Introduction
Vanishing Conjecture
Bounds
H ILBERT C OEFFICIENTS AND
R EDUCTION N UMBERS
Laura Ghezzi
New York City College of Technology
The City University of New York
Halifax, October 2014
Bibliography
Introduction
Vanishing Conjecture
Bounds
Abstract
Let (R, m) be a Noetherian local ring and let I be an
m-primary ideal. We discuss relationships between the
Hilbert coefficients of I and its reduction number.
Our general “philosophy” is to extend formulas from a
Cohen-Macaulay ring to a general Noetherian ring.
This is joint work with Goto, Hong, and Vasconcelos.
Bibliography
Introduction
Vanishing Conjecture
Bounds
Bibliography
Set-up and notation
Let (R, m) be a Noetherian local ring of dimension d > 0
with infinite residue field, and let I be an m-primary ideal.
It is well known that for n 0 the Hilbert-Samuel function
HI1 (n) = λ(R/I n+1 )
(where λ denotes length) is a polynomial in n of degree d,
n+d
n+d −1
n+1
λ(R/I ) = e0 (I)
−e1 (I)
+· · ·+(−1)d ed (I).
d
d −1
The integers e0 (I), e1 (I), . . . , ed (I) are called the Hilbert
coefficients of I.
Introduction
Vanishing Conjecture
Bounds
Bibliography
Set-up and notation
Let (R, m) be a Noetherian local ring of dimension d > 0
with infinite residue field, and let I be an m-primary ideal.
It is well known that for n 0 the Hilbert-Samuel function
HI1 (n) = λ(R/I n+1 )
(where λ denotes length) is a polynomial in n of degree d,
n+d
n+d −1
n+1
λ(R/I ) = e0 (I)
−e1 (I)
+· · ·+(−1)d ed (I).
d
d −1
The integers e0 (I), e1 (I), . . . , ed (I) are called the Hilbert
coefficients of I.
Introduction
Vanishing Conjecture
Bounds
Bibliography
Reductions
Recall that a subideal Q of an ideal I is a reduction of I if
I n+1 = QI n for some integer n ≥ 0.
The reduction number of I with respect to Q is the
minimum integer n such that I n+1 = QI n .
A reduction is minimal if it is minimal with respect to
inclusion.
The reduction number of I is
red(I) = min{redQ (I) | Q is a minimal reduction of I}.
Introduction
Vanishing Conjecture
Bounds
Bibliography
Hilbert coefficients of parameter ideals
Let Q be a parameter ideal of R, that is, Q is an
m-primary ideal generated by d elements.
For instance, if Q is a minimal reduction of the m-primary
ideal I, then Q is a parameter ideal.
If R is Cohen-Macaulay, then e0 (Q) = λ(R/Q) and
ei (Q) = 0 for i ≥ 1.
The value e1 (Q) occurs as a “correction term” in the
extensions of several well-known formulas in the
theory of Hilbert polynomials from a Cohen-Macaulay
ring to a general Noetherian ring (Corso,
Goto-Nishida, Rossi-Valla).
Introduction
Vanishing Conjecture
Bounds
Bibliography
Hilbert coefficients of parameter ideals
Let Q be a parameter ideal of R, that is, Q is an
m-primary ideal generated by d elements.
For instance, if Q is a minimal reduction of the m-primary
ideal I, then Q is a parameter ideal.
If R is Cohen-Macaulay, then e0 (Q) = λ(R/Q) and
ei (Q) = 0 for i ≥ 1.
The value e1 (Q) occurs as a “correction term” in the
extensions of several well-known formulas in the
theory of Hilbert polynomials from a Cohen-Macaulay
ring to a general Noetherian ring (Corso,
Goto-Nishida, Rossi-Valla).
Introduction
Vanishing Conjecture
Bounds
Bibliography
Hilbert coefficients of parameter ideals
Let Q be a parameter ideal of R, that is, Q is an
m-primary ideal generated by d elements.
For instance, if Q is a minimal reduction of the m-primary
ideal I, then Q is a parameter ideal.
If R is Cohen-Macaulay, then e0 (Q) = λ(R/Q) and
ei (Q) = 0 for i ≥ 1.
The value e1 (Q) occurs as a “correction term” in the
extensions of several well-known formulas in the
theory of Hilbert polynomials from a Cohen-Macaulay
ring to a general Noetherian ring (Corso,
Goto-Nishida, Rossi-Valla).
Introduction
Vanishing Conjecture
Bounds
Bibliography
History–The vanishing of e1(Q)
When d = 1, e1 (Q) = −λ(Hm0 (R)), so that
e1 (Q) = 0 ⇐⇒ R is Cohen Macaulay.
We have e1 (Q) ≤ 0 [GGHOPV, 2010], [MSV, 2011].
In general e1 (Q) = 0 =⇒ R is Cohen Macaulay does
not hold if d ≥ 2.
Example: Let R = k[[x, y , z]]/(z(x, y, z)).
R is unmixed, if dim R̂/p = d for all p ∈ Ass(R̂).
Introduction
Vanishing Conjecture
Bounds
Bibliography
History–The vanishing of e1(Q)
When d = 1, e1 (Q) = −λ(Hm0 (R)), so that
e1 (Q) = 0 ⇐⇒ R is Cohen Macaulay.
We have e1 (Q) ≤ 0 [GGHOPV, 2010], [MSV, 2011].
In general e1 (Q) = 0 =⇒ R is Cohen Macaulay does
not hold if d ≥ 2.
Example: Let R = k[[x, y , z]]/(z(x, y, z)).
R is unmixed, if dim R̂/p = d for all p ∈ Ass(R̂).
Introduction
Vanishing Conjecture
Bounds
Bibliography
History–The vanishing of e1(Q)
When d = 1, e1 (Q) = −λ(Hm0 (R)), so that
e1 (Q) = 0 ⇐⇒ R is Cohen Macaulay.
We have e1 (Q) ≤ 0 [GGHOPV, 2010], [MSV, 2011].
In general e1 (Q) = 0 =⇒ R is Cohen Macaulay does
not hold if d ≥ 2.
Example: Let R = k[[x, y , z]]/(z(x, y, z)).
R is unmixed, if dim R̂/p = d for all p ∈ Ass(R̂).
Introduction
Vanishing Conjecture
Bounds
Bibliography
History–The vanishing of e1(Q)
When d = 1, e1 (Q) = −λ(Hm0 (R)), so that
e1 (Q) = 0 ⇐⇒ R is Cohen Macaulay.
We have e1 (Q) ≤ 0 [GGHOPV, 2010], [MSV, 2011].
In general e1 (Q) = 0 =⇒ R is Cohen Macaulay does
not hold if d ≥ 2.
Example: Let R = k[[x, y , z]]/(z(x, y, z)).
R is unmixed, if dim R̂/p = d for all p ∈ Ass(R̂).
Introduction
Vanishing Conjecture
Bounds
Bibliography
History–The vanishing of e1(Q)
When d = 1, e1 (Q) = −λ(Hm0 (R)), so that
e1 (Q) = 0 ⇐⇒ R is Cohen Macaulay.
We have e1 (Q) ≤ 0 [GGHOPV, 2010], [MSV, 2011].
In general e1 (Q) = 0 =⇒ R is Cohen Macaulay does
not hold if d ≥ 2.
Example: Let R = k[[x, y , z]]/(z(x, y, z)).
R is unmixed, if dim R̂/p = d for all p ∈ Ass(R̂).
Introduction
Vanishing Conjecture
Bounds
Bibliography
e1(Q) encodes information about R
Let R be an unmixed Noetherian local ring with d ≥ 1.
Let Λ(R) = {e1 (Q) | Q is a parameter ideal of R}.
Theorem [GGHOPV, 2010]
(i) R is Cohen-Macaulay ⇐⇒ 0 ∈ Λ(R) (Vanishing
Conjecture);
(ii) R is generalized Cohen-Macaulay ⇐⇒ |Λ(R)| < ∞;
(iii) R Buchsbaum ⇐⇒ |Λ(R)| = 1.
Recently the authors extended the results to modules
[GGHOPV2, to appear].
Introduction
Vanishing Conjecture
Bounds
Bibliography
e1(Q) encodes information about R
Let R be an unmixed Noetherian local ring with d ≥ 1.
Let Λ(R) = {e1 (Q) | Q is a parameter ideal of R}.
Theorem [GGHOPV, 2010]
(i) R is Cohen-Macaulay ⇐⇒ 0 ∈ Λ(R) (Vanishing
Conjecture);
(ii) R is generalized Cohen-Macaulay ⇐⇒ |Λ(R)| < ∞;
(iii) R Buchsbaum ⇐⇒ |Λ(R)| = 1.
Recently the authors extended the results to modules
[GGHOPV2, to appear].
Introduction
Vanishing Conjecture
Bounds
Bibliography
e1(Q) encodes information about R
Let R be an unmixed Noetherian local ring with d ≥ 1.
Let Λ(R) = {e1 (Q) | Q is a parameter ideal of R}.
Theorem [GGHOPV, 2010]
(i) R is Cohen-Macaulay ⇐⇒ 0 ∈ Λ(R) (Vanishing
Conjecture);
(ii) R is generalized Cohen-Macaulay ⇐⇒ |Λ(R)| < ∞;
(iii) R Buchsbaum ⇐⇒ |Λ(R)| = 1.
Recently the authors extended the results to modules
[GGHOPV2, to appear].
Introduction
Vanishing Conjecture
Bounds
Bibliography
e1(Q) encodes information about R
Let R be an unmixed Noetherian local ring with d ≥ 1.
Let Λ(R) = {e1 (Q) | Q is a parameter ideal of R}.
Theorem [GGHOPV, 2010]
(i) R is Cohen-Macaulay ⇐⇒ 0 ∈ Λ(R) (Vanishing
Conjecture);
(ii) R is generalized Cohen-Macaulay ⇐⇒ |Λ(R)| < ∞;
(iii) R Buchsbaum ⇐⇒ |Λ(R)| = 1.
Recently the authors extended the results to modules
[GGHOPV2, to appear].
Introduction
Vanishing Conjecture
Bounds
Bibliography
e1(Q) encodes information about R
Let R be an unmixed Noetherian local ring with d ≥ 1.
Let Λ(R) = {e1 (Q) | Q is a parameter ideal of R}.
Theorem [GGHOPV, 2010]
(i) R is Cohen-Macaulay ⇐⇒ 0 ∈ Λ(R) (Vanishing
Conjecture);
(ii) R is generalized Cohen-Macaulay ⇐⇒ |Λ(R)| < ∞;
(iii) R Buchsbaum ⇐⇒ |Λ(R)| = 1.
Recently the authors extended the results to modules
[GGHOPV2, to appear].
Introduction
Vanishing Conjecture
Bounds
Bibliography
Generalized CM property
Recall that R is generalized Cohen-Macaulay (gCM) if
Hmi (R) is finitely generated for 0 ≤ i ≤ d − 1. By a result
of Goto and Nishida in a gCM ring we have that for all
parameter ideals Q,
d−1 X
d −2
e1 (Q) ≥ −
λ(Hmi (R)).
i −1
i=1
In particular Λ(R) = {e1 (Q) | Q is a parameter ideal of R}
is a finite set.
Assume that R is unmixed. We show that
Λ(R) is finite ⇐⇒ R is generalized Cohen-Macaulay.
Introduction
Vanishing Conjecture
Bounds
Bibliography
Buchsbaum property
Recall that R is Buchsbaum if every parameter ideal Q of
R is standard, that is,
P
d−1
λ(Hmi (R)).
λ(R/Q) − e0 (Q) = d−1
i=0
i
By a result of Schenzel in a Buchsbaum ring we have that
for all parameter ideals Q,
d−1 X
d −2
e1 (Q) = −
λ(Hmi (R)).
i −1
i=1
In particular Λ(R) = {e1 (Q) | Q is a parameter ideal of R}
is a constant set.
Assume that R is unmixed. We show that
Λ(R) is constant ⇐⇒ R is Buchsbaum.
Introduction
Vanishing Conjecture
Bounds
Bibliography
Buchsbaum property
Recall that R is Buchsbaum if every parameter ideal Q of
R is standard, that is,
P
d−1
λ(Hmi (R)).
λ(R/Q) − e0 (Q) = d−1
i=0
i
By a result of Schenzel in a Buchsbaum ring we have that
for all parameter ideals Q,
d−1 X
d −2
e1 (Q) = −
λ(Hmi (R)).
i −1
i=1
In particular Λ(R) = {e1 (Q) | Q is a parameter ideal of R}
is a constant set.
Assume that R is unmixed. We show that
Λ(R) is constant ⇐⇒ R is Buchsbaum.
Introduction
Vanishing Conjecture
Bounds
Bibliography
Variation of the first Hilbert coefficients
e1 occurs as a correction term in the extensions of several
well-known formulas in the theory of Hilbert polynomials.
For example, a classical result of Northcott asserts that if
R is Cohen-Macaulay, then
e1 (I) ≥ e0 (I) − λ(R/I).
For arbitrary Noetherian rings, if Q is a minimal reduction
of I, Goto and Nishida proved that
e1 (I) − e1 (Q) ≥ e0 (I) − λ(R/I).
Introduction
Vanishing Conjecture
Bounds
Bibliography
Variation of the first Hilbert coefficients
e1 occurs as a correction term in the extensions of several
well-known formulas in the theory of Hilbert polynomials.
For example, a classical result of Northcott asserts that if
R is Cohen-Macaulay, then
e1 (I) ≥ e0 (I) − λ(R/I).
For arbitrary Noetherian rings, if Q is a minimal reduction
of I, Goto and Nishida proved that
e1 (I) − e1 (Q) ≥ e0 (I) − λ(R/I).
Introduction
Vanishing Conjecture
Bounds
Bibliography
Variation of the first Hilbert coefficients
e1 occurs as a correction term in the extensions of several
well-known formulas in the theory of Hilbert polynomials.
For example, a classical result of Northcott asserts that if
R is Cohen-Macaulay, then
e1 (I) ≥ e0 (I) − λ(R/I).
For arbitrary Noetherian rings, if Q is a minimal reduction
of I, Goto and Nishida proved that
e1 (I) − e1 (Q) ≥ e0 (I) − λ(R/I).
Introduction
Vanishing Conjecture
Bounds
Bibliography
Variation of the first Hilbert coefficients
Question
What is an upper bound for e1 (I) − e1 (Q) when Q is a
(minimal) reduction of I ?
As a special case of a more general result we obtain
Theorem [GGHV, 2013]
Let (R, m) be a Noetherian local ring of dimension d ≥ 1,
let I = (Q, h1 , . . . , hm ), where Q is a minimal reduction of I.
Then
m+s
e1 (I) ≤ e1 (I) − e1 (Q) ≤ λ(R/(Q : I))·
−1 ,
s
where s is the reduction number of I with respect to Q.
Introduction
Vanishing Conjecture
Bounds
Bibliography
Variation of the first Hilbert coefficients
Question
What is an upper bound for e1 (I) − e1 (Q) when Q is a
(minimal) reduction of I ?
As a special case of a more general result we obtain
Theorem [GGHV, 2013]
Let (R, m) be a Noetherian local ring of dimension d ≥ 1,
let I = (Q, h1 , . . . , hm ), where Q is a minimal reduction of I.
Then
m+s
e1 (I) ≤ e1 (I) − e1 (Q) ≤ λ(R/(Q : I))·
−1 ,
s
where s is the reduction number of I with respect to Q.
Introduction
Vanishing Conjecture
Bounds
Bibliography
Main ideas
In general, to prove this kind of relationship we use the
fact that for n 0, λ(R/Q n ) − λ(R/I n ) is the difference of
two polynomials of degree d and with same leading
(binomial) coefficients e0 (Q) and e0 (I), therefore it is at
most a polynomial of degree d − 1 and leading coefficient
e1 (I) − e1 (Q). We also use filtrations involving powers of I
and Q and the fact that I s+1 = QI s .
Introduction
Vanishing Conjecture
Bounds
Bibliography
Upper bound for the reduction number
What is an upper bound for the reduction number of I ?
Theorem (GGHV)
Let (R, m) be a Noetherian local ring of dimension d ≥ 1.
If Q is a minimal reduction of I, we have that
red(I) ≤ max{d·λ(R/Q) − 2d + 1 , 0}.
This theorem extends a result of Vasconcelos for
Cohen-Macaulay rings. Using homological degrees we
can bound the reduction number of I in terms of I alone.
Introduction
Vanishing Conjecture
Bounds
Bibliography
Rossi’s bound
There are known bounds for the reduction number of an
ideal in terms of some of its Hilbert coefficients.
Theorem [R, 2000]
Let (R, m) be a Cohen-Macaulay local ring of dimension
at most 2. Then redQ (I) ≤ e1 (I) − e0 (I) + λ(R/I) + 1,
where Q is a minimal reduction of I.
Does it extend to higher dimensional CM rings (with a
correction term depending on the dimension)?
Which offsetting terms should be added in the non
Cohen-Macaulay case? For instance, in dimension 2
we ask whether the addition of −e1 (Q), a term that
can be considered a non Cohen-Macaulayness
“penalty”, would give a valid bound.
Introduction
Vanishing Conjecture
Bounds
Bibliography
Rossi’s bound
There are known bounds for the reduction number of an
ideal in terms of some of its Hilbert coefficients.
Theorem [R, 2000]
Let (R, m) be a Cohen-Macaulay local ring of dimension
at most 2. Then redQ (I) ≤ e1 (I) − e0 (I) + λ(R/I) + 1,
where Q is a minimal reduction of I.
Does it extend to higher dimensional CM rings (with a
correction term depending on the dimension)?
Which offsetting terms should be added in the non
Cohen-Macaulay case? For instance, in dimension 2
we ask whether the addition of −e1 (Q), a term that
can be considered a non Cohen-Macaulayness
“penalty”, would give a valid bound.
Introduction
Vanishing Conjecture
Bounds
Bibliography
Rossi’s bound
There are known bounds for the reduction number of an
ideal in terms of some of its Hilbert coefficients.
Theorem [R, 2000]
Let (R, m) be a Cohen-Macaulay local ring of dimension
at most 2. Then redQ (I) ≤ e1 (I) − e0 (I) + λ(R/I) + 1,
where Q is a minimal reduction of I.
Does it extend to higher dimensional CM rings (with a
correction term depending on the dimension)?
Which offsetting terms should be added in the non
Cohen-Macaulay case? For instance, in dimension 2
we ask whether the addition of −e1 (Q), a term that
can be considered a non Cohen-Macaulayness
“penalty”, would give a valid bound.
Introduction
Vanishing Conjecture
Bounds
Bibliography
Extension of Rossi’s bound
Question
Let (R, m) be a Noetherian local ring of dimension 2. Let
Q be a minimal reduction of I. Is
redQ (I) ≤ e1 (I)−e1 (Q) − e0 (I) + λ(R/I) + 1?
We get an affirmative answer if R is a Buchbaum unmixed
local ring of positive depth and I = m.
Such ring R has a “nice” presentation
0 −→ R −→ S −→ Hm1 (R) −→ 0,
where S = HomR (KR , KR ) is a Cohen-Macaulay ring of
dimension 2 and λ(Hm1 (R)) = −e1 (Q).
Introduction
Vanishing Conjecture
Bounds
Bibliography
Extension of Rossi’s bound
Question
Let (R, m) be a Noetherian local ring of dimension 2. Let
Q be a minimal reduction of I. Is
redQ (I) ≤ e1 (I)−e1 (Q) − e0 (I) + λ(R/I) + 1?
We get an affirmative answer if R is a Buchbaum unmixed
local ring of positive depth and I = m.
Such ring R has a “nice” presentation
0 −→ R −→ S −→ Hm1 (R) −→ 0,
where S = HomR (KR , KR ) is a Cohen-Macaulay ring of
dimension 2 and λ(Hm1 (R)) = −e1 (Q).
Introduction
Vanishing Conjecture
Bounds
Bibliography
Example
Let R = k[[x, y, z, w]]/(x, y ) ∩ (z, w). R is a Buchsbaum
ring of dimension 2, depth 1, and e1 (Q) = −1 for all
parameter ideals Q.
Let I = (x 4 , x 3 y, xy 3 , y 4 ) + (z, w)2 . Then
Q = (x 4 + z 2 , y 4 + w 2 ) is a minimal reduction of I.
Using CoCoA we have redQ (I) = 2, e0 (I) = 20, e1 (I) = 7,
λ(R/I) = 13, so that the bound is attained:
2 = redQ (I) ≤ e1 (I) − e1 (Q) − e0 (I) + λ(R/I) + 1 = 2.
Introduction
Vanishing Conjecture
Bounds
Bibliography
Bibliography
L. Ghezzi, S. Goto, J. Hong, K. Ozeki, T.T. Phuong
and W. V. Vasconcelos, Cohen–Macaulayness versus
the vanishing of the first Hilbert coefficient of
parameter ideals, J. London Math. Society 81 (2010),
679-695.
L. Ghezzi, S. Goto, J. Hong, K. Ozeki, T.T. Phuong
and W. V. Vasconcelos, The Chern numbers and
Euler characteristics of modules, Acta Mathematica
Vietnamica, to appear.
L. Ghezzi, S. Goto, J. Hong and W. V. Vasconcelos,
Variation of Hilbert coefficients, Proc. Amer. Math.
Soc. 141 (2013), 3037-3048.
Introduction
Vanishing Conjecture
Bounds
Bibliography
Bibliography
M. Mandal, B. Singh and J. K. Verma, On some
conjectures about the Chern numbers of filtrations, J.
Algebra 325 (2011), 147–162.
M. E. Rossi, A bound on the reduction number of a
primary ideal, Proc. Amer. Math. Soc. 128 (2000),
1325–1332.