aa r X i v : . [ m a t h . A C ] F e b GORENSTEIN BINOMIAL EDGE IDEALS
RENÉ GONZÁLEZ-MARTÍNEZ
Abstract.
We classify connected graphs G whose binomial edge ideal is Gorenstein.The proof uses methods in prime characteristic. Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. F-pure thresholds of graded rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33. F-pure thresholds of binomial edge ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64. Gorenstein Binomial Edge Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Introduction
Our main goal is to present homological properties of binomial edge ideals. Theseideals are a generalization of determinantal ideals and ideals generated by adjacent 2-minors in a × n generic matrix. The binomial edge ideals were introduced by Herzog,Hibi, Hreindóttir, Kahle, and Rauh [HHH +
10] and by Ohtani [Oht11] independently andabout the same time.Let G a simple graph (i.e. G has no loops and multiple edges) on the vertex set V ( G ) = [ n ] := { , . . . , n } with edge set E ( G ) . Let S = K [ x , . . . , x n , y , . . . , y n ] be thepolynomial ring on n variables over a field K . The binomial edge ideal J G of G is J G := ( f ij | { i, j } ∈ E ( G ) i < i ) , where f ij = x i y j − x j y i for i < j . The properties of binomial edge ideals have been studiedvastly by many researchers, for instance: • Cohen-Macaulayness [BNnB17, BMS18, EHH11, HHH +
10, KSM15, MR18, RR14,Rin13, Rin19, Zaf12], • Betti numbers and regularity [Bas16, CDI16, dAH18, EZ15, KSM14, KSM16,MM13, SMK12, SMK18, SZ14], • Gröbner bases [BBS17, CR11, HHH +
10, Oht11].
Mathematics Subject Classification.
Primary 05E40, 13D07, 05C75, 16W50, 13H10; Secondary13A35.
Key words and phrases.
Binomial Edge Ideal; Gorenstein ideal; Graded rings; Initial ideals; F -purethresholds. The author was partially supported by the CONACyT Grants 433381 and 284598 .
Herzog et al. characterized the graphs whose binomial edge ideal has quadratic Gröbnerbase. For a graph G , the generators f ij of J G form a quadratic Gröbner basis if andonly if for all edges { i, j } and { k, l } with i < j and k < l one has { j, l } ∈ E ( G ) if i = k , and { i, k } ∈ E ( G ) if j = l [HHH +
10, Theorem 1.1]. A graph G that satisfies theaforementioned condition is called closed with respect to the given labelling of the vertices .We say that a graph G is closed if there exists a labeling of its vertices such that G isclosed with respect to that labeling.Ene, Herzog and Hibi proved that if G is a closed graph, then S/ J G is Gorenstein ifand only if G is a path [EHH11, Corollary 3.4]. This motivated us to prove the mainresult of this paper. Theorem A.
Let G be a connected graph such that S/ J G is Gorenstein. Then, G is apath. This is achieved through the applications of methods in prime characteristic, in par-ticular, F -pure thresholds [TW04]. Along the way we compute the F -pure thresholdof binomial edge ideals associated to closed graphs and show that the F -pure thresholdof the binomial edge ideal coincide with the F -pure threshold of the initial ideal of thebinomial edge ideal for closed graphs. This follows the line of research which establishthat the binomial edge ideal and its initial ideal have similar properties for closed graphs[dAH18, EHH11]. 1. Background
In this section we recall some notions and known facts regarding binomial edge ideals.In this paper, all graphs are simple.
Definition 1.1.
Let G be a simple graph on [ n ] , and let i and j be two vertices of G with i < j . A path i = i , i , . . . , i r = j from i to j is called admissible , if(i) i k = i ℓ for k = ℓ ;(ii) for each k = 1 , . . . , r − one has either i k < i or i k > j ;(iii) for any proper subset { j , . . . , j s } of { i , . . . , i r − } , the sequence i, j , . . . , j s , j isnot a path.Given an admissible path π : i = i , i , . . . , i r = j from i to j , where i < j , we associate the monomial u π = Y i k >j x i k ! Y i ℓ
Theorem 1.2 ([HHH +
10, Theorem 2.1]) . Let G be a simple graph on [ n ] , and let < bethe lexicographic order on S = K [ x , . . . , x n , y , . . . , y n ] induced by x > x > · · · > x n >y > y > · · · > y n . Then the set of binomials G = [ i Since J G has a square-free Gröbner basis, we conclude that J G is a radical ideal[HHH + 10, Corollary 2.2]. Remark 1.3. Every monomial u π x i y j such that π is an admissible path from i to j isthe initial term of an element of G . From the fact that G is a reduced Gröbner basis, weconclude that the set In( G ) is the minimal generating set of the ideal In( J G ) .The binomial edge ideal of a path P n with n vertices is a complete intersection having n − generators of degree 2 and reg( S/ J P n ) = n − [EZ15]. The following results statethat regularity n − implies that the graph is a path. Theorem 1.4 ([MM13, Theorem 1.1]) . Let G be a graph on the set [ n ] of vertices. Then reg( S/ J G ) ≤ n − Theorem 1.5 ([KSM16, Theorem 3.4]) . Let G be a graph on [ n ] which is not a path.Then reg( S/ J G ) ≤ n − We now recall the Plücker relation for binomials. This plays an important role whilewe study the F -pure threshold for S/ J G . Proposition 1.6 (Plücker relation) . Let i < j < k < l be positive integers. Then, f ij f kl − f ik f jl + f il f jk = 0 . F-pure thresholds of graded rings In this section we introduce the basic definitions for methods in prime characteristic. Wealso list some properties for the F -pure threshold of standard graded rings. We considerthe F -pure threshold with respect to m , the maximal graded ideal. This invariant wasintroduced by Takagi and Watanabe [TW04]. The F -pure threshold is related to thelog-canonical threshold [BFS13, TW04], and roughly speaking measures the asymptoticsplitting order of m . Definition 2.1. Let R be a Noetherian ring of prime characteristic p . We say that R is F -finite if it is finitely generated R -module via the action induced by the Frobeniusendomorphism F : R → Rr r p . For e ∈ N , let F e : R → R the e -th iteration of the Frobenius endomorphism on R . If R is reduced, R /p e denotes the ring of p e -th roots of R . We often identify F e with theinclusion R ⊆ R /p e . In this case, R is F -finite if and only if R /p is a finitely generated R -module. For a standard graded K -algebra ( R, m , K ) , R is F -finite if and only if K is F -finite, that is, if and only if [ K : K p ] < ∞ . A ring R is called F -pure if F is apure homomorphism of R -modules, that is F ⊗ R ⊗ M → R ⊗ M is injective for all R -modules M . We say that R is called F -split if F is a split monomorphism. If R is an F -finite ring, R is F -pure if and only R is F -split [HR76, Corollary 5.3]. Let J ⊆ R anideal we write J [ p ] := ( x p | x ∈ J ) . Lemma 2.2 (Fedder’s Criterion for graded rings [Fed83, Theorem 1.12]) . Let K be afield of prime characteristic p , and S = K [ x , . . . , x n ] be a polynomial ring over K . Let m = ( x , . . . , x n ) be the irrelevant maximal ideal of S , and I ⊆ m be a homogeneous idealof S . Then S/I is F -pure if I [ p ] : I * m [ p ] . RENÉ GONZÁLEZ-MARTÍNEZ Definition 2.3 ([TW04, Definition 2.1]) . Let ( R, m , K ) be a standard graded K -algebrawhich is F -finite and F -pure, and let I ⊆ R be a homogeneous ideal. For a real num-ber λ ≥ , we say that ( R, I λ ) is F -pure if for every e ≫ , there exists an element f ∈ I ⌊ ( p e − λ ⌋ such that the inclusion of R -modules f /p e R ⊆ R /p e splits. The F -purethreshold of I is defined by fpt( I ) := sup { λ ∈ R ≥ | ( R, I λ ) is F pure } . If I = m , we denote the F -pure threshold by fpt( R ) . Definition 2.4. Let ( R, m , K ) be a standard graded K -algebra which is F -finite and F -pure, and J ⊆ m be and ideal. We define I e ( R ) := { r ∈ R | ϕ ( r /p e ) ∈ m for every ϕ ∈ Hom( R /p e , R ) } and b J ( p e ) := max { r | J r * I e ( R ) } . Proposition 2.5 ([DSNnB18, Proposition 3.10]) . Let ( R, m , K ) be a standard graded K -algebra which is F -finite and F -pure. Let J ⊆ R be a homogeneous ideal. Then fpt( J ) = lim e →∞ b J ( p e ) p e . Lemma 2.6 ([DSNnB18, Lemma 4.2]) . Let S = K [ x , . . . , x n ] be a polynomial ring overan F -finite field K . Let n = ( x , . . . , x n ) denote the maximal homogeneus ideal. Let I ⊆ S be an homogeneous ideal such that R := S/I is an F -pure ring, and let m = n R . Then, min (cid:26) s ∈ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:20) I [ p e ] : I + n [ p e ] n [ p e ] (cid:21) s = 0 (cid:27) = n ( p e − − b m ( p e ) . Theorem 2.7 ([DSNnB18, Theorem 7.3]) . Let S = K [ x , . . . , x n ] be a polynomial ringover an F -finite field K of prime characteristic p . Let I be a homogeneous ideal such that R = S/I is F -pure and Gorenstein. Then, reg S ( R ) = dim( R ) − fpt( R ) . Proposition 2.8. Let S = K [ x , . . . , x n , y , . . . , y m ] the ring of polynomials in n + m variables over a field K , and let J = a + b ⊆ R be an ideal such that a is an ideal in thevariables { x , . . . , x n } and b is an ideal in the variables { y , . . . , y m } . Then, fpt( S/J ) = fpt( K [ x , . . . , x n ] / a ) + fpt( K [ y , . . . , y m ] / b ) . Proof. We note that S/J ∼ = K [ x , . . . , x n ] / a ⊗ K [ y , . . . , y m ] / b by properties of tensor product. We now prove that b J ( p e ) = b a ( p e ) + b b ( p e ) . Set γ = b J ( p e ) . By the definition of b J ( p e ) , we have that J γ * I e ( S ) . This means that thereexists a generator r of J γ and also a morphism ϕ ∈ Hom( S /p e , S ) such that ϕ ( r /p e ) / ∈ m . This generator can be written as r = a i a i , . . . , a i s b j , . . . , b j t for some generators { a i a i , . . . , a i s } of a and some generators { b j , . . . , b j t } of b with γ = s + j . This elementcorrespond by(2.0.1) S ∼ = K [ x , . . . , x n ] ⊗ K [ y , . . . , y m ] ORENSTEIN BINOMIAL EDGE IDEALS 5 to a element α ⊗ β in the tensor product with α ∈ a s and β ∈ b t . We have the nextcomposition of morphisms: K [ x , . . . , x n ] /p e −→ S /p e ϕ −→ S ։ K [ x , . . . , x n ] α /p e r /p e ϕ ( r /p e ) ϕ ( r /p e ) , where the leftmost morphism send x /p e x /p e ⊗ β /p e and the rightmost is the naturalprojection. We have an element α ∈ a s and a morphism ϕ ∗ ∈ Hom( K [ x , . . . , x n ] /p e , K [ x , . . . , x n ]) (the composition of the morphisms above). Then, ϕ ∗ sends α /p e to ϕ ( r /p e ) . We notethat ϕ ( r /p e ) is not in the maximal ideal of K [ x , . . . , x n ] , because ϕ ( r /p e ) / ∈ m . Now wenote that b a ( p e ) = max( r | a r * I e ( K [ x , . . . , x n ])) ≥ s. By symmetric argument, we obtain that b b ( p e ) = max( r | b r * I e ( K [ y , . . . , y m ])) ≥ t proving that b J ( p e ) ≤ b a ( p e ) + b b ( p e ) . For the reverse inequality let s = b a ( p e ) , t = b b ( p e ) , α ∈ a s , β ∈ b t , ϕ ∈ Hom( K [ x , . . . , x n ] /p e , K [ x , . . . , x n ]) and ψ ∈ Hom( K [ y , . . . , y m ] /p e , K [ y , . . . , y m ]) such that ϕ ( α /p e ) and ψ ( β /p e ) are not in their respective maximal ideals. Then, byequation 2.0.1, α ⊗ β corresponds to an element αβ ∈ J t + s . Since ( ϕ ⊗ ψ )( α /p e ⊗ β /p e ) = ϕ ( α /p e ) ψ ( β /p e ) / ∈ m . Then b J ( p e ) ≥ t + s . (cid:3) Theorem 2.9 ([DSNnB18, Theorem 4.7]) . Let ( R, m , K ) a standard graded K -algebrawich is F -finite and F -pure, and let J ⊆ R a compatible ideal. Then, fpt( R ) ≤ fpt( R/J ) . In particular, fpt( R ) ≤ fpt( R/ p ) for every minimal prime ideal p of R . The next result shows us how to compute the F -pure threshold of a squarefree monomialideal. Proposition 2.10. Let I be a square-free monomial ideal of S = K [ x , . . . , x n ] . Then fpt( S/I ) is equal to the number of variables that do not appear in its minimal set ofgenerators.Proof. Let { x i , . . . , x i t } be the set of variables that appear in the minimal generating setof I . Then, x p e − i · · · x p e − i t ∈ ( I [ p e ] : I ) \ m [ p e ] . Set d := deg x p e − i · · · x p e − i t . RENÉ GONZÁLEZ-MARTÍNEZ For every monomial m of degree less than d , we have that m / ∈ ( I [ p e ] : I ) \ m [ p e ] . By Lemma2.6 b m ( p e ) = ( n − t )( p e − . Dividing both sides by p e and taking the limit as e go toinfinity yields the desired result. (cid:3) F-pure thresholds of binomial edge ideals Through this section K is a F − f inite field of characteristic p . For a sequence v , . . . , v s of natural numbers, we set f v ,...,v s := f p − v v · · · f p − v n − v n taking f ji := − f ij for j > i. Proposition 3.1. If { a, b } ∈ E ( G ) . Then, f v ,...,c,a,b,d,...,v s ≡ f v ,...,c,b,a,d,...,v s mod J G . Proof. By the Plücker relations, we have that f p − ca f p − ab f p − bd = f p − ab ( f cb f ad − f cd f ab ) p − = f p − ab p − X i =0 (cid:18) p − i (cid:19) ( − i f p − i − cb f p − i − ad f icd f iab = p − X i =0 ( − i (cid:18) p − i (cid:19) f p − i − cb f p − i − ad f icd f p + i − ab . By assumption f ab ∈ J G , so all the terms of the sum with i > are contained in J [ p ] G .This gives f v ,...,c,a,b,d,...,v s ≡ f p − v v · · · f p − ca f p − ab f p − bd · · · f p − v n − v n ≡ f p − v v · · · f p − cb f p − ad f p − ab · · · f p − v n − v n ≡ f v ,...,c,b,a,d,...,v s modulo J G . (cid:3) Theorem 3.2. Let G be a simple connected closed graph which is not the complete graphand let S := K [ x , . . . , x n , y , . . . , y n ] and m be the maximal homogeneous ideal. Then, S/ J G is F -pure, and min ( s ∈ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)" J [ p ] G : J G + m [ p ] m [ p ] s = 0 ) ≤ n − p − . Proof. By the Fedder’s criterion (Lemma 2.2), it suffices to show that f , ,...,n ∈ ( J [ p ] G : J G ) \ m [ p ] . First we prove that f ,...,n / ∈ m . Let < the lexicographic on S induced by x > · · · > x n >y > · · · > y n . Then In( f ,...,n ) = x p − · · · x p − n − y p − · · · y p − n / ∈ m [ p ] . Next we show that f ,...,n ∈ J [ p ] G : J G . It is enough to show that f ,...,n f ij ∈ J [ p ] G for all { i, j } ∈ E ( G ) . We assume that { i, j } ∈ E ( G ) . If j = i + 1 , then f ,...,n f ij ∈ J [ p ] G . We ORENSTEIN BINOMIAL EDGE IDEALS 7 assume that j > i + 1 . If j = n , then { k, j } ∈ E ( G ) for all k ∈ { i + 1 , . . . , j − } [Mat18,Propsition 1.3]. Hence, by using repeatedly Proposition 3.1, we obtain f , ,...,i,i +1 ,...,j − ,j,j +1 ,...,n ≡ f , ,...,i,j,i +1 ,...,j − ,j +1 ,...,n mod J [ p ] G . Since f p − i,j is a factor of the last expression, we have that f ,...,n ∈ J [ p ] G : J G .If j = n , then i = 1 . For if { , n } ∈ E ( G ) , then G is complete [Mat18, Proposition 1.3].By iterating Proposition 3.1, f , ,...,i,i +1 ,...,j − ,j,j +1 ,...,n ≡ f , ,...,i − ,i +1 ,...,i,n mod J [ p ] G . Then f p − in is a factor of the last expression, and f ,...,n ∈ J [ p ] G : J G . (cid:3) Corollary 3.3. Let G be a closed graph, let S := K [ x , . . . , x n , y , . . . , y n ] and let m bethe maximal homogeneous ideal. Then, fpt( S/ J G ) = 2 .Proof. If G is complete, then S/ J G is determinantal, and so, fpt( S/ J G ) = 2 [STV17,Proposition 4.3]. First we prove by induction on e that if f p − ∈ ( I [ p ] : I ) \ m [ p ] , then f p e − ∈ ( I [ p e ] : I ) \ m [ p e ] .The step base follows from our assumptions.For f p e − / ∈ m [ p e ] we have that ( m [ p e ] : f p e − ) ⊆ m ⇒ ( m [ p e +1 ] : f p e +1 − p ) ⊆ m [ p ] ⇒ (( m [ p e +1 ] : f p e +1 − p ) : f p − ) ⊆ ( m [ p ] : f p − ) ⊆ m ⇒ ( m [ p e +1 ] : f p e +1 − ) ⊆ m . This means that f p e +1 − / ∈ m [ p e +1 ] .If f p e − ∈ I [ p e ] : I , Then, f p − I ⊆ I [ p ] ⇒ ( f p − I ) [ p e ] ⊆ I [ p e +1 ] ⇒ f p e +1 − p e I [ p e ] ⊆ I [ p e +1 ] ⇒ f p e +1 − p e ( f p e − I ) ⊆ f p e +1 − p e ( I [ p e ] ) ⊆ I [ p e +1 ] ⇒ f p e +1 − I ⊆ I [ p e +1 ] , thus f p e +1 − ∈ I [ p e +1 ] : I . This means that ( x x · · · x n − y · · · y n ) p e − ∈ ( J [ p e ] G : J G ) \ m [ p e ] Using Lemma 2.6, we deduce that n ( p e − − b m ( p e ) ≤ n − p e − − b m ( p e ) ≤ − p e − b m ( p e ) p e ≥ p e − p e fpt( S/ J G ) = lim e →∞ b m ( p e ) p e ≥ lim e →∞ p e − p e = 2 . Since J K n is a minimal prime over J G , the reverse inequality is a consequence of 2.9 andthe fact that fpt J k n = 2 . (cid:3) Proposition 3.4. Let G be a connected graph on [ n ] . Then, x n and y are the onlyvariables that do not appear in the minimal generating set of In( J G ) . RENÉ GONZÁLEZ-MARTÍNEZ Proof. The set of monomials H = [ i Let G be a connected graph on [ n ] , then fpt( R/ In( J G )) = 2 .Proof. This is follows from Theorem 2.10 and Proposition 3.4. (cid:3) Remark 3.6. Let G be closed graph and S := K [ x , . . . , x n , y , . . . , y n ] . Then, fpt( S/ J G ) = fpt( S/ In( J G )) . Gorenstein Binomial Edge Ideals In this section we prove our main result. We start with a preparation theorem regarding F -injectivity of square Gröbner deformations. We first need to introduce notation. Notation 4.1. Let S = K [ x , . . . , x n ] be a polynomial ring over a field with maximalhomogeneous ideal m . Let I be an ideal and < a monomial order such that in( I ) issquare-free. There exists a vector w ∈ N n such that in < ( I ) = in w ( I ) [Stu96, Proposition1.11] . Let A = K [ t ] be a polynomial ring, L = frac( A ) , and T = A ⊗ K S . We set J = hom w ( I ) ⊆ T the homogenization of I , R = T /J , and R = R/xR. Remark 4.2. Under Notation 4.1, it is well known that(1) A → R is flat;(2) R/tR = S/ in < ( I ) ;(3) R/ ( t − a ) R = S/I for every a ∈ K \ { } ;(4) R ⊗ A L = S/I ⊗ K L ;The following result was obtained independently and simultaneously by Varbaro andKoley [KV]. Theorem 4.3. Let S = K [ x , . . . , x n ] be a polynomial ring over a field, K , of primecharacteristic. Let I be an ideal and < a monomial order such that in < ( I ) is square-free.Then, S/I is F -injective.Proof. We use Notation 4.1 and the facts in Remark 4.2 in this proof. We have that R/tR is an Stanley-Reisner ring, and so, F -pure. Since F -pure rings are F -full [SW07, Lemma2.5] and F -injective, R/tR satisfies these properties. Since t is a nonzero divisor, we havethat R is also F -full and F -injective [MQ18, Theorem 1.1]. Then, R ⊗ A L = S/I ⊗ K L are both F -injective and F -full, because these properties are preserved under localization.We note that S/I is a direct summand of S/I ⊗ K L . We note that m expands the maximalhomogeneous ideal in S/I ⊗ K L . Then, we have a commutative diagram ORENSTEIN BINOMIAL EDGE IDEALS 9 H i m ( S/I ) F S/I (cid:15) (cid:15) α / / H i m ( S/I ⊗ K L ) F S/I ⊗ KL (cid:15) (cid:15) H i m ( S/I ) α / / H i m ( S/I ⊗ K L ) ,where α denotes the maps induced by the inclusion S/I → S/I ⊗ K L . Since the horizontalmaps split, they are injective. Since S/I ⊗ K L is F -injective, we have that F S/I ⊗ K L ◦ α = α ◦ F S/I is injective. Hence, F S/I is injective, and the result follows. (cid:3) We are now ready to show our main result in prime characteristic. Theorem 4.4. Let S := K [ x , . . . , x n , y , . . . , y n ] . Suppose that char( K ) = p > . Let G be a connected graph such that S/ J G is Gorenstein. Then, G is a path.Proof. By Theorem 4.3, we have that S/ J G is F -injective. Since S/ J G is Gorenstein, wehave that S/ J G is F -pure [Fed83, Lemma 3.3].Since G is connected, J k n is a minimal prime over J G [HHH + 10] and its dimension is n + 1 . Then,(4.0.1) reg( R/ J G ) = dim( R/ J G ) − fpt( R/ J G ) ≥ ( n + 1) − n − where the inequality comes from the fact that if I ⊆ J then fpt( J G ) ≤ fpt( J K n ) = 2 .Hence, G is a path by Theorem 1.5. (cid:3) In the previous result we estimate the regularity of R/ J G using F -pure thresholds. Wepoint out that the extremal Betti numbers of R/ J G and R/ in( J G ) coincide, in particular, reg( R/ J G ) = reg( R/ in( J G )) [CV18, Corollary 2.7].We are now ready to prove the main result in this manuscript in characteristic zero. Theorem 4.5. Let S := K [ x , . . . , x n , y , . . . , y n ] . Suppose that char( K ) = 0 . Let G be aconnected graph such that S/ J G is Gorenstein. Then, G is a path.Proof. Since field extensions do not affect whether a ring is Gorenstein, without loss ofgenerality we can assume that K = Q .Let A = Z [ x , . . . , x n , y , . . . , y n ] and J = ( x i y j − x j y i : { i, j } ∈ G and i < j ) A. Then, reg S ( S/ J G ) = reg A ⊗ Z Q ( A ⊗ Z Q /J ⊗ Z Q ) = reg A ⊗ Z F p ( A/J ⊗ Z F p ) and A/J ⊗ Z F p is Gorenstein for p ≫ [HH, Theorem 2.3.5]. Then, reg S ( S/ J G ) ≥ n − from the proof of Theorem 4.4. Hence, G is a path by Theorem 1.5. (cid:3) Acknowledgments I thank Prof. Luis Núñez-Betancourt and Prof. Martha Takane for useful advice andsuggestions on this project. I thank Prof. Aldo Conca and Prof. Matteo Varbaro forpointing out a mistake in an earlier version of this manuscript. I thank Prof. AlessandroDi Stefani for helpful discussions. Finally, I thank the anonymous referee for helpfulcomments, and constructive remarks on this manuscript. References [Bas16] H. Baskoroputro. 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