aa r X i v : . [ m a t h . A C ] J a n F -INVARIANTS OF STANLEY-REISNER RINGS WÁGNER BADILLA-CÉSPEDES Abstract.
In prime characteristic there are important invariants that allow us to measuresingularities. For certain cases, it is known that they are rational numbers. In this article,we show this property for Stanley-Reisner rings in several cases.
Contents
1. Introduction 12. Stanley-Reisner Rings 33. F -Thresholds 44. The Ideal J e
65. Cartier Threshold of a with Respect to J a -Invariants and Regularity 16Acknowledgments 18References 181. Introduction
Throughout this manuscript R denotes a Noetherian ring of prime characteristic p . Incharacteristic zero, the log canonical threshold, lct( f ) , of a polynomial f with coefficients ina field, is an important invariant in birational geometry [BFS13]. This number measures thesingularities of f near to zero. In positive characteristic, the F -pure threshold of an element f ∈ R , denoted fpt( f ) , was defined by Takagi and Watanabe [TW04]. Roughly speaking,this measures the splitting order of f . It is defined by fpt( f ) = sup ( ap e | the inclusion Rf ape ⊆ R /p e is a split ) for f ∈ R .The F -pure threshold is considered as analogous to the log canonical threshold, and theyshare similar properties [TW04, MTW05]. In particular, if f is an element in Z [ x , . . . , x n ] ,then lim p →∞ fpt( f mod p ) = lct( f ) [HY03, MTW05].In this work, we study a general form of the F -pure threshold called the Cartier threshold.Given a and J ideals in R , the Cartier threshold of a with respect to J is defined as ct J ( a ) = Mathematics Subject Classification.
Primary 13A35, 13F55; Secondary 13D45, 14B05.
Key words and phrases.
Stanley-Reisner rings, F -thresholds, F -pure thresholds, a -invariants,Castelnuovo-Mumford regularity. Partially supported by CONACYT Grants and , and partially supported through LuisNúñez-Betancourt’s Cátedras Marcos Moshinsky Grant. lim e →∞ b J a ( p e ) p e , where b J a ( p e ) = max { t ∈ N | a t J e } & J e = { f ∈ R | ϕ ( f /p e ) ∈ J, ∀ ϕ ∈ Hom R ( R /p e , R ) } . These numbers have been studied in more depth in an upcoming work [DSHNnBW]. If weconsider ( R, m , K ) a local ring or a standard graded K -algebra which is F -finite and F -pure,then ct m ( a ) = fpt( a ) .In this manuscript, we focus on Stanley-Reisner rings. The combinatorial nature of theserings has been useful to study their structures in prime characteristic. For instance, in thiscase one can describe their algebras of Frobenius and Cartier operators [ÀlMBZ12, BZ19].In this work, we show that the Cartier threshold of a with respect to J in Stanley-Reisnerrings is a rational number when J is a radical ideal. Theorem A (see Theorem 5.15 and Corollary 5.16) . Let a , J be two ideals in a Stanley-Reisner ring R , such that a ⊆ J , and J is a radical ideal. Then, the Cartier threshold of a with respect to J is a rational number. In order to obtain Theorem A, we need to reduce the computation of ct J ( a ) to the casewhere J is a monomial ideal. For this trick, we need to replace R by the completion of asuitable localization. Then, the problem is reduced to the regular case by taking a quotientwith respect to the Cartier core (see Definition 4.12).We now recall the definition of the F -thresholds. They are numbers obtained by comparingordinary powers versus Frobenius powers. These were introduced in regular rings by Mustaţă,Takagi and Watanabe [MTW05], and their existence, in the general case, was proved by DeStefani, Núñez-Betancourt and Pérez [DSNnBP18]. These are defined as c J ( a ) = lim e →∞ ν J a ( p e ) p e ,where ν J a ( p e ) = max { m ∈ N | a m J [ p e ] } , and a , J ⊆ R are ideals.A recent line of research consists in understanding under which conditions the set of F -thresholds is a discrete set of rational numbers. This was proved by Blickle, Mustaţă, andSmith [BMS08] for an F -finite regular ring. Although the F -threshold is a rational numberin the regular case, this situation is unknown in general Noetherian rings. Trivedi [Tri18]showed that, in general, the set of F -thresholds with respect to the maximal ideal in a localring is not necessarily discrete. In this paper, we study the rationality of F -thresholds forStanley-Reisner rings. Theorem B (see Theorem 3.6) . Let a , J be two ideals in a Stanley-Reisner ring R , suchthat a ⊆ √ J , and J is a monomial ideal. Then, the F -threshold of a with respect to J is arational number. The key idea to prove Theorem B is to work modulo the minimal primes, which yields aregular ring. The result follows from comparing the F -thresholds of R versus these quotients.We point out that Theorem B is a key component of the proof of Theorem A.The Castelnuovo-Mumford regularity is an invariant that measures the complexity of thefree resolution of a standard graded K -algebra ( R, m , K ) . The growth of reg( R/J [ p e ] ) hasbeen intensively studied due to its relation to discreteness of F - jumping coefficients [KZ14,KSSZ14, Zha15], localization of tight closure [Kat98, Hun00], and existence of the generalizedHilbert-Kunz multiplicity [DS20, Vra16]. We recall that the Castelnuovo-Mumford regularitycan be computed in terms of the a -invariants introduced by Goto and Watanabe [GW78].In this manuscript, we provide a formula for the limits of reg( R/J [ p e ] ) . -INVARIANTS OF STANLEY-REISNER RINGS 3 Theorem C (see Theorem 6.7) . Let J be a homogeneous ideal in a Stanley-Reisner ring R .Then, lim e →∞ reg( R/J [ p e ] ) p e = max ≤ i ≤ dα ∈A ′ { a i ( S/ ( J α + J )) + | α |} , where A ′ = { α ∈ N n | ≤ α i ≤ for i = 1 , . . . , n } , J α = ( I : x α ) , and d = max { dim( S/ ( J α + J )) | α ∈ A ′ } . In particular, this limit is an integer number. Stanley-Reisner Rings
Throughout this section we use the following notation.
Notation 2.1.
We denote S = K [ x , . . . , x n ] with K an F -finite field of prime characteristic p . Let I be a squarefree monomial ideal of S . Let I = T li =1 p i such that p i p j for i = j and p , . . . , p l are generated by variables. We take R = S/I .These rings have mild singularities, for instance, they are F -pure. They also have combi-natorial structure given by simplicial complexes.Suppose that a ⊆ R is an ideal. We abuse notation and denote the inverse image of a ⊆ R under the natural projection S −→ S/I by a ⊆ S .We now characterize the ring of p -th roots of R in terms of ideal quotients. Proposition 2.2. If q = p e , where e is a nonnegative integer, then R /q = S /q /I /q ∼ = M ≤ i ≤ sα ∈A S/J α ( a i x α ) /q , with A = { α ∈ N n | ≤ α i ≤ q − for i = 1 , . . . , n } , B = { a i /q | i = 1 , . . . , s } is a base of K /q as K -vector space, and J α = ( I : x α ) .Proof. Each element r /q ∈ S /q can be written uniquely as r /q = M ≤ i ≤ sα ∈A r i,α ( a i x α ) /q , where r i,α ∈ S . We take ϕ : S /q −→ M ≤ i ≤ sα ∈A S/J α ( a i x α ) /q , defined by ϕ ( r /q ) = M ≤ i ≤ sα ∈A ( r i,α + J α )( a i x α ) /q . We have that ϕ is a surjective S -linear morphism.We claim that ker ϕ = I /q . First, we show ker ϕ ⊆ I /q . Let r /q ∈ ker ϕ . It is sufficient toconsider r /q = x θ ( a i x α ) /q for some θ ∈ N n , α ∈ A , and i ∈ { , . . . , s } . Hence, ϕ ( r /q ) =( x θ + J α )( a i x α ) /q . Thus, x θ ∈ J α . This implies that a i x α + θ ∈ I , and so, x θ/q ( a i x α ) /q ∈ I /q .It follows that r /q = x θ ( a i x α ) /q = ( x θ/q ) q ( a i x α ) /q ∈ I /q .Now, we show I /q ⊆ ker ϕ . Let r /q ∈ I /q . It is enough to consider r /q = x θ ( a i x α ) /q x β/q with θ ∈ N n , α ∈ A , i ∈ { , . . . , s } , and x β a generator of I . Since ≤ α j ≤ q − and ≤ β j ≤ for every ≤ j ≤ n , there exists γ ∈ { , } n such that α + β − qγ ∈ A . Let WÁGNER BADILLA-CÉSPEDES α ′ = α + β − qγ . We note that x θ + γ ( a i x α ′ ) ∈ I . As a consequence, x θ + γ ∈ J α ′ . Furthermore, r /q = x θ + γ ( a i x α ′ ) /q . Subsequently, ϕ ( r /q ) = ( x θ + γ + J α ′ )( a i x α ′ ) /q = 0 . Thus, r /q ∈ ker ϕ .It follows, R /q ∼ = M ≤ i ≤ sα ∈A S/J α ( a i x α ) /q as S -module. Therefore, they are isomorphic as R -modules. (cid:3) Remark 2.3.
We follow Notation 2.1. Let q be a prime ideal of S . Suppose that p , . . . , p r ⊆ q with r ≤ l , p j q for r < j , and ( x , . . . , x u ) = P ri =1 p i .Let e q , . . . , e q t ∈ Spec S q be such that ( x , . . . , x u ) S q = e q & e q & . . . & e q t . There exist q , . . . , q t ∈ Spec S , where q i ⊆ q and q i = e q i ∩ S . We have that (0) & ( x ) & ( x , x ) & . . . & ( x , . . . , x u ) = q & q & . . . & q t ⊆ q is a chain of prime ideals in S with length u + t , and so, u + t ≤ ht( q ) . Hence, t ≤ ht( q ) − u .Then, dim S q / ( x , . . . , x u ) S q ≤ ht( q ) − u . Therefore, ht( q ) − u = dim S q / ( x , . . . , x u ) S q . In particular, if we take A = b S q , we have that dim A − u = dim A/ ( x , . . . , x u ) A. Since A is a complete regular local ring, A ∼ = L [[ x , . . . , x u , y , . . . , y t ]] , where K ⊆ L is afield extension. Moreover, we have that IA = T li =1 p i A is squarefree monomial ideal of A invariables x , . . . , x u . We denote x θ = x θ · · · x uθ u y θ u +1 · · · y tθ u + t . We take B = A/IA and m its maximal ideal. Proposition 2.4. If q = p e , where e is a nonnegative integer, then B /q ∼ = M ≤ i ≤ sα ∈A A/J α ( a i x α ) /q , with A = { α ∈ N u + t | ≤ α i ≤ q − for i = 1 , . . . , u + t } , B = { a i /q | i = 1 , . . . , s } is a baseof L /q as L -vector space, and J α = ( IA : x α ) .Proof. The proof is analogous to Proposition 2.2. (cid:3) F -Thresholds The F -thresholds were introduced by Mustaţă, Takagi and Watanabe [MTW05] for F -finite regular local rings of prime characteristic. Subsequently, in work with Huneke [HMTW08],they defined F -thresholds in general rings of positive characteristic, provided the limit ex-ists. The existence of these invariants in the general case is proved in the work of De Stefani,Núñez-Betancourt and Pérez [DSNnBP18].Our main goal is to describe the F -thresholds of Stanley-Reisner rings, when we havemonomial ideals. -INVARIANTS OF STANLEY-REISNER RINGS 5 Definition and First Properties.
In this subsection R denotes a ring of prime char-acteristic p . We discuss properties related to F -thresholds. Definition 3.1.
Let R be a ring. Given a , J ideals inside R such that a ⊆ √ J , we define ν J a ( p e ) = max { m ∈ N | a m J [ p e ] } . Lemma 3.2 ([DSNnBP18, Lemma . ]) . Let R be a ring, and a , J ideals of R such that a ⊆ √ J . Then, ν J a ( p e + e ) p e + e − ν J a ( p e ) p e ≤ µ ( a ) p e for every e , e ∈ N . Theorem 3.3 ([DSNnBP18, Theorem . ]) . Let R be a ring, and a , J be two ideals in R such that a ⊆ √ J . Then, lim e →∞ ν J a ( p e ) p e exists. The previous theorem gives existence to the F -thresholds and we may define them. Definition 3.4 ([DSNnBP18]) . Let R be a ring. Given a , J ideals of R such that a ⊆ √ J ,we define the F -threshold of a with respect to J by c J ( a ) = lim e →∞ ν J a ( p e ) p e . Proposition 3.5 ([MTW05, Proposition . ] & [HMTW08, Proposition . ]) . Let R be aring, and let a , I, J be ideals in R . Then, the following hold. (1) If J ⊆ I , and a ⊆ √ J , then c I ( a ) ≤ c J ( a ) . (2) If a ⊆ √ J , then c J [ p ] ( a ) = p · c J ( a ) . F -Thresholds in Stanley-Reisner Rings. In this subsection, we focus on Stanley-Reisner rings. We denote S = K [ x , . . . , x n ] with K an F -finite field of prime characteristic p . Let I be a squarefree monomial ideal of S , and R = S/I .Suppose that a ⊆ R is an ideal. We abuse notation and denote the inverse image of a ⊆ R under the natural projection S −→ S/I by a ⊆ S .The following proposition is one of the main results of this paper, Theorem B. Using thefact that the quotient of R with each of its minimal prime ideals is a regular ring, we obtaina case where the F -threshold is a rational number. Theorem 3.6.
Let a , J be ideals of R , with a ⊆ √ J , and J monomial. Let p , . . . , p l bethe minimal prime ideals of R . Then, c JR ( a ) = max n c JS/ p i S/ p i ( a S/ p i ) o . In particular, c JR ( a ) ∈ Q .Proof. We know that I = T li =1 p i . Moreover, each p i is generated by variables. We claimthat c JR ( a ) ≥ max n c JS/ p i S/ p i ( a S/ p i ) o . Let e be a nonnegative integer. We take t i = ν JS/ p i a S/ p i ( p e ) .Then, a t i S/ p i J [ p e ] S/ p i . Hence, there exists r ∈ a t i such that r − c p i for every c ∈ J [ p e ] .Thus, r − c I , and so r J [ p e ] . As a consequence, a t i J [ p e ] . WÁGNER BADILLA-CÉSPEDES
We have that t i ≤ ν J a ( p e ) for all i . Then, ν JS/ p i a S/ p i ( p e ) p e ≤ ν J a ( p e ) p e . Thus, c JS/ p i S/ p i ( a S/ p i ) ≤ c JR ( a ) .Therefore, c JR ( a ) ≥ max n c JS/ p i S/ p i ( a S/ p i ) o .We now show that T li =1 ( J [ p e ] + p i ) ⊆ J [ p e ] + I . We proceed by contradiction. Let s be agenerator of T li =1 ( J [ p e ] + p i ) such that s J [ p e ] + I . Since J [ p e ] and each p i are monomialideals, we have that every J [ p e ] + p i is a monomial ideal too. Hence, T li =1 ( J [ p e ] + p i ) is amonomial ideal. We can take s as a monomial. Furthermore, s J [ p e ] and s I . Thus,there exists an i such that s p i . However, s ∈ J [ p e ] + p i . Since s is a monomial and p i isgenerated by variables, we conclude that s ∈ J [ p e ] , we get a contradiction. Thus, s ∈ J [ p e ] + I .We prove that c JR ( a ) ≤ max n c JS/ p i S/ p i ( a S/ p i ) o . Let e be a nonnegative integer. We take t = ν J a ( p e ) . Then, a t J [ p e ] . Hence, there exists r ∈ a t such that r − c I for every c ∈ J [ p e ] .As a consequence, r J [ p e ] + I , and so r T li =1 ( J [ p e ] + p i ) . Hence, r J [ p e ] + p i for some i .It follows that a t S/ p i J [ p e ] S/ p i .Consequently, we have t ≤ ν JS/ p i a S/ p i ( p e ) for some i . Then, ν J a ( p e ) p e ≤ max (cid:26) ν JS/ p i a S/ p i ( p e ) p e (cid:27) . There-fore, c JR ( a ) ≤ max n c JS/ p i S/ p i ( a S/ p i ) o . (cid:3) Remark 3.7.
Given e S = K [[ x , . . . , x n ]] with K an F -finite field of prime characteristic p .We take e I as a squarefree monomial ideal of e S , and e R = e S/ e I , same as in Theorem 3.6. Let e a , e J be two ideals of e R , with e a ⊆ p e J , and e J monomial. Then, c e J e R ( e a ) ∈ Q .4. The Ideal J e In this section we introduce the Cartier core of an ideal. This is related to the Cartieroperators. We also investigate the behavior of these ideals for Stanley-Reisner rings.4.1.
Cartier Contraction.
We begin this subsection introducing an ideal that allows thestudy of homomorphisms that do not give splittings.
Definition 4.1 ([AE05]) . Let ( R, m , K ) be a local ring or a standard graded K -algebra,which is F -finite and F -pure. We define I e ( R ) = { f ∈ R | ϕ ( f /p e ) ∈ m , for all ϕ ∈ Hom R ( R /p e , R ) } , where e ∈ N . Remark 4.2.
The set I e ( R ) is an ideal of R , and is called the e -th splitting ideal of R .Then, f I e ( R ) if and only if ϕ ( f /p e ) = 1 for some map ϕ ∈ Hom R ( R /p e , R ) .The ideal I e ( R ) is used to define the F -signature [Yao06]. Smith and Van den Berghin their work [SVdB97] showed existence of this invariant when the ring R is strongly F -regular and has finite Frobenius representation type. Later, Huneke and Leuschke [HL02]showed that this invariant exists if R is a complete local Gorenstein domain. For ringsthat are Gorenstein on the punctured spectrum, its existence was given by Yao [Yao06].Subsequently, Tucker [Tuc12] showed existence of the F -signature in R with full generality. -INVARIANTS OF STANLEY-REISNER RINGS 7 Definition 4.3 ([DSHNnBW]) . Let R be an F -finite F -pure ring, and J be an ideal in R .We define the Cartier contraction as J e = { f ∈ R | ϕ ( f /p e ) ∈ J, for all ϕ ∈ Hom R ( R /p e , R ) } , for e ∈ N . Remark 4.4.
The set J e is an ideal of R . If ( R, m , K ) is a local ring or a standard graded K -algebra, and m = J , we have I e ( R ) = J e . Proposition 4.5.
Let R be an F -finite F -pure ring, and J be an ideal of R . Then, for every e nonnegative integer J [ p e ] ⊆ J e ⊆ J .Proof. First, we show the inclusion J [ p e ] ⊆ J e . Let x be an element of J . For every ϕ ∈ Hom R ( R /p e , R ) , ϕ (( x p e ) /p e ) = ϕ ( x ·
1) = xϕ (1) ∈ J . Therefore, x p e ∈ J e .To show the other inclusion, we proceed by contrapositive. Let r J . Since R ⊆ R /p e is an R -module split, we can take β ∈ Hom R ( R /p e , R ) such that β | R = 1 R . It follows that β (( r p e ) /p e ) = β ( r ) = r J . Hence, r p e J e , and so, r J e . (cid:3) The equality J e = J holds under certain conditions. This is done in Proposition 4.9 below.The following proposition shows that the construction of the ideals J e commutes witharbitrary intersections. Proposition 4.6.
Let R be an F -finite F -pure ring, and { J i } i be a family of ideals in R .Then, ( T i J i ) e = T i ( J i ) e for every e nonnegative integer.Proof. For every ϕ ∈ Hom R ( R /p e , R ) , we have that x ∈ \ i J i ! e ⇔ ϕ ( x /p e ) ∈ \ i J i ⇔ ϕ ( x /p e ) ∈ J i for every i ⇔ x ∈ ( J i ) e for every i ⇔ x ∈ \ i ( J i ) e . (cid:3) Proposition 4.7.
Let R be an F -finite F -pure ring, and q be a prime ideal of R . Then q e is a q -primary ideal for every e ∈ N .Proof. We show that √ q e = q . By Proposition 4.5, q [ p e ] ⊆ q e ⊆ q , and so, q = √ q = p q [ p e ] ⊆ √ q e ⊆ √ q = q . We now show that q e is primary. Suppose that there exist a, b ∈ R such that a q e and b q . There is ϕ ∈ Hom R ( R /p e , R ) satisfying ϕ ( a /p e ) q . As q is a prime ideal, ϕ (( b p e a ) /p e ) = ϕ ( ba /p e ) = bϕ ( a /p e ) q . Hence, b p e a q e , and so, ab q e . Therefore, q e isa q -primary ideal of R. (cid:3) We now recall the definition of uniformly compatible. Our goal is to study the biggestuniformly compatible ideal contained in other given ideal.
Definition 4.8 ([Sch10]) . Let R be an F -finite ring, and J be an ideal of R . We say that J is uniformly F -compatible if ϕ ( J /p e ) ⊆ J for every e > and every ϕ ∈ Hom R ( R /p e , R ) . WÁGNER BADILLA-CÉSPEDES
Proposition 4.9.
Let R be an F -finite F -pure ring. Let J be an ideal of R . Then, J e = J for every e nonnegative integer if and only if J is uniformly F -compatible.Proof. We suppose that J e = J for every e ≥ . We have that ϕ ( J /p e ) ⊆ J for every ϕ ∈ Hom R ( R /p e , R ) by Definition 4.3.For the other direction, it is enough to see that J ⊆ J e for every e > . In fact, byDefinition 4.8, ϕ ( J /p e ) ⊆ J for all ϕ ∈ Hom R ( R /p e , R ) . Therefore, J ⊆ J e . (cid:3) Lemma 4.10.
Let R be an F -finite F -pure ring. Let J be an ideal of R . Then, T s ∈ N J s isuniformly F -compatible.Proof. We proceed by contradiction. We suppose that ϕ (cid:16)(cid:0)T s ∈ N J s (cid:1) /p e (cid:17) T s ∈ N J s for some e > and ϕ ∈ Hom R ( R /p e , R ) , and so, we have an f ∈ T s ∈ N J s such that ϕ ( f /p e ) T s ∈ N J s .Thus, ϕ ( f /p e ) J s for some s ∈ N . Consequently, there exists φ ∈ Hom R ( R /p s , R ) suchthat φ ( ϕ ( f /p e ) /p s ) J .If we take ψ : R /p e + s −→ R /p s such that ψ ( r /p e + s ) = ϕ ( r /p e ) /p s , we have that ψ is R -linear. As a consequence, σ = φ ◦ ψ ∈ Hom R ( R /p e + s , R ) . Then, σ ( f /p e + s ) = φ ◦ ψ ( f /p e + s ) = φ ( ϕ ( f /p e ) /p s ) J. Therefore, f J e + s , and we reach a contradiction. (cid:3) Proposition 4.11.
Let R be an F -finite F -pure ring. Let J be an ideal of R . Then, T s ∈ N J s is the biggest uniformly F -compatible ideal contained in J .Proof. Let I ⊆ J be an uniformly F -compatible ideal. By Proposition 4.9, I = I e ⊆ J e forevery e ≥ . Therefore, I ⊆ T s ∈ N J s . (cid:3) Motivated by the splitting prime ideal [AE05] and differential core [BJNnB19], we intro-duce the Cartier core.
Definition 4.12.
Let R be an F -finite F -pure ring. Given J an ideal of R , we define theCartier core of J as P ( J ) = T s ∈ N J s . Remark 4.13.
Let ( R, m , K ) be a local ring or a standard graded K -algebra, and m = J .Then, the ideal P ( J ) coincides with the splitting prime of R , denoted P ( R ) , and introducedby Aberbach and Enescu [AE05].In Proposition 4.15, we see a characterization of the Cartier core. This plays an importantrole in Subsection 4.2 to describe the ideal q e for Stanley-Reisner rings. Remark 4.14.
Let R be an F -finite F -pure ring, and J be an ideal of R . For every r ∈ p P ( J ) , r p e ∈ P ( J ) for some e ∈ N . Since R ⊆ R /p e is an R -module split, thereexists β ∈ Hom R ( R /p e , R ) such that β | R = 1 R . Moreover, r = ( r p e ) /p e ∈ ( P ( J )) /p e . Thus, r = β ( r ) ∈ P ( J ) by Lemma 4.10. Therefore, the Cartier core of J is a radical ideal.Since J s +1 is not necessarily contained in J s , we need to show that T s ≥ e J s is the Cartiercore for any e . Proposition 4.15.
Let R be an F -finite F -pure ring, and J be an ideal of R . Then, P ( J ) = T s ≥ e J s for every nonnegative integer e .Proof. We must show that T s ≥ e J s ⊆ P ( J ) . Let x ∈ T s ≥ e J s . Thus x ∈ J by Proposition4.5. Hence, x p s ∈ J [ p s ] for every s ≤ e . As a consequence, x p e ∈ J [ p s ] . As x p e ∈ T s ≥ e J s , wehave that x p e ∈ P ( J ) . Thus, x ∈ p P ( J ) . Therefore, x ∈ P ( J ) by Remark 4.14. (cid:3) -INVARIANTS OF STANLEY-REISNER RINGS 9 The Ideal q e in Stanley-Reisner Rings. Throughout this subsection, we denote S = K [ x , . . . , x n ] with K an F -finite field of prime characteristic p . Let I be a squarefreemonomial ideal of S , R = S/I , and p , . . . , p l are the minimal prime ideals of R . We wantto compute the ideal q e , when q is a monomial prime ideal of R . Lemma 4.16.
Let J be a monomial ideal in R , and e be a nonnegative integer. Then, J e and P ( J ) are monomial ideals.Proof. We note that R is N n -graded, and R /p e is p e N n -graded R -module. As R ⊆ R /p e , wecan view R as an p e N n -graded R -module. To show that J e is a monomial ideal, it sufficesto prove that J e is a homogeneous ideal with the N n grading. Let r = r α + · · · + r α t ∈ J e ,with r α i ∈ R of degree α i ∈ N n . Let ϕ ∈ Hom R ( R /p e , R ) . Since R /p e is a finitely generated R -module, every homomorphism R /p e −→ R is a finite sum of graded homomorphisms.Thus, we can take ϕ homogeneous of degree β ∈ p e N n . Then, ϕ ( r /p e ) = ϕ ( r /p e α ) + · · · + ϕ ( r /p e α t ) ∈ J, and each ϕ ( r /p e α i ) has degree p e α i + β . As J is a homogeneous ideal, ϕ ( r /p e α i ) ∈ J . Then, r α i ∈ J e for all i ∈ { , . . . , t } . Therefore, J e is a homogeneous ideal.Since J e is a monomial ideal, P ( J ) is a monomial ideal because intersection of monomialideals is monomial. (cid:3) Proposition 4.17.
Given q a monomial prime ideal of R , then q e = q [ q ] + P ( q ) for every e ∈ N , and q = p e .Proof. We must show q e ⊆ q [ q ] + P ( q ) . We proceed by contradiction. Let r be an element in q e . We suppose that r q [ q ] + P ( q ) . From Lemma 4.16, q e is a monomial ideal of R . Then,we can take r = x β , with β ∈ N n .Thus, x β q [ q ] , and x β
6∈ P ( q ) . By Proposition 4.15, x β T s ≥ e q s , and so, there exists e ′ ≥ e such that x β q e ′ .Let A = { α ∈ N n | ≤ α i ≤ q − for i = 1 , . . . , n } , A ′ = { α ′ ∈ N n | ≤ α ′ i ≤ p e ′ − for i = 1 , . . . , n } , B = { a i /q | i = 1 , . . . , s } be a base of K /q as K -vector space, and B ′ = { ( a ′ i ) /p e ′ | i = 1 , . . . , s ′ } be a base of K /p e ′ as K -vector space. We may suppose that a = a ′ = 1 .Moreover, x β/p e = x θ x α/p e and x β/p e ′ = x θ ′ x α ′ /p e ′ , with θ, θ ′ ∈ N n , α ∈ A , and α ′ ∈ A ′ . As p e ′ ≥ p e , then α i ≤ α ′ i and θ i ≥ θ ′ i for every i . Thus, there exists τ i ∈ N such that θ i = θ ′ i + τ i .Furthermore, J α = ( I : x α ) ⊆ ( I : x α ′ ) = J α ′ . Hence, we take a morphism φ ∈ Hom R (( S/J α ) x α/p e , ( S/J α ′ ) x α ′ /p e ′ ) such that φ ( x α/p e ) = x α ′ /p e ′ .Since x β q e ′ , there exists ψ ∈ Hom R (( S/J α ′ ) x α ′ /p e ′ , R ) such that ψ ( x θ ′ x α ′ /p e ′ ) q byProposition 2.2.We have an R -linear map ϕ : R /q −→ M ≤ i ≤ sα ∈A S/J α ( a i x α ) /q such that ϕ ( r /q ) = M ≤ i ≤ sα ∈A ( r i,α + J α )( a i x α ) /q , where r /q = M ≤ i ≤ sα ∈A r i,α ( a i x α ) /q . Taking γ = ψ ◦ φ ◦ π α ◦ ϕ , we have γ ∈ Hom R ( R /q , R ) , and γ ( x β/p e ) = ψ ( x θ x α ′ /p e ′ ) = ψ ( x τ x θ ′ x α ′ /p e ′ ) = x τ ψ ( x θ ′ x α ′ /p e ′ ) .In addition, x β = x qθ x α = x qτ x qθ ′ x α . As x β q [ q ] , we get that x τ q . Since x β ∈ q e ,it follows that x τ ψ ( x θ ′ x α ′ /p e ′ ) = γ ( x β/p e ) ∈ q . We get a contradiction, because q is a primeideal in R , and x τ , ψ ( x θ ′ x α ′ /p e ′ ) q . (cid:3) Proposition 4.18.
Let e be a nonnegative integer, q = p e , R = R/ P ( q ) with q a monomialprime ideal in R , and f ∈ R . Then, the following hold. (1) If f ∈ q e , then f ∈ ( q ) e ; (2) f ∈ q [ q ] if and only if f ∈ q e .Proof. We show Part (1) . We can assume that f a monomial, because q e and ( q ) e aremonomial ideals by Lemma 4.16.We have that f ∈ q e = q [ q ] + P ( q ) by Proposition 4.17. Since f is a monomial, it followsthat f ∈ q [ q ] or f ∈ P ( q ) . If f ∈ P ( q ) , then f = 0 ∈ ( q ) e . Moreover, if f ∈ q [ q ] , then f ∈ q [ q ] ⊆ ( q ) e .Now, we show Part (2) . From Proposition 4.17, we see that f ∈ q [ q ] = q [ q ] ⇔ f − g ∈ P ( q ) for some g ∈ q [ q ] ⇔ f ∈ q [ q ] + P ( q ) = q e . (cid:3) Proposition 4.19.
Suppose A as in Remark 2.3 and B = A/IA . Given q a monomialprime ideal of B , then q e = q [ q ] + P ( q ) for every e ∈ N , and q = p e .Proof. The proof is analogous to Proposition 4.17. (cid:3)
Proposition 4.20.
Suppose A as in Remark 2.3 and B = A/IA . Let e be a nonnegativeinteger, q = p e , B = B/ P ( q ) with q a monomial prime ideal in B , and f ∈ B . Then, thefollowing hold. (1) If f ∈ q e , then f ∈ ( q ) e ; (2) f ∈ q [ q ] if and only if f ∈ q e .Proof. The proof is analogous to Proposition 4.18. (cid:3) Cartier Threshold of a with Respect to J In this section we prove another one of our main results, Theorem A. In order to obtainthis, we recall the definition of the Cartier threshold of a with respect to J . We give someproperties of this and show that it is preserved under localization and completion. Westudy its relation with the F -thresholds. We also compare this number with its analog in R = R/ P ( J ) . -INVARIANTS OF STANLEY-REISNER RINGS 11 Definition and First Properties.
In this subsection R denotes a ring of prime char-acteristic p . We give properties related to Cartier thresholds. Definition 5.1 ([DSHNnBW]) . Let R be an F -finite F -pure ring. Given a , J two ideals in R such that a ⊆ √ J , we define b J a ( p e ) = max { t ∈ N | a t J e } . We define the Cartier threshold of a in R with respect to J by ct J ( a ) = lim e →∞ b J a ( p e ) p e . If ( R, m , K ) is a local ring or a standard graded K -algebra and m = J , the Cartierthreshold ct J ( a ) coincides with the F -pure threshold fpt( a ) . When a = m , fpt( m ) is denotedby fpt( R ) .Using Proposition 4.6, it follows that ct J ( a ) also commutes with arbitrary intersections. Proposition 5.2.
Let R be an F -finite F -pure ring. Let { q i } i be a family of ideals in R ,and J = T i q i . Let a be an ideal in R such that a ⊆ √ J . Then, ct J ( a ) = sup { ct q i ( a ) } .Proof. By Proposition 4.6, we have that J e = T i ( q i ) e for every nonnegative integer e . Then, t ≥ b J a ( p e ) ⇔ a t +1 ⊆ J e ⇔ a t +1 ⊆ ( q i ) e for every i ⇔ t ≥ b q i a ( p e ) for every i ⇔ t ≥ sup { b q i a ( p e ) } . Hence, b J a ( p e ) p e = sup n b q i a ( p e ) p e o . Therefore, ct J ( a ) = sup { ct q i ( a ) } . (cid:3) Since q e is a q -primary ideal by Proposition 4.7, we have that ct J ( a ) is preserved underlocalization. This fact, we prove it in Proposition 5.4 below. Lemma 5.3.
Let R be an F -finite F -pure ring, q be a prime ideal of R , and f ∈ R . Then, f ∈ I e ( R q ) if and only if f ∈ q e .Proof. We focus on the first direction. Let ψ ∈ Hom R ( R /p e , R ) . Since ( R /p e ) q ∼ = R q /p e as R q -module, ψ q ∈ Hom R q ( R q /p e , R q ) , and so, ψ ( f /pe )1 = ψ q ( f /pe ) = ψ q (( f ) /p e ) ∈ q R q .Hence, as q is a prime ideal, ψ ( f /p e ) ∈ q . Therefore, f ∈ q e .We now show the other direction. Let ψ ∈ Hom R q ( R q /p e , R q ) . Since Hom R q ( R q /p e , R q ) ∼ =Hom R ( R /p e , R ) q , we have that ψ = s ϕ q for some ϕ ∈ Hom R ( R /p e , R ) and s ∈ R \ q . As aconsequence, ψ (( f ) /p e ) = ψ ( f /pe ) = ϕ ( f /pe ) s ∈ q R q . Therefore, f ∈ I e ( R q ) . (cid:3) Proposition 5.4.
Let R be an F -finite F -pure ring. Let a , q be two ideals of R with q aprime ideal, and a ⊆ q . Then, ct q ( a ) = fpt( a R q ) . Proof.
By Lemma 5.3, we observe that, b qa ( p e ) = max { t ∈ N | a t q e } = max { t ∈ N | a t R q I e ( R q ) } = max { t ∈ N | ( a R q ) t I e ( R q ) } = b q R q a R q ( p e ) . Therefore, ct q ( a ) = fpt( a R q ) . (cid:3) Consider a local ring ( R, m , K ) . Let a ⊆ √ J be two ideals of R . We claim that the Cartierthreshold of a with respect to J does not vary under completion. To show this, we comparethe ideal J e versus ( J b R ) e . Lemma 5.5.
Let ( R, m , K ) be an F -finite F -pure local ring, f ∈ R , and J be an ideal in R .Then, f ∈ J e if and only if f ∈ ( J b R ) e .Proof. We suppose that f ∈ J e . Let ϕ ∈ Hom b R ( b R /p e , b R ) . Since R is an F -finite ring and b R /p e ∼ = [ R /p e as b R -module, we have Hom b R ( b R /p e , b R ) ∼ = Hom R ( R /p e , R ) ∼ = Hom R ( R /p e , R ) ⊗ R b R. Hence, ϕ = P ni =1 ϕ i ⊗ r i with ϕ i ∈ Hom R ( R /p e , R ) and r i ∈ b R . Then, ϕ ( f /p e ) = P ni =1 r i ϕ i ( f /p e ) . However, f ∈ J e , in consequence ϕ i ( f /p e ) ∈ J , thus ϕ ( f /p e ) ∈ J b R .Therefore, f ∈ ( J b R ) e .We now suppose that f ∈ ( J b R ) e . Let ϕ ∈ Hom R ( R /p e , R ) . Since b R /p e ∼ = [ R /p e as b R -module, we have b ϕ ∈ Hom b R ( b R /p e , b R ) . Then, b ϕ ( f /p e ) ∈ J b R , and so, ϕ ( f /p e ) ∈ J .Therefore, f ∈ J e . (cid:3) Proposition 5.6.
Suppose that ( R, m , K ) is an F -finite F -pure local ring. Let a , J be twoideals in R such that a ⊆ √ J . Then, ct J ( a ) = ct J b R ( a b R ) .Proof. By Lemma 5.5, we observe that b J a ( p e ) = max { t ∈ N | a t J e } = max { t ∈ N | a t b R ( J b R ) e } = max { t ∈ N | ( a b R ) t ( J b R ) e } = b J b R a b R ( p e ) . Therefore, ct J ( a ) = ct J b R ( a b R ) . (cid:3) Given J an ideal in R , we consider the ring R = R/ P ( J ) . Let a be an ideal in R suchthat a ⊆ √ J . Our goal is to compare the Cartier threshold of a with respect to J versus theCartier threshold of a with respect to J . Lemma 5.7.
Let R be an F -finite F -pure ring, J be an ideal of R , R = R/ P ( J ) , and f ∈ R .Then, f ∈ ( J ) e implies that f ∈ J e . -INVARIANTS OF STANLEY-REISNER RINGS 13 Proof.
For every ϕ ∈ Hom R ( R /p e , R ) , we take ϕ : R /p e −→ R such that ϕ ( x /p e ) = ϕ ( x /p e ) .By Lemma 4.10, it follows that ϕ is well defined.Since ϕ ∈ Hom R ( R /p e , R ) , it follows that ϕ ∈ Hom R ( R /p e , R ) . As f ∈ ( J ) e , then ϕ ( f /p e ) = ϕ ( f /p e ) ∈ J . Hence, there exists y ∈ J such that ϕ ( f /p e ) − y ∈ P ( J ) ⊆ J , andso ϕ ( f /p e ) ∈ J . Therefore, f ∈ J e . (cid:3) Proposition 5.8.
Let R be an F -finite F -pure ring. Let a , J be two ideals in R such that a ⊆ √ J , and R = R/ P ( J ) . Then, ct J ( a ) ≤ ct J ( a ) . In particular, if ( R, m , K ) is a local ringor a standard graded K -algebra, then fpt( a ) ≤ fpt( a ) .Proof. From Lemma 5.7, we have that b J a ( p e ) = max { t ∈ N | a t J e }≤ max { t ∈ N | a t ( J ) e } = b J a ( p e ) . Therefore, ct J ( a ) = lim e →∞ b J a ( p e ) p e ≤ lim e →∞ b J a ( p e ) p e = ct J ( a ) . (cid:3) Relation Between c J ( a ) and ct J ( a ) . In this subsection we give a characterization of ct J ( a ) using F -thresholds. Remark 5.9.
Suppose that R is an F -finite F -pure ring. Let a , J be two ideals in R suchthat a ⊆ √ J . Since J [ p e ] ⊆ J e , we have that b J a ( p e ) = max { t ∈ N | a t J e }≤ max { t ∈ N | a t J [ p e ] } = ν J a ( p e ) . Therefore, ct J ( a ) ≤ c J ( a ) .The following Remark relates F -pure rings and Frobenius powers. Remark 5.10.
Suppose that R is an F -pure ring. Let J be an ideal in R and r ∈ R . Then, r p ∈ J [ p ] if and only if r ∈ J [Fed83]. Let a be an ideals in R such that a ⊆ √ J . Then, wehave that a ν J a ( p e ) * J [ p e ] . As a consequence, ( a ν J a ( p e ) ) [ p ] * J [ p e +1 ] . Hence, a p · ν J a ( p e ) * J [ p e +1 ] ,and so, p · ν J a ( p e ) ≤ ν J a ( p e +1 ) . Therefore, the sequence n ν J a ( p e ) p e o e ≥ is non-decreasing.The following propositions are an extension of the work done by De Stefani, Núñez-Betancourt and Pérez [DSNnBP18, Theorem . ]. Proposition 5.11.
Let R be an F -finite F -pure ring. Let J be an ideal in R . Then, J [ p ] e ⊆ J e +1 for every e ∈ N .Proof. Let f be an element in J e . Let ϕ ∈ Hom R ( R /p e +1 , R ) . As R /p e ⊆ R /p e +1 , we havethat ϕ | R /pe ∈ Hom R ( R /p e , R ) . Thus, ϕ (( f p ) /p e +1 ) = ϕ | R /pe ( f /p e ) ∈ J . Hence, f p ∈ J e +1 ,and so, J [ p ] e ⊆ J e +1 . (cid:3) Proposition 5.12.
Let R be an F -finite F -pure ring, and a , J be two ideals in R suchthat a ⊆ √ J . The sequence n c Je ( a ) p e o e ≥ is non-increasing and bounded below by zero. Inparticular, its limit exists. Proof.
Let e be nonnegative integer, J [ p ] e ⊆ J e +1 . Thus, c J e +1 ( a ) ≤ c J [ p ] e ( a ) = p · c J e ( a ) byProposition 3.5. Therefore, c Je +1 ( a ) p e +1 ≤ c Je ( a ) p e . (cid:3) The following proposition gives us a relation between the Cartier thresholds and F -thresholds. Specifically, we can obtain the Cartier threshold as a limit F -thresholds. Proposition 5.13.
Let R be an F -finite F -pure ring. Let a , J be two ideals in R such that a ⊆ √ J . Then, ct J ( a ) = lim e →∞ c Je ( a ) p e .Proof. Let e be nonnegative integer. We note that b J a ( p e ) = max { t ∈ N | a J e } = max { t ∈ N | a J [ p ] e } = ν J e a ( p ) . For every nonnegative integer s , we have ν J e a ( p s ) p s − ν J e a ( p ) p ≤ µ ( a ) p by Lemma 3.2.The sequence n ν Je a ( p s ) p s o s ≥ is non-decreasing by Remark 5.10. As a consequence, ≤ ν J e a ( p s ) p s − ν J e a ( p ) ≤ µ ( a ) . Thus, ≤ ν J e a ( p s ) p s − b J a ( p e ) ≤ µ ( a ) . We take limit over s to get ≤ c J e ( a ) − b J a ( p e ) ≤ µ ( a ) , dividing by p e gives ≤ c J e ( a ) p e − b J a ( p e ) p e ≤ µ ( a ) p e . Taking limit over e we conclude that ct J ( a ) = lim e →∞ c J e ( a ) p e . (cid:3) Corollary 5.14.
Let R be an F -finite F -pure ring. Let a , J be two ideals in R such that a ⊆ √ J . Then, ct J ( a ) = c J ( a ) if and only if c J e ( a ) = c J [ pe ] ( a ) for every e ∈ N .Proof. We focus on the first direction, it suffices to show c J [ pe ] ( a ) ≤ c J e ( a ) . As the sequence n c Je ( a ) p e o e ≥ is non-increasing and bounded below, it converges to its infimum. By Proposition5.13, c J ( a ) ≤ c Je ( a ) p e . As a consequence, c J [ pe ] ( a ) = p e · c J ( a ) ≤ c J e ( a ) . -INVARIANTS OF STANLEY-REISNER RINGS 15 We now show the other direction, ct J ( a ) = lim e →∞ c Je ( a ) p e = lim e →∞ c J [ pe ] ( a ) p e = lim e →∞ p e · c J ( a ) p e = c J ( a ) . (cid:3) Cartier Thresholds in Stanley-Reisner Rings.
Throughout this subsection, wedenote S = K [ x , . . . , x n ] with K an F -finite field of prime characteristic p . Let I be asquarefree monomial ideal of S , R = S/I , and p , . . . , p l are the minimal prime ideals of R . Theorem 5.15.
Suppose A as in Remark 2.3 and B = A/IA . Let a , q be two ideals in B with q a prime monomial ideal, such that a ⊆ q , and B = B/ P ( q ) . Then, the following hold: (1) ct q ( a ) = ct q ( a ) ; (2) ct q ( a ) = c q ( a ) ; (3) ct q ( a ) is a rational number.In particular, fpt( a ) is a rational number.Proof. We show Part (1) . From Proposition 4.20 and Lemma 5.7, we have b qa ( p e ) = max { t ∈ N | a t q e } = max { t ∈ N | a t ( q ) e } = b qa ( p e ) . Therefore, ct q ( a ) = ct q ( a ) .Now, we show Part (2) . We claim that c q ( a ) ≤ ct q ( a ) . From Proposition 4.20, it followsthat ν qa ( p e ) = max { t ∈ N | a t q [ q ] }≤ max { t ∈ N | a t q e } = b qa ( p e ) . Thus, c q ( a ) = lim e →∞ ν qa ( p e ) p e ≤ lim e →∞ b qa ( p e ) p e = ct q ( a ) .By Part (1) and Remark 5.9, we have c q ( a ) ≤ ct q ( a ) = ct q ( a ) ≤ c q ( a ) . Therefore, c q ( a ) =ct q ( a ) .We show Part (3) . Since q is a monomial ideal, P ( q ) is also a monomial ideal by Lemma4.16. In addition, P ( q ) is a radical ideal by Remark 4.14. Thus, P ( q ) is squarefree monomialideal. Consequently, B is a power series ring modulo a squarefree monomial ideal. Since q is a monomial ideal in B , c q ( a ) is a rational number by Remark 3.7. Therefore, ct q ( a ) is arational number by Part (2) .The last statement follows, since ct m ( a ) = fpt( a ) and m is a monomial prime ideal in B . (cid:3) Since ct J ( a ) is preserved under localization and completion, Theorem 5.15 allows us toobtain one of the main results of this work. Corollary 5.16.
Let a , J be two ideals in R with J radical ideal, such that a ⊆ J . Then, ct J ( a ) is a rational number. In particular, fpt( a ) is a rational number.Proof. Let q ⊆ R be a prime ideal such that a ⊆ q . We have that ct q ( a ) = fpt( a c R q ) byPropositions 5.4 and 5.6. Thus, ct q ( a ) is a rational number by Theorem 5.15. Since J is a radical ideal, we have that J = T mi =1 q i where q , . . . , q m are the minimal primeideals of J . From Proposition 5.2, ct J ( a ) = max { ct q i ( a ) } . Therefore, ct J ( a ) is a rationalnumber. (cid:3) a -Invariants and Regularity In this section we focus on standard graded K -algebras. We study the a -invariants andregularity in rings modulo Frobenius powers of an ideal. We also investigate their behaviorwith the Castelnuovo-Mumford regularity in Stanley-Reisner rings.6.1. Definitions and Property.
Suppose that ( R, m , K ) is a standard graded K -algebra,and let I be a homogeneous ideal of R . We recall that if M is a graded R -module, its i -th local cohomology H iI ( M ) is a graded module. Moreover, if M is finitely generated, themodule H i m ( M ) is Artinian. Therefore, one can define the following number. Definition 6.1 ([GW78]) . Let ( R, m , K ) be a standard graded K -algebra. Let M be an p e N -graded finitely generated R -module. If H i m ( M ) = 0 , we define the i -th a -invariant of M by a i ( M ) = max (cid:26) s ∈ p e Z | H i m ( M ) s = 0 (cid:27) . If H i m ( M ) = 0 , we set a i ( M ) = −∞ . Definition 6.2.
Let ( R, m , K ) be a standard graded K -algebra. Let M be an p e N -gradedfinitely generated R -module. We define the regularity of M by reg( M ) = max i ∈ Z { a i ( M ) + i } . Next theorem gives us conditions for the regularity in rings modulo Frobenius power ofideals.
Theorem 6.3 ([DSNnBP18, Theorem . ]) . Let ( R, m , K ) be a standard graded K -algebrathat is F -finite and F -pure. Suppose that J is a homogeneous ideal of R . If there exists aconstant C such that reg( R/J [ p e ] ) ≤ Cp e for all e ≫ , then lim e →∞ reg( R/J [ p e ] ) p e exists, and it is bounded below by max i ∈ N { a i ( R/J ) } + fpt( R ) . Regularity in Stanley-Reisner Rings.
Throughout this subsection, we denote S = K [ x , . . . , x n ] with K an F -finite field of prime characteristic p . Let I be a squarefreemonomial ideal of S , R = S/I . Definition 6.4.
Let N ⊆ { , . . . , n } . We define x N = Y i ∈ N x i . Now, we define the support of an element in N n . Definition 6.5.
Let α ∈ N n . The support of α is defined by Supp( α ) = { i ∈ { , . . . , n } | α i = 0 } . -INVARIANTS OF STANLEY-REISNER RINGS 17 Lemma 6.6.
Given α ∈ N n , then ( I : x α ) = ( x Supp( λ ) \ Supp( α ) | x λ minimal generator of I ) .In particular, if α, β ∈ N n are such that Supp( α ) = Supp( β ) , then ( I : x α ) = ( I : x β ) .Proof. Since I is a monomial ideal, it follows that ( I : x α ) is a monomial ideal as well. Wehave ( x Supp( λ ) \ Supp( α ) | x λ minimal generator of I ) ⊆ ( I : x α ) . Indeed, for every x λ minimalgenerator of I , x Supp( λ ) \ Supp( α ) x α ∈ I .We show that ( I : x α ) ⊆ ( x Supp( λ ) \ Supp( α ) | x λ minimal generator of I ) . Let x θ be agenerator of ( I : x α ) . Thus x θ x α ∈ I . Hence, x λ | x θ x α for some x λ minimal generatorof I . Then, Supp( λ ) \ Supp( α ) ⊆ Supp( θ ) , and so, x Supp( λ ) \ Supp( α ) | x θ . Therefore, x θ ∈ ( x Supp( λ ) \ Supp( α ) | x λ minimal generator of I ) . (cid:3) Now, we prove Theorem C.
Theorem 6.7.
Let J be a homogeneous ideal of R . Then, lim e →∞ reg( R/J [ p e ] ) p e = max ≤ i ≤ dα ∈A ′ { a i ( S/ ( J α + J )) + | α |} , where A ′ = { α ∈ N n | ≤ α i ≤ for i = 1 , . . . , n } , J α = ( I : x α ) , and d = max { dim( S/ ( J α + J )) | α ∈ A ′ } . In particular, this limit is an integer number.Proof. Without loss of generality, we can take K a perfect field. Let e be a nonnegativeinteger and A = { α ∈ N n | ≤ α i ≤ p e − for i = 1 , . . . , n } . Then, R /p e ∼ = M α ∈A ( S/J α ) x α/p e , where J α = ( I : x α ) by Proposition 2.2. Applying − ⊗ R R/J , we obtain that ( R/J [ p e ] ) /p e ∼ = R /p e /J R /p e ∼ = M α ∈A ( S/ ( J α + J )) x α/p e , and so H i m (( R/J [ p e ] ) /p e ) ∼ = M α ∈A H i m (( S/ ( J α + J )) x α/p e ) . Hence, we have a i ( R/J [ p e ] ) p e = a i (( R/J [ p e ] ) /p e )= max α ∈A { a i (( S/ ( J α + J )) x α/p e ) } = max α ∈A (cid:26) a i ( S/ ( J α + J )) + | α | p e (cid:27) . From Lemma 6.6, we have a i ( R/J [ p e ] ) p e = max α ∈A ′ (cid:26) a i ( S/ ( J α + J )) + | α | ( p e − p e (cid:27) . Thus, lim e →∞ reg( R/J [ p e ] ) p e = lim e →∞ max i ∈ Z (cid:26) a i ( R/J [ p e ] ) p e + ip e (cid:27) = lim e →∞ max i ∈ Z (cid:26) max α ∈A ′ { a i ( S/ ( J α + J )) + | α | ( p e − p e } + ip e (cid:27) = lim e →∞ max ≤ i ≤ dα ∈A ′ (cid:26) a i ( S/ ( J α + J )) + | α | ( p e − p e + ip e (cid:27) = max ≤ i ≤ dα ∈A ′ (cid:26) lim e →∞ a i ( S/ ( J α + J )) + | α | ( p e − p e + ip e (cid:27) = max ≤ i ≤ dα ∈A ′ { a i ( S/ ( J α + J )) + | α |} . (cid:3) Acknowledgments
I would like to thank Alessandro De Stefani, Daniel J. Hernández, Luis Núñez-Betancourt,and Emily E. Witt for sharing their work [DSHNnBW]. I also thank them, and the refereefor helpful comments, and suggestions.
References [AE05] Ian M. Aberbach and Florian Enescu. The structure of F-pure rings.
Math. Z. , 250(4):791–806,2005. 6, 8[ÀlMBZ12] Josep Àlvarez Montaner, Alberto F. Boix, and Santiago Zarzuela. Frobenius and Cartier alge-bras of Stanley-Reisner rings.
J. Algebra , 358:162–177, 2012. 2[BFS13] Angélica Benito, Eleonore Faber, and Karen E. Smith. Measuring singularities with Frobenius:the basics. In
Commutative algebra , pages 57–97. Springer, New York, 2013. 1[BJNnB19] Holger Brenner, Jack Jeffries, and Luis Núñez Betancourt. Quantifying singularities with dif-ferential operators.
Adv. Math. , 358:106843, 89, 2019. 8[BMS08] Manuel Blickle, Mircea Mustaţă, and Karen E. Smith. Discreteness and rationality of F -thresholds. Michigan Math. J. , 57:43–61, 2008. Special volume in honor of Melvin Hochster.2[BZ19] Alberto F. Boix and Santiago Zarzuela. Frobenius and Cartier algebras of Stanley-Reisnerrings (II).
Acta Math. Vietnam. , 44(3):571–586, 2019. 2[DS20] Hailong Dao and Ilya Smirnov. On the generalized Hilbert-Kunz function and multiplicity.
Israel J. Math. , 237(1):155–184, 2020. 2[DSHNnBW] Alessandro De Stefani, Daniel J. Hernández, Luis Núñez Betancourt, and Emily E. Witt. σ -modules and σ -jumping numbers. In preparation . 2, 7, 11, 18[DSNnBP18] Alessandro De Stefani, Luis Núñez Betancourt, and Felipe Pérez. On the existence of F -thresholds and related limits. Trans. Amer. Math. Soc. , 370(9):6629–6650, 2018. 2, 4, 5, 13,16[Fed83] Richard Fedder. F -purity and rational singularity. Trans. Amer. Math. Soc. , 278(2):461–480,1983. 13[GW78] Shiro Goto and Keiichi Watanabe. On graded rings. I.
J. Math. Soc. Japan , 30(2):179–213,1978. 2, 16[HL02] Craig Huneke and Graham J. Leuschke. Two theorems about maximal Cohen-Macaulay mod-ules.
Math. Ann. , 324(2):391–404, 2002. 6 -INVARIANTS OF STANLEY-REISNER RINGS 19 [HMTW08] Craig Huneke, Mircea Mustaţă, Shunsuke Takagi, and Kei-ichi Watanabe. F-thresholds, tightclosure, integral closure, and multiplicity bounds.
Michigan Math. J. , 57:463–483, 2008. Specialvolume in honor of Melvin Hochster. 4, 5[Hun00] Craig Huneke. The saturation of Frobenius powers of ideals.
Comm. Algebra , 28(12):5563–5572, 2000. Special issue in honor of Robin Hartshorne. 2[HY03] Nobuo Hara and Ken-Ichi Yoshida. A generalization of tight closure and multiplier ideals.
Trans. Amer. Math. Soc. , 355(8):3143–3174, 2003. 1[Kat98] Mordechai Katzman. The complexity of Frobenius powers of ideals.
J. Algebra , 203(1):211–225,1998. 2[KSSZ14] Mordechai Katzman, Karl Schwede, Anurag K. Singh, and Wenliang Zhang. Rings of Frobeniusoperators.
Math. Proc. Cambridge Philos. Soc. , 157(1):151–167, 2014. 2[KZ14] Mordechai Katzman and Wenliang Zhang. Castelnuovo-Mumford regularity and the discrete-ness of F -jumping coefficients in graded rings. Trans. Amer. Math. Soc. , 366(7):3519–3533,2014. 2[MTW05] Mircea Mustaţă, Shunsuke Takagi, and Kei-ichi Watanabe. F-thresholds and Bernstein-Satopolynomials. In
European Congress of Mathematics , pages 341–364. Eur. Math. Soc., Zürich,2005. 1, 2, 4, 5[Sch10] Karl Schwede. Centers of F -purity. Math. Z. , 265(3):687–714, 2010. 7[SVdB97] Karen E. Smith and Michel Van den Bergh. Simplicity of rings of differential operators inprime characteristic.
Proc. London Math. Soc. (3) , 75(1):32–62, 1997. 6[Tri18] Vijaylaxmi Trivedi. Nondiscreteness of F -thresholds. arXiv:1808.07321 , 2018. 2[Tuc12] Kevin Tucker. F -signature exists. Invent. Math. , 190(3):743–765, 2012. 6[TW04] Shunsuke Takagi and Kei-ichi Watanabe. On F-pure thresholds.
J. Algebra , 282(1):278–297,2004. 1[Vra16] Adela Vraciu. An observation on generalized Hilbert-Kunz functions.
Proc. Amer. Math. Soc. ,144(8):3221–3229, 2016. 2[Yao06] Yongwei Yao. Observations on the F -signature of local rings of characteristic p . J. Algebra ,299(1):198–218, 2006. 6[Zha15] Wenliang Zhang. A note on the growth of regularity with respect to Frobenius. arXiv:1512.00049 , 2015. 2
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