Bounds for the Minimum Distance Function
aa r X i v : . [ m a t h . A C ] D ec BOUNDS FOR THE MINIMUM DISTANCE FUNCTION
LUIS N ´U ˜NEZ-BETANCOURT a , YURIKO PITONES b , AND RAFAEL H. VILLARREAL c Abstract.
Let I be a homogeneous ideal in a polynomial ring S . In this paper, weextend the study of the asymptotic behavior of the minimum distance function δ I of I and give bounds for its stabilization point, r I , when I is an F -pure or a square-freemonomial ideal. These bounds are related with the dimension and the Castelnuovo–Mumford regularity of I . Keywords:
Minimum distance, Castelnuovo–Mumford regularity, monomial ideal.
Contents
1. Introduction 12. Preliminaries 23. Asymptotic behavior of the minimum distance function 44. Stanley–Reisner ideals associated to a shellable simplicial complex 75. Results related to F -purity 8Acknowledgments 10References 101. Introduction
In this manuscript we study the minimum distance function δ I of a homogeneousideal I contained in a polynomial ring S = K [ x , . . . , x n ] over a field K . This minimumdistance function for ideals was introduced by the second-named and third-named au-thors together with Mart´ınez-Bernal [MBPV17] to obtain an algebraic formulation of theminimum distance of projective Reed–Muller-type codes over finite fields.If I is an unmixed radical graded ideal and its associate primes are generated by linearforms, then δ I is non-increasing [MBPV17]. In our first result, we extend this propertyto any radical ideal. Mathematics Subject Classification.
Primary 13D40; Secondary 13H10, 13P25.
Key words and phrases.
Minimum distance, Castelnuovo–Mumford regularity, monomial ideal. a The first author was partially supported by CONACyT Grant 284598, C´atedras Marcos Moshinsky,and SNI, Mexico. b The second author was partially supported by CONACyT Grant 427234 and CONACyT PostdoctoralFellowship 177609. c The third author was partially supported by SNI, Mexico.
Theorem A (Theorem 3.4) . Suppose that I ⊆ S is a radical ideal. Then, δ I ( d ) is anon-increasing function. The previous result allow us to define the regularity index of I , r I , as the valuewhere δ I stabilizes. If dim( S/I ) = 1, previous work shows that r I ≤ reg( S/I ) [GSRTR02,RMSV11], where reg(
S/I ) is the Castelnuovo–Mumford regularity of
S/I . This motivatedthe authors to conjecture that this relation holds in greater generality.
Conjecture B ([NnBPV18]) . Let I ⊆ S be a radical homogeneous ideal whose associatedprimes are generated by linear forms. Then, r I ≤ reg( S/I ) . This conjecture was previously showed for edge ideals associated to Cohen–Macaulaybipartite graphs [NnBPV18] and if dim(
S/I ) = 1 [GSRTR02, RMSV11]. However, theconjecture does not hold in general. Jaramillo and the third-named author provided anexample of a monomial edge ideal I such that r I > reg( S/I ) [JV21]. In this work, we findbounds for r I for square-free monomial ideals. Theorem C (Theorem 4.4 & 5.7 ) . Let I ⊆ S be a square-free monomial ideal. Then, r I ≤ dim( S/I ) . Moreover, if I is shellable or Gorenstein, then r I ≤ reg( S/I ) . We also have prove a similar result for ideals such that
S/I is a F -pure ring. Theseclass of rings play an important role in the study of singularities in prime characteristic[HR76]. Theorem D (Theorem 5.5 & 5.6) . Suppose that K is a field of prime characteristic. Let I ⊆ S be an ideal such that S/I if F -pure. Then, r I ≤ dim( S/I ) . Moreover, if I isGorenstein, then r I ≤ reg( S/I ) . Preliminaries
In this section we recall some well known notion and results that are needed through-out this manuscript.Let S = K [ x , . . . , x n ] = L ∞ t =0 S t be a polynomial ring over a field K with the standardgrading and let I = (0) be a homogeneous ideal of S . Let d denote the Krull dimensionof R = S/I .The
Hilbert function of S/I , denoted H I , is given by H I ( t ) = dim K ( R ≤ t ) = dim K ( S ≤ t /I ≤ t ) = dim K ( S ≤ t ) − dim K ( I ≤ t ) , where I ≤ t = I ∩ S ≤ t . By a classical theorem of Hilbert there is a unique polynomial h I ( t ) ∈ Q [ t ] of degree d such that H I ( t ) = h I ( t ) for t ≫ Hilbert-Samuel multiplicity or degree of R , denoted e( R ), is the positive integerdefined by e( R ) = d ! lim t →∞ H I ( t ) /t d .Given an integer t ≥
1, let F t be the set of all zero-divisors of S/I not in I of degree t ≥
1. That is F t = { f ∈ S t | f I, ( I : f ) = I } . We note that ( I : f ) = I is equivalent to f ∈ p for some p ∈ Ass S ( S/I ), Ass S ( S/I ) is theset of associated primes of
S/I . INIMUM DISTANCE 3
Definition 2.1.
The minimum distance function of I is the function δ I : N + → Z givenby δ I ( t ) = (cid:26) e( S/I ) − max { e( S/ ( I, f )) | f ∈ F t } if F t = ∅ , e( S/I ) if F t = ∅ . Definition 2.2.
Let I ⊆ S be a graded ideal and let F ⋆ be the minimal graded freeresolution of S/I as an S -module: F ⋆ : 0 → M j S ( − j ) β gj → · · · → M j S ( − j ) β j → S → S/I → . The
Castelnuovo–Mumford regularity of S/I , regularity of S/I for short, is defined asreg(
S/I ) = max { j − i | β ij = 0 } . The following result show the asymptotic behavior of δ I for a particular case of gradedideals.An ideal I ⊆ S is called unmixed if all its associated primes have the same height, inother case I is mixed . Theorem 2.3 ([MBPV17, Theorem 3.8]) . Let I ⊆ S be an unmixed radical homogeneousideal. If all the associated primes of I are generated by linear forms, then there is aninteger r ≥ such that δ I (1) > δ I (2) > · · · > δ I ( r ) = δ I ( d ) = 1 for d ≥ r . The integer r where the stabilization occurs is called the regularity index of δ I andis denoted by r I . In Section 3, we show that one can define this index for any radicalideal. Local cohomology.
Let R be a commutative Noetherian ring with identity and let I bea homogeneous ideal generated by the forms f , . . . , f ℓ ∈ R . Consider the ˇCech complex,ˇC ⋆ ( ¯ f ; R ): 0 → R → M i R f i → M i,j R f i f j → · · · → R f ,...,f ℓ → . where ˇC i ( ¯ f ; R ) = L ≤ j ≤ ... ≤ j i ≤ ℓ R f j ,...,f ji and the homomorphism in every summand is alocalization map with appropriate sign. Definition 2.4.
Let M be a graded R -modue. The i -th local cohomology of M withsupport in I is defined as H iI ( M ) = H i ( ˇC ⋆ ( ¯ f ; R ) ⊗ R M ). Remark 2.5.
Since M is a graded R -module and I is homogeneous the local cohomologymodule H iI ( M ) is graded. Remark 2.6. If φ : M → N is a homogeneous R -module homomorphism of degree t ,then the induced R -module map H iI ( M ) → H iI ( N ) is homogeneous of degree t . Theorem 2.7 (Grothendieck’s Vanishing Theorem) . Let M be an R -module of dimension d . Then, H iI ( M ) = 0 for all i > d . LUIS N ´U ˜NEZ-BETANCOURT, YURIKO PITONES, AND RAFAEL H. VILLARREAL
Theorem 2.8 (Grothendieck’s Non-Vanishing Theorem) . Let M be a finitely generated R -module of dimension d . Then, H dI ( M ) = 0 . Definition 2.9.
Let M be an R -module with dimension d . The a i -invariants , a i ( M ), for i = 0 , . . . , d are defined as follows. If H i m ( M ) = 0, a i ( M ) = max { α | H i m ( M ) α = 0 } ,for 0 ≤ i ≤ d , where H i m ( M ) denotes the local cohomology module with support in themaximal ideal m . If H i m ( M ) = 0, we set a i ( M ) = −∞ . If d = dim( M ), then, a d ( M ), is often just called the a -invariant of M .The a -invariant, is a classical invariant [GW78], and is closely related to the Castelnuovo-Mumford regularity. Definition 2.10.
Let R be a positively graded ring and let M be a finitely generated R -module. The Castelnuovo–Mumford regularity of M , reg( M ), is defined asreg( M ) = max { a i ( M ) + i | ≤ i ≤ d } . Remark 2.11. If M is a standard graded module of dimension d , then the a -invariant isrelated to the Castelnuovo–Mumford regularity, via the inequeality a ( M ) + d ≤ reg( M )which is equality in the case Cohen–Macaulay. Definition 2.12.
Suppose that R has prime characteristic p . The Frobenius map F : R → R is defined by r r p . Remark 2.13. If R is reduced, R /p e the ring of the p e -th roots of R is well defined, and R ⊆ R /p e .3. Asymptotic behavior of the minimum distance function
In this section we prove that the minimum distance function δ I is non-increasing.Then, the notion of regularity index of δ I is well defined. We also find what is the stablevalue of the minimum distance function. We start this section establishing notation. Notation 3.1.
Given an ideal I ⊆ S , we set A ( I ) = { p ∈ Ass S ( S/I ) | dim( S/I ) = dim( S/ p ) } ; V ( I ) = { p ∈ Spec( S ) | I ⊆ p } ; D ( I ) = Spec( S ) \ V ( I ) . Remark 3.2.
Given an ideal I ⊆ S , thene( S/I ) = X p ∈A ( I ) λ S p ( S p /IS p ) e( S/ p ) . In particular, if I is radical, e( S/I ) = X p ∈A ( I ) e( S/ p ) . INIMUM DISTANCE 5
Lemma 3.3.
Suppose that I is a radical ideal. Let f ∈ F t such that dim( S/ ( I, f )) =dim(
S/I ) . Then, A (( I, f )) = A ( I ) ∩ V ( f ) . Furthermore, e( S/ ( I, f )) = X p ∈A ( I ) ∩V ( f ) e( S/ p ) . Proof.
Let J = ( I, f ). Let Q be an associated prime of J . Since I ⊆ J , there exists anassociated prime p of I such that p ⊆ Q. If dim(
S/Q ) = dim(
S/J ) = dim(
S/I ) , then p = Q . Thus, A ( J ) ⊆ A ( I ) ∩ V ( f ).Let Q ∈ A ( I ) ∩V ( f ). Then, J ⊆ Q and dim( S/Q ) = dim(
S/J ) . Then, Q is a minimalprime of S/J . Thus, Q ∈ Ass S ( S/J ), and so, Q ∈ A ( J ) . We note that J is not necessarily radical. However, J S p = IS p for every p ∈ A ( I ) ∩V ( f ) . Thus, λ S p ( S p /IS p ) = 1 for every p ∈ A ( I ) ∩ V ( f ) . Then,e( S/ ( I, f )) = X p ∈A ( I ) ∩V ( f ) e( S/ p )by the additivity formula. (cid:3) We now show that the minimum distance function is non-increasing.
Theorem 3.4.
Suppose that I is a radical ideal. Then, δ I ( d ) is a non-increasing function.Proof. If F t = ∅ for every t ≥
1, then δ I ( t ) = e( S/I ), which is the maximum value. Wenote that this case is equivalent to I being a prime ideal.We now assume that F t = ∅ for some t ∈ N . We note that in this case dim( S/I ) = 0,otherwise, I = m and so F t = ∅ . Let f ∈ F t such that δ I ( t ) = e( S/I ) − e( S/ ( I, f )). Then, δ I ( t ) = e( S/I ) − e( S/ ( I, f ))= X p ∈A ( I ) e( S/ p ) − X p ∈A (( I,f )) e( S/ p )= X p ∈A ( I ) e( S/ p ) − X p ∈A ( I ) ∩V ( f ) e( S/ p )= X p ∈A ( I ) ∩D ( f ) e( S/ p ) . Since I is radical and dim( S/I ) >
0, we have that m is not an associated prime. Then, m f I because f I . We conclude that there exists i = 1 , . . . , n such that x i f I . Inparticular, x i f ∈ F t +1 and F t +1 = ∅ . Then, δ I ( t + 1) ≤ X p ∈A ( I ) ∩D ( x i f ) e( S/ p )= X p ∈A ( I ) ∩D ( x i ) ∩D ( f ) e( S/ p ) ≤ X p ∈A ( I ) ∩D ( f ) e( S/ p ) = δ I ( t ) . LUIS N ´U ˜NEZ-BETANCOURT, YURIKO PITONES, AND RAFAEL H. VILLARREAL (cid:3)
Thanks to the previous theorem we have that the minimum distance function even-tually stabilizes, and it has a regularity index.
Definition 3.5.
Suppose that I is a radical ideal. The regularity index of I , denoted by r I , is defined by r I = min { s ∈ N | δ I ( s ) = lim t →∞ δ I ( t ) } . Proposition 3.6.
Suppose that I is a radical ideal. Then, δ I ( t ) = min { e( S/ p ) | p ∈ Ass S ( S/I ) } for t ≫ if I is unmixed and δ I ( t ) = 0 for t ≫ otherwise.Proof. We first assume that I is mixed. Let J be the intersection of the minimal primesof I of dimension dim( S/I ) and let J be the intersection of the minimal primes of I ofdimension smaller than dim( S/I ). Let f ∈ J \ I . Let α = deg( f ). In particular, f ∈ F α . We note that dim(
S/I ) = dim( S/ ( I, f )) and A ( I ) = A ( I, f ), and so, e( S/ ( I, f )) =e(
S/I ). We conclude that δ I ( t ) = 0 . Since δ I is nondecresing by Theorem 3.4, we obtainthat δ I ( t ) = 0 for t ≥ α. We now assume that I is unmixed. Then, A ( I ) = Ass S ( S/I ) . If I is a prime ideal,then δ I ( t ) = e( S/I ) for every t ∈ N , and our claim follows. We assume that dim( S/I ) > I is not a prime ideal. For every f ∈ F t , there exists a prime ideal p such that f I . Then,e( S/I ) − e( S/ ( I, f )) ≥ e( S/ p ) ≥ min { e( S/ p ) | p ∈ Ass S ( S/I ) } . We conclude that δ I ( t ) ≥ min { e( S/ p ) | p ∈ Ass S ( S/I ) } . Let p , . . . , p ℓ denote theassociated primes of I in an order such that e ( S/ p i ) ≤ e( S/ p j ) for i ≥ j . Let f ∈ p ∩ . . . ∩ p ℓ \ I. Let α = deg( f ). We have that f ∈ F α . Then, δ I ( α ) ≤ e( S/I ) − e( S/ ( I, f )) =e( S/ p ) . Since δ is nondecresing by Theorem 3.4, we obtain that δ I ( t ) ≤ e( S/ p ) for t ≥ α. We conclude that δ I ( t ) = e( S/ p ) for t ≥ α. (cid:3) Proposition 3.7.
Let I be a mixed radical ideal. Let J be the intersection of the minimalprimes of I of dimension dim( S/I ) and let J be the intersection of the minimal primesof I of dimension smaller than dim( S/I ) . Then, r I = min { t | [ J /I ] t = 0 } . Proof.
As in the proof of Proposition 3.6, we have that δ I ( t ) = 0 for t ≥ min { t | [ J /I ] t =0 } . We conclude that r I ≤ min { t | [ J /I ] t = 0 } . Let f ∈ (cid:16)S p ∈ Ass S ( S/I ) p (cid:17) \ I of degree strictly less than min { t | [ J /I ] t = 0 } . Then, f J , and so, there exists a prime ideal p such that f p and dim( S/ p ) < dim( S/I ). Then, e ( S/I ) − e( S/ ( I, f )) ≥ e( S/ p ). We conclude that δ I ( t ) > . Then, r I ≥ min { t | [ J /I ] t =0 } . (cid:3) Proposition 3.8.
Suppose that I is an unmixed radical ideal with associated primes p , . . . , p r and q , . . . , q s such that e( S/ p i ) = min { e( S/Q ) | Q ∈ Ass S ( S/I ) } . Let J i = (cid:16)T j = i p j (cid:17) T (cid:16)T sj =1 q j (cid:17) . Then, r I = min { t | ∃ i such that [ J i /I ] t = 0 } . INIMUM DISTANCE 7
Proof.
We set e = e( S/ p i ) . As in the proof of Proposition 3.6, we have that δ I ( t ) = e for t ≥ min { t | ∃ i such that [ J i /I ] t = 0 } . We conclude that r I ≤ min { t | ∃ i such that [ J i /I ] t =0 } . Let f ∈ (cid:16)S p ∈ Ass S ( S/I ) p (cid:17) \ I of degree strictly less than min { t | ∃ i such that [ J i /I ] t =0 } . Then, f J i for every i , and so, either f q j or f does not belong to to differentprimes p i nor p j . In both cases, dim(
S/I ) = dim( S/ ( I, f )) . In the first case, e(
S/I ) − e( S/ ( I, f )) ≥ e( S/ q j ) > e . In the second case, e( S/I ) − e( S/ ( I, f )) ≥ > e . Weconclude that δ I ( t ) ≥ e . Then, r I ≥ min { t | ∃ i such that [ J i /I ] t = 0 } . (cid:3) Stanley–Reisner ideals associated to a shellable simplicial complex
In this section we use the shellability condition to relate the regularity index ofa Stanley–Reisner ideal of a shellable simplicial complex, I ∆ , with the Castelnuovo–Mumford regularity. Definition 4.1. A simplicial complex on a vertex set X = { x , x , . . . , x n } is a collectionof subsets of X , called faces , satisfying that { x i } ∈ ∆ for every i ∈ [ n ] and, if σ ∈ ∆ and θ ⊆ σ then θ ∈ ∆. A face of ∆ not properly contained in another face of ∆ is called a facet .A face σ ∈ ∆ of cardinality | σ | = i + 1 has dimension i and is called an i - face of ∆.The dimension of ∆ is dim ∆ = max { dim σ | σ ∈ ∆ } , or if ∆ = {} is the void complex,which has no faces. We say that ∆ is pure if all its facets have the same dimension.Let ∆ be a simplicial complex of dimension d with the vertex set [ n ] = { , , . . . , n } ,and let K be a field. The square-free monomial ideal I ∆ in the polynomial ring S = K [ x , . . . , x n ] is generated by the monomials x σ = Q i ∈ σ x i which σ is a non-face in ∆.The simplicial complex ∆ is said Cohen-Macaulay when the quotient ring K [∆] = S/I ∆ , called Stanley–Reisner ring of ∆, is Cohen-Macaulay. Definition 4.2.
A pure simplicial complex ∆ of dimension d is shellable if the facets of∆ can be order σ , . . . , σ s such that ¯ σ i \ ( i − [ j =1 ¯ σ j )is pure of dimension d − i ≥
2. Here ¯ σ i = { σ ∈ ∆ | σ ⊆ σ i } . If ∆ is pure shellable, σ , . . . , σ s is called a shelling. Theorem 4.3 ([Vil15, Theorem 6.3.23]) . Let ∆ be a simplicial complex. If ∆ is pureshellable, then ∆ is Cohen–Macaulay over any field K . We come to one of our main results.
Theorem 4.4.
Let I = I ∆ be the Stanley–Reisner ideal of a shellable simplicial complex,with dim( S/I ∆ ) = d . Then r I ≤ reg( S/I ) . LUIS N ´U ˜NEZ-BETANCOURT, YURIKO PITONES, AND RAFAEL H. VILLARREAL
Proof.
Since ∆ is shellable,
S/I ∆ is a Cohen–Macaulay ring by Theorem 4.3. Let p , . . . , p ℓ denote the associate primes of I . For 1 ≤ i ≤ ℓ , we set R i = S/ p ∩ p ∩ · · · ∩ p i and J i = p ∩ · · · ∩ p i . We have that R i is Cohen-Macaulay of dimension d , because J i is ashelling of I ∆ .We have the following short exact sequence;0 −→ J i − /J i −→ R i −→ R i − −→ ≤ i ≤ ℓ .Note that dim( R i − ) = d , then H i m = 0 for all i > d , thus the short exact sequenceinduces a long exact sequence as follows:0 → H m ( J i − /J i ) → H m ( R i ) → H m ( R i − ) → · · · → H d m ( J i − /J i ) → H d m ( R i ) → H d m ( R i − ) → . Since R i − and R i are Cohen–Macaulay rings, we havereg( R i − ) = a d ( R i − ) + d and reg( R i ) = a d ( R i ) + d. From the exact sequence we get reg( J i − /J i ) ≤ reg( R i ). By Proposition 3.8, r I ≤ reg( R i ).Then r I ≤ reg( R ℓ ), and so, r I ≤ reg( S/I ∆ ). (cid:3) Results related to F -purity Definition 5.1.
Let R be a Noetherian ring of prime characteristic p , and F : R → R bethe Frobenius map. We say that R is F -pure if for every R -module, M , we have that M ⊗ R R M ⊗ R F / / M ⊗ R R is injective. We say that R is F -finite if R is finitely generated as R p -module. Definition 5.2.
Suppose that K has prime characteristic, K is F -finite, and that I is aradical ideal. Then, we set • m e = { f ∈ S/I | φ ( F e ∗ f ) ∀ φ : F e ∗ S/I → S/I } [AE05]. • b e = max { t | m t m e } . • fpt( S/I ) = lim e →∞ b e p e [TW04]. Theorem 5.3 ([DSNnB18, Theorem B]) . Suppose that K has prime characteristic. If S/I is a F -pure ring, then a i ( S/I ) ≤ − fpt( S/I ) . Furthermore, if S/I is a Gorensteinring, then reg(
S/I ) = d − fpt( S/I ) . Remark 5.4.
Suppose that K has prime characteristic, K is F -finite, and that S/I is a F -pure ring. Let p , . . . , p ℓ be the minimal primes of I , and J i = T i = j p j . Then, S/J i is F -pure [Sch10, Corollary 4.8]. Furthermore, fpt( S/I ) ≤ fpt( S/J i ) [DSNnB18, Theorem4.7], because J i · S/I is a compatible ideal for
S/I . Theorem 5.5.
Suppose that K has prime characteristic. If S/I is a F -pure ring, then r I ≤ dim( S/I ) . INIMUM DISTANCE 9
Proof.
Let p , . . . , p ℓ be the minimal primes of I . For i = 1 , . . . , ℓ , we set J i = T i = j p j .We have a short exact sequence0 → J i /I → S/J i → S/I → . This induces a long exact sequence0 → H m ( J i /I ) → H m ( S/J i ) → H m ( S/I ) → H m ( J i /I ) → . . . . Since both
S/J i and S/I are F -pure, we have that a j ( S/J i ) ≤ a j ( S/I ) ≤ j . Then, a j ( J i /I ) ≤ j [HR76, Proposition 2.4]. Then,min { t | [ J i /I ] t = 0 } ≤ max { ℓ | β ,ℓ = 0 } ( J i /I ) ≤ reg( J i /I )= max { a j ( J i /I ) + j }≤ dim( S/I ) . (cid:3) Theorem 5.6.
Suppose that K has prime characteristic. If S/I is a F -pure ring, then r I ≤ reg( S/I ) .Proof. We first assume that K is F -finite. Let p , . . . , p ℓ be the minimal primes of I . For i = 1 , . . . , ℓ , we set J i = T i = j p j . We have a short exact sequence0 → J i /I → S/J i → S/I → . This induces a long exact sequence0 → H m ( J i /I ) → H m ( S/J i ) → H m ( S/I ) → H m ( J i /I ) → . . . . Since both
S/J i and S/I are F -pure, we have that a j ( S/J i ) ≤ − fpt( S/J i ) and a j ( S/I ) ≤− fpt( S/I ) for every j . Then, a j ( J i /I ) ≤ max {− fpt( S/J i ) , − fpt( S/I ) } ≤ − fpt( S/I )for every j by Theorem 5.3. Then,min { t | [ J i /I ] t = 0 } ≤ max { ℓ | β ,ℓ ( J i /I ) = 0 }≤ reg( J i /I )= max { a j ( J i /I ) + j } = reg( S/I ) . The result for non F -finite fields follows from taking the product ⊗ K K , because thenumerical invariants do not change after field extensions. In addition, F -purity is stablefor field extensions. (cid:3) Theorem 5.7.
Let K be any field and I is a square-free monomial ideal. Then, r I ≤ dim( S/I ) . If S/I is a Gorenstein ring, then r I ≤ reg( S/I ) . Proof. If K has prime characteristic, the result follows from Theorems 5.5 and 5.6.We now assume that K has characteristic zero. Since field extensions do not affectwhether a ring is Gorenstein and their dimension, without loss of generality we can assumethat K = Q . Let A = Z [ x , . . . , x n ] and I A the monomial ideal generated by the monomialsin I . We have that r I = r I A ⊗ Z F p by Propositions 3.7 and 3.8, since dim( S/I ) = dim( A ⊗ Z Q /I A ⊗ Z Q ) = dim( F p [ x , . . . , x n ] /I A ⊗ Z F p ). Then, r I = r I A ⊗ Z F p ≤ dim( F p [ x , . . . , x n ] /I A ⊗ Z F p ) = reg( S/I )by Theorem 5.6, because Stanley–Reisner rings in prime characteristic are F -pure.We have thatreg S ( S/I ) = reg A ⊗ Z Q ( A ⊗ Z Q /I A ⊗ Z Q ) = reg A ⊗ Z F p ( A/I A ⊗ Z F p )and A/J ⊗ Z F p is Gorenstein for p ≫ F -pure. (cid:3) Acknowledgments
We thank Carlos Espinosa-Vald´ez for comments on an earlier draft.
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INIMUM DISTANCE 11
Luis N´u˜nez-Betancourt, Centro de Investigaci´on en Matem´aticas, Guanajuato, Gto.,M´exico.
Email address : [email protected] Yuriko Pitones, Centro de Investigaci´on en Matem´aticas, Guanajuato, Gto., M´exico.
Email address : [email protected] Rafael H. Villarreal, Departamento de Matem´aticas, Centro de Investigaci´on y deEstudios Avanzados del IPN, Apartado Postal 14–740, 07000 Mexico City, D.F.
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