On the Symmetric Algebra of Graded Modules and Torsion Freeness

ON THE SYMMETRIC ALGEBRA OF GRADED MODULES AND TORSION FREENESS

MONICA LA BARBIERA

Communicated by Constantin N˘ast˘asescu

Let R be a commutative noetherian graded ring. We study the torsion freeness of the symmetric algebra of a ﬁnitely generated graded R-module. AMS 2010 Subject Classiﬁcation: 13A02, 13C15, 13A30, 13P10.

∗ Key words: graded rings, ( Gq)-condition, torsion freeness.

INTRODUCTION

Let R be a commutative noetherian ring with unit, E be a finitely gen- L∞ erated R-module and SymmR(E) = t=0 Symmt(E) be its symmetric algebra. It is known that the symmetric algebra SymmR(E) has a presentation R[Y1,...,Ym]/J, where m is the number of generators of E and J is the relation Pm ideal generated by the linear forms ai = j=1 ajiYj, 1 ≤ i ≤ n in the variables Y1,...,Ym. Theoretical properties of SymmR(E) (such as integrality, regu- larity, being Cohen-Macaulay) were studied by various authors. For a finitely generated module E, this topic developed and culminated in the study of the approximation complex of E. For best results, the projective resolution of E may be examined, when it is finite. In [4] the Ischebeck-Auslander’s condition was considered to study the q-torsion freeness of finitely generated R-modules. A noetherian ring that satisfies the Ischebeck-Auslander’s condition is said to be a Gq-ring. In [8] it is proved that if a finitely generated R-module E satisfies the Samuel’s condition, then E is q-torsion free, when R is a Gq-ring. In [2] the q-torsion freeness of the symmetric powers of a finitely generated R-module is studied in terms of a condition on Fitting ideals of E, for a module of projective dimension ≤ 1. The aim of this note is to investigate such properties in the graded case. If R is a graded ring, similar properties can be established and studied. The definitions and the results in this paper are inspired by known results in the non-graded case. The graded versions are important as well, since many good properties can be established for graded algebras, for example in the local case [3, 5, 6].

MATH. REPORTS 16(66), 4 (2014), 503–511 504 Monica La Barbiera 2

The paper is organized as follows. In Section 1, we present briefly some notions and properties concerning graded rings which will be used throughout the paper. In Section 2, for an integer q > 0, we consider the Ischebeck- ∗ ∗ Auslander’s ( Gq) condition, because of its link to the q-torsion freeness of graded modules (see [5]). We study the ∗q-torsion freeness of the symmetric ∗ algebra of a finitely generated graded module E over a Gq-ring and we state the graded versions of some classical results about finitely generated modules. In Section 3, we investigate the torsion freeness of the symmetric algebra of a finitely generated graded module E, because this property is linked to the integrality of the symmetric algebra (see [10]). We consider the symmetric algebra of a class of monomial modules that are ideals, namely the ideals of Veronese-type Iq,s (see [12]). Finally, we state a necessary and sufficient condition for SymmR(Iq,s) to be torsion free or, equivalently, for SymmR(Iq,s) to be a domain.

1. PRELIMINARIES AND NOTATIONS

Let R be a commutative noetherian graded ring. In [3] there are some deﬁnitions related to graded ideals of R. Deﬁnition 1.1. Let I ⊂ R be an ideal. I∗ is the graded ideal of R generated by all the homogeneous elements of I (the homogeneous elements of I of degree j are f ∈ Ij = I ∩ Rj). The ideal I∗ is the largest graded ideal contained in I. If I is a graded ideal, then I = I∗. 2 Example 1.1. Let I = (X1 − 2,X1X2 − 2) ⊂ R = K[X1,X2]. We have:

I0 = I ∩ R0 = (0)

I1 = I ∩ R1 = (0) 2 I2 = I ∩ R2 = hX1 − X1X2i, 2 2 in fact f = X1 − 2 − (X1X2 − 2) = X1 − X1X2 ∈ I2. In general, for i > 2: 2 Ii = I ∩ Ri = hg(X1 − X1X2)i, ∗ 2 with deg(g) = i − 2. It follows that I = (X1 − X1X2) ⊂ I. Remark 1.1. If ℘ ⊂ R is a prime ideal, then ℘∗ is also a prime ideal ([3], 1.5.6). The following deﬁnitions are the graded versions of the classical deﬁnitions of the conditions (Sq) and (Gq) [9, 11]. 3 On the symmetric algebra of graded modules and torsion freeness 505

Definition 1.2. Let R be a commutative noetherian graded ring, and ∗ q ≥ 0 be an integer. R satisfies Serre’s condition ( Sq) if for all prime ideals ℘ of R depthR℘∗ ≥ min{q, dimR℘∗ }. Definition 1.3. Let R be a commutative noetherian graded ring and q > 0 ∗ be an integer. R satisfies the ( Gq) condition of Ischebeck-Auslander (or R is ∗ a Gq-ring) if: ∗ 1) R satisfies the condition ( Sq); 2) for all prime ideal ℘ of R such that dimR℘∗ < q, R℘∗ is a Gorenstein ring.

Example 1.2. Let R = K[X1,...,Xn] be the polynomial ring over a ﬁeld ∗ K. R is a graded ring with standard gradation. R is a Gq-ring for all q > 0 because it is a regular ring.

2. ∗q-TORSION FREENESS OF GRADED MODULES

∗ ∗ The Ischebeck-Auslander’s condition ( Gq) is linked to the q-torsion freeness of graded R-modules. In the non-graded case the q-torsion freeness was studied in [11]. We give the following deﬁnitions for graded rings.

Definition 2.1. A sequence x1, . . . , xn of homogeneous elements of R is called ∗regular sequence on R (or a homogeneous R-sequence) if the following conditions are satisfied: 1) (x1, . . . , xn) is a proper graded ideal; 2) xi+1 is a regular homogeneous element in R/(x1, . . . , xi) for i = 0,..., n − 1. Definition 2.2. Let R be a graded noetherian ring, and E be a finitely generated graded R-module. A sequence x = {x1, . . . , xn} of homogeneous elements of R is called E-∗regular sequence (or a homogeneous E-sequence) if the following conditions are satisfied: 1) xE 6= E; 2) xi is a nonzero-divisor in E/(x1, . . . , xi−1)E, for i = 1, . . . , n. Definition 2.3. Let R be a graded ring, E be a finitely generated graded R-module and q > 0 be an integer. E is ∗q-torsion free if each homogeneous R-sequence of length q is a homogeneous E-sequence. Definition 2.4. Let R be a commutative noetherian graded ring, q > 0 be an integer and E be a finitely generated graded R-module. E satisfies Samuel’s ∗ ∗ condition ( aq) (or E is an aq-module) if each homogeneous R-sequence of 506 Monica La Barbiera 4 length less or equal than q made of non invertible elements is a homogeneous E-sequence. Definition 2.5. Let R be a commutative noetherian graded ring and q > 0 be an integer. Let E be a finitely generated graded R-module. E is a ∗q-th module of syzygies if there exists an exact sequence:

0→E→P1 → · · · → Pq, where each Pi is a graded projective R-module. ∗ Theorem 2.1 ([6]). Let R be a Gq-ring, E be a graded finitely generated R-module and q > 0 be an integer. The following conditions are equivalent: ∗ 1) E satisfies Samuel’s condition ( aq); 2) E is a ∗q-th module of syzygies; 3) E is ∗q-torsion free. An application of the previous theorem is the following result. ∗ Proposition 2.1. Let R be a Gq-ring. If 0→E0→E → E00 → 0 is an exact sequence of finitely generated graded R-modules with E0 and E00 ∗q-torsion free, then E is ∗q-torsion free. Proof. We use induction on q. Let q = 1. Since E0 and E00 are ∗q-torsion ∗ free, they satisfy the Samuel condition ( a1) by Theorem 2.1. It is enough to ∗ prove that E satisfies ( a1). Let f g 0 → E0 → E → E00 → 0 be the exact sequence. Let a be a homogeneous R-sequence and x ∈ E such that ax = 0. Then g(ax) = 0 implies g(x) = 0, and x ∈ Imf =∼ E0 implies x = 0. Hence, a is a homogeneous E-sequence. Suppose q > 2 and the property is true for q − 1. We show that E verifies ∗ ( aq). Let (a1, . . . , aq) be a homogeneous R-sequence. From the case q = 1 and the Snake Diagram, it follows that the sequence 0 0 00 00 0 → E /a1E →E/a1E→E /a1E → 0 0 0 00 00 ∗ is exact. E /a1E and E /a1E are graded R/a1R-modules that satisfy ( aq−1). ∗ R/a1R is a Gq−1-ring ([6], Proposition 3.7). (a2, . . . , aq) is a homogeneous R/a1R-sequence and by the induction hypothesis it is a homogeneous E/a1E- ∗ sequence. Hence, E satisfies ( aq) and by Theorem 2.1 the conclusion follows. 5 On the symmetric algebra of graded modules and torsion freeness 507

Now we apply these results to the symmetric algebra of a graded module. Let E = Rf1 +···+Rfm be a ﬁnitely generated graded R-module, dj = deg(fj) L for j = 1, . . . , m, and SymmR(E) = t≥0 Symmt(E) be its symmetric algebra. If E has a free graded presentation: n m M ϕ M ψ R(−ci) → R(−dj) → E → 0, i=1 j=1 with ϕ = (aji), then the kernel J of the surjective homomorphism

Symm(ψ): R[Y1,...,Ym] → SymmR(E) → 0 induced by ψ is generated by linear forms in the variables Yj m X ai = ajiYj, i = 1, . . . , n, j=1 Pm such that j=1 ajifj = 0, 1 ≤ i ≤ n. Hence, the symmetric algebra SymmR(E) has a presentation R[Y1,...,Ym]/J. ∗ Our aim is to study the q-torsion freeness of SymmR(E) as a graded R-module. L ∗ Definition 2.6. SymmR(E) = t≥0 Symt(E) is said to be q-torsion free ∗ if each symmetric power Symmt(E) is q-torsion free. ∗ Theorem 2.2. Let R be a Gq-ring, and E be a finitely generated graded R-module. The following conditions are equivalent: ∗ 1) SymmR(E) satisfies Samuel’s condition ( aq); ∗ 2) SymmR(E) is a q-th module of syzygies; ∗ 3) SymmR(E) is q-torsion free. ∗ ∗ ∗ Proof. 1) ⇒ 2) Each aq-module on a Gq-ring is a q-th module of syzygies ([4], 4.6). ∗ ∗ 2) ⇒ 3) R is a Gq-ring if and only if each (q + 1)-th module of syzy- ∗ gies of SymmR(E)(Syzq+1(SymmR(E))) is (q + 1)-torsion free ([11], 4.3). If ∗ ∗ Syzq+1(SymmR(E)) is (q+1)-torsion free, then Syzj(SymmR(E)) is j-torsion ∗ free for all j = 1, . . . , q + 1 ([11], 4.2). It follows that Syzq(SymmR(E)) is q- ∗ torsion free. By hypothesis SymmR(E) is a q-th module of syzygies, hence ∗ SymmR(E) is q-torsion free. For a generic graded ring R we prove that 3) ⇒ 2) ⇒ 1). 3) ⇒ 2) The two conditions are equivalent for graded modules of finite projective dimension ([1], 4.25). ∗ ∗ 2) ⇒ 1) Each q-th graded module of syzygies is an aq-module ([4], 4.4). 508 Monica La Barbiera 6

3. THE SYMMETRIC ALGEBRA OF IDEALS OF VERONESE-TYPE AND TORSION FREENESS

This section is dedicated to the symmetric algebra of a class of monomial modules over the ring R = K[X1,...,Xn] that are ideals, more precisely the ideals of Veronese type [7, 12]. We recall the following deﬁnition.

Deﬁnition 3.1. Let R = K[X1,...,Xn] be the polynomial ring over a ﬁeld K. The ideal of Veronese-type of degree q is the monomial ideal Iq,s generated by the set: n ai1 ain X X ··· Xn ai = q, 0 ≤ ai ≤ s, s ∈ {1, . . . , q} 1 j j j=1

Remark 3.1. In general Iq,s ⊆ Iq, where Iq is the Veronese ideal of degree q of R which is generated by all the monomials in the variables X1,...,Xn of q degree q: Iq = (X1,...,Xn) [13]. If q = 1, 2 or s = q, then Iq,s = Iq.

Example 3.1. If R = K[X1,X2,X3], then: 2 2 2 2 2 2 I3,2 = (X1 X2,X1 X3,X1X2 ,X2 X3,X1X3 ,X2X3 ,X1X2X3) ⊂ I3 3 3 3 2 2 2 2 2 2 I3,3 = (X1 ,X2 ,X3 ,X1 X2,X1 X3,X1X2 ,X2 X3,X1X3 ,X2X3 ,X1X2X3) = I3. We recall from [10] two important results about the integrality of the symmetric algebra of a ﬁnitely generated module. Theorem 3.1. Let A be a domain, E be a ﬁnitely generated A-module. Then SymmA(E) is a domain if and only if Symmt(E) is torsion free, for all t ≥ 0. In particular, for an ideal I of a domain A: Theorem 3.2. Let A be a domain and I be an ideal of A. The following conditions are equivalent: 1) SymmA(I) is a domain; 2) SymmA(I) is torsion free; ∼ 3) I is of linear type, i.e. SymmA(I) = <(I) where <(I) is the Rees algebra of I.

In order to study the torsion freeness of SymmR(Iq,s), we use the previous results stating the condition for Iq,s to be of linear type. fi If Iq,s = (f1, . . . , ft) ⊂ R, then for all 1 i < j t we set fij = 6 6 GCD(fi,fj ) and gij = fijTj −fjiTi. Hence, the relation ideal J of SymmR(Iq,s) is generated by {gij}16i

Iq,st, where t is an indeterminate on R. Let us consider the epimorphism of graded R-algebras:

ϕ : R[T1,...,Tt] → <(Iq,s) = R[f1t, . . . , ftt] deﬁned by ϕ(Ti) = fit, i = 1, . . . , t. The ideal N = kerϕ of R[T1,...,Tt] is the ideal of presentation of <(Iq,s). Our aim is to investigate in which cases, for the ideal Iq,s, the linear relations gij form a system of generators for the ideal N.

Theorem 3.3. Let R = K[X1,...,Xn] be the polynomial ring over a ﬁeld K with n > 2. Iq,s is of linear type if and only if q = sn − 1.

Proof. ⇒ Let Iq,s = (f1, f2, . . . , ft), where f1 ≺ f2 ≺ · · · ≺ ft with respect to the monomial order ≺Lex on the variables of R. We assume that Iq,s is of linear type, i.e. the ideal of presentation N of <(Iq,s) is generated by linear relations: N = (gij = fijTj − fjiTi|1 ≤ i < j ≤ t). This means that all the relations among the generators of Iq,s are linear relations (in the variables Ti). By the construction of the set of monomial generators of Iq,s this fact happens when f1j = f2j = ... = fn−1,j = Xn−j+1 for j = 2, . . . , n. Hence, we can deduce the minimal set of generators of Iq,s that satisfies the hypothesis: s s s s s−1 f1 = X1X2 ··· Xn−2Xn−1Xn , s s s s−1 s f2 = X1X2 ··· Xn−2Xn−1Xn, s s s−1 s s f3 = X1 ··· Xn−3Xn−2Xn−1Xn, ...... s s−1 s s s s fn−1 = X1X2 X3 ··· Xn−2Xn−1Xn, s−1 s s s fn = X1 X2 ··· Xn−1Xn. Then q = sn − 1. s s s−1 ⇐ Let q = sn−1, then Iq,s = (f1, f2, . . . , ft) where f1 = X1 ··· Xn−1Xn , s s s−1 s s s−1 s s s−1 s s f2 = X1 ··· Xn−2Xn−1Xn,..., fn−1 = X1X2 X3 ··· Xn, fn = X1 X2 ··· Xn. We prove that the linear relations gij = fijTj − fjiTi, 1 ≤ i < j ≤ n, form a Gröbnerbasis of N with respect to a monomial order ≺ on the polynomial ring R[T1,...,Tn]. Denote F = (fijTj : 1 ≤ i < j ≤ n). To show that gij form a Gröbnerbasis of N, we suppose that the claim is false. Since the binomial relations are known to be a Gröbnerbasis of N, there exists a binomial aT α −bT β, a1 an b1 bn α α1 αn β β1 βn where a = X1 ··· Xn , b = X1 ··· Xn , T = T1 ··· Tn , T = T1 ··· Tn , and the initial monomial of aT α − bT β is not in F . More precisely, we assume that T α,T β have no common factors and that both aT α and bT β are not in F . α β Let i be the smallest index such that Ti appears in T or in T . Since α β β aT − bT ∈ N, then fi divides bϕ(T ), where ϕ(Ti) = fit. If fi|b, then let β β Tj be any of the variables of T . One has fijTj|fiTj|bT for i < j. This β contradicts our assumption (because bT ∈/ F ). Hence, fi - b. Let ik be the 510 Monica La Barbiera 8

aik minimum of the indices such that Xi does not divide b, aik ∈ {s, s − 1}. a k Since X ik divides bϕ(T β) (because f |bϕ(T β)), then there exists j such that ik i β Tj appears in T and Xik |fj.

By the structure of the generators f1, . . . , fn of Iq,s if Xik |fi and Xik |fj a ai with j such that T is in T β then f |X i1 ··· X k−1 , a , . . . , a ∈ {s, s − 1} j ij i1 ik−1 i1 ik−1 (in fact if a variable of fij is of degree D in the monomial fh, with h 6= i, j, then a such variable of degree N belongs to any other generators fl for all l > h and β l 6= j). Hence, fij|b and, as a consequence, fijTj|bT , which is a contradiction β (because bT ∈/ F ). It follows that N = (gij : 1 ≤ i < j ≤ n) = J. Hence, Isn−1,s is of linear type.

Example 3.2. If R = K[X1,X2,X3], then 4 4 3 4 3 4 3 4 4 I11,4 = (X1 X2 X3 ,X1 X2 X3 ,X1 X2 X3 ) 4 4 3 4 3 4 3 4 4 <(I11,4) = R[I11,4t] = R[X1 X2 X3 t, X1 X2 X3 t, X1 X2 X3 t] 4 4 3 4 3 4 3 4 4 ϕ : R[T1,T2,T3] → R[X1 X2 X3 t, X1 X2 X3 t, X1 X2 X3 t] 4 4 3 T1 → X1 X2 X3 t 4 3 4 T2 → X1 X2 X3 t 3 4 4 T3 → X1 X2 X3 t

Kerϕ = (X2T2 − X1T3,X3T1 − X1T3) = J. I11,4 is of linear type. By Theorems 3.1 and 3.2 we can state:

Corollary 3.1. SymmR(Isn−1,s) is a domain.

Corollary 3.2. SymmR(Isn−1,s) is torsion free.

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Received 26 August 2012 University of Messina, Department of Mathematics, Viale Ferdinando Stagno d’Alcontres, 31, 98166 Messina, Italy [email protected]