( T ∆ ) are thezero map for t ≥ HSL( T ∆ ), which implies f tT ∆ is the zero map. Consequently, F-depth( T /f ) ∆ ≥ F-depth T ∆ − q = F-depth T ∆ − H q +1 m ( T ( − d, − e ) ∆ ) may still be nilpotent due to the linear shift in index. The localcohomology of T ( − d, − e ) ∆ has a K¨unneth-type decomposition as shown in [KSSW09, Lem. 2.1]. We apply a similaranalysis as in the proof of Theorem 4.10 and Theorem 4.14 to determine vanishing of the map F t | . First, note thatif H q m ( T ∆ ) is nilpotent then certainly F t | H q m ( T ( − d, − e ) ∆ ) is zero for t ≫ H q m ( T ( − d, − e ) ∆ ) = A q ⊕ B q ⊕ C q , where:[ A q ] n = [ H q a ( A )] − d + gn ⊗ k [ B ] − e + hn [ B q ] n = [ A ] − d + gn ⊗ k [ H q b ( B )] − e + hn C q = M q + q = q +1 C q ,q [ C q ,q ] n = [ H q a ( A )] − d + gn ⊗ k [ H q b ( B )] − e + hn and the image of A q under the restriction of F t on H q m ( T ∆ ) is inside the corresponding summand of the decompo-sition of H q m ( T ( − dp t , − ep t ) ∆ ), similarly for the images of B q , and C q ,q . Here we will say A q etc. are nilpotent ornilpotent in a certain degree if the restriction of F t is the zero map for some t .Set d A = dim A and d B = dim B . Like in the proof of Theorem 4.10, we discuss the nilpotence of each summandin turn. If b q ( A ) = 0 for some q < d A , then A q is not nilpotent only when − d + gn = 0 and − e + hn ≥
0, whichforces d/g ∈ Z and e/h ≤ d/g ; notably, these bounds are independent of q . Consequently, if b q ( A ) = 0 for some q < d A , then A q is nilpotent when either d/g Z or d/g < e/h . We get symmetric bounds for B q , i.e. if q < d B and b q ( B ) = 0 then B q is nilpotent if e/h Z or e/h < d/g .The analysis for q = d A is more subtle since H q m ( A ) is not generalized nilpotent. For − d + gn > b d A ( A ), H q m ( A )is nilpotent in degree − d + gn , and if − e + hn < B vanishes in degree − e + hn . So, if there are no integers n so that − d + gn ≤ b d A ( A ) and − e + hn ≥
0, then A d A is nilpotent. Manipulating the inequalities, we see that A d A and, symmetrically, B d B ) are nilpotent when: (cid:22) b d A ( A ) + dg (cid:23) < eh and, symmetrically (cid:22) b d B ( B ) + eh (cid:23) < dg . Now we consider C q ,q with q < d A and q < d B . If b q ( A ) = −∞ or b q ( B ) = −∞ then C q ,q is nilpotent.However, if b q ( A ) = b q ( B ) = 0, then C q ,q is not nilpotent in degree n only if − d + gn = − e + hn = 0, whichrequires d/g = e/h ∈ Z . When q = d A but q < d B , then we have C d A ,q = [ H d A a ( A )] − d + gn ⊗ k [ H q b ( B )] − e + hn is only possibly non-nilpotent when n = e/h ∈ Z and − d + gn ≤ b d A ( A ), so C d A ,q is nilpotent when e/h Z orwhen e/h > ( b d A ( A ) + d ) /g . Applying the same reasoning to B shows that C q ,d B with q < d A is nilpotent when d/g Z or when d/g > ( b d B ( B ) + e ) /h .Now we assume that e/h, d/g Z , and further that ⌊ ( b d A ( A ) + d ) /g ⌋ < e/h and ⌊ ( b d B ( B ) + e ) /h ⌋ < d/g . Theabove analysis shows that this forces A q , B q to be nilpotent for all q and C q to be nilpotent for q < d A + d B − F t | H q m ( T ( − d, − e ) ∆ ) is the zero map for q < dim( T ∆ ) and t ≫
0. Consequently, f tT ∆ is the zero map, whichas noted in the proof of Theorem 4.21 implies that H q m (( T /f ) ∆ ) is nilpotent for 0 ≤ q < F-depth T ∆ . Thus,F-depth( T /f ) ∆ ≥ F-depth T ∆ . Remark 4.22.
Note that the theorem above and examples generated thereby can be viewed as similar to counter-examples to the deformation of F -nilpotent singularities. Srinivas-Takagi showed that F -nilpotence does not deformeven in the Gorenstein case, see [ST17, Example 2.7]. Polstra-Quy also noted that weak F -nilpotence does notdeform by showing that there are difficulties at the H level, see [PQ19, Section 4]. Example 4.23.
Suppose k is an algebraically closed field of characteristic 7. Let A = k [ x , x , x ] / ( x + x − x )and B = k [ y , y , y ] / ( y + y − y ), notably both Cohen-Macaulay normal domains of dimension 2. By Lemma 4.16we see that b ( A ) = ∞ and by [Bli01, Lemma 5.27], b ( B ) = 2. We also have a ( A ) = 1 and a ( B ) = 0.Let T = A ⊗ k B . For all ∆ ′ = ( g, h ) ∈ N with g > a ( A ( g ) ) = a ( B ( h ) ) = 0, so dim( T ∆ ′ ) = dim( A ( g ) B ( h ) ) =3 and F-depth T ∆ ′ = 2 so that T ∆ ′ is not weakly F -nilpotent. Thus, if we take ∆ = (2 ,
2) and f = x ⊗ y ∈ T ,then the numerical conditions in the previous theorem are satisfied, and hence ( T /f ) ∆ is weakly F -nilpotent.We conclude with an example combining the various graded constructions from this section, which furtherdemonstrates the behavior of F -depth between T and T /f outlined in the previous theorem.
ENERALIZED F -DEPTH AND GRADED NILPOTENT SINGULARITIES 17 Example 4.24.
Suppose k is an algebraically closed field of characteristic 7 and set ∆ = (2 , A = k [ x , x , x , x ] / ( x + x + x − x ) and B = k [ y , · · · , y ] /I , where I is the defining ideal of the Segre prod-uct C = k [ r, s ] k [ c, d, e ] / ( c + d − e ), i.e. I is the kernel of the map B → C by y rc , y rd , · · · , y sd .Then A and B are dimension 3 normal domains, and T = A ⊗ k B is dimension 6 normal domain. We showed inExample 4.15 that b ( B ) = 0 since 7 ≡ b ( T ∆ ) = 0. This also implies that F-depth T ∆ = 2 <
6, sothat T ∆ is not weakly F -nilpotent.Take f = x x x ⊗ y y y ∈ T , homogeneous of multidegree ( d, e ) = (3 , a ( B (2) ) ≤ − a ( k [ r, s ]) = −
2, and consequently b ( B (2) ) ≤ −
2. Furthermore, b ( A ) < (cid:22) b ( A ) + 32 (cid:23) ≤ (cid:22) (cid:23) <
52 and (cid:22) b ( B ) + 52 (cid:23) ≤ (cid:22) (cid:23) < , so we see the numerical conditions are satisfied. Consequently, we have shown all the conditions of Theorem 4.21are satisfied, so we must have F-depth( T /f ) ∆ = F-depth T ∆ = 2.5. Frobenius Test Exponents
In this section, we describe effective bounds on Frobenius test exponents for the various constructions of ringswith nilpotent singularities outlined earlier. As noted after Definition 2.10, weakly F -nilpotent and generalizedweakly F -nilpotent local rings are known to have finite Frobenius test exponents by [Quy19] and [Mad19]. Thesebounds are in terms of the Hartshorne-Speiser-Lyubeznik numbers. We show how our theorems provide upperbounds on the Hartshorne-Speiser-Lyubeznik numbers of the result and consequently provide upper bounds forFrobenius test exponents. Throughout this section, R will be a ring of prime characteristic p > parameter ideal is one whose height isequal to its minimal number of generators. The Frobenius test exponent , if it exists, for a local ring ( R, m ) isthe smallest e ∈ N such that for all parameter ideals q ⊂ R , we have ( q F ) [ p e ] = q [ p e ] . We write e = Fte( R ). Upperbounds for Fte( R ) are known in the weakly F -nilpotent and generalized weakly F -nilpotent cases, generalizingprevious results known in the Cohen-Macaulay and generalized Cohen-Macaulay cases. We recall these theoremsbelow. Remark 5.1.
Suppose ( R, m ) is a local ring of dimension d . Let h j = HSL( H j m ( R )). • Suppose R is Cohen-Macaulay. Then, Fte( R ) = HSL( R ) = h d . [KS06, Thm. 2.4] • Suppose R is weakly F -nilpotent. Then Fte( R ) ≤ P dj =0 (cid:0) dj (cid:1) h j . [Quy19, Main Thm.] • Suppose R is generalized weakly F -nilpotent. Let e be the minimum e ∈ N such that p e ≥ d − N , where N is the smallest n such that m n H j m ( R ) ⊂ FH j m ( R ) for 0 ≤ j < d . Then, Fte( R ) ≤ e + P dj =0 (cid:0) dj (cid:1) h j . [Mad19,Thm. 3.6]Few other classes of ideals are known to have finite Frobenius test exponents. Huong-Quy very recently showedin [HQ21] that bounds on generalized F -depth can provide finite Frobenius test exponents for the class of idealsgenerated by filter regular sequences of bounded length, generalizing the main result of [Mad19].To study these finiteness theorems in the graded setting, we note that the literature on effective bounds ofFrobenius test exponents is written mostly in the local setting and is dependent on filter regular sequences. We let R be a standard graded ring over R = k a field. Definition 5.2.
The homogeneous Frobenius test exponent of R is the smallest e , if it exists, so that for anyhomogeneous ideal p which is primary to the irrelevant maximal ideal, we have ( p F ) [ p e ] = p [ p e ] . We write Fte ∗ ( R )for this number.A key technique in studying the finiteness of Frobenius test exponents of a local ring ( R, m ) is the use of filterregular sequences and the fact that m -primary ideals are generated by such, [Quy19, Rmk. 2.5]. This essentiallyfollows by prime avoidance, thus homogeneous prime avoidance guarantees the graded analogue, namely existenceof homogeneous filter regular sequences and the graded analogue of [Quy19, Rmk. 2.5]. The rest of the proof reliesonly on homological algebra techniques available in the graded setting, in particular the Nagel-Schenzel theorem.This is also available in the graded setting, as can be easily adapted by the proof given in [Huo17, Thm. 1.1].So we may apply Quy’s bound in part 2 of Remark 5.1 for the homogeneous Frobenius test exponents in weakly F -nilpotent rings, i.e. if R is graded ring which is weakly F -nilpotent, then Fte ∗ ( R ) ≤ P dj =0 HSL( H j m ( R )) where H j m ( R ) are the graded local cohomology modules of R . We can attain similar bounds for the homogeneous Frobeniustest exponents of a generalized weakly F -nilpotent graded ring as in [Mad19] also follow with minimal adjustmentto the graded setting.Now, to calculate upper bounds on the Frobenius test exponents of the constructions produced elsewhere in thepaper, we first need to bound their Hartshorne-Speiser-Lyubeznik numbers. Lemma 5.3.
Let the following be an exact sequence of R [ F ] -modules. · · · L M N · · · α β If HSL( L ) and HSL( N ) are finite, and ker( α ) is a nilpotent R [ F ] -module, then HSL( M ) ≤ HSL( L ) + HSL( N ) .Proof. We use ρ for the Frobenius action on each place. Set ℓ = HSL( L ) and n = HSL( N ). If ξ ∈ ρM , then for some e ≫ ρ e ( ξ ) = 0. Then β ( ρ e ( ξ )) = ρ e ( β ( ξ )) = 0, so β ( ξ ) ∈ ρN , and consequently ρ n ( β ( ξ )) = β ( ρ n ( ξ )) = 0.Hence, ρ n ( ξ ) ∈ ker( β ) = im( α ), so there is a ξ ′ ∈ L such that α ( ξ ′ ) = ρ n ( ξ ). Since ξ is nilpotent, for some e ′ ≫ ρ e ′ ( α ( ξ ′ )) = ρ e ′ + n ( ξ ) = 0, consequently ρ e ′ ( ξ ′ ) ∈ ker( α ).By assumption, ker( α ) is nilpotent so we know ρ ℓ ( ξ ′ ) = 0, hence ρ ℓ + n ( ξ ) = ρ ℓ ( α ( ξ )) = α ( ρ ℓ ( ξ )) = 0. This showsHSL( M ) ≤ ℓ + n as required. We note that we can replace the condition that HSL( L ) < ∞ and HSL( N ) < ∞ withHSL(ker( β )) < ∞ and HSL(im( β )) < ∞ with no adjustment to the proof. This implies the slightly sharper boundHSL( M ) ≤ HSL(ker( β )) + HSL(im( β )).When the sequence is split exact, we get an improved equality. Corollary 5.4.
Consider a split exact sequence of R [ F ] -modules, L M N, α β so that M ≃ ker( β ) ⊕ im( β ) . If HSL(ker( β )) and HSL(im( β )) are finite, then HSL( M ) = max { HSL(ker( β )) , HSL(im( β )) } . Proof.
Set HSL(im( β )) = e , HSL(ker( β )) = e , and e = max { e , e } . Since e and e are finite, in particular thereare elements ξ ∈ ρ im( β ) and ξ ∈ ρ ker( β ) such that ρ e − ( ξ ) = 0 and ρ e − ( ξ ) = 0. Thus, ( ξ , ξ ) ∈ ρM and ρ e − ( ξ , ξ ) = 0 but ρ e ( ξ , ξ ) = 0. Since 0 ρM = 0 ρ im( β ) ⊕ ρ ker( β ) , we are finished.5.1. Frobenius test exponents for gluing.
We now apply the finiteness theorems in Remark 5.1 and theHartshorne-Speiser-Lyubeznik bounds just outlined to bound the Frobenius test exponent of the glued schemeproduced in Theorem 3.2. As in the Section 3, we fix the following notation throughout. Let ( R, m ) be a local ringof dimension d > a and a be ideals of R such that a ∩ a = 0, and we set a + a = b . Theorem 5.5. If j ≤ F-depth R/ b , we have HSL( H j m ( R )) ≤ HSL( H j − m ( R/ b )) + max { HSL( H j m ( R/ a )) , HSL( H j m ( R/ a )) } . In particular, suppose dim( R ) = dim( R/ a i ) = d for i = 1 , and that R/ a , R/ a , and R/ b are weakly F -nilpotent.Then, if dim( R/ b ) = dim( R ) − , we have Fte( R ) ≤ d · d − X j =0 (cid:18) d − j (cid:19) HSL( H j m ( R/ b )) + d X j =0 (cid:18) dj (cid:19) max { HSL( H j m ( R/ a ) , HSL( H j m ( R/ a ) } . If dim( R/ b ) = dim( R ) , we have: Fte( R ) ≤ d · d − X j =0 (cid:18) dj (cid:19) HSL( H j m ( R/ b )) + d X j =0 (cid:18) dj (cid:19) max { HSL( H j m ( R/ a ) , HSL( H j m ( R/ a ) } . Proof.
Let j ≤ F-depth R/ b . From the short exact sequence0 → R → R/ a ⊕ R/ a → R/ b → , we get the exact sequence · · · H j − m ( R/ b ) H j m ( R ) H j m ( R/ a ) ⊕ H j m ( R/ a ) · · · ENERALIZED F -DEPTH AND GRADED NILPOTENT SINGULARITIES 19 to which we may apply Lemma 5.3, since j − < F-depth( R/ b ) to obtain the first term and Corollary 5.4 to obtainthe second.Under either dimension hypothesis on R/ b , Theorem 3.2 shows that R is also weakly F -nilpotent. We can applyQuy’s bound for R and the Hartshorne-Speiser-Lyubeznik bound just shown for all 0 ≤ j < d .For j = d , first assume dim( R/ b ) = d −
1. Then, by Lemma 3.1 we have H d m ( R ) ≃ H d m ( R/ a ) ⊕ H d m ( R/ a ) as R [ F ]-modules, and so HSL( H d m ( R )) = max { HSL( H d m ( R/ a )) , HSL( H d m ( R/ a )) } =: e similar to Lemma 5.7 (3).Using Quy’s bound, we get:Fte( R ) ≤ P dj =0 (cid:0) dj (cid:1) HSL( H j m ( R )) ≤ P d − j =0 (cid:0) dj (cid:1) (cid:16) HSL( H j − m ( R/ b )) + max { HSL( H j m ( R/ a )) , HSL( H j m ( R/ a )) } (cid:17) + e ≤ P d − j =0 (cid:0) dj (cid:1) HSL( H j − m ( R/ b )) + P dj =0 (cid:0) dj (cid:1) max { HSL( H j m ( R/ a )) , HSL( H j m ( R/ a )) } . We can re-index the first term to get P d − j =0 (cid:0) dj +1 (cid:1) HSL( H j m ( R/ b )) and use the binomial identity (cid:0) dj +1 (cid:1) = dj +1 (cid:0) d − j (cid:1) .We then replace dj +1 with its maximal value d to provide the term d · P d − j =0 (cid:0) d − j (cid:1) HSL( H j m ( R/ b )).If dim( R/ b ) = dim( R ), the proof is the same except it is not clear that im( H d − m ( R/ b ) → H d m ( R )) vanishes, sowe must involve the Harthsorne-Speiser-Lyubeznik number of H d − m ( R/ b ). We then get the term: d X j =1 (cid:18) dj (cid:19) HSL( H j − m ( R/ b )) = d − X j =0 (cid:18) dj + 1 (cid:19) HSL( H j m ( R/ b )) ≤ d − X j =0 d (cid:18) dj (cid:19) HSL( H j m ( R/ b )) , using the binomial identity (cid:0) dj +1 (cid:1) = d − jj +1 (cid:0) dj (cid:1) and replacing d − jj + 1 with its maximal value d .We finish this subsection by computing the upper bounds on the Frobenius test exponent of a glued generalizedweakly F -nilpotent ring in terms of the input data. Despite only assuming that the lower local cohomology modulesare generalized nilpotent, we must still impose the condition that the kernel of H j − m ( R/ b ) → H j m ( R ) is nilpotentto apply Lemma 5.3 for all j ≤ dim( R ), which as before is guaranteed if dim( R/ b ) ≥ dim( R ) − R/ b is weakly F -nilpotent. Theorem 5.6.
Suppose R is equidimensional and dim( R ) = dim( R/ a i ) = d , with dim( R/ b ) ≥ d − , and that R/ a , R/ a , and R/ b are generalized weakly F -nilpotent. Further, suppose for each j ≤ d that ker( H j − m ( R/ b ) → H j m ( R )) is nilpotent.Let N be the smallest n ∈ N such that m n H j m ( R/ b ) is nilpotent for all ≤ j < d − and similarly let N be thesmallest n ∈ N such that m n ( H j m ( R/ a ) ⊕ H j m ( R/ a )) is nilpotent for all ≤ j < d . If e be the smallest e ∈ N such that p e ≥ ( N + N )2 d − , we have Fte( R ) ≤ e + e , where e is the bound given for Fte( R ) in Theorem 5.5.Proof. We may still apply Lemma 5.3 to obtain the upper bound for the Hartshorne-Speiser-Lyubeznik numbersof R used in Theorem 5.5 in this case, but we note that if dim( R/ b ) = d −
1, we do not need to assume thatker( H d − m ( R/ b ) → H d m ( R )) is nilpotent as the image vanishes.Finally, the proof of Lemma 2.15 clearly shows that if N and N are as in the statement of the theorem, then m N + N H j m ( R ) is nilpotent for all 0 ≤ j < d . Now the theorem follows by [Mad19, Thm. 3.1].5.2. Frobenius test exponents for Segre products.
Now, fix R and S graded rings of positive dimension d R and d S respectively with R = S = k a field, and we let T be the Segre product T = R S , which hasdim( T ) = d T = d R + d S −
1. To apply the finiteness theorems to Segre products which have nilpotent singularitytypes, we first explicate the relationship between the Hartshorne-Speiser-Lyubeznik numbers for T and those for R and S . Lemma 5.7.
Let M and M ′ be Z -graded R [ F ] -modules and let N be a Z -graded S [ F ] -module. (1) Suppose that
HSL( M ) < ∞ and M is nilpotent, then HSL( M N ) ≤ HSL( M ) . Similarly, if HSL( N ) < ∞ and N is nilpotent, HSL( M N ) ≤ HSL( N ) . (2) When both M and N are nilpotent, HSL( M N ) ≤ min { HSL( M ) , HSL( N ) } . (3) We have
HSL( M ⊕ M ′ ) = max { HSL( M ) , HSL( M ′ ) } . (4) If M is concentrated in finite degree, say M t = 0 for t ≥ t and t ≤ − t , then: HSL( M ) ≤ max { HSL( M ) , min { e ∈ N | p e ≥ t }} . Proof.
First, suppose HSL( M ) = e < ∞ . We have ρ e : M → M is zero, which implies ρ e : M N → M N is alsozero, as the action is diagonal. This argument is clearly symmetric in M and N . When both M and N are nilpotent, if either HSL( M ) or HSL( N ) are finite, we can apply the previous case toobtain HSL( M N ) ≤ min { HSL( M ) , HSL( N ) } . If neither HSL( M ) nor HSL( N ) are finite, the claim is vacuous.The action on M ⊕ M ′ is the direct sum of the actions on each summand. If both a = HSL( M ) and b = HSL( M ′ )are finite, we can take elements m and m ′ nilpotent in M and M ′ such that ρ a − M ( m ) = 0 and ρ b − M ′ ( m ′ ) = 0, then( m, m ′ ) is nilpotent and requires ρ max { a,b } M ⊕ M ′ to vanish. Since 0 ρM ⊕ M ′ = 0 ρM ⊕ ρM ′ , the result follows. If one ofHSL( M ) , HSL( M ′ ) are infinite, we assume without loss of generality that HSL( M ) = ∞ . There must be a sequenceof elements { m n } n ∈ N nilpotent in M such that ρ e ( m n ) = 0 for e < n . Consequently ρ e ( m n , = 0 for e < n . So,HSL( M ⊕ M ′ ) = ∞ as well.The final claim is obvious, since ρ ( M t ) ⊂ M pt .We now apply this lemma to the summands in the K¨unneth formula. Recall when R and S are of depth at leasttwo we have H j m T ( T ) ≃ H j m R ( R ) S ⊕ R H j m S ( S ) ⊕ M r + s = j +1 H r m R ( R ) H s m S ( S ) . We use the following notational convenience. Set F-exp j ( R ) = min { e ∈ N | p e > a j ( R ) } . This number is anobvious upper bound for the required iterate of F to uniformly annihilate [ H j m R ( R )] > ⊂ H j m R ( R ), and dependsonly on the graded structure of the ring. Theorem 5.8.
Let R and S be of depth at least two, and suppose that R and S are weakly F -nilpotent. Then, wehave the following bound on the Hartshorne-Speiser-Lyubeznik number of H j m T ( T ) . HSL( H j m T ( T )) ≤ max HSL([ H j m R ( R )] ) , HSL([ H j m S ( S )] ) , F-exp j ( R ) , F-exp j ( S ) , max { HSL( H r m R ( R )) , HSL( H s m S ( S )) | r + s = j + 1 } Proof.
We analyze each summand of the Kunneth formula above. We then combine several maxima.First, since S is supported in degree ≥
0, we have H j m R ( R ) S is also supported in degree ≥
0. If a j ( R ) ≤ j ( R ) = 0 which we may remove from the max. If a j ( R ) >
0, then 0 FH j m R ( R ) S = 0 F [ H j m R ( R )] ⊕ Q where Q is only possibly supported in degrees n with 0 < n ≤ a j ( R ). Consequently, F F-exp j ( Q ) = 0. We then applyLemma 5.7. The complementary summand R H j m S ( S ) is handled identically.For the final summand, for any combination of r and s , each outcome of Lemma 5.7 is bounded above by themaximum shown. Remark 5.9.
Note that as we successively compute the data required bound HSL( H j m T ( T )), we do not actuallyneed to know HSL( H j m R ( R )) or HSL( H j m S ( S )), only HSL([ H j m R ( R )] ) ≤ HSL( H j m R ( R )) and HSL([ H j m S ( S )] ) ≤ HSL( H j m S ( S )), as well as a j ( R ) and a j ( S ).We also note that above bound is not optimal. We give this form as it is compact. One can improve it byincluding the data of the Hartshorne-Speiser-Lyubeznik number for Q , since F-exp j ( R ) is only a coarse upperbound for HSL( Q ). Similarly, for certain ranges of ( r, s ), we may replace the maximum of HSL( H r m R ( R )) andHSL( H s m S ( S )) with a smaller value using Lemma 5.7. We may also use HSL( R ) and HSL( S ) as very coarse upperbounds to the terms in the maximum above. Corollary 5.10.
If we set H j the maximum appearing on the right hand side of the estimate in Theorem 5.8, thenwhen R and S are weakly F -nilpotent and b ( R ) = b ( S ) = ∞ , Fte ∗ ( T ) ≤ P d T j =0 (cid:0) dj (cid:1) H j . In particular, Fte ∗ ( T ) ≤ d T max { HSL( R ) , HSL( S ) } . Remark 5.11.
Suppose R and S are of depth at least two, and let j < gF-depth R + gF-depth S − ≤ gF-depth T .We may use the ideas in the proof of Theorem 4.14 to understand the lengths of H j m T ( T ). For notational convenience,we set G j ( U ) = (cid:2) H j m U ( U ) (cid:3) / h FH j m U ( U ) i for U ∈ { R, S, T } .By the K¨unneth formula and the fact that length is additive on short exacts sequences, we have λ T ( H j m T ( T ) / FH j m T ( T ) ) = λ R ( G j ( R )) + λ S ( G j ( S )) + X r + s = j +1 λ T (cid:16) H r m R ( R ) H s m S ( S ) / FH r m R ( R ) H s m S ( S ) (cid:17) . ENERALIZED F -DEPTH AND GRADED NILPOTENT SINGULARITIES 21 The last collection of summands is most easily understood when R and S are of depth at least two, generalizedCohen-Macaulay, and standard graded, so that T is also generalized Cohen-Macaulay and standard graded. In thiscase, we see λ T (cid:16) H r m R ( R ) H s m S ( S ) / FH r m R ( R ) H s m S ( S ) (cid:17) = λ R ( G r ( R )) · λ S ( G s ( S )) = dim k ( G r ( R )) · dim k ( G s ( S )) . Corollary 5.12.
Let R and S be standard graded, of depth at least two, and generalized weakly F -nilpotent. Let N be the smallest n ∈ N such that for all ≤ j < d R and ≤ i < d S m nR H j m R ( R ) ⊂ FH j m R ( R ) and m nS H i m S ( S ) ⊂ FH i m S ( S ) . Further, let e be the smallest e ∈ N such that p e ≥ ( N + 1)2 d T − . We have Fte ∗ ( T ) ≤ e + P d T j =0 (cid:0) d T j (cid:1) H j , where H j are the bounds on the Hartshorne-Speiser-Lyubeznik numbers of T given in Theorem 5.8.Proof. Note first that m R [ H j m R ( R )] is nilpotent for all j , including when j = d R as H j m R ( R ) vanishes in high degree,and similarly for S . Consequently, m T ( H j m R ( R ) S ) and m T ( R H j m S ( S ) are nilpotent for all j . It may be the casethat T is weakly F -nilpotent so that the e may seem superfluous. The role of the N + 1 factor in the statementensures H d R m R ( R ) S and R H d S m S ( S ) are sent into the nilpotent part of H d R m T ( T ) and H d S m T ( T ) respectively.Now we need only show m NT ( H r m R ( R ) H s m S ( S )) is nilpotent for 0 ≤ j < d T and r + s = j + 1. However, since R, S, and T are all standard graded, if x y is a homogeneous element in m NT , we must have that x ∈ m NR and y ∈ m NS . Consequently, for r < d R and s < d S , x y sends H r m R ( R ) H s m S ( S ) into 0 FH j m T ( T ) .5.3. Frobenius test exponents for Veronese subrings.
We conclude this article by bounding the homogeneousFrobenius test exponent for a Veronese subring of a graded ring with a nilpotent singularity. We let R be a standardgraded ring of dimension d defined over R = k . Theorem 5.13.
Fix n ∈ N and let R ′ = R ( n ) . One has HSL( H j m R ′ ( R ′ )) ≤ HSL( H j m R ( R )) . Consequently, if R isweakly F -nilpotent, then Fte ∗ ( R ′ ) ≤ P dj =0 (cid:0) dj (cid:1) HSL( H j m R ( R )) . Furthermore, if R is generalized weakly F -nilpotentand N is the smallest m ∈ N such that m mR H j m R ( R ) is nilpotent for all ≤ j < d and e is the smallest e ∈ N suchthat p e ≥ N d − , then Fte ∗ ( R ′ ) ≤ e + P dj =0 (cid:0) dj (cid:1) HSL( H j m R ( R )) .Proof. We recall the diagram from the proof of Theorem 4.19, from which the first claim follows immediately. h H j m R ( n ) ( R ( n ) ) i t h H j m R ( R ) i nt h H j m R ( n ) ( R ( n ) ) i pt h H j m R ( R ) i pnt ∼ F F ∼ Now, given an x ∈ m NR ′ , viewed in R we also have x ∈ m NR . Then, since xH j m R ( R ) is nilpotent, xH j m R ( R ) isnilpotent in degree nt for all t ∈ Z , completing the claim. References [Bli01] M. Blickle,
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