The gorensteiness of modules over a noetherian local ring by specializations

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In this paper, we prove that the Gorensteiness, the injective resolution, the injective envelope and Gorenstein dimension of modules are preserved by specializations.

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Nội dung Text: The gorensteiness of modules over a noetherian local ring by specializations

JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2013, Vol. 58, No. 7, pp. 66-71 This paper is available online at http://stdb.hnue.edu.vn THE GORENSTEINESS OF MODULES OVER A NOETHERIAN LOCAL RING BY SPECIALIZATIONS Dam Van Nhi1 and Luu Ba Thang2 1 School for Gifted Student, Hanoi National University of Education 2 Faculty of Mathematics, Hanoi National University of Education Abstract. In this paper, we prove that the Gorensteiness, the injective resolution, the injective envelope and Gorenstein dimension of modules are preserved by specializations. Keywords. Specialization, injective dimension, Gorensteiness. 1. Introduction Specialization is a technique which considers algebraic equations with generic coefficients and substitute the generic coefficients by elements of the base field. This technique is usually used to show the existence of algebraic structures with a given property. The theory of specialization of ideals was introduced by W. Krull [3] where it was shown that the property of being a prime ideal is preserved by almost all specializations. Using specializations of finitely generated free modules and homomorphisms between them, D.V. Nhi and N.V. Trung defined in [5] the specializations of finitely generated modules over a local ring. They showed that the basic properties of modules are preserved by specializations. Developing the ideas of [5], we show that the Gorensteiness, the injective resolution of a module are also preserved by specializations. Notice that in [4] Minh and Nhi proved that the Gorensteiness is preserved by total specializations. 2. Specializations of RP -modules Throughout this paper we alway assume that k is an arbitrary perfect infinite field and K is an extension field of k. We denote the polynomial rings in n variables x1 , . . . , xn over k(u) and k(α) by R = k(u)[x] and by Rα = k(α)[x], respectively, where u = (u1 , . . . , um ) is a family of parameters and α = (α1 , . . . , αm ) ∈ K m is a family of elements of an infinite field K. Received October 12, 2013. Accepted November 5, 2013. Contact Luu Ba Thang, e-mail address: thanglb@hnue.edu.vn 66
The gorensteiness of modules over a noetherian local ring by specializations p(u, x) Each element a(u, x) of R can be written in the form a(u, x) = with q(u) p(u, x) ∈ k[u, x], q(u) ∈ K[u] \ {0}. For any α with q(α) ̸= 0 we define p(α, x) a(α, x) = . q(α) We shall say that a property holds for almost all α if it holds for all points of a Zariski-open non-empty subset of Ωm . For convenience we shall often omit the phrase "for almost all α" in the proofs of the results of this paper. Let P be an arbitrary prime ideal of R. Then, the specialization Pα of P is a radical unmixed ideal by ([2], Satz 14). Denote an associated prime ideal of Pα by p. For short, we will put S = RP and Sα = (Rα )p . Denote P S and pSα by m and mα . The local ring (Sα , mα ) is called a specialization of S with respect to α. The specialization of ideals can be generalized to modules. Now, we will recall the definition of a specialization of a finitely generated S-module. Let F be a free S-module of finite rank. The specialization Fα of F is a free Sα -module of the same rank. Let ϕ : F → G be a homomorphism of free S-modules. ( a (u, x) ) ij Fixing the bases F and G, we can represent ϕ by a matrix A = . Let Aα = bij (u, x) ( a (α, x) ) ij , then Aα is well-defined for almost all α. The specialization ϕα : Fα −→ Gα bij (α, x) of ϕ is given by the matrix Aα provided that ϕα is well-defined. Note that the definition of ϕα depends on the chosen bases of Fα and Gα . ϕ Definition 2.1. [5] Let L be an S-module, F1 −→ F0 −→ L −→ 0 be a finite free presentation of L and ϕα : (F1 )α −→ (F0 )α be a specialization of ϕ. Then, Lα := Coker ϕα is called a specialization of L (with respect to ϕ). If we choose a different finite free presentation F1′ −→ F0′ −→ L −→ 0, we can get a different specialization L′α of L, but Lα and L′α are canonically isomorphic. Hence Lα is uniquely determined up to isomorphisms and we have the following result: Remark 2.1. With the finite free presentation 0 → S → RP → 0 of S-module RP we get the finite free presentation 0 → Sα → (RP )α → 0. Then, we obtain (RP )α = Sα . Lemma 2.1. ([5], Theorem 2.6) Let L be a finitely generated S-module. Then, for almost all α, we have: (i) Ann Lα = (Ann L)α . (ii) dim Lα = dim L. Lemma 2.2. ([5], Theorem 3.1) Let L be a finitely generated S-module. For almost all α, we have: depth Lα = depth L. 67
Dam Van Nhi and Luu Ba Thang Lemma 2.3. ([5], Proposition 3.3) Let L and M be finitely generated S-modules. Then, for almost all α, there are ExtiSα (Lα , Mα ) ∼ = ExtiS (L, M )α , i ≥ 0. 3. Preservation of some basis properties of modules by specializations 3.1. Gorensteiness and injective resolution of modules In this section, we show that specializations preserve the injective resolution, the injective envelope and the Gorensteiness of modules. Lemma 3.1. [4] Let L be a finitely generated S-module. Then, for almost all α, we have: inj. dim Lα = inj. dim L. In particular, if L is an injective module, then Lα is also an injective module. Proposition 3.1. Let L be a finitely generated S-module and let I• : 0 → L → I 0 → I 1 → · · · → I r be an injective resolution of L. Then, for almost all α, the sequence (I• )α : 0 → Lα → Iα0 → Iα1 → · · · → Iαr is an injective resolution of Lα . Proof. Since I j is an injective S-module, therefore Iαj is an injective module, too, by Lemma 3.1. From the exact sequence I• : 0 → L → I 0 → I 1 → · · · → I r , by ([5], Theorem 2.2) we deduce that the sequence (I• )α : 0 → Iα0 → Iα1 → · · · → Iαr is exact. Hence (I• )α : 0 → Lα → Iα0 → Iα1 → · · · → Iαr is an injective resolution of Lα for almost all α. We will now recall the definition of the Gorenstein module. Definition 3.1. [8] Let (A, n) be a Noetherian local ring of dimension d and L be a Cohen-Macaulay module over A. L is called a Gorenstein module if inj. dim L = dim L = d. We have the following result: Theorem 3.1. Let L be a finitely generated S-module. If L is a Gorenstein module over the ring S then Lα is again a Gorenstein module over the ring Sα for almost all α. Proof. Assume that L is a finitely genarated Gorenstein module over the ring S. Then, L is a Cohen-Macaulay S-module and inj. dim L = dim S = d by ([5], Theorem 3.11). By Lemma 2.1 and Lemma 2.2, Lα is also a Cohen-Macaulay Sα -module. Since dim Sα = dim S and inj. dim Lα = inj. dim L by Lemma 3.1, we get dim Sα = dim S = inj. dim L = inj. dim Lα . Hence Lα is also a Gorenstein Sα -module. 68
The gorensteiness of modules over a noetherian local ring by specializations The problem to be considered in the rest of this paper is that the specialization of the injective envelope of the residue class field of S. ( ) Lemma 3.2. We have S/m α ∼ = Sα /mα for almost all α. ( ) ( ) Proof. We have S/m α = RP /P RP α ∼ = (RP )α /(P RP )α by ([5], Lemma 2.2) and ( ) ∼ ([5], Lemma 2.5). Hence S/m α = Sα /mα for almost all α. Corollary 3.1. ([6], Corollary 4.3) ) an (injective) envelope of module L by E(L). ( Denote Then, for almost all α, we have E S/m α ∼= E Sα /mα . ( ) ( ) Proof. Since S/m α ∼ = Sα /mα by Lemma 3.2 and E S/m ∼ = S/m by ([8], Proposition ( ) ( ) ( ) 3.2.11), there is E S/m α ∼ = S/m α ∼ = Sα /mα ∼ = E Sα /mα . Remark 3.1. Denote by E(L) the injective envelope of the S-module L. We have an unanswered question: Is it true E(L)α ∼ = E(Lα ) for almost all α? 3.2. Gorenstein dimension Let L be a finitely generated S-module. The ith Bass numbers of L, see [1], denoted by µiS (L), are defined as µiS (L) = dimS/m ExtiS (S/m, L), i ≥ 0. The notation of Gorenstein dimension of modules was introduced by Auslander (see [7]). We say that L has Gorenstein dimension 0 and write G. dim L = 0 if it satisfies the following conditions: (i) The natural homomorphism L → HomS (HomS (L, S), S) is an isomorphism. (ii) ExtiS (L, S) = 0 for all i > 0. (iii) ExtiS (HomS (L, S), S) = 0 for all i > 0. With a non-negative integer n we say that S-module L has Gorenstein dimension at most n and write G. dim L ≤ n if there is an exact sequence 0 → Gn → Gn−1 → · · · → G1 → G0 → L → 0 with G. dim Gi = 0 for every i = 0, 1, . . . , n. Proposition 3.2. Let L be a finitely generated S-module. Then, for almost all α, we have µiS (L) = µiSα (Lα ), i ≥ 0. Proof. ( Since L is finitely generated , all integers µiS (L) are finite. We have µiS (L) = ℓ ExtS (S/m, L)). By Lemma 2.3, there are ExtiSα (Lα , Mα ) ∼ i =( ExtiS (L, M )α , i ≥ 0. Since ( Pα is a radical ideal, ( from ([5], Proposition 2.8) it follows ℓ ExtiSα (Sα /mα , Lα )) = ℓ ExtiS (S/m, L)α ) = ℓ ExtiS (S/m, L)). Hence µiS (L) = µiSα (Lα ), i ≥ 0. 69
Dam Van Nhi and Luu Ba Thang Lemma 3.3. Let L be a finitely generated S-module. If G. dim L = 0 then G. dim Lα = 0, too, for almost all α. Proof. Assume that G. dim L = 0. Then L ∼ = HomS (HomS (L, S), S), ExtiS (L, S) = i 0 and ExtS (HomS (L, S), S) = 0 for all i > 0. By ([5], Theorem 2.2) and Lemma 2.3 we get have Lα ∼ = HomSα (HomSα (Lα , Sα ), Sα ), ExtiSα (Lα , Sα ) = 0 and ExtiSα (HomSα (Lα , Sα ), Sα ) = 0 for all i > 0. Thus, G. dim Lα = 0 for almost all α. Proposition 3.3. Let L be a finitely generated S-module. If G. dim L ≤ n then G. dim Lα ≤ n, too, for almost all α. Proof. Assume that G. dim L ≤ n. There is an exact sequnce of the form 0 → Gn → Gn−1 → · · · → G1 → G0 → L → 0 with G. dim Gi = 0 for every i = 0, 1, . . . , n. Since the squence 0 → (Gn )α → (Gn−1 )α → · · · → (G1 )α → (G0 )α → Lα → 0 by ([5], Theorem 2.2) and G. dim(Gi )α = 0 for every i = 0, 1, . . . , n by Lemma 3.3, therefore G. dim Lα ≤ n. Now, we recall the definition of a Gorenstein injective resolution of a module. A Gorenstein injective resolution of a module L is an exact sequence 0 → L → G0 → G1 → · · · → Gr → · · · such that Gi is Gorenstein injective for all i ≥ 0. We say that L has Gorenstein injective dimension less than or equal to r, Gid L ≤ r, if L has a Gorenstein injective resolution 0 → L → G0 → G1 → · · · → Gr → 0. Proposition 3.4. Let L be a finitely generated S-module and let G• : 0 → L → G0 → G1 → · · · → Gr → 0 be a Gorenstein injective resolution of L. Then, for almost all α, the sequence (G• )α : 0 → Lα → G0α → G1α → · · · → Grα → 0 is a Gorenstein injective resolution of Lα and we have also Gid Lα = Gid L. Proof. Since Gj is a Gorenstein injective S-module, therefore Gjα is a Gorenstein injective Sα -module, too, by Theorem 3.1. The exactnees of of the sequence (G• )α : 0 → Lα → G0α → G1α → · · · → Grα → 0 by [[5], Theorem 2.2]. Hence (G• )α : 0 → Lα → G0α → G1α → · · · → Grα → 0 is a Gorenstein injective resolution of Lα for almost all α. Then, Gid Lα = Gid L. Corollary 3.2. For almost all α, we have: ( ) ( ) sup{i| ExtiS E(S/m), L ̸= 0} = sup{i| ExtiSα E(Sα /mα ), Lα ̸= 0}. Proof. It is well-known that ( ) Gid Lα = sup{i| ExtiSα E(Sα /mα ), Lα ̸= 0} 70
The gorensteiness of modules over a noetherian local ring by specializations and ( ) Gid L = sup{i| ExtiS E(S/m), L ̸= 0}. Since Gid Lα = Gid L by Proposition 3.4, we get ( ) ( ) sup{i| ExtiS E(S/m), L ̸= 0} = sup{i| ExtiSα E(Sα /mα ), Lα ̸= 0} for almost all α, 4. Conclusion In this paper, we completed the proof of some properties of modules by specializations such as the injective resolution, the injective envelope, the Gorensteiness and the Gorenstein dimension. In a future work, we will study the preservation of inverse limits of modules by specializations which is an interesting problem. REFERENCES [1] W. Bruns and J. Herzog, 1993. Cohen-Macaulay rings. Cambridge University Press. [2] D. Eisenbud and S. Goto, 1984. Linear free resolutions and minimal multiplicity. J. Algebra 88, pp. 89-133. [3] W. Krull, 1948. Parameterspezialisierung in Polynomringen. Arch. Math. 1, pp. 56-64. [4] D. N. Minh and D. V. Nhi, 2009. The total specialization of modules over a local ring. VNU Journal of Science, Mathematics-Physics 25, pp. 39-45. [5] D.V. Nhi and N.V. Trung, 2000. Specialization of modules over a local ring. J. Pure Appl. Algebra 152, pp. 275-288. [6] D. V. Nhi, 2007. Specializations of direct limits and of local cohomology modules. Proc. Edinburgh Math. Soc. 50, pp. 1-17. [7] Sh. Payrovi, 2009. On the Bass numbers of modules with finite Gorenstein dimension. International Journal of Algebra Vol. 3, No. 11, pp. 531-537. [8] R. Y. Sharp, 1970. Gorenstein Modules. Math. Z. 115, pp. 117-139. [9] J. R. Strooker, 1990. Homological Questions in Local Algebra. Cambridge University Press. [10] K. Yamagishi, 1992. Recent aspect of the theory of Buchsbaum modules. Series of Mathematics Tokyo Metropolitan Univ, Japan. 71