ON A CLASS OF MONOMIAL IDEALS GENERATED BY s-SEQUENCES

BY s-SEQUENCES

MONICA LA BARBIERA

The symmetric algebra of classes of monomial ideals in a polynomial ring in two sets of variables is studied. In certain cases, the theory ofs-sequences permits to compute standard invariants.

AMS 2000 Subject Classification: 13A99, 13P10.

Key words: monomial ideal, Gr¨obner base, symmetric algebra.

INTRODUCTION

Let R = K[X₁, . . . , X_n;Y₁, . . . , Y_m] be a polynomial ring in two sets of variables. Let r, k ≥ 1 be integers, then I_r,s is the Veronese-type ideal generated on degree r by the set {X₁^ai1. . . Xn^ain |P_n

j=1a_ij =r, 0 ≤a_ij ≤s, s ∈ {1, . . . , r}} and J_k,s is the Veronese-type ideal generated on degree k by the set {Y₁^bi1. . . Ym^bim | Pm

j=1bij = k, 0 ≤ bij ≤ s, s ∈ {1, . . . , k}}. In [10] the author introduced the Veronese bi-type ideals Lq,s =P

r+k=qIr,sJ_k,s generated in the same degree q.

For s = 2 the Veronese bi-type ideals are the ideals of the walks of a bipartite graph with loops. In [9] the author studies the combinatorics of the integral closure and the normality of L_q,2. More in general, in [10] the same problem is studied for Lq,s for all s.

In this paper we are interested to study the symmetric algebra of these classes of monomial ideals. In order to compute the standard invariants we investigate in which cases these monomial ideals are generated bys-sequences.

In [6] the notion of s-sequences has been employed to compute the in- variants of the symmetric algebra of finitely generated modules. The proposal is to compute standard invariants of the symmetric algebra in terms of the corresponding invariants of special quotients of the ring R. This computation can be obtained for finitely generatedR-modules generated by ans-sequence.

In Section 1 we consider the ideals of Veronese-type I_q,s. We give the conditions such that Iq,s is generated by an s-sequence. Then we compute standard algebraic invariants of the symmetric algebra of I_q,s.

MATH. REPORTS12(62),3 (2010), 201–216

The computation of these invariants can be obtained when the idealIq,s

is generated by an s-sequence in terms of its annihilator ideals.

In Section 2 we determine the subclasses of Veronese bi-type idealsLq,s

generated by s-sequences. We establish the theorem: Lq,s is generated by an s-sequence if and only ifq =s(n+m)−1. The technic used to prove this result is the characterization of the monomials-sequences by the Gr¨obner bases. For s= 1 and q = 2,3 the ideals obtained are the mixed product ideals I₁J₁ and I1J2+I2J1, studied in [11]. Then we give the structure of the annihilator ideals of Veronese bi-type ideals generated by s-sequences and we achieve formu- las for the dimension dim_R(Sym_R(L_q,s)), the multiplicity e(Sym_R(L_q,s)) and bounds for the Castelnuovo-Mumford regularity reg_R(Sym_R(Lq,s)) in terms of the annihilator ideals as described in [6]. As an application, we verify the Eisenbud-Goto inequality in the formulation given in [12], for the symmetric algebra of these ideals generated bys-sequences. In Section 3 we consider the ideals of the walks of a bipartite graph with loops Iq(G) = Lq,2. We prove that the idealsI_q(G) generated by ans-sequence are a class of monomial ideals with linear quotients and we investigate some their algebraic invariants.

1. IDEALS OF VERONESE-TYPE

We recall the theory of s-sequences in order to apply it to our classes of monomial ideals. Let R be a noetherian ring, M be a finitely generated R-module andf1, . . . , ftbe the generators ofM. For everyi= 1, . . . , t, we set Mi−1 =Rf1+· · ·+Rfi−1 and let I_i =Mi−1 :_R fi be the colon ideal. Since M_i/Mi−1 ' R/I_i, so I_i is the annihilator of the cyclic module R/I_i. I_i is called an annihilator ideal of the sequence f1, . . . , ft.

Let (a_ij), for i = 1, . . . , t, j = 1, . . . , p, be the relation matrix of M. The symmetric algebra Sym_R(M) has a presentation R[T1, . . . , Tt]/J, with J = (g₁, . . . , g_p) whereg_j =Pt

i=1a_ijT_i forj= 1, . . . , p. LetS=R[T₁, . . . , T_t] be the polynomial ring and let ≺ be a monomial order on the monomials of S in the variablesT_i such that T₁ ≺T₂≺ · · · ≺T_t. With respect to this term order, if f =P

a_αT^α, where T^α =T₁^α1· · ·T_t^αt and α= (α₁, . . . , α_t)∈N^t, we put in≺(f) =aαT^α, where T^α is the largest monomial inf such thataα6= 0.

If we assign degree 1 to each variable T_i and degree 0 to the elements of R, we have the following facts:

1) J is a graded ideal.

2) The natural epimorphism S →Sym_R(M) is a graded homomorphism of graded algebras on R.

Set the monomial ideal in≺(J) = (in≺(f)|f ∈J). In general (I₁T1,I₂T2, . . . ,I_tTt)⊆in≺(J)

and the two ideal coincide in the linear case.

Definition 1.1. A sequencef₁, . . . , f_tis an s-sequence forM if (I₁T₁,I₂T₂, . . . ,I_tT_t) = in≺(J).

If I₁⊆ I₂⊆ · · · ⊆ I_t, the sequence is astrong s-sequence.

If R = K[X1, . . . , Xn] is the polynomial ring over a field K, we can use the Gr¨obner bases theory to compute in≺(J). Let ≺ any term order on K[X1, . . . , Xn;T1, . . . , Tt] with T1 ≺ T2 ≺ · · · ≺ Tt, Xi ≺ Tj for all i and j.

Then for any Gr¨obner basis G of J ⊂K[X₁, . . . , X_n, T₁, . . . , T_t] with respect to≺, we have in≺(J) = (in≺(f)|f ∈G). If the elements ofGare linear in the Ti, thenf1, . . . , ft is ans-sequence for M.

Let M = I = (f₁, . . . , f_t) a monomial ideal of R =K[X₁, . . . , X_n]. Set fij = _[f^fi

i,fj] for i 6= j, where [fi, fj] is the greatest common divisor of the monomials fi and fj. J is generated by gij =fijTj−fjiTi for 1≤i < j ≤t.

The monomial sequencef₁, . . . , f_tis ans-sequence if and only ifg_ij for 1≤i <

j ≤tis a Gr¨obner basis for J for any term order inK[X1, . . . , Xn;T1, . . . , Tt] withT₁≺T₂ ≺ · · · ≺T_t,X_i ≺T_j for alli,j. Notice that the annihilator ideals of the monomial sequencef1, . . . , ft are the idealsIi = (f1i, f2i, . . . , fi−1,i) for i= 1, . . . , t ([6]).

Remark 1.1 ([6, Lemma 1.4]). From the theory of Gr¨obner bases, if f1, . . . , ftis a monomials-sequence with respect to some admissible term order

≺, thenf₁, . . . , f_t is as-sequence for any other admissible term order.

The first section of this paper is dedicated to the symmetric algebra of a class of monomial modules over the polynomial ring R=K[X₁, . . . , X_n] that are ideals. We recall the following definition.

Definition 1.2 ([13]). LetR=K[X₁, . . . , X_n] be the polynomial ring over a field K. The ideal of Veronese-type of degree q is the monomial ideal I_q,s generated by the set

X₁^ai1 · · ·Xn^ain

n

X

j=1

aij =q, 0≤aij ≤s

.

Remark 1.2. In generalI_q,s⊆I_q, whereI_qis theVeronese ideal of degree q ofR which is generated by all the monomials in the variablesX₁, . . . , X_n of degree q: Iq = (X1, . . . , Xn)^q ([14]). Ifq = 1,2 ors=q, thenIq,s =Iq.

Example 1.1. R = K[X1, X2, X3], I3,2 = (X₁²X2, X₁²X3, X1X₂², X₂²X3, X1X₃², X2X₃², X1X2X3)⊂I3, I3,3 = (X₁³, X₂³, X₃³, X₁²X2, X₁²X3, X1X₂², X₂²X3, X1X₃², X2X₃², X1X2X3) =I3.

We give the condition such that the ideal Iq,s is generated by an s- sequence. First of all we observe that the property to be an s-sequence may depend on the order of the sequence.

Example1.2 ([6]). R=K[X₁, X₂],I_2,2 = (X₁², X₁X₂, X₂²);X₁², X₂², X₁X₂ is an s-sequence, but X₁², X₁X₂, X₂² is not an s-sequence. Let Sym_R(I_2,2) = R[T₁, T₂, T₃]/J, J = (X₂T₁−X₁T₂, X₁T₂−X₂T₃). Fix T₁ ≺ T₂ ≺ T₃ and X1 ≺ X2 ≺ Ti, we have in≺(J) = ((X₁²)T2,(X1, X2)T3) and I₁ = (0), I₂ = (X₁²), I₃ = (X₁, X₂). Hence X₁², X₁X₂, X₂² is a strong s-sequence.

Instead X₁², X1X2, X₂² is not an s-sequence because in this case in≺(J) = ((X₁)T₃,(X₁)T₂,(X₂)T₁T₃). It means thatJ does not admit a linear Gr¨obner basis for any term order in K[X1, X2;T1, T2, T3] with Xi ≺Tj for alli, j and T₁ ≺ T₂ ≺ T₃. In fact there are S-pairs S(g_ij, g_hl) that have not a standard expression with respect G={g_ij =fijTj−fjiTi |1≤i < j ≤3}with remain- der 0: S(g12, g23) = _[f^f12f32

12,f23]T₂²−_[f^f21f23

12,f23]T1T3=X2T₂²−X2T1T3, withf1=X₁², f2 =X1X2,f3 =X₂². There is no gij ∈G whose initial term divides a term of S(g₁₂, g₂₃).

Hence in the sequel we will suppose Iq,s = (f1, f2, . . . , ft) where f1 ≺ f₂ ≺ · · · ≺ f_t with respect to the monomial order ≺_Lex on the variables X_i and X1≺X2 ≺ · · · ≺Xn.

Lemma 1.1. Let R =K[X₁, . . . , X_n]be the polynomial ring over a field K with n > 2 and Iq,s ⊂ R, 2 ≤ q ≤ sn. If [fij, fhl] = 1 for i < j, h < l, i6=h, j6=l withi, j, h, l∈ {1, . . . , t}, then q =sn−1.

Proof. By hypotheses q ≤ sn. Let f1, . . . , ft be the generators of Iq,s. Set fij = _[f^fi

i,fj] for fi, fj with i < j. We have fij = X_i^a₁ⁱ¹ · · ·X_i^a_c^ic and fhl = X_h^bh1

1 · · ·X_h^bhd

d forfh,flwithh < l,i6=h,j6=l. By the hypothesis [fij, fhl] = 1 for i < j, h < l, i 6= h, j 6= l, we have X_ij 6= X_hp for all j = 1, . . . , c and p= 1, . . . , d. This means that there are no other generatorsf_h,f_l ofI_q,s such that f_hl contains some variablesXi1, . . . , Xic (that are infij). It follows that if a variable of f_ij is of degree kin the monomialf_h, with h6=i, j, then such variable in the same degree k belongs to any other generators fl for alll > h and l 6= j. Hence we deduce the structure of the monomials that generate Iq,s and satisfying the hypotheses of the lemma. Ifq =sn−1 we have f1 = X₁^sX₂^s· · ·X_n−1^s X_n^s−1, f₂ = X₁^sX₂^s· · ·X_n−1^s−1X_n^s, f₃ = X₁^sX₂^s· · ·X_n−2^s−1X_n−1^s X_n^s, . . . ,fn−1=X₁^sX₂^s−1· · ·X_n−2^s X_n−1^s X_n^s,f_n=X₁^s−1X₂^s· · ·X_n−2^s X_n−1^s X_n^s.

We compute f12 = Xn−1, f13 = Xn−2, . . ., f1n = X1, f23 = Xn−2, . . ., f_2n = X₁, and so on. Hence f_1j = f_2j = . . . = f_nj = Xn−j+1 for all j = 2, . . . , n. Then fij 6=fhl because j6=l and [fij, fhl] = 1 as required.

Letq < sn−1. If we consider the generators fi =X₁^sX₂^s· · ·X_n−3^s−2X_n−1^s , fj =X₁^sX₂^s· · ·X_n−4^2s−2X_n^s,fh =X₁^sX₂^s· · ·X_n−3^s−2X_n−2^s ,fl =X₁^sX₂^s· · ·X_n−4^2s−2X_n−1^s ,

then fij =X_n−3^s−2X_n−1^s and fhl =X_n−3^s−2X_n−2^s . Hence [fij, fhl]6= 1, a contradic- tion. It follows that q =sn−1.

Theorem1.1. LetR=K[X₁, . . . , X_n]be the polynomial ring over a field K with n >2. Iq,s is generated by ans-sequence if and only if q=sn−1.

Proof. Let Iq,s = (f1, f2, . . . , ft) and suppose that f1, f2, . . . , ft is an s- sequence. We prove that [f_ij, f_hl] = 1 for i < j, h < l, i 6= h, j 6= l with i, j, h, l∈ {1, . . . , t}.

The s-sequence property implies that G = {g_ij = f_ijT_j −f_jiT_i | 1 ≤ i < j ≤t} is a Gr¨obner basis for J. In particular, S(gij, ghl) has a standard expression with respect to Gwith remainder 0. We have

S(gij, g_hl) = fijflh

[f_ij, f_hl]TjT_h− fhlfji

[f_ij, f_hl]TiT_l.

We suppose that i < j, h < l, i 6= h, j 6= l. As S(g_ij, g_hl) has a stan- dard expression with respect to G there exists grp such that in≺(grp) divides in≺(S(g_ij, g_hl)).

I) Ifl > j then in≺(g_rp)| _[f^fhlfji

ij,fhl]. The first case is f_hl | _[f^fhlfji

ij,fhl], then [f_ij, f_hl] | f_ji. But, as we have [[f_ij, f_hl], f_ji] = 1, it follows [f_ij, f_hl] = 1.

The second case is frl | _[f^fhlfji

ij,fhl], where frl = in≺(grl) with r < j and r < h. We can write

S(g_ij, g_hl) =− f_jif_hl

f_rl[f_ij, f_hl]g_rlT_i+ f_ijf_lh

[f_ij, f_hl]T_jT_h− f_jif_hlf_lr f_rl[f_ij, f_hl]T_iT_r. Then _[f^fijflh

ij,fhl]T_jT_h is divided by f_ij. Hence [f_ij, f_hl] | f_lh. But, as we have [[f_ij, f_hl], f_lh] = 1, it follows [f_ij, f_hl] = 1.

II) Ifl < j then in≺(grp)| _[f^fijflh

ij,fhl]. The first case is fij | _[f^fijflh

ij,f_hl], then [fij, fhl] | flh. But, as we have [[fij, fhl], flh] = 1, it follows [fij, fhl] = 1.

The second case isf_rh | _[f^fijflh

ij,fhl], wheref_rh = in≺(g_rh). We can write S(g_ij, g_hl) = fijf_lh

f_rh[f_ij, f_hl]T_jg_rh− f_hlfji

[f_ij, f_hl]T_iT_l+ fijf_lhf_hr f_rh[f_ij, f_hl]T_jT_r. Then _f^fijflhfhr

rh[fij,fhl]T_jT_r is divided by f_ij. Hence f_rh[f_ij, f_hl] | f_lhf_hr. But as we have [[f_ij, f_hl], f_lh] = 1, f_rh | f_lh and [f_ij, f_hl] | f_hr. By the structure of f1, . . . , ft, if [fij, fhl]|fhr withr < h then [fij, fhl] = 1.

Hence in any case we have [f_ij, f_hl] = 1 in the hypothesis i < j, h < l, i6=h,j 6=l withi, j, h, l∈ {1, . . . , t}. It follows q=sn−1 by Lemma 1.1.

Conversely, let q =sn−1. Iq,s is generated by nmonomials f1, . . . , fn: f1 =X₁^sX₂^s· · ·X_n−1^s X_n^s−1,f2 =X₁^sX₂^s· · ·X_n−1^s−1X_n^s, f3 =X₁^sX₂^s· · ·X_n−2^s−1X_n−1^s X_n^s, . . ., fn−1 = X₁^sX₂^s−1· · ·X_n−2^s X_n−1^s X_n^s, fn = X₁^s−1X₂^s· · ·X_n−2^s X_n−1^s X_n^s. We compute f12 = Xn−1, f13 = Xn−2, . . ., f1n = X1; f23 = Xn−2, . . ., f_2n=X₁, and so on. It follows thatf_ij 6=f_hlbecausej 6=l. Then [f_ij, f_hl] = 1 fori < j,h < l,i6=h,j6=lwithi, j, h, l∈ {1, . . . , n}. By [6, Proposition 1.7], f₁, . . . , f_n is ans-sequence.

Example 1.3. R = K[X₁, X₂, X₃], I_7,3 = (X₁³X₂³X₃, X₁³X₂²X₃², X₁²X₂³ X₃², X₁³X₂X₃³, X₁²X₂²X₃³, X₁X₂³X₃³). Set f₁ = X₁³X₂³X₃, f₂ = X₁³X₂²X₃², f₃ = X₁²X₂³X₃², f4 = X₁³X2X₃³, f5 = X₁²X₂²X₃³, f6 = X1X₂³X₃³, where f1 ≺ · · · ≺ f₆ with respect to the Lex order and X₁ ≺ X₂ ≺ X₃. Let G = {g_ij = fijTj −fjiTi | 1 ≤ i < j ≤ 6}; f1, . . . , f6 is not an s-sequence because J does not admit a linear Gr¨obner basis for any term order in R[T1, . . . , T6] with X_i ≺ T_j for all i, j and T₁ ≺ · · · ≺ T₆. In fact there are S-pairs S(gij, g_hl) that have not a standard expression with respect G with remain- der 0: S(g25, g36) = _[f^f25f63

25,f36]T3T5−_[f^f36f52

25,f36]T2T6 =X3T3T5−X3T2T6. There is no g_ij ∈Gwhose initial term divides a term ofS(g₂₅, g₃₆).

Remark 1.3. A particular case is s= 1: I_q,1 is the square-free Veronese ideal of degree q generated by all the square-free monomials in the variables X₁, . . . , X_nof degreeq.I_q,1is generated by ans-sequence if and only ifq=n−1 as proved in [11].

Now we solve the problem to compute standard algebraic invariants of the symmetric algebra of Veronese-type ideals generated by s-sequences.

Proposition1.1. LetR=K[X₁, . . . , X_n]be the polynomial ring over a field K andIsn−1,s = (f1, . . . , fn). Then the annihilator ideals off1, . . . , fnare

I₁ = (0), I_i = (Xn−i+1) for i= 2, . . . , n.

Proof. Let Isn−1,s = (f₁, . . . , f_n) with f₁ ≺ . . . ≺ f_n: f₁ = X₁^sX₂^s· · · X_n−1^s X_n^s−1, f₂ =X₁^sX₂^s· · ·X_n−1^s−1X_n^s, f₃ =X₁^sX₂^s· · ·X_n−2^s−1X_n−1^s X_n^s, . . . , fn−1 = X₁^sX₂^s−1· · ·X_n−2^s X_n−1^s X_n^s,f_n=X₁^s−1X₂^s· · ·X_n−2^s X_n−1^s X_n^s. Hence we observe thatIsn−1,sis generated bymh1, . . . , mhn, wherem=X₁^s−1X₂^s−1· · ·X_n−1^s−1X_n^s−1 and h_i = X₁· · ·X\n+1−i· · ·X_n for i = 1, . . . , n. Then In−1 = (h₁, . . . , h_n) is the square free Veronese ideal of degree n−1. Hence the annihilator ideals of the sequence f₁, . . . , f_n are the same of the sequence h₁, . . . , h_n [11, Proposi- tion 2.1].

Theorem 1.2. Let R = K[X1, . . . , Xn] be the polynomial ring over a field K andIns−1,s. Then

1) dim(Sym_R(Ins−1,s)) =n+ 1;

2) e(Sym_R(Ins−1,s)) =Pn−2 j=1

n−1 j

+ 2;

3) reg(Sym_R(Ins−1,s))≤n−1.

Proof. 1) By Proposition 1.1 in≺(J) = ((Xn−1)T2,(Xn−2)T3, . . . ,(X1)Tn) and it is generated by a regular sequence. We obtain dim(Sym_R(Ins−1,s)) = n+n−(n−1) =n+ 1.

2) By [6, Proposition 2.4] e(Sym_R(Ins−1,s)) = P

1≤i₁<···<i_r≤ne(R/(I_i1, . . . ,I_ir)) with dim(R/(I_i1, . . . ,I_ir)) =d−r whered= dim(Sym_R(Ins−1,s)) = n + 1 and 1 ≤ r ≤ n. By Proposition 1.1, the annihilator ideals of the generators of Ins−1,s are the same of In−1. Then by [11, Theorem 2.1], we obtain e(Sym_R(Ins−1,s)) =Pn−2

j=1 n−1

j

+ 2.

3) reg(Sym_R(Ins−1,s)) = reg(R[T1, . . . , Tn]/J) ≤ reg(R[T1, . . . , Tn]/

in≺(J)), where in≺(J) = ((Xn−1)T₂,(Xn−2)T₃, . . . ,(X₁)T_n). By Proposition 1.1 in≺(J) is generated by a regular sequence of elements of degree 2, then R[T₁, . . . , T_n]/in≺(J) has a 2-linear resolution and projective dimensionn−1 equal to the number of the generators of in≺(J) ([5]): 0 → S^bn−1(−2(n− 1)) → · · · → S^b3(−6) → S^b2(−4) → S^b1(−2) → S → S/in≺(J) → 0, where S =R[T₁, . . . , T_n]. Then reg(R[T₁, . . . , T_n]/in≺(J)) =n−1.

2. IDEALS OF VERONESE BI-TYPE

In this section we consider the class of monomial ideals of Veronese bi- type in the polynomial ring R=K[X1, . . . , Xn;Y1, . . . , Ym].

Definition 2.1 ([10]). Let R = K[X₁, . . . , X_n;Y₁, . . . , Y_m] be the poly- nomial ring over a field K in two sets of variables. We define the ideals of Veronese bi-type of degreeq as the monomial ideals ofR

Lq,s = X

r+k=q

Ir,sJk,s, r, k≥1,

whereI_r,s is the ideal of Veronese-type of degreer in the variablesX₁, . . . , X_n and Jk,s is the ideal of Veronese-type of degreekin the variables Y1, . . . , Ym.

Example 2.1. LetR=K[X1, X2;Y1, Y2] be a polynomial ring. 1)L2,2= I_1,2J_1,2 = I₁J₁ = (X₁Y₁, X₁Y₂, X₂Y₁, X₂Y₂); 2) L_4,2 = I_3,2J_1,2 +I_1,2J_3,2 + I_2,2J_2,2 = I_3,2J₁ +I₁J_3,2+I₂J₂ = (X₁²X₂Y₁, X₁²X₂Y₂, X₁X₂²Y₁, X₁X₂²Y₂, X₁ Y₁²Y₂, X₂Y₁²Y₂, X₁Y₁Y₂², X₂Y₁Y₂², X₁²Y₁², X₁²Y₁Y₂, X₁²Y₂², X₂²Y₁², X₂²Y₂², X₂²Y₁Y₂, X1X2Y₁², X1X2Y₂², X1X2Y1Y2).

Remark 2.1. Fors= 1 andq= 2,3 we haveL_q,1 =P

r+k=qI_r,1J_k,1,r, k≥ 1, a square-free monomial ideal, more precisely a mixed product ideal ([14]).

Now our aim is to investigate in which cases L_q,s is generated by an s-sequence. In the sequel we will suppose L = (f1, f2, . . . , ft) where f1 ≺

f2 ≺ · · · ≺ ft with respect to the monomial order ≺_Lex on the variables X₁, . . . , X_n;Y₁, . . . , Y_m andX₁ ≺X₂≺ · · · ≺X_n≺Y₁≺Y₂ ≺ · · · ≺Y_m.

Lemma 2.1. Let R = K[X1, . . . , Xn;Y1, . . . , Ym] be the polynomial ring over a field K andL_q,s ⊂R. If[f_ij, f_hl] = 1for i < j,h < l,i6=h,j 6=l with i, j, h, l∈ {1, . . . , t}, then q =s(n+m)−1.

Proof. Let L_q,s = (f₁, . . . , f_t). Set f_ij = _[f^fi

i,fj] forf_i, f_j with i < j. We have f_ij = X_i^ai1

1 · · ·X_i^aiw

w Y_i^bi1

1 · · ·Y_i^biz

z . In the same way for f_h, f_l with h < l, i 6= h, j 6= l we have fhl = X_i^c₁ⁱ¹· · ·X_i^c_x^ixY_i^d₁ⁱ¹· · ·Y_i^d_y^iy. By the hypothesis [f_ij, f_hl] = 1 for i < j, h < l, i6=h,j 6=l. Then it follows that X_ij 6=X_hp for all j= 1, . . . , w,p = 1, . . . , x, and Yij 6=Y_hp for allj = 1, . . . , z,p = 1, . . . , y.

This means that there are no other generators f_h, f_l of L_q,s such that f_hl contains one of the variables of fij. It follows that if a variable of fij is in degree N in the monomial f_h, with h 6=i, j, then such variable in degree N belongs to any other generators fl for alll > h and l6=j. In the same way of the Lemma 1.1 we deduce the structure of the monomials that generate Lq,s:

f1 =X₁^sX₂^s· · ·X_n−2^s X_n−1^s X_n^sY₁^sY₂^s· · ·Y_m−1^s Y_m^s−1, f₂ =X₁^sX₂^s· · ·X_n−2^s X_n−1^s X_n^sY₁^sY₂^s· · ·Y_m−1^s−1Y_m^s, f3 =X₁^sX₂^s· · ·X_n−2^s X_n−1^s X_n^sY₁^sY₂^s· · ·Y_m−2^s−1Y_m−1^s Y_m^s,

· · ·

fn+m−1 =X₁^sX₂^s−1· · ·X_n−2^s X_n−1^s X_n^sY₁^sY₂^s· · ·Y_m−1^s Y_m^s, f_n+m =X₁^s−1X₂^s· · ·X_n−2^s X_n−1^s X_n^sY₁^sY₂^s· · ·Y_m−1^s Y_m^s.

Theorem2.1. LetR=K[X1, . . . , Xn;Y1, . . . , Ym]be the polynomial ring over a fieldK. L_q,s is generated by ans-sequence if and only ifq=s(n+m)−1.

Proof. Let Lq,s = (f1, f2, . . . , ft) and suppose that f1, f2, . . . , ft is an s-sequence. We prove that [f_ij, f_hl] = 1 for i < j, h < l, i 6= h, j 6= l with i, j, h, l∈ {1, . . . , t}.

The s-sequence property implies that G={g_ij =fijTj−fjiTi |1≤i <

j ≤ t} is a Gr¨obner basis for J. This means that S(g_ij, g_hl) has a standard expression with respect to Gwith remainder 0. We have

S(g_ij, g_hl) = fijf_lh

[f_ij, f_hl]T_jT_h− f_hlfji

[f_ij, f_hl]T_iT_l.

We suppose that i < j, h < l, i 6= h, j 6= l. As S(gij, ghl) has a stan- dard expression with respect to G, there existsg_rp such that in≺(g_rp) divides in≺(S(gij, ghl)).

I) Ifl > j then in≺(grp)| _[f^fhlfji

ij,fhl]. The first case isfhl| _[f^fhlfji

ij,f_hl], then [fij, fhl]|fji. But, as we have [[fij, fhl], fji] = 1, it follows [fij, fhl] = 1.

The second case is f_rl | _[f^fhlfji

ij,fhl], where f_rl = in≺(g_rl) with r < j and r < h. We can write

S(g_ij, g_hl) =− f_jif_hl

f_rl[f_ij, f_hl]g_rlT_i+ f_ijf_lh

[f_ij, f_hl]T_jT_h− f_jif_hlf_lr f_rl[f_ij, f_hl]T_iT_r. Then _[f^fijflh

ij,fhl]T_jT_h is divided by f_ij. Hence [f_ij, f_hl] | f_lh. But, as we have [[f_ij, f_hl], f_lh] = 1, it follows [f_ij, f_hl] = 1.

II) Ifl < j then in≺(g_rp)| _[f^fijflh

ij,f_hl]. The first case isf_ij | _[f^fijflh

ij,fhl], then [f_ij, f_hl]|f_lh. But, as we have [[f_ij, f_hl], f_lh] = 1, it follows [f_ij, f_hl] = 1.

The second case isfrh | _[f^fijflh

ij,fhl], wherefrh = in≺(grh). We can write S(gij, ghl) = f_ijf_lh

frh[fij, fhl]Tjgrh− f_hlf_ji

[fij, fhl]TiTl+ f_ijf_lhf_hr frh[fij, fhl]TjTr. Then _f^fijflhfhr

rh[fij,fhl]T_jT_r is divided by f_ij. Hence f_rh[f_ij, f_hl] | f_lhf_hr. But as we have [[fij, fhl], flh] = 1, frh | flh and [fij, fhl] | fhr. By the structure of f1, . . . , ft , if [fij, f_hl] | f_hr with r < h, then [fij, f_hl] = 1. Hence in any case we have [f_ij, f_hl] = 1 in the hypothesisi < j, h < l, i 6=h, j 6= l with i, j, h, l∈ {1, . . . , t}. It followsq =s(n+m)−1 by Lemma 2.1.

Conversely, letq =s(n+m)−1. The generators ofL_q,s are f₁ =X₁^sX₂^s· · ·X_n−2^s X_n−1^s X_n^sY₁^sY₂^s· · ·Y_m−1^s Y_m^s−1, f2 =X₁^sX₂^s· · ·X_n−2^s X_n−1^s X_n^sY₁^sY₂^s· · ·Y_m−1^s−1Y_m^s, f₃ =X₁^sX₂^s· · ·X_n−2^s X_n−1^s X_n^sY₁^sY₂^s· · ·Y_m−2^s−1Y_m−1^s Y_m^s,

· · ·

fn+m−1 =X₁^sX₂^s−1· · ·X_n−2^s X_n−1^s X_n^sY₁^sY₂^s· · ·Y_m−1^s Y_m^s, fn+m =X₁^s−1X₂^s· · ·X_n−2^s X_n−1^s X_n^sY₁^sY₂^s· · ·Y_m−1^s Y_m^s.

We compute f₁₂=Ym−1, f₁₃=Ym−2,. . ., f_1m =Y₁,f_1,m+1 =X_n,f_1,m+2 = Xn−1, . . ., f1,n+m =X1, f23 = Ym−2, . . ., f2m = Y1, f2,m+1 = Xn, f2,m+2 = Xn−1, . . ., f_2,n+m = X₁ and so on. Hence f_ij 6= f_hl because j 6= l. Then [fij, fhl] = 1 for i < j, h < l, i6=h, j 6=l with i, j, h, l∈ {1, . . . , n+m}. By [6, Proposition 1.7] it follows that f₁, . . . , f_n+m is ans-sequence.

Example2.2. R=K[X1, X2;Y1, Y2],L3,2= (X₁²Y1, X1X2Y1, X₂²Y1, X1Y₁², X2Y₁², X₁²Y2, X1X2Y2, X₂²Y2, X1Y1Y2,,X2Y1Y2, X1Y₂², X2Y₂²). Let G={g_ij =

fijTj −fjiTi | 1 ≤ i < j ≤ 12}; f1, . . . , f12 is not an s-sequence because J does not admit a linear Gr¨obner basis for any term order in R[T₁, . . . , T₁₂] with Xi ≺ Tj, Yi ≺ Tj for all i, j and T1 ≺ · · · ≺ T12. In fact, there are S-pairs S(gij, g_hl) that have not a standard expression with respect G with remainder 0: S(g16, g27) = _[f^f16f72

16,f27]T2T6 − _[f^f61f27

16,f27]T1T7 = Y2T2T6 −Y2T1T7. There is no gij ∈Gwhose initial term divides a term ofS(g16, g27).

As in Section 1 we use the theory of s-sequences to compute standard invariants of the symmetric algebra of the monomials idealsLq,s generated by an s-sequence.

Proposition 2.1. Let R = K[X1, . . . , Xn;Y1, . . . , Ym] be a polynomial ring over a field K in two sets of variables andL_q,s forq =s(n+m)−1. Then the annihilator ideals of the generators of Lq,s areI₁ = (0),I_i = (Ym−i+1) for i= 2, . . . , m, I_i= (Xn+m−i+1) for i=m+ 1, . . . , m+n.

Proof. Letq =s(n+m)−1. Lq,s = (f1, . . . , fn+m) withf1≺ · · · ≺fn+m

and

f₁ =X₁^sX₂^s· · ·X_n−2^s X_n−1^s X_n^sY₁^sY₂^s· · ·Y_m−1^s Y_m^s−1, f2 =X₁^sX₂^s· · ·X_n−2^s X_n−1^s X_n^sY₁^sY₂^s· · ·Y_m−1^s−1Y_m^s, f₃ =X₁^sX₂^s· · ·X_n−2^s X_n−1^s X_n^sY₁^sY₂^s· · ·Y_m−2^s−1Y_m−1^s Y_m^s,

· · ·

fn+m−1 =X₁^sX₂^s−1· · ·X_n−2^s X_n−1^s X_n^sY₁^sY₂^s· · ·Y_m−1^s Y_m^s, f_n+m =X₁^s−1X₂^s· · ·X_n−2^s X_n−1^s X_n^sY₁^sY₂^s· · ·Y_m−1^s Y_m^s.

The annihilator ideals of the sequencef1, . . . , fn+mareI_i = (f1i, f2i, . . . , fi−1,i) fori= 1, . . . , n+m. Fori= 1 we haveI₁ = (0). By the structure of the mono- mials f1, . . . , fn+m we have I₂ = (f12) = (Ym−1), I₃ = (f13, f23) = (Ym−2), . . ., I_m = (f_1m, . . . , fm−1,m) = (Y₁), I_m+1 = (f_1,m+1, . . . , f_m,m+1) = (X_n), . . . , I_n+m= (f1,n+m, . . . , fn+m−1,n+m) = (X1).

Theorem2.2. LetR=K[X₁, . . . , X_n;Y₁, . . . , Y_m]be the polynomial ring over a field K in two sets of variables and Lq,s for q=s(n+m)−1. Then

1) dim(Sym_R(L_q,s)) =n+m+ 1;

2) e(Sym_R(Lq,s)) =Pn+m−2 j=1

n+m−1 j

+ 2;

3) reg(Sym_R(Lq,s))≤n+m−1.

Proof. 1) By Proposition 2.1, we have I₁ = (0), I_i = (Ym−i+1) for i = 2, . . . , m, I_i = (Xn+m−i+1) for i = m+ 1, . . . , m+n. Then in≺(J) is generated by a regular sequence. We obtain dim(Sym_R(L_q,s)) =n+m+n+ m−(n+m−1) =n+m+ 1.