We study the torsion freeness of the symmetric algebra of a finitely generated gradedR-module

AND TORSION FREENESS

MONICA LA BARBIERA

Communicated by Constantin N˘ast˘asescu

LetR be a commutative noetherian graded ring. We study the torsion freeness of the symmetric algebra of a finitely generated gradedR-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- erated R-module and Symm_R(E) = L∞

t=0Symm_t(E) be its symmetric alge- bra. It is known that the symmetric algebra SymmR(E) has a presentation R[Y₁, . . . , Y_m]/J, wheremis the number of generators ofEandJ is the relation ideal generated by the linear forms a_i =Pm

j=1a_jiY_j, 1≤i≤nin 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 theq-torsion freeness of finitely generatedR-modules.

A noetherian ring that satisfies the Ischebeck-Auslander’s condition is said to be aGq-ring. In [8] it is proved that if a finitely generatedR-moduleEsatisfies the Samuel’s condition, thenE isq-torsion free, whenRis aG_q-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 ofE, 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. REPORTS16(66),4(2014), 503–511

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 (^∗G_q) 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 forSymm_R(Iq,s) to be torsion free or, equivalently, forSymm_R(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 ofR gener- ated by all the homogeneous elements of I (the homogeneous elements ofI of degree j are f ∈I_j =I∩R_j).

The ideal I^∗ is the largest graded ideal contained in I. If I is a graded ideal, then I =I^∗.

Example 1.1.Let I = (X₁²−2, X1X2−2)⊂R =K[X1, X2]. We have:

I₀ =I∩R₀ = (0) I₁ =I∩R₁ = (0) I₂=I ∩R₂=hX₁²−X₁X₂i,

in fact f =X₁²−2−(X1X2−2) =X₁²−X1X2 ∈I2. In general, fori >2:

I_i =I∩R_i =hg(X₁²−X₁X₂)i, with deg(g) =i−2. It follows that I^∗ = (X₁²−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 (S_q) and (G_q) [9, 11].

Definition 1.2. Let R be a commutative noetherian graded ring, and q ≥0 be an integer. RsatisfiesSerre’s condition (^∗S_q)if for all prime ideals℘ of R

depthR℘^∗ ≥min{q,dimR℘^∗}.

Definition 1.3.LetRbe a commutative noetherian graded ring andq >0 be an integer. R satisfies the (^∗G_q) condition of Ischebeck-Auslander (or R is a ^∗G_q-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 ^∗G_q-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 free- ness 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)x_i+1 is a regular homogeneous element inR/(x₁, . . . , x_i) fori= 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, . . . , x_n} of homogeneous elements of R is calledE-^∗regular sequence (or ahomogeneous E-sequence) if the following conditions are satisfied:

1)xE 6=E;

2)xi is a nonzero-divisor in E/(x1, . . . , xi−1)E, fori= 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 homogeneousE-sequence.

Definition 2.4. LetR be a commutative noetherian graded ring,q >0 be an integer andE be a finitely generated gradedR-module. EsatisfiesSamuel’s condition (^∗a_q) (or E is an ^∗a_q-module) if each homogeneous R-sequence of

length less or equal than q made of non invertible elements is a homogeneous E-sequence.

Definition 2.5.LetRbe a commutative noetherian graded ring andq >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 eachP_i is a graded projective R-module.

Theorem2.1 ([6]). Let Rbe a^∗G_q-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 ^∗G_q-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. Letq= 1. SinceE⁰ andE⁰⁰ are^∗q-torsion free, they satisfy the Samuel condition (^∗a₁) by Theorem 2.1. It is enough to prove that E satisfies (^∗a₁). Let

0→E⁰ →^f E →^g 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 homogeneousE-sequence.

Supposeq >2 and the property is true forq−1. We show thatE verifies (^∗aq).

Let (a₁, . . . , a_q) be a homogeneousR-sequence. From the caseq = 1 and the Snake Diagram, it follows that the sequence

0→E⁰/a₁E⁰→E/a₁E→E⁰⁰/a₁E⁰⁰→0

is exact. E⁰/a₁E⁰andE⁰⁰/a₁E⁰⁰are gradedR/a₁R-modules that satisfy (^∗aq−1).

R/a₁R is a ^∗Gq−1-ring ([6], Proposition 3.7). (a₂, . . . , a_q) is a homogeneous R/a1R-sequence and by the induction hypothesis it is a homogeneous E/a1E- sequence. Hence, E satisfies (^∗a_q) and by Theorem 2.1 the conclusion fol- lows.

Now we apply these results to the symmetric algebra of a graded module.

LetE =Rf₁+· · ·+Rf_mbe a finitely generated gradedR-module,d_j = deg(f_j) forj= 1, . . . , m, and SymmR(E) =L

t≥0Symmt(E) be its symmetric algebra.

If E has a free graded presentation:

n

M

i=1

R(−c_i)→^ϕ

m

M

j=1

R(−d_j)→^ψ E→0,

withϕ= (a_ji), then the kernelJ of the surjective homomorphism Symm(ψ) :R[Y1, . . . , Ym]→SymmR(E)→0 induced by ψis generated by linear forms in the variables Yj

a_i =

m

X

j=1

a_jiY_j, i= 1, . . . , n,

such thatPm

j=1a_jif_j = 0, 1≤i≤n. Hence, the symmetric algebraSymm_R(E) has a presentationR[Y₁, . . . , Y_m]/J.

Our aim is to study the ^∗q-torsion freeness of SymmR(E) as a graded R-module.

Definition 2.6.Symm_R(E) =L

t≥0Sym_t(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) Symm_R(E) is a ^∗q-th module of syzygies;

3) Symm_R(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 ^∗G_q-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, thenSyzj(SymmR(E)) is^∗j-torsion free for all j = 1, . . . , q+ 1 ([11], 4.2). It follows that Syz_q(Symm_R(E)) is ^∗q- torsion free. By hypothesis Symm_R(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 ^∗a_q-module ([4], 4.4).

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.LetR=K[X₁, . . . , X_n] be the polynomial ring over a field K. Theideal of Veronese-type of degreeqis the monomial ideal Iq,sgenerated by the set:

X₁^ai1· · ·Xn^ain

n

X

j=1

a_ij =q, 0≤a_ij ≤s, s∈ {1, . . . , q}

Remark 3.1. In generalI_q,s⊆I_q, whereI_q is the Veronese ideal of degree q ofR which is generated by all the monomials in the variablesX₁, . . . , X_n of degree q: Iq= (X1, . . . , Xn)^q [13]. Ifq = 1,2 ors=q, thenIq,s=Iq.

Example 3.1.If R=K[X₁, X₂, X₃], then:

I3,2 = (X₁²X2, X₁²X3, X1X₂², X₂²X3, X1X₃², X2X₃², X1X2X3)⊂I3

I_3,3 = (X₁³, X₂³, X₃³, X₁²X₂, X₁²X₃, X₁X₂², X₂²X₃, X₁X₃², X₂X₃², 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 Symm_A(E) is a domain if and only if Symm_t(E) is torsion free, for all t≥0.

In particular, for an ideal I of a domainA:

Theorem 3.2. Let A be a domain and I be an ideal ofA. 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. Symm_A(I)∼=<(I) where<(I)is the Rees algebra of I.

In order to study the torsion freeness ofSymm_R(I_q,s), we use the previous results stating the condition for I_q,s to be of linear type.

IfIq,s= (f1, . . . , ft)⊂R, then for all 16i < j6twe setfij = _GCD(f^fi

i,fj)

andgij =fijTj−f_jiTi. Hence, the relation idealJofSymmR(Iq,s) is generated by{g_ij}_16i<j6t. The Rees algebra<(I_q,s) is defined to be theR-graded algebra L

i≥0(I_q,s)ⁱ. It can be identified with the R-subalgebra of R[t] generated by

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

ϕ:R[T1, . . . , Tt]→ <(I_q,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 <(I_q,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 idealN.

Theorem3.3. LetR =K[X₁, . . . , X_n]be the polynomial ring over a field K with n >2. I_q,s is of linear type if and only if q=sn−1.

Proof. ⇒LetIq,s= (f1, f2, . . . , ft), wheref1 ≺f2 ≺ · · · ≺ftwith respect to the monomial order ≺_Lex on the variables of R. We assume that I_q,s is of linear type, i.e. the ideal of presentation N of <(I_q,s) is generated by linear relations: N = (gij = fijTj −fjiTi|1 ≤ i < j ≤ t). This means that all the relations among the generators ofI_q,s are linear relations (in the variablesT_i).

By the construction of the set of monomial generators ofI_q,sthis 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,sthat satisfies the hypothesis:

f₁ =X₁^sX₂^s· · ·X_n−2^s X_n−1^s X_n^s−1, f2 =X₁^sX₂^s· · ·X_n−2^s X_n−1^s−1X_n^s, f3 =X₁^s· · ·X_n−3^s X_n−2^s−1X_n−1^s X_n^s, ...

fn−1=X₁^sX₂^s−1X₃^s· · ·X_n−2^s X_n−1^s X_n^s, fn=X₁^s−1X₂^s· · ·X_n−1^s X_n^s.

Then q =sn−1.

⇐Letq =sn−1, thenIq,s= (f1, f2, . . . , ft) wheref1=X₁^s· · ·X_n−1^s X_n^s−1, f2 =X₁^s· · ·X_n−2^s X_n−1^s−1X_n^s, . . . ,fn−1 =X₁^sX₂^s−1X₃^s· · ·X_n^s,fn=X₁^s−1X₂^s· · ·X_n^s. We prove that the linear relations g_ij =f_ijT_j −f_jiT_i, 1 ≤i < j ≤n, form a Gr¨obner basis ofN 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 Gr¨obner basis ofN, we suppose that the claim is false. Since the binomial rela- tions are known to be a Gr¨obner basis ofN, there exists a binomialaT^α−bT^β, where a=X₁^a1· · ·X_n^an, b =X₁^b1· · ·X_n^bn, T^α =T₁^α1· · ·T_n^αn, T^β =T₁^β1· · ·Tn^βn, and the initial monomial ofaT^α−bT^β is not inF. More precisely, we assume thatT^α, T^β have no common factors and that bothaT^α andbT^β are not inF. Let ibe the smallest index such that T_i appears in T^α or in T^β. Since aT^α−bT^β ∈ N, then fi divides bϕ(T^β), where ϕ(Ti) = fit. If fi|b, then let T_j be any of the variables of T^β. One has f_ijT_j|f_iT_j|bT^β for i < j. This contradicts our assumption (because bT^β ∈/ F). Hence, f_i -b. Let i_k be the

minimum of the indices such that X_i^aik

k does not divide b, a_ik ∈ {s, s−1}.

Since X_i^aik

k dividesbϕ(T^β) (because f_i|bϕ(T^β)), then there exists j such that Tj appears inT^β and Xik|f_j.

By the structure of the generators f₁, . . . , f_n of I_q,s if X_ik|f_i and X_ik|f_j withj such thatT_j is inT^β thenf_ij|X_i^ai1

1 · · ·X_i^aik−1

k−1 ,a_i1, . . . , a_ik−1 ∈ {s, s−1}

(in fact if a variable off_ij is of degreeDin the monomialf_h, withh6=i, j, then a such variable of degreeN belongs to any other generatorsflfor alll > hand l6=j). Hence,f_ij|band, as a consequence,f_ijT_j|bT^β, which is a contradiction (because bT^β ∈/ F). It follows that N = (g_ij : 1 ≤ i < j ≤ n) = J. Hence, Isn−1,s is of linear type.

Example 3.2.If R=K[X1, X2, X3], then

I11,4= (X₁⁴X₂⁴X₃³, X₁⁴X₂³X₃⁴, X₁³X₂⁴X₃⁴)

<(I_11,4) =R[I11,4t] =R[X₁⁴X₂⁴X₃³t, X₁⁴X₂³X₃⁴t, X₁³X₂⁴X₃⁴t]

ϕ:R[T1, T2, T3]→R[X₁⁴X₂⁴X₃³t, X₁⁴X₂³X₃⁴t, X₁³X₂⁴X₃⁴t]

T₁ →X₁⁴X₂⁴X₃³t T₂ →X₁⁴X₂³X₃⁴t T3 →X₁³X₂⁴X₃⁴t

Kerϕ= (X2T2−X1T3, X3T1−X1T3) =J.

I_11,4 is of linear type.

By Theorems 3.1 and 3.2 we can state:

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

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

REFERENCES

[1] M. Auslander and M. Bridger,Stable module theory. Soc. 94, Providence, R.I., 1969.

[2] L. Avramov, Complete intersection and symmetric algebra. J. of Algebra 73 (1981), 248–263.

[3] W. Bruns and J. Herzog, Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics, 1998.

[4] F. Ischebeck, Eine Dualit¨at zwischen den Funktoren Ext und Tor. J. Algebra 11, 510 (1969).

[5] M. La Barbiera,ConditionsSq andGq on graded rings. Atti dell’Accademia Peloritana dei Pericolanti, Vol LXXXIV, C1A0601007, (2006).

[6] M. La Barbiera, On graded ^∗Gq-rings. International Mathematical Forum5 (2010), 191–198.

[7] M. La Barbiera,On a class of monomial ideals generated bys-sequence. Math. Reports 12(62),3, (2010), 201–216.

[8] M. Malliavin, Conditionaq de Samuel etq−torsion. Bull. Soc. Math. France 96, 193 (1968).

[9] H. Matsumura,Commutative Algebra. Benjamin-Cummings, Reading, MA, 1980.

[10] A. Micali, Sur les algebr`es universelles. Ann. Inst. Fourier14(1964), 33–88.

[11] M. Paugam, ConditionGq de Ischebeck-Auslander et conditionSq de Serre. Th`ese de Doctorat d’´etat Universit´e de Monpellier (1973).

[12] B. Sturmfels, Groebner Bases and Convex Polytopes. American Mathematical Society, Providence, Rhode Island, 1991.

[13] R.H. Villarreal, Monomial Algebras. Monographs and Textbooks in Pure and Applied Mathematics, 238, Marcel Dekker, Inc., New York, 2001.

Received 26 August 2012 University of Messina, Department of Mathematics, Viale Ferdinando Stagno d’Alcontres, 31,

98166 Messina, Italy monicalb@unime.it