REGULARITY BOUNDS IN TERM OF PARTIAL ELIMINATION
IDEALS AND APPLICATIONS
THANH VU
Abstract. Let X be a non-degenerate projective variety over an algebraically
closed field k. Let RX be the coordinate ring of X. In this paper, we prove that
the Castelnuovo-Mumford regularity of RX is bounded by deg X − codim X
under the assumption that RX satisfies property Ncodim X−1 . To accomplish
this, we give a bound of the regularity of an arbitrary homogeneous ideal in
terms of its partial elimination ideals.
1. Introduction
Let X ⊂ PN be a non-degenerate projective variety over an algebraically closed
field k. Let RX , S be the coordinate rings of X and PN respectively. For each
finitely generated graded S-module M , the Castelnuovo-Mumford regularity of M
is defined to be
regS M = sup{t : TorSi (M, k)i+t 6= 0 for some i}.
The Castelnuovo-Mumford regularity of M is among the most important numerical
invariant of M . It measures the complexity of M . The Castelnuovo-Mumford
regularity of RX , is a bound for the Castelnuovo-Mumford regularity of X, which
is the least integer r such that H i (X, OX (m − i)) = 0 for all i ≥ 0 and all m > r.
Bounding the regularity of X or RX is a fundamental problem in algebraic geometry.
When X is a smooth curve, in the beautiful paper [3], Gruson, Lazarsfeld and
Peskine proved that reg RX ≤ deg X − codim X. Motivating by this result, in [1],
Eisenbud and Goto made the following conjecture:
Conjecture 1 (Eisenbud-Goto). Let X be a non-degenerate projective variety over
an algebraically closed field k with coordinate ring RX . Then
reg RX ≤ deg X − codim X.
Besides the case of curves, which was generalized for any connected curve by
Giaimo [2], the conjecture is known for smooth surfaces by Lazarsfeld [7], and
smooth three folds in P5 by Kwak [6]. Very good bound for general smooth three
folds and smooth four folds are given by Kwak [5]. The conjecture is open in
general. In this paper, we prove
Theorem 1.1. Let X ⊂ PN be a non-degenerate projective variety over an algebraically closed field k with coordinate ring R. Furthermore, assume that RX
Date: September 1, 2014.
2010 Mathematics Subject Classification. Primary 13D02, 13F55, 05E40.
Key words and phrases. Castelnuouvo-Mumford regularity, Eisenbud-Goto conjecture, Partial
elimination ideals.
1
2
THANH VU
satisfies property Ncodim X−1 . Then
reg RX ≤ deg X − codim X.
For example, any variety of codimension 2 defined by quadrics satisfies the
Eisenbud-Goto conjecture. As another example, any variety embedded by a high
enough degree very ample line bundle satisfies the Eisenbud-Goto conjecture.
The idea of the proof of Theorem 1.1 is the use of partial elimination ideals,
which was introduced by Green in his lecture note [2]. Recently, a more careful
analysis of these ideals have been succesfully used by Han and Kwak [4] to give
lower bounds on the betti numbers, the Np property of projections.
In this paper, we will carry the analysis of partial elimination ideals for regularity
problems. We have the following general setting. Let S = k[x0 , . . . , xn ] be a
polynomial ring over an arbitrary field k. Let R = k[x1 , . . . , xn ]. Let I be a
homogeneous ideal of S. We will define the partial elimination ideals in section 2
and prove:
Theorem 1.2. Let s be the stablization index. We have
regS I ≤ max (regR Ki (I) + i) .
i=0,...,s
As an application, in section 3, we prove Theorem 1.1.
2. Regularity bounds in term of regularity of partial elimination
ideals
In this section we recall the definition of the partial elimination ideals, and some
of their properties. The main result of the section is the bound of the regularity of
an ideal I in term of the regularity of the partial elimination ideals.
Let R = k[x1 , · · · , xn ] ⊂ S = k[x0 , · · · , xn ] be polynomial rings over an arbitrary
field k. Let I ⊂ S be a homogeneous ideal of S. Fix an elimination monomial term
order with respect to x0 on S. For each f ∈ I, let d0 (f ) be the x0 -degree of the
initial term of f .
Definition 2. The ith R-module approximation of I is
e i (I) = {f ∈ I : d0 (f ) ≤ i}.
K
e i (I) is
Note that I is an infinitely generated graded R-module, while for each i, K
a finitely generated graded R-module. We will see that the S-homological invariants
e i (I). For each f ∈
of I are reflected nicely in the R-homological invariants of K
i ¯
e
Ki (I), we may write uniquely f = x0 f + g, where f is a homogeneous form in R
(could be zero), and g is a homogeneous form in S such that d0 (g) < i.
e i (I) under
Definition 3. The ith partial elimination ideal of I is the image of K
the map f → f¯. We denote the ith partial elimination ideal of I by Ki (I).
Remark 4.
(1) The 0th partial elimination ideal of I is the actual elimination
ideal of I with respect to the variable x0 .
(2) The ideals Ki (I) form an ascending chain in R, thus stabilize at some finite
step. We call s the stabilization index of I, i.e. the smallest integer s such
that Ki (I) = Ki+1 (I) for all i ≥ s.
REGULARITY OF PEIS
3
For simplicity of notation, since we fix an ideal I in this section, we will write Ki
e i for K
e i (I). Also, sometimes we will use T S (−) for TorS (−, k)i+j
for Ki (I) and K
i
i,j
R
and Ti,j
(−) for TorR
i (−, k)i+j to make a long exact sequence or a large commutative
diagram easier to see. Here are some simple properties of these partial elimination
ideals:
Lemma 2.1. The following sequences of R-modules are exact for every i ∈ Z:
(2.1)
e i−1 (I) → K
e i (I) → Ki [−i] → 0
0→K
(2.2)
0→
e i−1 (I)
e i (I)
K
K
Ki (I)
·x0
[−1] −→
→
[−i] → 0
e i−2 (I)
e i−1 (I)
Ki−1 (I)
K
K
To connect the syzygies of an S-module M with those of the R-module approximations of M we will use the following:
Proposition 2.2 (Elimination mapping cone sequence). Let M be a graded Smodule which is not necessarily finitely generated. Then we have a natural long
exact sequence
R
· · · → TorSi (M, k)i+j → TorR
i−1 (M, k)i−1+j → Tori−1 (M, k)i−1+j+1
→ TorSi−1 (M, k)i−1+j+1 → · · ·
Proof. Note that the Koszul complex K(x0 , . . . , xn ; M ) is the mapping cone of the
map of complexes:
x
0
K(x1 , . . . , xn ; M )[−1] −→
K(x1 , . . . , xn ; M ).
The proposition follows by taking the long exact sequence of the mapping cone
sequence and the fact that Tor modules can be computed from the homology groups
of the Koszul complexes.
Proposition 2.3 (Approximation of syzygies). For any given i, j ≥ 0, we have
R e
∼
TorR
i (I, k)i+j = Tori (Kd , k)i+j
for any d ≥ j − 1.
Proof. From exact sequence (2.1), we have a long exact sequence for any i, j:
R
R e
R e
R
Ti+1,j−d−2
(Kd+1 ) → Ti,j
(Kd , k) → Ti,j
(Kd+1 , k) → Ti,j−d−1
(Kd+1 ).
Given that d ≥ j − 1, then i + j − d − 1 ≤ i, the first and the last term vanish.
Therefore, we have
R e
∼
e
TorR
i (Kh (I), k)i+j = Tori (Kh+1 (I), k)i+j
for any h ≥ j−1. Since Tor commutes with direct limits the Proposition follows.
Lemma 2.4 (First Isomorphism of Tor). Let s be the stabilization index. For any
a > b ≥ s, we have the following isomorphisms for any i and j
(2.3)
·x
0
R e
e e
e
TorR
i (Ka /Kb , k)i+j −→ Tori (Ka+1 /Kb+1 , k)i+j+1
Proof. We will prove by induction on a − b. Assume that a − b = 1. In this case,
the conclusion follows from the exact sequence (2.2) applied to i = a, and the fact
that Ki /Ki−1 ∼
= 0.
4
THANH VU
Now assume that the statement is true for a − b. We will now prove the claim
for a + 1 and b. Consider the following commutative diagram of graded R-modules:
e a /K
e b [−1] −−−−→ K
e a+1 /K
e b [−1] −−−−→ K
e a+1 /K
e a [−1] −−−−→ 0
0 −−−−→ K



·x
·x
·x
y 0
y 0
y 0
e a+1 /K
e b+1 −−−−→ K
e a+2 /K
e b+1 −−−−→ K
e a+2 /K
e a+1 −−−−→ 0.
0 −−−−→ K
From that, we have induced commutative diagrams for any i and j:
R e
eb)
Ti,j
(Ka /K

·x
y 0
R e
eb)
Ti,j
(Ka+1 /K

·x
y 0
−−−−→
−−−−→
R e
ea)
Ti,j
(Ka+1 /K

·x
y 0
R
e a+1 /K
e b+1 ) −−−−→ T R (K
e a+2 /K
e b+1 ) −−−−→ T R (K
e a+2 /K
e a+1 )
Ti,j+1
(K
i,j+1
i,j+1
By induction, the left and the right vertical arrows are isomorphisms, therefore the
middle arrow is an isomorphism. The lemma follows.
Lemma 2.5 (Second Isomorphism of Tor). Let s be the stabilization index. Denote
e s−1 , regS Ks +s). For any d ≥ s−1, and j ≥ t+1, the multiplication
t = max(regS K
map
·x0
R e
e
TorR
i (Kd , k)i+j −→ Tori (Kd+1 , k)i+j+1
(2.4)
is an isomorphism for any i.
Proof. We will prove by induction on d. For the base case, assume that d = s − 1.
e s−1 + 1, the first term is zero. Moreover, from the
In this case, since j ≥ regR K
exact sequence (2.1) applied to i = s − 1, we see that
e s ≤ max regR K
e s−1 , regR Ks + s = t.
regR K
Therefore, when j ≥ t + 1, the second term is also zero, the claim is trivial. For
the induction step, assume that the conclusion is true for d ≥ s − 1. Consider the
commutative diagram:
e d [−1] −−−−→ K
e d+1 [−1] −−−−→ Kd+1 [−d − 2] −−−−→ 0
0 −−−−→ K



x
x
ϕ
y 0
y 0
y
e d+1 −−−−→
0 −−−−→ K
e d+2
K
−−−−→ Kd+2 [−d − 2] −−−−→ 0,
where the induced map ϕ is an isomorphism. For each i and j, we have an induced
commutative diagram on Tor:
e
TorR
i (Kd , k)i+j

x
y 0
−−−−→
e
TorR
i (Kd+1 , k)i+j

x
y 0
−−−−→ TorR
i (Kd+1 , k)i+j−d−2

ϕ
y
R e
R
e
TorR
i (Kd+1 , k)i+j+1 −−−−→ Tori (Kd+2 , k)i+j+1 −−−−→ Tori (Kd+2 , k)i+j−d−2
The left vertical arrow is an isomorphism by induction, the right vertical arrow is
an isomorphism since ϕ is an isomorphism. Therefore, the middle vertical arrow is
an isomorphism. The lemma follows.
We are now ready for the main technical result of the section:
REGULARITY OF PEIS
5
Theorem 2.6 (Comparison of regularity). Let s be the stabilization index. Then
e s−1 , regR Ks + s .
regS I ≤ max regR K
e s−1 , regR Ks + s). By the appoximation of syzygies
Proof. Denote t = max(regR K
(Proposition 2.3) and Lemma 2.5, the induced multiplication map on Tor
·x
0
R
TorR
i (I, k)i+j −→ Tori (I, k)i+j+1
is an isomorphism for all i and j ≥ t + 1.
From the elimination mapping cone sequence applied to I, it suffices now to
prove that for all i the multiplication map
·x
0
R
TorR
i (I, k)i+t −→ Tori (I, k)i+t+1
is a surjection.
By the approximation of syzygies, we proceed as in the proof of Lemma 2.5. We
will prove by induction on d that the multiplication map
·x0
R e
e
(2.5)
TorR
i (Kd , k)i+t −→ Tori (Kd+1 , k)i+t+1
is a surjection for any i and any d ≥ s − 1.
For the base case, assume that d = s − 1. In this case, as in the proof of Lemma
e s + 1. Therefore, the second module is zero, the claim is
2.5, we have j ≥ regR K
trivial. For the induction step, assume that the conclusion is true for d ≥ s − 1.
Consider the commutative diagram:
e d [−1] −−−−→ K
e d+1 [−1] −−−−→ Kd+1 [−d − 2] −−−−→ 0
0 −−−−→ K



x
x
ϕ
y 0
y 0
y
e d+1 −−−−→ K
e d+2 −−−−→ Kd+2 [−d − 2] −−−−→ 0,
0 −−−−→ K
where the induced map ϕ is an isomorphism. For each i, we have an induced
commutative diagram on Tor:
R
R e
R e
R
Ti+1,t+d−2
(Kd+1 ) −−−−→
Ti,t
(Kd )
−−−−→ Ti,t
(Kd+1 ) −−−−→ Ti,t+d−2
(Kd+1 )




ϕ
·x
·x
ϕ
y
y 0
y 0
y
R
R
e d+1 ) −−−−→ T R (K
e d+2 ) −−−−→ T R
Ti+1,t+d−2
(Kd+1 ) −−−−→ Ti,t+1
(K
i,t+1
i,t+d−2 (Kd+2 )
The left most and right most vertical arrows are isomorphism since ϕ is an isomorphism. The second vertical arrow is a surjection by induction, therefore, the third
vertical arrow is a surjection by the four lemma. The theorem follows.
For the purpose of application, it is better to reduce the comparison to the
regularity of the partial elimination ideals Ki as stated in Theorem 1.2. We are
now ready for its proof.
Proof of Theorem 1.2. From the exact sequence (2.1), we have
e i ) ≤ max regR K
e i−1 , regR Ki + i .
regR (K
The proof is completed by induction on i and Theorem 2.6.
For our purpose of application later, the following improvement on the bound of
the stabilization index s will be useful:
6
THANH VU
Proposition 2.7 (Stabilization index). Fix a minimal system of generators h1 , . . . , hp
of I. Then the stabilization index satisfies:
s ≤ max (d0 (hi )) .
i=1,...,p
Proof. Let t = maxi=1,...,p (d0 (hi )). We will show that for any u ≥ t, we have
Ku+1 = Ku . Assume that there exists an u ≥ t such that there exists f ∈ Ku+1
but f is not in Ku . We may assume that f is such an element with smallest leading
term (we may always assume that the coefficient of the leading term of f is 1). Let
f˜ = xu+1 f + g
be an element of I such that d0 (g) < u + 1. Since f˜ ∈ I, we can write
f˜ = x0 · q̃ +
p
X
ai · hi
i=1
with the property that q̃ ∈ I, and moreover, no term of ai has x0 for any i = 1, . . . , p.
Since d0 (hi ) ≤ t, we have
in(f˜) = in(x0 · q̃)
xu+1
0
xu0
· in(f ) = x0 · in(q̃)
· in(f ) = in(q̃).
Since q̃ ∈ I, we can write
q̃ = xu0 · q + r
such that d0 (r) < u. By definition q ∈ Ku . Moreover, in(f ) = in(q). Therefore, we
have
f˜ − x0 · q̃ = xu+1
(f − q) + (g − x0 · r).
0
Since f ∈
/ Ks , we have f − q 6= 0. Since d0 (g − x0 · r) < u + 1, and the fact that
f˜ − x0 · q̃ ∈ I, this implies that f − q ∈ Ks+1 . Moreover, since in(f ) = in(q), we
have in(f − q) < in(f ). By the choice of f , this implies that f − q ∈ Ks . Since
q ∈ Ks , this implies that f ∈ Ks which is a contradiction.
3. Regularity bounds for algebraic varieties
n
Let X ⊂ P be a non-degenerate projective algebraic variety, i.e. irreducible and
integral, over an algebraically closed field k. Let S = k[x0 , . . . , xn ] be the coordinate
ring of Pn and I be the defining ideal of X. Assume that q = (1 : 0 : · · · : 0) ∈ X by
suitable change of coordinate. In this case, the defining ideal of the inner projection
of X from q is the elimination ideal of I with respect to x0 (see [4] for more detail).
Since k is algebraically closed, so a general point on X exists. The theory of section
2 applies to give a proof of Theorem 1.1. Also, recall that RX is said to satisfies
property Np if RX is defined by quadrics and that the resolution of RX is linear up
to pth step.
Proof of Theorem 1.1. We prove by induction on the codimension of X. When
X has codimension 1, we have equality. Now, assume that the theorem is true
for codimension c − 1 and that X is a projective variety of codimension c. Let I
be the defining ideal of X. Let q be a general point of X. Let Y be the inner
projection of X corresponding to the point q ∈ X. Under the assumption that the
REGULARITY OF PEIS
7
coordinate ring of X satisfies property Nc−1 , we have by [4, Proposition 2.5 (b)]
the stablization index is 1, and K1 (I) is a linear ideal. By Theorem 1.2, we have
reg I ≤ max(reg K0 (I), reg K1 (I) + 1) ≤ reg K0 (I).
Since K0 (I) is the defining ideal of Y , and moreover the coordinate ring of Y
satisfies property Nc−2 , by induction on codimension we have
reg RX ≤ reg RY ≤ deg Y − codim Y.
Moreover, we have codim X = codim Y + 1, and deg X ≥ deg Y + 1. Therefore,
reg RX ≤ deg Y − codim Y ≤ deg X − codim X.
The theorem follows.
Acknowledgement
The paper was initiated while the author visited Korea Institute for Advanced
Study in Seoul in November 2013. He is greatful for the support and hospitality
from KIAS .
References
[1] D. Eisenbud, S. Goto, Linear free resolutions and minimal multiplicity., J. Algebra 88 (1984),
no. 1, 89-133.
[2] D. Giaimo, On the Castelnuovo-Mumford regularity of connected curves., Trans. Amer. Math.
Soc. 358 (2006), no. 1, 267-284.
[3] L. Gruson, R. Lazarsfeld, C. Peskine, On a theorem of Castelnuovo and the equations defining
projective varieties., Invent. Math. 72 (1983), 491-506.
[4] K. Han, S. Kwak, Analysis on some infinite modules, inner projections, and applications.,
Trans. Amer. Math. Soc. 364 (2012), no. 11, 5791–5812.
[5] S. Kwak, Castelnuovo regularity for smooth subvarieties of dimensions 3 and 4., J. Algebraic
Geom. 7 (1998), no. 1, 195-206.
[6] S. Kwak, Castelnuovo-Mumford regularity bound for smooth three folds in P5 and extremal
examples., J. Reine Angew. Math. 509 (1999), 21-34.
[7] R. Lazarsfeld, A sharp Castelnuovo bound for smooth surfaces., Duke Math. J. 55 (1987), no.
2, 423-429.
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska
68588, United States of America
E-mail address: [email protected]