khóa luận tốt nghiệp tiếng anh :THE ASYMPTOTIC LINEARITYOF CASTELNUOVO-MUMFORD REGULARITY

VIETNAM NATIONAL UNIVERSITY
UNIVERSITY OF SCIENCE
FACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS
Nguyen Duy Khanh
THE ASYMPTOTIC LINEARITY
OF CASTELNUOVO-MUMFORD REGULARITY
Undergraduate Thesis.
Advanced Undergraduate Program in Mathematics.
Hanoi - 2012
VIETNAM NATIONAL UNIVERSITY
UNIVERSITY OF SCIENCE
FACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS
Nguyen Duy Khanh
THE ASYMPTOTIC LINEARITY
OF CASTELNUOVO-MUMFORD REGULARITY
Undergraduate Thesis.
Advanced Undergraduate Program in Mathematics.
Thesis Advisor: Prof.Dr.Sc. Ngo Viet Trung.
Hanoi - 2012
Acknowledgements
I would like to thank my advisor, Prof. Ngo Viet Trung, for his limitless patience in
making abstract mathematics so easy to be perceived, and for his supporting my work on
this thesis. I also would like to thank Prof. Ha Huy Tai for his very helpful conversation
on Castelnuovo-Mumford regularity. Finally, I would like to thank my family for always
believing in me, supporting me, and encouraging me spiritually and mentally.
1
Contents
1. Introduction 3
2. Castelnuovo-Mumford regularity 3
2.1. Graded rings and graded modules 3
2.2. Castelnuovo-Mumford regularity 4

3. Bigraded module and regularity 7
4. Asymptotic Linearity 9
5. Open problem 10
6. Conclusion 13
References 14
2
1. Introduction
Castelnuovo-Mumford regularity is an important invariant in commutative algebra. It
was first defined by D.Mumford-who attributes the idea to G.Castelnuovo-for coherent
sheaves on projectives space. In 1982, Ooshi[O] and in 1984, D.Eisenbud, and S.Goto[EG]
both turned Castelnuovo-Mumford regularity into algebraic side. Ooishi characterized this
regularity by local cohomology and Eisenbud and Goto made explicite the link between
algebraic regularity of a graded module over a polynomial ring and its minimal free reso-
lution.
Denote by reg(M) the Castelnuvo-Mumford regularity of M. The behaviour of reg(I
n
)
where I is a homogeneous ideal in a polynomial ring over a field interests many mathemati-
cians. Bertram, Ein, and Lazarsfeld[BEL] discovered firstly that if I is the defining ideal
of a smooth complex variety, reg(I
n
) is bounded by a linear function. Later, Chandler[C]
has conjectured that: reg(I
n
) ≤ nreg(I). Swanson[S] prove that for any homogenous ideal
I and integer n ≥ 1 there exists a number D such that : reg(I
n
) ≤ nD. This result sup-
ported Chandler’s conjecture, however the method of the proof there makes it difficult to
find such a constant. Finally, Cutkosky et al [CHT] and Kodiyalam[Kod] found out that

reg(I
n
) is asymptotically a linear function, in particular, Kodiyalam pointed out that the
slope of that funtion is the least maximal degreee of reductions of I. However, the free
coefficient is now still a mysterious, there are only some special results in some special
situations of I.
The main task of this writing is proving the following result, some other related results
are also derived.
Theorem: Let R be a standard graded algebra over a commutative Noetherian ring with
unity and I is a graded ideal of R. Define:
d(I) := max{degf|f belongs to a minimal generating set of I}
ρ
M
(I) = min{d(J)|J is an M reduction of I}
Let M be a finitely generated graded R-module, (M) denote the smallest degree of the
homogeneous element of M. Then there exists an integer e ≥ (M) such that for n  0,
reg(I
n
M) = ρ
M
(I)n + e
2. Castelnuovo-Mumford regularity
2.1. Graded rings and graded modules. In this section we study the bigraded struc-
ture, which will play the important role in the proof for the main theorem. We start with
a definition.
3
Definition 2.1. Let (G, +) be an abelian group. A ring R is called G graded if there
exists a family of Z-modules R
g
, g ∈ G such that R =


g∈G
R
g
as a Z module with
R
g
R
h
∈ R
g+h
for all g, h ∈ R. Let R be a graded ring. A R-module M is called G−
graded if there exists a family of Z-modules M
g
, g ∈ G such that M =

g∈G
M
g
as a
Z-module with R
g
M
h
∈ M
g+h
for any g, h ∈ G.
The element x ∈ M is called homogeneous of degree g if x ∈ M
g
for some g ∈ G

and we set d(x) = g. M
g
is called the homogeneous component of M, an ideal I of R
is G-graded if I =

g∈G
I
g
where I
g
= I ∩ R
g
. From this definition, it is easy to check
that if I is graded and finitely generated then the degree of generators of I is uniquely
determined by I.
Definition 2.2. Let R be a G-graded ring, M be a G-graded R-module, and g ∈ G. We
define the M(g) to be the G-graded R-module M by shifting its grading g steps, i.e :
M(g)
h
= M
g+h
for all h ∈ G. This module M(g) is isomorphic to M as a module and
called the g
th
twist of M. For a N-graded module M, the maximal non-vanishing degree
of M is defined to be the maximal number g such that M
g
= 0.
If G is Z or Z
2

we say that R is a graded or bigraded ring and M is a graded or
bigraded R-module.
In particular, the following examples will be considered in this writing:
i- Let R = k[x
1
, , x
n
] be a polynomial ring of n-variable over a field k. Then R has a
graded structure by setting d(x
i
) = 1 for every i = 1, , n.
ii- Let S = k[x
1
, , x
n
, y
1
, , y
m
] be the polynomial ring in n + m variables. Then S has
bigraded structure by setting d(x
i
) = (1, 0) and d(y
j
) = (0, 1) for any 1 ≤ i ≤ n, 1 ≤ j ≤
m.
The above polynomial rings are usually called the standard graded/bigraded polynomial
ring. Note that there are other grading structures that can be defined on them.
iii- Let R = k[x
1

, , x
n
], I is a graded ideal of R. Define
R[It] = {
n

i=0
a
i
t
i
|n ∈ N, a
i
∈ I
i
} =

n≥0
I
n
t
n
This is called the Rees algebra of I. There is a bigraded structure on this algebra defined
by two degree, one by elements of I, one by the variable t.
2.2. Castelnuovo-Mumford regularity. In this section, we will define the Castelnuvo-
Mumford regularity in two way: one in term of local cohomology and one in term of filter
regular sequence. We begin with a definition
4
Definition 2.3. Let A be a commutative Noetherian ring with unity, R be a standard
graded algebra over A, R

+
be the ideal generated by the element of positive degree. For
any finitely generated graded R-module M, define:
Γ
R
+
(M) = {m ∈ M|mR
t
+
= 0 for some t ∈ N}
This is a functor from the category of R-module to R-module and it is left exact. The
right derived funtor of Γ
R
+
is called the local cohomology functor with respect to R
+
and
denoted by H
R
+
.
Definition 2.4. Denote by a(M) the maximal non-vanishing degree of M, for i ≥ 0 we
denote by H
i
R
+
(M) to be the i
th
local cohomology module of M with respect to R
+

. The
Castelnuvo-Mumford regularity of M is the invariant defined to be :
reg(M) := max{a(H
i
R
+
(M)) + i|i ≥ 0}
Let z
1
, , z
s
be linear form in R. This set of element is called M-filter-regular sequence
if z
i
/∈ p for any associated prime p  R
+
of (z
1
, , z
i−1
)M for i = 1, , s. Then the
Castelnuovo-Mumford regularity can be characterized as follows:
Proposition 2.1. Let z
1
, , z
s
be an M-filter regular sequence of linear forms which gen-
erate an M-reduction of R
+
. Then :

reg(M) = max{a((z
1
, , z
i
)M : R
+
/(z
1
, , z
i
)M)|i = 1, , s}
In order to prove this proposition, we need a lemma:
Lemma 2.2. Let z ∈ R
1
be an M-filter regular element.
Define a
i
(M) = max{r|H
i
R
+
(M)
r
= 0}. Then, for all k ≥ 0, we have
a
k+1
(M) + 1 ≤ a
k
(M/zM) ≤ max{a
k

(M), a
k+1
(M) + 1}.
Proof. Consider the exact sequence: 0 → (M/0 : z)(−1)
z
−→ M → M/zM → 0. This
exact sequence induces the following exact sequence :
H
k
R
+
(M)
i
→ H
k
R
+
(M/zM)
i
→ H
k+1
R
+
(M)
i−1
→ H
k+1
R
+
(M)

i
→ H
k+1
R
+
(M/zM)
i
→
From this exact sequence we get : a
k
(M/zM) ≤ max{a
k
(M), a
k+1
(M) + 1}
Note that for all i > a
k
(M/zM) we have :
H
k+1
R
+
(M)
i−1
→ H
k+1
R
+
(M)
i

→ H
k+1
R
+
(M)
i+1
→ → 0
Then, we get a
k+1
(M) < a
k
(M/zM). This completes the proof for lemma. 
Now we are ready to prove the proposition.
5
Proof. Note that from the definition of a
i
(M) we have reg(M) = max{a
i
(M) + i|i ≥ 0}.
By using lemma successively we get
a
i
(M) + i ≤ a
0
(M/(z
1
, . . . , z
i
)M) ≤ max{a
j

(M) + j| j = 0, . . . , i}.
This implies that
max{a
i
(M) + i| i = 0, . . . , t} = max{a
0
(M/(z
1
, . . . , z
i
)M)| i = 0, . . . , t}
for all t ≤ s. Now we identify H
0
R
+
(M/(z
1
, . . . , z
i
)M)) with

n≥0
(z
1
, , z
i
)M : R
n
+
/(z

1
, , z
i
)M.
Set a = a
0
(M/(z
1
, . . . , z
i
)M). Then we have
H
0
R
+
(M/(z
1
, . . . , z
i
)M))
a
⊆ (z
1
, , z
i
)M : R
+
/(z
1
, , z

i
)M ⊆ H
0
R
+
(M/(z
1
, . . . , z
i
)M))
and therefore a((z
1
, , z
i
)M : R
+
/(z
1
, , z
i
)M) = a. 
The following lemma show that any M-reduction of R
+
can be generated by an M-filter
regular sequence in a flat extension of A.
Lemma 2.3. Let q be an M-reduction generated by the linear form x
1
, , x
s
. For i =

1, , s let z
i
=

s
j=1
u
ij
x
j
, where U = {u
ij
|i, j = 1, , s} is a matrix of indeterminates.Put
: A

= A[U, det(U)
−1
], R

= R ⊗
A
A

, M

= M ⊗ A

. Then if we consider R

as a standard

graded algebra over A

and M

as a graded R

module, then z
1
, , z
s
is an M

filter regular
sequence.
Proof. From the independence of indeterminates, it suffices to show that z
1
/∈ p for any
associated prime p  R
+
of M

. From the definition of R

, we see that this prime ideal
p must have have the form qR

for some associated prime q  R
+
of M. If Q ⊆ q then
since M/qM is a quotient module of M/QM we have (M/qM)

n
= 0 for all n  0 . It
follows that there is a number t such that R
t
+
M ⊆ qM. But ann(M) ⊆ q we get R
+
⊆ q
which is a contradiction. Therefore Q  q. Since Q = (x
1
, , x
s
), this implies that :
z
1
= u
11
x
1
+ + u
1s
x
s
/∈ qR

= p 
The following corollary is crucial in proving the main result :
Corollary 2.4. The maximal degree of the generator of M does not exceed the regularity
of M.
Proof. Applying Lemma 2.2 with notice that reg(M) = max{a

k
(M)+k|k ≥ 0} we have:
a
k
(M/zM) + k ≤ max{a
k
(M) + k, a
k+1
(M) + k + 1}
↔ reg(M/zM) ≤ reg(M)
6
Denote by dim(M) the Krull dimension of M. If d = 0 then dim(R/ann
R
(M)) = 0 thus
M is annihilated by some power of R
+
therefore M = Γ
R
+
(M). Hence H
i
R
+
(M) = 0 for
every i ∈ N and so reg(M) = a(M).
Assume that a(Q) ≤ reg(Q) for every module Q of dimension less than dim(M), denote
by r the regularity of M. Let N be the submodule of M generated by all the elements of
degree less than r.
Note that : dim(M/zM) ≤ dim(M) then by the induction assumption we have :
a(M/zM) ≤ reg(M/zM) ≤ reg(M)

Hence : M = N + zM. Applying the graded version of Nakayama lemma, we get N = M.
Therefore the maximal degree of the generator of M does not exceed the regularity of M.

3. Bigraded module and regularity
In this section we will apply the bigraded structure to study the behaviour of the
regularity. Let S = A[X
1
, . . . , X
s
, Y
1
. . . , Y
v
] be a polynomial ring over a commutative
Noetherian ring A with unity. Then we can view S as a bigraded ring by define degX
i
=
(1, 0), i = 1, . . . , s, and deg(Y
j
) = (d
j
, 1), j = 1, . . . , v for a given sequence d
1
, . . . , d
v
of
non-negative integers. Without loss of generality, assume that d
v
= max{d
i

|i = 1, . . . , v}.
Let M be a finitely generated bigraded module over A[X
1
, . . . , X
s
, Y
1
. . . , Y
v
]. For a fixed
number n define
M
n
:=

a≥0
M
(a,n)
.
Then M
n
is a finitely generated graded module over the naturally graded polynomial
ring A[X
1
. . . , X
s
]. We will show that reg(M
n
) is asymptotically a linear function. If
s = 0, then reg(M

n
) = max{a|M
(a,n)
= 0} := a(M
n
). In [CHT-Theorem 3.4] they has
already consider this case for the case of bigraded module over polynomial ring over a
field. However the proof there can not be used to prove the general case.
Proposition 3.1. Let M be a finitely generated bigraded module over the bigraded
polynomial ring A[Y
1
. . . , Y
v
]. Then a(M
n
) is a linear function with slope ≤ d
v
for n  0.
Proof. Since the case v = 0 is trivial then we only need to consider the case v ≥ 1.
Consider the exact sequence of bigraded A[Y
1
. . . , Y
v
]-modules:
0 −→ [0
M
: Y
v
]
(a,n)

−→ M
(a,n)
Y
v
−→ M
(a+d
v
,n+1)
−→ [M/Y
v
M]
(a+d
v
,n+1)
−→ 0.
Assume that a([0
M
: Y
v
]
n
) and a([M/Y
v
M]
n
) are asymptotically linear functions with
slopes ≤ d
v
, then we will prove the proposition inductively on n. We have
a([0

M
: Y
v
]
n
) + d
v
≥ a([0
M
: Y
v
]
n+1
)
7
a([M/Y
v
M]
n
) + d
v
≥ a([M/Y
v
M]
n+1
)
for all large n. The proof is completed if we show that a(M
n
) = a([0
M

: Y
v
]
n
) for all
large n. Since a(M
n
) ≥ a([0
M
: Y
v
]
n
) for all n, it suffices to consider the case that there
exists an infinite sequence of integers m such that a(M
m
) ≥ a([0
M
: Y
v
]
m
). Applying this
condition into the exact sequence above we have :
a(M
m+1
) = max{a(M
m
) + d
v

, a([M/Y
v
M]
m+1
)}.
On the other hand, we have :
a(M
m
) + d
v
≥ a([M/Y
v
M]
m
) + d
v
≥ a([M/Y
v
M]
m+1
).
Then for all n ≥ m, a(M
n
) > a([0
M
: Y
v
]
n
), hence a(M

n+1
= a(M
n
)) + d
v
. Therefore
a(M
n
) is asymptotically a linear function with slope d
v
in this case.
The proof completed. 
Using the above result and the technique of flat extension, we can prove the following
lemma:
Theorem 3.2 Let M be a finitely generated bigraded module over the bigraded poly-
nomial A[X
1
, . . . , X
s
, Y
1
. . . , Y
v
]. Then reg(M
n
) is a linear function with slope ≤ d
v
for
n  0.
Proof. The case s = 0 is the Proposition 3.1. Now assume that s ≥ 1.

Let U = {u
ij
|i, j = 1, . . . , s} is a matrix of indeterminates, let z
i
=

s
j=1
u
ij
X
j
, i =
1, . . . , s.
Define A

:= A[U, det(U)
−1
] and M

:= M
n
⊗
A
A

. Then M

is a finitely generated bi-
graded module over A


[X
1
, . . . , X
s
, Y
1
, . . . , Y
v
]. Since M

n
= M
n
⊗
A
A

, then reg(M
n
) =
reg(M

n
) for all n ≥ 0. According to Lemma 2.3 , we have z
1
, . . . , z
s
is an M


n
-filter-
regular sequence. Let R = A

[X
1
, . . . , X
s
]. Since (z
1
, . . . , z
s
) = (X
1
, . . . , X
s
) = R
+
,
applying Proposition 3.1 we get
reg(M

n
) = max{a((z
1
, . . . , z
n
)M

n

: R
+
/(z
1
, . . . , z
i
)M

n
)|i = 0, . . . , s}.
We have
(z
1
, . . . , z
n
)M

n
: R
+
/(z
1
, . . . , z
i
)M

n
) = [(z
1
, . . . , z

n
)M

n
: R
+
/(z
1
, . . . , z
i
)M

n
)].
On the other hand note that (z
1
, . . . , z
i
)M

: R
+
/(z
1
, . . . , z
i
)M

can be viewed as a
graded module over the bigraded polynomial ring A


[Y
1
, . . . , Y
v
] then by Proposition
3.1, a([(z
1
, . . . , z
i
)M

: R
+
/(z
1
, . . . , z
i
)M

]
n
) is asymptotically a linear function with slope
≤ d
v
for i = 0, . . . , s.
Hence reg(M
n
) is asymptotically a linear function with slope ≤ d
v

. 
8
4. Asymptotic Linearity
Let A be a commutative Noetherian ring with unity. Let R be a standard graded
algebra over A and I a graded ideal of R, M be a finitely generated graded R-module,
let ε(M) be the smallest degree of a homogeneous element of M. An ideal J ⊆ I is called
an M-reduction of I if I
n+1
M = JI
n
M for some n ≥ 0. Define:
d(I) := max{degf|f belongs to a minimal generating set of I}
ρ
M
(I) = min{d(J)|J is an M reduction of I}
We will use the results in the previous section to prove the main theorem. Firstly we prove
a lemma that represents an inequality between d(I
n
M) and the invariants ρ
M
(I), ε(M)
which was prove in the case of ideal of polynomial ring over a field in [Kod-Proposition
4], but that proof used Nakayama’s lemma that can not handle the general case.
Lemma 4.1. d(I
n
M) ≥ ρ
M
(I)n + ε(M) for all n ≥ 0.
Proof. Let J and K be the ideals generated by the homogeneous elements of I of degree
< ρ

M
(I) and ≥ ρ
M
(I), respectively. Then I = J +K and therefore I
M
= JI
n−1
M +K
n
M.
Since K
n
M is generated by homogeneous elements of degree ≥ ρ
M
(I)n + ε(M), then
I
n
M = JI
n−1
M. Hence J is an M-reduction of I. But from the definition of J, d(J) <
ρ
M
(I), which is a contradiction. This completes the proof. 
Now we are ready to prove the main theorem :
Theorem 4.2. Let R be a standard graded ring over a commutative Noetherian ring with
unity and I a graded ideal of R. Let M be a finitely generated graded R-module. Then
there exists an integer e ≥ ε(M) such that for all large n,
reg(I
n
M) = ρ

M
(I)n + e.
Proof. Let J be an M-reduction of I with d(J) = ρ
M
(I).
Let R[Jt] =

n0
J
n
t
n
be the Rees algebra of R with respect to J and M =

n≥0
I
n
M.
Since I
n+1
M = JI
n
M for all large n, we may consider M as a finitely generated graded
module over R[Jt].
Assume that R
1
is generated by the linear forms x
1
, . . . , x
s

and J is generated by the
forms y
1
, . . . , y
v
with degy
j
= d
j
, j = 1, . . . , v. Then we can view R[Jt] as a quotient ring
of the bigraded polynomial ring A[X
1
, . . . , X
s
, Y
1
. . . , Y
v
] with degX
i
= (1, 0), i = 1, . . . , s,
and degY
j
= (d
j
, 1), j = 1, . . . , v. Hence M is a finitely generated bigraded module over
A[X, Y ] with M
n
= I
n

M.
According to Theorem 3.2, we have reg(I
n
M) is asymptotically a linear function with
slope ≤ max d
1
, . . . , d
v
= d(J). Let dn + e be this linear function. Combining with
9
Proposition 2.4 and Lemma 4.1 we have
reg(I
n
M) ≥ d(I
n
M) ≥ ρ
M
(I)n + ε(M)
for all n. Therefore, d ≥ ρ
M
(I). Since d ≤ d(J) = ρ
M
(I), then d = ρ
M
(I) and e ≥
ε(M). 
In particular, if M = R then we have the following corollary, which was proved inde-
pendently in [CHT] and [Kod] :
Corollary 4.3. Let R be a standard graded ring over a commutative Noetherian ring with
unity and I a graded ideal of R. Then there exists an integer e ≥ 0 such that for all large

n,
reg(I
n
) = ρ
R
(I)n + e.
In particular, we can use Theorem 3.2 to prove that regI
n+1
the regularity of integral
closure of I
n
is asymptotically linear. This result was also derived in [CHT] for the case
R is a polynomial ring over a field however in there the slope was not determined.
Corollary 4.4. Let R be a standard graded domain over over a commutative Noetherian
ring with unity and I a graded ideal of R such that I
n+1
= II
n
for n sufficiently large.
Then reg(I
n
) is a linear function with slope ρ
R
(I) for n  0.
Proof. Since
I
n+1
= II
n
, there is an integer n

0
such that I
n+1
= I
¯
I
n
for all n ≥ n
0
. Let
M = I
n
0
. Then I
n
M = I
n+n
0
for all n ≥ 0. Therefore J ⊆ I is an M-reduction if and
only if
¯
J =
¯
I or J is an R-reduction of I and hence ρ
M
(I) = ρ(I). Applying Theorem 3.2
we get the result. 
5. Open problem
5.1 Asymptotic linearity of maximal degree of power of ideal Recall that if I
is an ideal in a finitely generated standard graded algebra A over a Noetherian ring A,

then dt + ε(M) ≤ d(I
t
) ≤ reg(I
t
) ≤ dt + e for every t ∈ N and some positive integer
e. Therefore it is reasonable to ask the question about the behaviour of d(I
t
) for t large
enough : Is it asymptotically a linear function ? This question is still open now, but we
can prove it in a particular situation :
Theorem 5.1. Let I be a homogeneous ideal in the polynomial ring over a field R =
k[x
1
, , x
s
]. Then d(I
n
) is asymptotically a linear function in slope d, i.e : d(I
n
) = dn +e
for n  0.
Proof. The linearity part was implicitly appeared in [CHT], we will recall and make it
preciser here.
Define reg
i
(I) := max{a|Tor
i
(k, L)
a
= 0} − i. Theorem 3.1 in [CHT] asserts that reg

i
(I
n
)
10
is linear for n  0. In particular, reg
0
(I
n
) is linear for n  0.
Note that Tor
0
(k, I
n
) is nothing but k ⊗
R
I
n
. Since all the element of k has degree 0, it is
easy to check that the maximal number a such that (k ⊗
R
I
n
)
a
= 0 is the maximal degree
of generator of d(I
n
). Therefore d(I
n

) is linear for n  0. We know that dt + ε(M) ≤
d(I
t
) ≤ dt + e, then divide both side of these inequalities by t and taking the limit as
t → ∞, we can conclude that the slope of that linear function is d. 
5.2 The free coefficent in case of regularity
The problem of finding the constant e in case of regularity is very hard and in fact, there
are only several results in some special situation.In [EH], D.Eisenbud and J.Harris has
characterized Castelnuovo-Mumford regularity in more geometrical way, they consider the
ideal of a projective scheme X in P
n
, the information of regularity is received under the
fiber of morphism X → P
n
. Using this result, they give a explicit formula for the free
coefficient e in the case of polynomial ring in 2 variables. We recall here a theorem in
that paper, which though we find the number e but the hypothesis are very strong.
Theorem 5.2. Let M be a finitely generated graded module over a polynomial ring over
a field S generated in degree 0 and let I ⊂ S be a homogeneous ideal generated by form
of degree d. If M/IM has finite length, but M does not, then we may write:
i-reg(M/I
t
M) = dt + f
t
− 1 with f
1
≥ f
2
≥ ≥ 0.
ii-reg(I

t
M) = dt + e
t
with e
1
≥ e
2
≥ ≥ 0.
Moreover, e = inf {e
t
}, then reg(I
t
M) = dt + e for t  0.
In order to prove the above theorem, we need a lemma:
Lemma 5.3. Let 0 → L → M → N → 0 be a short exact sequence of graded R-modules,
then :
i- regL ≤ max{regM, regN + 1}. Equality holds if regM = regN.
ii-regM ≤ max{regL, regN} . Equality holds if regL − 1 = regN, or L
n
= 0 for n  0.
We will not prove the lemma here. For reader who interested in, we refer to [E-Corollary
20.19].
Sketch of the proof of theorem Firstly we prove that reg(M/I
t
M) = dt + f
t
− 1 with
f
1
≥ f

2
≥ ≥ 0.
From the hypothesis, we know that M/I
t
M has finite length, hence, using the character-
ization of regularity under local cohomology module, we have reg(M/I
t
M = a(M/I
t
M).
Let f
t
be the minimal number such that I
t
M constains all the homogeneous element of
M of degree at least dt + f
t
. Then a(M/I
t
M) ≤ dt + f
t
.
Let m be the maximal graded ideal of R. Now note that I generated by single degree d
11
then I ⊆ m
d
. Thus :
m
f
t

I
t
M = m
f
t
II
t−1
M ⊆ m
d+f
t
I
t−1
M ⊆ m
dt+f
t
M
Hence : m
f
t
I
t
M/I
t
M ⊆ m
dt+f
t
M/I
t
M. By the definition of f
t

and taking into account
that reg(M/I
t
M) ≤ dt + f
t
we get m
f
t
I
t
M = m
dt+f
t
M. And hence : m
f
t
I
t+1
M =
m
d(t+1)+f
t
M, so f
t
≥ f
t+1
. Since I is generated by element of degree d, then it is easy to
check that f
t
≥ 0.

Now we turn to the second part. Let N be the largest submodule of finite length in M.
If N = 0, consider the exact sequence :
0 → I
t
M → M → M/I
t
M → 0
Applying the lemma, and note that M/I
t
M has finite length but M does not we have :
reg(I
t
M) = max{reg(M), reg(M/I
t
M) + 1}
Since reg(M/I
t
M) tends to infinity if t → ∞, but regM is finite, then for t large enough,
we have reg(I
t
M) = dt + f
t
and e
t
= f
t
follows.
The general case can be deduced by taking the exact sequence :
0 → I
t

M ∩ N → I
t
M → I
t
(M/N) → 0
Since I
t
M ∩ N has finite length while I
t
(M/N) has no finite length submodule except 0
then
reg(I
t
M) = max{reg(I
t
M ∩ N), reg(I
t
(M/N))}
If t is replaced by t + 1 then it is easy to see that reg(I
t
M ∩ N) does not increase, but
reg(I
t
(M/N)) increases at most d, then e
t
≥ e
t+1
. Since reg(I
t
(M/N)) grows without

bounds, it eventually dominates, therefore e
t
= f
t
for t  0. The proof is completed. 
The above theorem requires a lot of conditions for both I and M. Unfortunately, these
conditions are strict, i.e the theorem will not hold if M = S, if dim I ≥ 2, the ground
field has characteristic diffenrent from 2, then the ideal I associated to the triangulation
of the projective plane has regularity reg(I) = 3 however reg(I
2
) = 7.
The minimal number t
0
such that the function reg(I
t
) is linear for all t ≥ t
0
also be
interested, but we also know a little of it. Eisenbud and Ulrich[EU] prove that as in the
case I is equigenerated in degree d and m-primary, then we have a lower bound for t
0
:
t ≥ max {1 +
1 + reg(M)
d
, N}, in which N is the regularity of the Rees module R(I, M)
with respect to ideal of the Rees ring R(I) generated by the variables corresponding to
generators of I. Once again the role of the Rees algebra is very important. The problem
of finding out the number e and t
0

in general is now still open.
12
6. Conclusion
In this thesis, I presented the asymptotic linearity of Castelnuovo-Mumford regularity,
based almost on the paper [TW]. I proved the Proposition 1.3 without using the idea
in that paper(the authors did not prove it, they left the idea of the proof only), and also
made other proofs more precise. In the section Open Problems, the result of Theorem
5.1 is received implicitly in [CHT], but I made it explicitly and pointed out the slope of
that linear function. I also sketch the proof of Theorem 5.2(already proved in [EH]) in
more details.
13
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14