Frobenius test exponent for ideals generated by filter regular sequences
aa r X i v : . [ m a t h . A C ] J a n FROBENIUS TEST EXPONENT FOR IDEALS GENERATED BY FILTERREGULAR SEQUENCES
DUONG THI HUONG AND PHAM HUNG QUY
Dedicated to Professor Nguyen Tu Cuong on the occasion of his 70th birthday
Abstract.
Let ( R, m ) be a Noetherian local ring of prime characteristic p >
0, and t aninteger such that H j m ( R ) / FH j m ( R ) has finite length for all j < t . The aim of this paper is toshow that there exists an uniform bound for Frobenius test exponents of ideals generatedby filter regular sequences of length at most t . Introduction
Throughout this paper, let R be a Noetherian commutative ring of prime characteristic p >
0, and I an ideal of R . The key ingredient in study ring of prime characteristic is usingthe Frobenius endomorphism F : R → R ; x x p and its e -th iterations F e , e ≥
1. The e -th Frobenius power of I is the extension of I via F e , I [ p e ] = ( x p e | x ∈ I ). The Frobeniusclosure of I is I F = { x | x p e ∈ I [ p e ] for some e ≥ } . Hence I F is the set of all nilpotentelements under Frobenius endomorphism modulo I . By the Noetherianess of R there is aninteger e , depending on I , such that ( I F ) [ p e ] = I [ p e ] . The Frobenius test exponent of I isdefined by Fte( I ) = min { e | ( I F ) [ p e ] = I [ p e ] } . Under mild conditions, R is F -pure if and onlyif Fte( I ) = 0 for all I by Hochster [6]. In general, we can not expect an upper bound for allFte( I ) by Brenner [1]. Restricting to the class of parameter ideals, for any local ring ( R, m )of dimension d we define the Frobenius test exponent for parameter ideals
Fte( R ) as followsFte( R ) = min { e | ( q F ) [ p e ] = q [ p e ] for all parameter ideals q } , and Fte( R ) = ∞ if we have no such integer. Katzman and Sharp asked whether Fte( R ) < ∞ for any (equidimensional) local ring. Moreover they [11] proved that it is the case whenthe ring is Cohen-Macaulay. The main idea in [11] is connecting Fte( R ) with an invari-ant defined by the Frobenius actions on the local cohomology modules H i m ( R ), namely the Hartshorne-Speiser-Lyubeznik number of H i m ( R ). Recall that the Frobenius endomorphisminduces natural Frobenius actions on the local cohomology modules H i m ( R ) for all i ≥ Frobenius closure of zero submodule of H i m ( R ), denoted by 0 FH i m ( R ) , consists nilpotentelements of H i m ( R ) under the Frobenius action. The Hartshorne-Speiser-Lyubeznik number Key words and phrases.
The Frobenius test exponent, The Hartshorne-Speiser-Lyubeznik number, Localcohomology, Filter regular sequence.2020
Mathematics Subject Classification : 13A35, 13D45.The authors are partially supported by a fund of Vietnam National Foundation for Science and TechnologyDevelopment (NAFOSTED) under grant number 101.04-2020.10. of H i m ( R ) is defined byHSL( H i m ( R )) = min { e | FH i m ( R ) = Ker( H i m ( R ) F e −→ H i m ( R )) } . It is proved that HSL( H i m ( R )) always exists. The Hartshorne-Speiser-Lyubeznik number of R is HSL( R ) = max { HSL( H i m ( R )) | i = 0 , . . . , d } . In fact Katzman and Sharp proved thatFte( R ) = HSL( R ) provided R is Cohen-Macaulay. In general, we have Fte( R ) ≥ HSL( R )for any local ring by [9]. Beyond the Cohen-Macaulay case, Huneke, Katzman, Sharp,and Yao [8] showed that Fte( R ) < ∞ for generalized Cohen-Macaulay local rings by usingseveral concepts and techniques from commutative algebra, namely unconditioned strongd-sequences, cohomological annihilators, and modules of generalized fractions. In 2019,the second author [16] not only simplified the proof for generalized Cohen-Macaulay ringsbut also proved Fte( R ) < ∞ for weakly F-nilpotent rings, i.e. H i m ( R ) = 0 FH i m ( R ) for all i < d . Recently, Maddox [13] extended this result for generalized weakly F-nilpotent rings,i.e. H i m ( R ) / FH i m ( R ) has finite length for all i < d .Suppose Fte( R ) < ∞ , and let x , . . . , x d be a system of parameters of R . Then for any t ≤ d we have Fte(( x , . . . , x t )) ≤ Fte( R ) by using Krull’s intersection theorem. If we do notknow about the finiteness of Fte( R ), in which conditions we have Fte(( x , . . . , x t )) < ∞ ?For a regular sequence x = x , . . . , x t , Katzman and Sharp [11, Corollary 4.3] showed thatthere exists an integer C x such that Fte(( x n , . . . , x n t t )) ≤ C x for all n , . . . , n t ≥
1. Since C x depends on the choice of x , the previous result does not answer for the following question. Question 1.
Let ( R, m ) be a local ring of prime characteristic and of dimension d , and t ≤ depth( R ) an integer. Does there exist a positive integer C such that for any regularsequence x , . . . , x t we have Fte(( x , . . . , x t )) ≤ C ? Inspired by known results about the finiteness of Fte( R ), we give an affirmative answerfor the above question in a more general form. It covers all aforementioned results. Theorem 1.1.
Let ( R, m ) be a Noetherian local ring of dimension d and of prime charac-teristic p > , and t ≤ d a positive integer such that H j m ( R ) / FH j m ( R ) has finite length forall j < t . Then there exists a positive integer C such that for any filter regular sequence x , . . . , x t we have Fte(( x , . . . , x t )) ≤ C . In the next section we recall the basic notions and relevant materials. We prove the maintheorem in the last section. 2.
Preliminaries
Filter regular sequences.
In this subsection, the assumption that R is of primecharacteristic is unnecessary. Definition 2.1.
Let ( R, m ) be a Noetherian local ring, and x , . . . , x t a sequence of elementsof R . Then we say that x , . . . , x t is a filter regular sequence if the following conditions hold:(1) ( x , . . . , x t ) ⊆ m ,(2) x i / ∈ p for all p ∈ Ass (cid:0) M ( x ,...,x i − ) M (cid:1) \ { m } , i = 1 , . . . , t . ROBENIUS TEST EXPONENTS 3
The notion of filter regular sequences was introduced by Cuong, Schenzel, and Trung in[2]. The following is well-known.
Lemma 2.2.
Let ( R, m ) be a Noetherian local ring and x , . . . , x t ⊆ m . The following areequivalent(1) x , . . . , x t is a filter regular sequence.(2) For each ≤ i ≤ t the quotient ( x , . . . , x i − ) : R x i ( x , . . . , x i − ) is an R -module of finite length.(3) For each ≤ i ≤ t the sequence x , x , . . . , x i forms an R p -regular sequence for every p ∈ Spec( R/ ( x , . . . , x i )) \ { m } .(4) The sequence x n , . . . , x n t t is a filter regular sequence for all n , . . . , n t ≥ .(5) The sequence x , . . . , x t is a filter regular sequence of b R , where b R is the m -adic com-pletion of R . The Frobenius action on local cohomology.
In this subsection, let R be a Noe-therian ring containing a field of characteristic p >
0. Let F : R → R, x x p denote theFrobenius endomorphism. If we want to notationally distinguish the source and target ofthe e -th Frobenius endomorphism F e : R x x pe −−−−→ R , we will use F e ∗ ( R ) to denote the target. F e ∗ ( R ) is an R -bimodule, which is the same as R as an abelian group and as a right R -module,that acquires its left R -module structure via the e -th Frobenius endomorphism F e . By defi-nition the e -th Frobenius endomorphism F e : R → F e ∗ ( R ) sending x to F e ∗ ( x p e ) = x · F e ∗ (1)is an R -homomorphism. We say R is F -finite if F ∗ ( R ) is a finite R -module. Definition 2.3.
Let I be an ideal of R , we define(1) The e -th Frobenius power of I is I [ p e ] = ( x p e | x ∈ I ).(2) The Frobenius closure of I , I F = { x | x p e ∈ I [ p e ] for some e ≥ } . Definition 2.4.
Let I be an ideal of R . By the Noetherianess of R there is an integer e ,depending on I such that ( I F ) [ p e ] = I [ p e ] . The smallest number e satisfying the condition iscalled the Frobenius test exponent of I , and denoted by Fte( I ),Fte( I ) = min { e | ( I F ) [ p e ] = I [ p e ] } . A problem of Katzman and Sharp [11, Introduction] asks in its strongest form: doesthere exist an integer e , depending only on the ring R , such that for every ideal I we have( I F ) [ p e ] = I [ p e ] . A positive answer to this question, together with the actual knowledge of abound for e , would give an algorithm to compute the Frobenius closure I F . Unfortunately,Brenner [1] gave two-dimensional normal standard graded domains with no Frobenius testexponent. In contrast, Katzman and Sharp showed the existence of Frobenius test exponentif we restrict to class of parameter ideals in a Cohen-Macaulay ring. Therefore it is naturalto ask the following question. DUONG THI HUONG AND PHAM HUNG QUY
Question 2.
Let ( R, m ) be an (equidimensional) local ring of prime characteristic p . Thendoes there exist an integer e such that for every parameter ideal q of R we have ( q F ) [ p e ] = q [ p e ] ? We define the
Frobenius test exponent for parameter ideals of R , Fte( R ), the smallestinteger e satisfying the above condition and Fte( R ) = ∞ if we have no such e . It shouldbe noted that the authors recently used the finiteness of Fte( R ), if have, to find an upperbound of the multiplicity of a local ring [10].For any ideal I = ( x , . . . , x t ), the Frobenius endomorphism F : R → R and its localizationsinduce a natural Frobenius action on local cohomology F : H iI ( R ) → H iI [ p ] ( R ) ∼ = H iI ( R ) forall i ≥
0. In general, let A be an Artinian R -module with a Frobenius action F : A → A .Then we define the Frobenius closure FA of the zero submodule of A is the submodule of A consisting all elements z such that F e ( z ) = 0 for some e ≥
0. Hence 0 FA is the nilpotent partof A under the Frobenius action. By [5, Proposition 1.11], [12, Proposition 4.4] and [17] wehave the following. Theorem 2.5.
Let ( R, m ) be a local ring of prime characteristic p > , and A an Artinian R -module with a Frobenius action F : A → A . Then there exists a non-negative integer e such that FA = Ker( A F e −→ A ) . Definition 2.6. (1) Let A be an Artinian R -module with a Frobenius action F . The Hartshorne-Speiser-Lyubeznik number of A is denoted by HLS( A ) and is defined tobe HSL( A ) = min { e | FA = Ker( A F e −→ A ) } . (2) Notice that H i m ( R ) is always Artinian for all i ≥
0. We define the
Hartshorne-Speiser-Lyubeznik number of a local ring ( R, m ) as followsHSL( R ) := min { e | FH i m ( R ) = Ker( H i m ( R ) F e −→ H i m ( R )) for all i = 0 , . . . , d } . As mentioned in the introduction, Question 2 was answered affirmatively in the followingcases:(1) (Katzman-Sharp) R is Cohen-Macaulay ring. Moreover, Fte( R ) = HSL( R ).(2) (Huneke-Katzman-Sharp-Yao) R is generalized Cohen-Macaulay ring, i.e. H i m ( R ) hasfinite length for all i < d .(3) (Quy) R is weakly F -nilpotent ring, i.e. H i m ( R ) = 0 FH i m ( R ) for all i < d .(4) (Maddox) R is generalized weakly F -nilpotent ring, i.e. H i m ( R ) / FH i m ( R ) has finitelength for all i < d .The main idea of the proofs of the affirmative cases is to make a connection between theFrobenius test exponent for parameter ideals with Hartshorne-Speiser-Lyubeznik numbersof the local cohomology modules. For example, if R is weakly F -nilpotent, then the secondauthor [16] used the notion of relative Frobenius action on local cohomology to proveFte( R ) ≤ d X i =0 (cid:18) di (cid:19) HSL( H i m ( R )) . ROBENIUS TEST EXPONENTS 5
The relative Frobenius action on local cohomology.
In this subsection, we recallthe notion of relative Frobenius action on local cohomology which was introduced in [14]by Polstra and Quy in study F -nilpotent rings. Let K ⊆ I be ideals of R . The Frobeniusendomorphism F : R/K → R/K can be factored as composition of two natural maps:
R/K → R/K [ p ] ։ R/K, where the second map is the natural project map. We denote the first map by F R : R/K → R/K [ p ] , F R ( a + K ) = a p + K [ p ] for all a ∈ R . The homomorphism F R induces the relativeFrobenius actions on local cohomology F R : H iI ( R/K ) → H iI ( R/K [ p ] ) via ˇCech complexes. Definition 2.7. (1) We define the relative Frobenius closure of the zero submodule of H iI ( R/K ) with respect to R as follows0 F R H iI ( R/K ) = { η | F eR ( η ) = 0 ∈ H iI ( R/K [ p e ] ) for some e ≫ } . (2) If there is an integer e such that0 F R H iI ( R/K ) = Ker( H iI ( R/K ) F eR −→ H iI ( R/K [ p e ] )) , then we call the smallest of such integers the Hartshorne-Speiser-Lyubeznik numberof H iI ( R/K ) with respect to R , denoted by HSL R ( H iI ( R/K )). And convention thatHSL R ( H iI ( R/K )) = ∞ if we have no such integer. Lemma 2.8.
Let ( R, m ) be a Noetherian local ring, and I an ideal of R . Then F R H m ( R/I ) = I F ∩ ( I : m ∞ ) /I. Proof.
Pick a + I ∈ F R H m ( R/I ) . We have a ∈ I : m ∞ and there exists an integer e such that F e ( a + I ) = a p e + I [ p e ] = 0 ∈ R/I [ p e ] . Equivalently, a + I ∈ I F ∩ ( I : m ∞ ) /I . (cid:3) Main result
Throughout this section, let ( R, m ) be a Noetherian local ring of prime characteristic p > d , and t ≤ d an integer such that H j m ( R ) / FH j m ( R ) has finite length for all j < t . Let x , . . . , x t be a filter regular sequence. Set I i = ( x , . . . , x i ) for all i ≤ t and I = ( x , . . . , x t ). Lemma 3.1.
Let ( R, m ) be a local ring of prime characteristic p > and of dimension d ,and t ≤ d an integer such that H j m ( R ) / FH j m ( R ) has finite length for all j < t . Let n be annon-negative integer such that m n H j m ( R ) / FH j m ( R ) = 0 for all j < t . Then for every filterregular sequence x , . . . , x t we have m i n H j m ( R/I i ) ⊆ F R H j m ( R/I i ) for all i ≤ t and all j < t − i . DUONG THI HUONG AND PHAM HUNG QUY
Proof.
We proceed by induction on i . The case i = 0 is nothing to do. Suppose i > i −
1. For each e ≥ −→ R/ ( I i − : x i ) x i −−−→ R/I i − −−−→ R/I i −−−→ ( F eR ) ′ y F eR y F eR y ( ⋆ )0 −→ R/ ( I [ p e ] i − : x p e i ) x pei −−−→ R/I [ p e ] i − −−−→ R/I [ p e ] i −−−→ R/ ( I i − : x i ) F eR −→ R/ ( I i − : x i ) [ p e ] ։ R/ ( I [ p e ] i − : x p e i ) . Because x p e , . . . , x p e t is a filter regular sequence, ( I [ p e ] i − : x p e i ) /I [ p e ] i − has finite length. Thus H j +1 m ( R/ ( I [ p e ] i − : x p e i )) ∼ = H j +1 m ( R/ ( I i − : x i ) [ p e ] ) ∼ = H j +1 m ( R/I [ p e ] i − )for all j ≥ e ≥
0. Therefore the diagram ( ⋆ ) induces the following commutativediagram with exact rows · · · x i −→ H j m ( R/I i − ) −−−→ H j m ( R/I i ) δ −−−→ H j +1 m ( R/I i − ) F eR y F eR y F eR y · · · x pei −−→ H j m ( R/I [ p e ] i − ) α −−−→ H j m ( R/I [ p e ] i ) β −−−→ H j +1 m ( R/I [ p e ] i − ) . Pick any u ∈ H j m ( R/I i ), and x, y ∈ m i − n . Then δ ( u ) ∈ H j +1 m ( R/I i − ). By the inductivehypothesis xδ ( u ) ∈ F R H j +1 m ( R/I i − ) , so there is an integer e such that0 = F eR ( xδ ( u )) = β ( F eR ( xu )) . Hence there exists v ∈ H j m ( R/I [ p e ] i − ) such that α ( v ) = F eR ( xu ). Moreover, y p e ∈ m i − n so y p e v ∈ F R H j m ( R/I [ pe ] i − ) by using the induction for the sequence x p e , . . . , x p e i − . Therefore, α ( y p e v ) = y p e .F eR ( xu ) = F eR ( yxu ) ∈ F R H j m ( R/I [ pe ] i ) . Leading to xyu ∈ F R H j m ( R/I i ) . Hence xy ∈ Ann R ( H j m ( R/I i ) / F R H j m ( R/I i ) ), and so m i n ⊆ Ann R ( H j m ( R/I i ) / F R H j m ( R/I i ) )for all i ≤ t and j < t − i . The proof is complete. (cid:3) Remark 3.2.
Using the notation in Lemma 3.1, and suppose x . . . , x t ∈ m t n . Then wehave Im( H j m ( R/I i − : x i ) x i −→ H j m ( R/I i − )) ⊆ F R H j m ( R/I i − ) for all i ≤ t and j ≤ t − i . Indeed the case 1 ≤ j ≤ t − i follows from Lemma 3.1 andthe fact H j m ( R/I i − : x i ) ∼ = H j m ( R/I i − ). For j = 0 we use Lemma 3.1 and the surjectivemap H m ( R/I i − ) ։ H m ( R/I i − : x i ). In particular, every relative nilpotent element of theinduced Frobenius action on Coker( H j m ( R/I i − : x i ) x i −→ H j m ( R/I i − )) is an image of someelement in 0 F R H j m ( R/I i − ) . ROBENIUS TEST EXPONENTS 7
Using the above Remark and the argument of [16, Proof of the main theorem] and [13,Theorem 3.1] we obtain the following whose proof is left to the reader.
Proposition 3.3.
Let ( R, m ) be a local ring of prime characteristic p > and of dimension d , and t ≤ d an integer such that H j m ( R ) / FH j m ( R ) has finite length for all j < t . Then thereexists a non-negative integer e such that for every filter regular sequence x , . . . , x t we have HSL R ( H m ( R/ ( x , . . . , x t ))) ≤ t X k =0 (cid:18) tk (cid:19) HSL( H k m ( R )) + e . Remark 3.4. If x , . . . , x t is a regular sequence then H j m ( R ) = 0 for all j < t , and the number e in the previous result can be chosen as zero. Thus we have HSL R ( H m ( R/ ( x , . . . , x t ))) ≤ HSL( H t m ( R )).The following is known to experts. Lemma 3.5.
Let ( R, m ) be an F -finite local ring of dimension d . Let p ∈ Spec( R ) a primeideal. Then for all i ≤ d we have HSL( H i − dim R/ pp R p ( R p )) ≤ HSL( H i m ( R )) . In particular, HSL( R p ) ≤ HSL( R ) .Proof. Since ( R, m ) is F -finite, it is an image of some regular local ring ( S, n ) by Gabber[4]. Suppose that dim S = n . For each i ≤ d , by the definition we have HLS( H i m ( R )) is thesmallest non-negative integer e such thatKer( H i m ( R ) F e −→ H i m ( F e ∗ R )) = Ker( H i m ( R ) F e +1 −−−→ H i m ( F e +1 ∗ R )) . By the local duality theorem we haveCoker(Ext n − iS ( F e ∗ R, S ) ( F e ) ∨ −−−→ Ext n − iS ( R, S )) = Coker(Ext n − iS ( F e +1 ∗ R, S ) ( F e +1 ) ∨ −−−−→ Ext n − iS ( R, S )) . Let P be the preimage of p in S . Taking localization we haveCoker(Ext n − iS P ( F e ∗ R p , S P ) → Ext n − iS P ( R p , S P )) = Coker(Ext n − iS P ( F e +1 ∗ R p , S P ) → Ext n − iS P ( R p , S P )) . Applying the local duality theorem for ( R p , p R p ) we haveKer( H i − dim R/ pp R p ( R p ) F e −→ H i − dim R/ pp R p ( F e ∗ R p )) = Ker( H i − dim R/ pp R p ( R p ) F e +1 −−−→ H i − dim R/ pp R p ( F e +1 ∗ R p )) . Hence HSL( H i − dim R/ pp R p ( R p )) ≤ e . The proof is complete. (cid:3) We are ready to prove the main result of this paper.
Theorem 3.6.
Let ( R, m ) be a local ring of dimention d and of prime characteristic p > ,and t ≤ d an integer such that H j m ( R ) / FH j m ( R ) has finite length for all j < t . Then there existsan integer C such that for any filter regular sequence x , . . . , x t we have Fte( x , . . . , x t ) ≤ C .Proof. By passing to the completion we can assume that ( R, m ) is complete. We next use theΓ-construction of Hochster and Huneke [7] to obtain a faithfully flat extension R → R Γ suchthat R Γ is F -finite and m R Γ is the maximal ideal of R Γ . Notice that the induced Frobeniusactions on H j m ( R ) / FH j m ( R ) are injecitve for all j ≥
0. By [3, Lemmas 2.9 and 4.3] we can choosea sufficiently small Γ such that the Frobenius actions on R Γ ⊗ R H j m ( R ) / FH j m ( R ) are injecitve DUONG THI HUONG AND PHAM HUNG QUY for all j ≥
0. Thus 0 FH j m ( R Γ ) ∼ = R Γ ⊗ R FH j m ( R ) . Since ( x , . . . , x t ) F R Γ ⊆ (( x , . . . , x t ) R Γ ) F , it isenough to prove the requirement for R Γ . Therefore we can assume henceforth that ( R, m ) is F -finite.Let c be the upper bound in Proposition 3.3 and h := HSL( R ). Let I = ( x , . . . , x t ), andpick any a ∈ I F \ I . Suppose dim R/ ( I : a ) = s . We will prove dim R/ ( I [ p hs ] : a p hs ) ≤ s . There is nothing to do if s = 0. Suppose s >
0, and let p be any minimalprime of R/ ( I : a ) with dim R/ p = s . Then aR p ∈ ( IR p ) F by [15, Lemma 3.3]. By theminimality we have a + IR p ∈ H p R p ( R p /IR p ). Since x , . . . , x t becomes a regular sequencein R p we have HSL R p ( H p R p ( R p /IR p )) ≤ HSL( H t p R p ( R p )) ≤ h by Remark 3.4 and Lemma3.5. Therefore F hR p ( a + IR p ) = 0 ∈ R p /I [ p h ] R p , and so a p h R p ∈ I [ p h ] R p . Hence R p / ( I [ p h ] : a p h ) R p = 0 for all p ∈ Min( R/ ( I : a )) with dim R/ p = s . Thus dim R/ ( I [ p h ] : a p h ) ≤ s − I [ p h ] and a p h .On the other hand s ≤ d − t , so we always have dim R/ ( I [ p ( d − t ) h ] : a p ( d − t ) h ) ≤
0. Moreover,if dim R/ ( I : a ) ≤
0, then ¯ a = a + I ∈ I F ∩ ( I : m ∞ ) /I = 0 F R H m ( R/I ) by Lemma 2.8.By Proposition 3.3 we have F cR (¯ a ) = 0. Thus a p c ∈ I [ p c ] . Putting all together we have a p ( d − t ) h + c ∈ I [ p ( d − t ) h + c ] for all a ∈ I F , so Fte( I ) ≤ ( d − t ) h + c . The proof is complete. (cid:3) Acknowledgement .
The authors are grateful to the referee for carefully reading of the paperand valuable suggestions and comments. The first author was funded by Vingroup JointStock Company and supported by the Domestic Master/ PhD Scholarship Programme ofVingroup Innovation Foundation (VINIF), Vingroup Big Data Institute (VINBIGDATA).
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Department of Mathematics, Thang Long University, Hanoi, Vietnam
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