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- L1 erated R-module and SymmR(E) = t=0 Symmt(E) be its symmetric alge- bra. 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 gener- ated by all the homogeneous elements of I (the homogeneous elements of I of degree j are f 2 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 2 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}∗ ≥ minfq; dimR}∗ g: 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 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) 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 = fx1; : : : ; xng 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 2 E such that ax = 0. Then g(ax) = 0 implies g(x) = 0, and x 2 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 fol- lows. 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 ' M R(−ci) ! R(−dj) ! E ! 0; i=1 j=1 with ' = (aji), then the kernel J of the surjective homomorphism Symm( ): R[Y1;:::;Ym] ! SymmR(E) ! 0 induced by 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.