Tài liệu Báo cáo " The total specialization of modules over a local ring" ppt

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

α
= {f(α, x)| f(u, X) ∈ I ∩ k[u, x]}
of the polynomial ring R
α
= k(α)[x]. For almost all α ∈ K
m
, I
α
inherits most of the basic properties
of I. Let p
u
be a separable prime ideal of R. In [2], we introduced and studied the specializations
of finitely generated modules over a local ring R
p
u
at an arbitrary associated prime ideal of p
α
(For
specialization of modules, see [3]). Now, we will introduce the notation about the total specializations
of modules. We showed that the Cohen-Macaulayness, the Gorensteiness and Buchsbaumness of a
module are preserved by the total specializations.
2. Specializations of prime separable ideals
Let p
u
be an arbitrary prime ideal of R. The first obstacle in defining the specialization of R
p
u
is that the specialization p
α
of p

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:
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

k(u, ξ
1
, . . ., ξ
n
). We enlarge k(u) by adjoining (v) . We form d + 1 linear forms
y
i
= −
n

j=1
v
ij
x
j
, i = 0, 1, . . ., d.
Then pk(u, v)[x] ∩ k(u, v)[y] = (f(u, v; y
0
, . . ., y
d
)) is a principal ideal. We put λ
i
=
n

j=0
v
ij
ξ
j

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


i=1
(R
α
\ p
i
).
Lemma 2.2. For almost all α, we have (R
α
)
T
is a semi-local ring.
Proof. Note that T is a multiplicative subset of R
α
. We show that (R
α
)
T
is a semi-local ring. Indeed,
let m be a maximal ideal of (R
α
)
T
. Then, there is a prime ideal q of R
α
such that m = q(R
α
)
T
.

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

1
⊂ p
j
, contradiction.
Hence m = p
1
(R
α
)
T
.
The natural candidate for the total specialization of R
p
u
is the semi-local ring (R
α
)
T
.
Definition We call (R
α
)
T
a total specialization of R
p
u
with respect t o α. For short we will put
S = R
p
u

, 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

element does not depend on the choice of p(u, x) and q(u) for almost all α. Now, for every fraction
a = f/g, f, g ∈ R, g = 0, we define a
α
:= f
α
/g
α
if g
α
= 0. Then a
α
is uniquely determined for
almost all α.
The following lemma shows that the above definition of S
T
reflects the intrinsic substitution
u → α of elements of R.
Lemma 3.1. Let a be an arbitrar y element of S. Then a
α
∈ S
T
for almost all α.
Proof. Since p
u
is a separable prime ideal of R , p
α
= R
α
for almost all α. Let a = f/g with f, g ∈ R,
g /∈ p

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.

D.N. Minh, D.V. Nhi / VNU Journal of Science, Mathematics - Physics 25 (2009) 39-45 41
By Lemma 3.1, the matrix A
α
:= ((a
ij
)
α
) has all its entries in (R
α
)
p
for almost all α. Let F
α
and
G
α
be free (R
α
)
p
-modules of the same rank as F and G, respectively.
Definition. [2] For fixed bases of F
α
and G
α
, the homomorphism φ
α
: F
α
→ G

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

:=Cokerφ
α
is called a specialization of L (with respect
to φ).
Then, we have the following results.
Lemma 3.2. [2, Theorem 2.2] Let 0 → L → M → N → 0 be an exact sequence of finitely
generated S-modules. Then 0 → L
α
→ M
α
→ N
α
→ 0 is exact f or almost all α.
Lemma 3.3. [2, Theorem 2.6] Let L be a finitely generated S-module. Then, for almost all α, we
have
(i) (Ann L)
α
=Ann (L
α
).
(ii ) dim L = dim L
α
.
Lemma 3.4. [2, Theorem 3.1] Let L be a finitely generated S-module. Then, for almost all α, we
have
(i) projL
α
= projL.
(ii ) depthL
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

T
-module L
T
:= Cokerφ
T
is called a total specialization of L (w ith
respect to φ). The module L
T
depends on the chosen presentation of L, but L
T
is uniquely determined
up to isomorphisms. Hence the finite free presentation of L will be chosen in the form S
s
φ
→ S
r
→
L → 0.
Lemma 3.5. Let L be a finitely generat ed S-module. Suppose that p = p
1
. Th en (L
T
)
p
1T
∼
=
L
α
for

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

T
]
p
T
→ [(R
α
)
T
]
r
p
T
→
[L
T
]
p
T
→ 0. By an easy computation A
α
=

(a
ij
)
α

=

(f

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

p
= S
α
, we have a commutative diagram
[(R
α
)
T
]
s
p
T
φ
α
−−−−→ [(R
α
)
T
]
r
p
T
−−−−→ [L
T
]
p
T
−−−−→ 0




∼
=



∼
=



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

, and an isomorphism (L
T
)
p
T
→ L
α
.
Hence (L
T
)
p
T
∼
=
L
α
for almost all α.
Proposition 3.6. Let L be a finitely generated S-module. For almost al l α, we have
(i) (AnnL)
α
= Ann(L
T
)
p
T
.
(ii ) dim L = dim L
T
.

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

)
p
T
for almost all α.
(ii) We have dim L = dim L
α
by Lemma 3.3. Then dim L = dim(L
T
)
p
T
. Semilarly, dim L =
dim(L
T
)
p
iT
for i = 1, . . . , s. Hence dim L = dim L
T
for almost all α.
Theorem 3.7. Let 0 → L → M → N → 0 be an exact sequence of finitely generated S-modules.
Then 0 → L
T
→ M
T
→ N
T
→ 0 is exact f or almost all α.
Proof. Since 0 → L → M → N → 0 is an exact sequence, the sequence 0 → L
α

→ 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
,
(ii ) depthL = depthL
T
.
Proof. (i) Since projL = projL
α
for almost all α by Lemma 3.4, there is projL
T
= sup
m∈sup(S
T
)
{proj(L
α
)
m
} = projL
α
= projL.
(ii) By [ 4, Lemma 18.1], there is a maximal ideal m of S
T
such that depthL
T
= depth(L
T
)
m
=
dim(L

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

)
m
) by [5, 3.
Theorem 12], there is ℓ(L
T
) = sℓ(L).
Proposition 3.10. Let L be a fin itely gen erated S-module of dimension d and q = (a
1
, . . ., a
d
)S a
parameter ideal on L. Then, we have e(q
T
, L
T
) = se(q, L) for almost all α, where e(q
T
, L
T
) and
e(q, L) are the multiplici ties of L
T
and L with respect to q
T
and q, respectively.
Proof. First, we will to show that e(q
α
, L
α
) = e(q, L). Indeed, Since a

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

1
, . . ., a
d
)L = 0. Then (a
1
)
α
, . . ., (a
d
)
α
is a system of parameters on L
α
. The multi-
plicity symbol of a
1
, . . ., a
d
with respect to L will be denoted by e(a
1
, . . ., a
d
|L), and the multiplicity
symbol of (a
1
)
α
, . . ., (a
d
)

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

d
|L).
We need only show that e(a
1
, . . ., a
d
|L) = e((a
1
)
α
, . . ., (a
d
)
α
|L
α
). This claim will be proved by
induction on d. For d = 0, by applying [2, Proposition 2.8], there is
e(∅|L
α
) = ℓ(L
α
) = ℓ(L) = e(∅|L).
D.N. Minh, D.V. Nhi / VNU Journal of Science, Mathematics - Physics 25 (2009) 39-45 43
Now we assume that d ≥ 1, and the claim is true for all S-modules with the dimension ≤ d − 1. By
[2, Lemma 2.3 and Lemma 2.5 ], there are
L
α
/(a
1

)
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
d
)
α
|L
α
/(a
1
)
α
L
α
) = e(a
2
, . . ., a
d
|L/a
1
L)
e((a
2
)
α
, . . ., (a
d
)
α
|0
L

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

)
T
|L
T
) =

m∈

(R
T
)
e
(R
T
)
m
(Φ
m
(a
1
)
T
, . . ., Φ
m
(a
d
)
T
|(L
T

)
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
)

p
iT
-module if L is a
(maximal) Cohen-Macaulay S-module.
(i) Assume that L is a Cohen-Macaulay S-module. Therefore dim L = depthL. Since dim L = dim L
α
by Lemma 3.3 and depthL = depthL
α
by Lemma 3.4, we get dim L
α
= depthL
α
. Hence L
α
is
also a Cohen-Macaulay S
α
-module for almost all α. Since L
α
= (L
T
)
p
iT
, it follows that L
T
is a
Cohen-Macaulay S
T
-module for almost all α.

(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

(S/m, L), β
i
(L) = dim
S/m
Tor
S
i
(S/m, L), ∀ i ≥ 0.
Lemma 4.2. Let L be finitely generated S-modules. Then, f or almost all α, we have
µ
i
S
α
(L
α
) = µ
i
S
(L), β
i
(L
α
) = β
i
(L), ∀ i ≥ 0.
Proof. Since L and L
α
are the finitely generated modules, all integers µ
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

(S
α
/m
α
, L
α
)

.
44 D.N. Minh, D.V. Nhi / VNU Journal of Science, Mathematics - Physics 25 (2009) 39-45
By [2, Proposition 3.3], there is Ext
i
S
α
(S
α
/m
α
, L
α
)
∼
=
Ext
i
S
(S/m, L)
α
. Since p
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)


.
Hence µ
i
S
(L) = µ
i
S
α
(L
α
). Similar, we obtain β
i
(L) = β
i
(L
α
).
Before invoking Lemma 4.2 to reprove Corollary 3.8 in [2], we will define a quasi-Buchsbaum
module. A finitely generated module over a Noetherian commutative ring is said to be a quasi-
Buchsbaum module if its localization at every maximal ideal is a surjective Buchsbaum.
Corollary 4.3. Let L be finitely generated S-modules. Then, for a llmost all α, we ha ve
(i) If L is a surjective Buchsbaum S-module, then L
α
is also a surjective Buchsbaum S
α
-module.
(ii ) If L is a quasi-Buchsbaum S-module, t hen L
T
is also a quasi-Buchsbaum S

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

m
(L)) by [2, Theorem 3.6]. Now the proof is
immedialtely from Lemma 4.2.
(ii) It is easily seen that the localization of L
T
at every maximal ideal is a surjective Buchsbaum,
Hence L
T
is also a quasi-Buchsbaum S
T
-module.
We will now recall the definition of the Gorenstein module. A non-zero and finitely generated
L is said to be a Gorenstein module if and only if the cousin complex for L provides a injective
resolution for L, see [7]. Before proving the preservation of Gorensteiness of module, we will show
that the injective dimension of module L is not change by specialization.
Lemma 4.4. Let L be finitely generated S-modules. Then, f or almost all α, we have
inj.dim(L
α
) = i nj.dim (L).
In p articula r, if L is an inj ective module, then L
α
is also an injective module.
Proof. Since S and S
α
have finite global dimensions, therefore inj.dimL and inj.dimL
α
are finite.
From [8, Theorem 3.1.17] we obtain the following relations
inj.dimL
α

= 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

α
= dim L
α
. Hence
(L
T
)
p
T
is again a Gorenstein ( S
T
)
p
T
-module for almost all α.
Corollary 4.6. Let I be an ideal of S. If S/I is a Gorenstein ring, then S
T
/I
T
is again a Gorenstein
ring for almost all α.
Proof. We first will recall the definition about the Gorenstein ring. A Noetherian ring is a Gorenstein
ring if its localization at every maximal ideal is a Gorenstein local ring. Since the localization of
D.N. Minh, D.V. Nhi / VNU Journal of Science, Mathematics - Physics 25 (2009) 39-45 45
S
T
/I
T
at every maximal ideal is also a Gorenstein ring by Theorem 4.5, therefore S
T

/I
T
is again a
Gorenstein ring for almost all α.
References
[1] W. Krull, Parameterspezialisierung in Polynomringen II, Grundpolynom, Arch. Math. 1 (1948) 129.
[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.