VNU Journal of Science, Mathematics - Physics 25 (2009) 39-45
The total specialization of modules over a local ring
Dao Ngoc Minh
∗
, Dam Van Nhi
Department of Mathematics, Hanoi National University of Education
136 Xuan Thuy Road, Hanoi, Vietnam
Received 23 March 2009
Abstract. In this paper we introduce the total specialization of an finitely generated module
over local ring. This total specialization preserves the Cohen-Macaulayness, the Gorensteiness
and Buchsbaumness of a module. The length and multiplicity of a module are studied.
1. Introduction
Given an object defined for a family of parameters u = (u
1
, . . ., u
m
) we can often substitute u
by a family α = (α
1
, . . ., α
m
) of elements of an infinite field K to obtain a similar object which is
called a specialization. The new object usually behaves like the given object for almost all α, that is,
for all α except perhaps those lying on a proper algebraic subvariety of K
m
. Though specialization is
a classical method in Algebraic Geometry, there is no systematic theory for what can be “specialized”.
The first step toward an algebraic theory of specialization was the introduction of the special-
ization of an ideal by W. Krull in [1 ]. Given an ideal I in a polynomial ring R = k(u)[x], where k is
a subfield of K, he defined the specialization of I as the ideal
I

u
need not to be a prime ideal. By [1], p
α
=
s

i=1
p
i
is an unmixed
ideal of R
α
.
∗
Corresponding author. E-mail: [email protected]
39
40 D.N. Minh, D.V. Nhi / VNU Journal of Science, Mathematics - Physics 25 (2009) 39-45
Assume that dim p
u
= d and ( ξ) is a generic point of p
u
over k. Without loss of generality,
we may suppose that this is normalised so that ξ
0
= 1. Denote by (v) = (v
ij
) with i = 0, 1, . . ., d,
j = 1, . . . , n, a system of (d + 1)n new indeterminates v
ij
, which are algebraically independent over

with
i = 0, 1, . . ., d. Then λ
0
, . . ., λ
d
satisfies f(u, v; λ
0
, . . ., λ
d
) = 0 and is called the ground-form of
p
u
. The prime ideal p
u
is called a separable prime ideal if it’s ground-form is a separable polynomial.
We have the following lemma:
Lemma 2.1.[1, Satz 14] A specializat ion of a prime separable ideal i s an intersection of a finite pri m e
ideals for almos t all α.
Let the prime ideal p
u
be separable. Assume that p
α
=
s

i=1
p
i
and set T =
s

Suppose that m ⊃ p
1
(R
α
)
T
, m = p
1
(R
α
)
T
. We have q ⊃ p
1
, q = p
1
. Since m = q( R
α
)
T
is a maximal
ideal, q ∩T = ∅. Hence q ⊂
s

i=1
p
i
. Therefore, it exists j such that q ⊆ p
j
. Then p

, S
α
= (R
α
)
p
and S
T
= (R
α
)
T
, where p is one of the p
i
. Then there is (S
T
)
p
T
= S
α
.
3. The total specialization of R
p
u
-modules
Let f be an arbitrary element of R. We may write f = p(u, x) /q(u), p(u, x) ∈ k[u, x], q( u) ∈
k[u] \ {0}. For any α such that q(α) = 0 we define f
α
:= p(α, x)/q(α). It is easy to check that this

u
. Since p is prime, p
u
: g = p
u
. By [1, Satz 3], p
α
= (p
u
: g)
α
= p
α
: g
α
. Hence g
α
∈ T. Then
a
α
∈ S
α
for almost all α.
First we want to recall the definition of specialization of finitely generated S-module by [2].
Let F, G be finitely generated free S-modules. Let φ : F → G be an arbitrary homomorphism of free
S-modules of finite ranks. With fixed bases of F and G, φ is given by a matrix A = (a
ij
), a
ij
∈ S.

α
given by the
matrix A
α
is called the specialization of φ with respect to α.
The definition of φ
α
does not depend on the choice of the bases of F, G in the sense that if B
is the matrix of φ with respect to other bases of F, G, then there are bases of F
α
, G
α
such that B
α
is
the matrix of φ
α
with respect to these bases.
Definition. [2] Let L be a finitely generated S-module and F
1
φ
→ F
0
→ L → 0 a finite free
presentation of L. The (R
α
)
p
-module L
α

= depthL.
Now we will define the total specialization of an arbitrary finitely generated S-module as follows. As
above, the matrix A
α
:= ((a
ij
)
α
) has all its entries in S
T
for almost all α. Let F
T
and G
T
be free
S
T
-modules of the same rank as F and G, respectively, and B
α
is the matrix of φ
T
with respect to
these bases.
Definition. Let L be a finitely generated S-module and F
1
φ
→ F
0
→ L → 0 a finite free
presentation of L. The S

almost all α.
Proof. Let S
s
φ
→ S
r
→ L → 0 be a finite free presentation of L. There exists an exact sequence
(R
α
)
s
T
φ
T
→ (R
α
)
r
T
→ L
T
→ 0. This will induces also an exact sequence [(R
α
)
T
]
s
p
T
[φ

ij
)
α
/1
(g
ij
)
α
/1

, it follows that (φ
T
)
p
T
= φ
α
.
Since [(R
α
)
T
]
p
T
∼
=
(R
α
)


∼
=



∼
=



S
s
α
ψ
α
−−−−→ S
r
α
−−−−→ L
α
−−−−→ 0,
42 D.N. Minh, D.V. Nhi / VNU Journal of Science, Mathematics - Physics 25 (2009) 39-45
where to rows are finite free presentations of [L
T
]
p
T
and L
α

Proof. (i) Since (L
T
)
p
T
∼
=
L
α
by Lemma 3.5, there is Ann(L
T
)
p
T
= Ann((L
T
)
p
T
) = Ann(L
α
).
Since Ann(L)
α
= Ann(L
α
) by Lemma 3.3, therefore Ann(L)
α
= Ann(L
T

→ Mα → N
α
→ 0
is also exact by Lemma 3.2, or the sequence 0 → (L
T
)
p
T
→ (M
T
)
p
T
→ (N
T
)
p
T
→ 0 is exact for
every maximal ideal p
T
. Hence 0 → L
T
→ M
T
→ N
T
→ 0 is exact for almost all α.
Proposition 3.8. Let L be a finitely generated S-module. For almost al l α, we have
(i) projL = projL

T
)
p
T
. Then depthL
T
= depthL
α
= depthL by Lemma 3.4.
Proposition 3.9. Let L be a S-module of finite length. Then L
T
is a S
T
-module of finite length for
almost all α. Moreover, ℓ(L
T
) = sℓ(L).
Proof. Since ℓ( L
α
) = ℓ(L) by [2, Proposition 2.8] and ℓ (L
T
) =

m∈

(R
T
)
ℓ((L
T

1
, . . ., a
d
∈ pS, for almost all
α there are (a
1
)
α
, . . ., (a
d
)
α
∈ p
α
S
α
. By Lemma 3.2 and by Lemma 3.3, dim L
α
/((a
1
)
α
, . . ., (a
d
)
α
)
L
α
= dim L/( a

α
with respect to Lα by e( (a
1
)
α
, . . ., (a
d
)
α
|L
α
). Then we have
e(q
α
; L
α
) = e((a
1
)
α
, . . ., (a
d
)
α
|L
α
)
e(q; L) = e(a
1
, . . ., a

)
α
L
α
∼
=
(L/a
1
L)
α
and 0
L
α
: (a
1
)
α
∼
=
(0
L
: a
1
)
α
.
Since the dimensions of these modules ≤ d − 1, therefore
e((a
2
)

α
: (a
1
)
α
) = e(a
2
, . . ., a
d
|0
L
: a
1
).
The statment follows from the definition of the multiplicity.
Now we prove the result e(q
T
, L
T
) = se(q, L) Since
e(q
T
, L
T
) = e((a
1
)
T
, . . ., (a
d

)
m
)
by [5, 7.8. Theorem 15], there is e(q
T
, L
T
) = se(q, L) for almost all α.
4. Preservation of some properties of modules
By virtue of Proposition 3.10 one can show that preservartion of Cohen-Macaulayness by total
specializations.
Theorem 4.1. Let L be a finitely generated S-modul e. For almost all α, we have
(i) L
T
is a Cohen-Macaulay S
T
-module if L is a Cohen-Macaulay S-module.
(ii ) L
T
is a maximal Cohen-Macaulay S
T
-module if L is a maximal Cohen-Macaulay S-module.
Proof. We need only show that (L
T
)
p
iT
is a (maximal) Cohen-Macaulay (S
T
)

(ii) Assume that L is a maximal Cohen-Macaulay S-module. Therefore dim L = dim S. Since
dim L
α
= dim L and dim S
α
= dim S, it follows that dim L
α
= dim S
α
. Hence L
T
is a maxi-
mal Cohen-Macaulay S
T
-module.
The ith Bass and ith Betti numbers of L, which are denoted by µ
i
S
(L) and β
i
(L) respectively,
are defined as follows:
µ
i
S
(L) = dim
S/m
Ext
i
S

(L) and µ
i
S
α
(L
α
) are finite.
We have
µ
i
S
(L) = ℓ

Ext
i
S
(S/m, L)

, µ
i
S
α
(L
α
) = ℓ

Ext
i
S
α

is a radical ideal,
from [2, Proposition 2.8] it follows that
ℓ

Ext
i
S
α
(S
α
/m
α
, L
α
)

= ℓ

Ext
i
S
(S/m, L)
α

= ℓ

Ext
i
S
(S/m, L)

T
-module.
Proof. (i) Put d = dim L. By Lemma 3.3, dim L
α
= d. Since S is a regular ring, by [6, Chapter 2.
Theorem 4.2] we known that L is a surjective S-module if and only if
µ
i
S
(L) =
i

j=0
β
i−j
(S/m)ℓ(H
j
m
(L)), i = 0, . . ., d − 1.
Since ℓ(H
j
m
(L)) < ∞, therefore ℓ(H
j
m
α
(L
α
)) = ℓ(H
j

= depthS
α
= depthS = inj.dimL.
If L is an injective module, then inj. dimL = 0. Hence inj.dimL
α
= 0, and therefore L
α
is also an
injective module.
Theorem 4.5. Let L be finitely generated S-modules. If L is a G orenstein S-module, then (L
T
)
p
T
is
again a Gorenstein (S
T
)
p
T
-module for almost all α.
Proof. Assume that L is a Gorenstein S-module of dimension d. Then L is a Cohen-Macaulay S-
module and dim S = inj.dimL = d by [7, Theorem 3. 11]. Since dim L
α
= dim L = d by Lemma
3.3 and inj.dim(L
α
) = inj.dim(L) by Lemma 4.2, therefore dim S
α
= inj.dimL

/I
T
is again a
Gorenstein ring for almost all α.
References
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[2] D.V. Nhi, N.V. Trung, Specialization of modules over a local ring, J. Pure Appl. Algebra 152 (2000) 275.
[3] D.V. Nhi, N.V. Trung, Specialization of modules, Comm. Algebra 27 (1999) 2959.
[4] D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Springer-Verlag, 1995.
[5] D.G. Northcott, Lessons on rings, modules and multiplicities, Cambridge at the University Press 1968.
[6] K. Yamagishi, Resent aspect of the theory of Buchsbaum modules, College of Liberal Arts. Himeji Dokkyo Uni-
versity.
[7] R.Y. Sharp, Gorenstein Modules, Math. Z. 115 (1970) 117.
[8] W. Bruns, J. Herzog, Cohen-Macaulay rings, Cambridge University Press, 1993.