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.

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JOURNAL OF SCIENCE OF HNUE
Mathematical and Physical Sci., 2013, Vol. 58, No. 7, pp. 66-71
This paper is available online at
THE GORENSTEINESS OF MODULES OVER A NOETHERIAN LOCAL RING
BY SPECIALIZATIONS
Dam Van Nhi1 and Luu Ba Thang2
1School for Gifted Student, Hanoi National University of Education
2Faculty 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) ∈ Km 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
Each element a(u, x) of R can be written in the form a(u, x) =
p(u, x)
q(u)
with
p(u, x) ∈ k[u, x], q(u) ∈ K[u] \ {0}. For any α with q(α) ̸= 0 we define
a(α, x) =
p(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 PS 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.
Fixing the bases F and G, we can represent ϕ by a matrix A =
(aij(u, x)
bij(u, x)
)
. Let Aα =(aij(α, x)
bij(α, x)
)
, then Aα is well-defined for almost all α. The specialization ϕα : Fα −→ Gα
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.
Definition 2.1. [5] Let L be an S-module, F1
ϕ−→ 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 F ′1 −→ F ′0 −→ 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) AnnLα = (AnnL)α.
(ii) dimLα = dimL.
Lemma 2.2. ([5], Theorem 3.1) Let L be a finitely generated S-module. For almost all α,
we have: depthLα = depthL.
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
ExtiS(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. dimLα = inj. dimL.
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 → I0 →
I1 → · · · → Ir be an injective resolution of L. Then, for almost all α, the sequence
(I•)α : 0→ Lα → I0α → I1α → · · · → Irα is an injective resolution of Lα.
Proof. Since Ij is an injective S-module, therefore Ijα is an injective module, too, by
Lemma 3.1. From the exact sequence I• : 0 → L → I0 → I1 → · · · → Ir , by ([5],
Theorem 2.2) we deduce that the sequence (I•)α : 0 → I0α → I1α → · · · → Irα is exact.
Hence (I•)α : 0 → Lα → I0α → I1α → · · · → Irα 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. dimL =
dimL = 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. dimL = dimS = d by ([5], Theorem
3.11). By Lemma 2.1 and Lemma 2.2, Lα is also a Cohen-Macaulay Sα-module. Since
dimSα = dimS and inj. dimLα = inj. dimL by Lemma 3.1, we get dimSα = dimS =
inj. dimL = inj. dimLα. Hence Lα is also a Gorenstein Sα-module.
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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/PRP
)
α
∼= (RP )α/(PRP )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) Denote an injective envelope of module L by E(L).
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 Ext
i
S(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. dimL = 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 writeG. dimL ≤ n if there is an exact sequence 0→ Gn → Gn−1 → · · · → G1 →
G0 → L→ 0 with G. dimGi = 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) = µ
i
S
(Lα), i ≥ 0.
Proof. Since L is finitely generated , all integers µiS(L) are finite. We have µ
i
S(L) =
ℓ
(
ExtiS(S/m, L)). By Lemma 2.3, there are Ext
i
S(Lα,Mα)
∼= ExtiS(L,M)α, i ≥ 0.
Since Pα is a radical ideal, from ([5], Proposition 2.8) it follows ℓ
(
ExtiS(Sα/mα, Lα)) =
ℓ
(
ExtiS(S/m, L)α) = ℓ
(
ExtiS(S/m, L)). Hence µ
i
S(L) = µ
i
S
(Lα), i ≥ 0.
69
Dam Van Nhi and Luu Ba Thang
Lemma 3.3. LetL be a finitely generated S-module. IfG. dimL = 0 thenG. dimLα = 0,
too, for almost all α.
Proof. Assume that G. dimL = 0. Then L ∼= HomS(HomS(L, S), S), ExtiS(L, S) =
0 and ExtiS(HomS(L, S), S) = 0 for all i > 0. By ([5], Theorem 2.2) and
Lemma 2.3 we get have Lα ∼= HomS(HomS(Lα, Sα), Sα), ExtiS(Lα, Sα) = 0 and
ExtiS(HomS(Lα, Sα), Sα) = 0 for all i > 0. Thus, G. dimLα = 0 for almost all α.
Proposition 3.3. Let L be a finitely generated S-module. If G. dimL ≤ n then
G. dimLα ≤ n, too, for almost all α.
Proof. Assume that G. dimL ≤ n. There is an exact sequnce of the form 0 → Gn →
Gn−1 → · · · → G1 → G0 → L→ 0 with G. dimGi = 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. dimLα ≤ 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, GidL ≤ 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 GidLα = GidL.
Proof. SinceGj is a Gorenstein injective S-module, thereforeGjα 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,
GidLα = GidL.
Corollary 3.2. For almost all α, we have:
sup{i|ExtiS
(
E(S/m), L
) ̸= 0} = sup{i|ExtiS (E(Sα/mα), Lα) ̸= 0}.
Proof. It is well-known that
GidLα = sup{i|ExtiS
(
E(Sα/mα), Lα
) ̸= 0}
70
The gorensteiness of modules over a noetherian local ring by specializations
and
GidL = sup{i|ExtiS
(
E(S/m), L
) ̸= 0}.
Since GidLα = GidL by Proposition 3.4, we get
sup{i|ExtiS
(
E(S/m), L
) ̸= 0} = sup{i|ExtiS (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.
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