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

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 =

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

, , 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

. 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 :

(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)

→ 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

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

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

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

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

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
: 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

, 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

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

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

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

. 
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)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

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

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,

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

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

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

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

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

[S]-I. Swanson, Powers of ideals, primary decompositions, ArtinRees lemma and regularity. Math.
Ann. 307, 299313, (1997)
[TW]-N. V .Trung and H J. Wang, On the asymptotic linearity of Castelnuovo-Mumford regularity,
J.Pure Appl. Algebra 201, no. 1-3, 42-48, (2005).
[Ro]-T.Romer, Homological properties of bigraded algebras, Illinois J. Math. 45, o 4, 1361-1376, (2001)
14