Séminaire Dubreil.
Algèbre
DAVID E ISENBUD
Notes on an extension of Krull’s principal ideal theorem
Séminaire Dubreil. Algèbre, tome 28, no 1 (1974-1975), exp. no 20, p. 1-4.
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20-01
Séminaire P. DUBREIL
(Algèbre)
15 avril 1975
.
1974~75 ~
annee,
28e
n° 20, 4 p.
NOTES ON AN EXTENSION OF KRULL’S PRINCIPAL IDEAL THEOREM
by David EISENBUD
(*)
generalization of Krull’s principal ideal theorem
which we can prove to be correct, for example, for rings containing a field. We
will then sketch an application to the theory of determinantal ideals ; roughly
In this note,
we
will propose
a
implies a more precise version of the theorems of MACAULAY
and EAGON on the heights of determinantal ideals. We also mention a speculative
connection between our conjecture and the intersection conjectures of SERRE and
speaking,
theorem
our
PESKINE-SZPIRO. Details will appear elsewhere.
1. The
ideal theorem.
generalized principal
Throughout
Krull’s
rings will
this paper, all
principal
ideal
theorem [5]
be assumed commutative and noetherian.
states that
an
element
in the maximal
a
height at most one (the apparently
sharper statement that the minimal primes all have height at most one follows trivially by localization). Regarding a as a homomorphism R -~.> R , and noting that
the rank of R , as an R-module, is 1, one might be lead to conjecture that
something "similar" can be said about homomorphisms from an arbitrary module into
ideal of
R . To be
Since
we
more
generates
R
local ring
a
precise,
quire that
an
should have rank
As with all notions of rank,
(k
its
+
a
module
1)th-exterior power has
notion of the rank of
Definition. - Let
M
right notion of the rank of a module.
the ring, it is not unreasonable to re0
should have
rank $
0 . These conditions
can
be
more
simply put
if, and only if,
k
uniquely specify
a
follows :
as
R . The rank of
generators of
is the minimal number of
M
0~
if, and only if,
be the set of nonzerodivisors of
U
R-module
tely generated
rank
module, which
a
to
homomorphisms
M
R-module
ideal of
needs first the
one
wish to work with
an
M~
a
finiR-
as an
module.
Of course, if
our
R
is
domain, this is the usual notion of rank. We
a
can now
state
conjecture :
Generalized principal
ideal
J , and let
M
conjecture. - Let R be a local ring with maximal
finitely generated R-module of rank n . Let
ideal
be
a
M~ ~ Hom(M , R) ,
and let
cp
be
an
element ef
JM~ .
Then the
height of
’
the ideal
cp(M)
is at most
n .
#
Jr, and myself. Both authors
This is a joint work of E. Graham
the
for
N.S.P’.
teful to the Sloan Foundation and
partial support,
are
It is easy to
Krull
can
prove
and
"depth",
weakned form of the
a
we
can
is
I s R
THEOREM 1. - Let
be
conjecture becomes the version of the
that the height of a proper n-generator
is the
be
R
of
length
’’height’’
replaced by
is
itself in many cases.
ideal, and
an
in which
conjecture,
conjecture
prove the
( I , N)
depth
this
n .
Recall that if
then
Rn ,
=
ideal theorem which states
principal
ideal is at most
We
M
that, if
see
is
N
maximal
a
R-module,
finitely generated
a
I.
in
N-sequence
J . Let
noetherian local ring with maximal ideal
a
finitely generated R-module, and let ~p E
(a) If N is a finitely generated R-module,
M
a
(b) If
sense
(possibly
has
R
[ 4],
of HOCHSTER
not
hypothesis of
conjectured to be true
cp(M) ~
of
(t)
is known to be fulfilled if
in
be
M
a
our
conjecture
part
of
a
minimal
(R , J)
Let
module of rank
that, for every prime ideal
is
field, and
a
be reformulated
can
an
element of
a
giving
as
a
module to be
system of generators :
minimal
Conjecture (second version). and let
contains
R
[ 4].
general
perhaps amusing to note that
a
modules in the
rank M .
condition, in terms of the punctered spectrum, for
part
rank M .
then
The extra
It is
generated) Cohen-Macaulay
finitely
height
is
N) $
then
J,
a
local ring of dimension
d ,
Suppose that a is an element of M such
generates a free summand of M . Then a
d .
P ~
re
a
system of generators for M.
2. Determinantal ideals.
One of the earliest
[6]
of MACAULAY
minors of
a
p
that
x
of the
generalizations
(for
ideal theorem
principal
the
polynomial rings)
height
of the ideal of
matrix, if the ideal is proper, is at most
q
the theorem
was
x
p
p
q - p + 1 . This
was
generalized by EAGON in 1960, who showed
(for
a
general noetherian
the
height
a
x
q matrix, if the ideal is
of the ideal of
(p -
proper, is at most
in
[ 3~).
k
k
x
k
minors of
l)(q -
+
On the basis of the
k +
p
1) (There
is
a
made in the last
conjecture
very
ring)
that
elegant proof of this
section,
this result to say something about what happens to the ideal of
we
k
x
can
k
extend
minors when
extra column is added to the matrix.
an
Before
nique
of
stating
our
result
on a
result that
can
be
proved by
the tech-
[ 3].
PROPOSITION. - Let 03C6
(t +
remark
we
(l + 1)
be
minors of
a
p
cp
x
q
are
matrix
all
over a
ring
R , and suppose that the
0 . Then the ideal
generated
by the l
l
height at most
has
of (p
minors
the maximal ideal, to
Suppose that
k
ideal of
k
the
x
x
a
p x q
k
minors
column,
new
so one
obtains
The next theorem shows that
p
x
mat rix
q
ove r
obtained from (p
Then the
As
(For
conjecture.
a
R
k
k
"rigidity"
a
lized the result that if
ideal of height
are
elements
n
COROLLARY. - Suppose that
R
f~,
is
a
is
... ,
x
q
in the maximal ideal, such that the ide al of
of cp
s
x
t
the
k
is at most
we can
of
fn
an
cp , the
matrix
p - k + 1 .
a
prove
local
a
ideal of
result which
ring generate
height
an
k :
satisfying the generalized princimatrix
x
k
over
mino rs
largest possible value. Then
submatrix cp of
a
a
which, in turn, genera-
generate
p
be
be
in the maximal ideal.
are
l
local ring
a
time : ) :
field.) Let (p
0 , and let
all
proposition,
of them
k
pal ideal conjecture, and that 03C6
and eve ry
a
theorem of BUCHSBAUM and
n , then any
(p-k + l~~q ... k + 1~ ,
contains
minors
k
x
consequence of the theorem and the
generalizes
least much of the
column whose entries
a
of the ideal of
height
proposition :
(at
R
mino rs
k
x
entries
p
satisfying the generalized principal
local ring
a
suppose that
example,
by adjoining
do better
can
is
who se
bound from the
a
one
R
THEOREM. - Suppose that
ideal
generated by the
and also in the ideal
minors
of the
we
be ? Of course, it is contained in the ideal of
of 03C6’
minors
k
adjoin a new column, with entries in
matrix 03C6 , obtaining a p x (q + 1 ) matrix cp’ .
of cp are all 0 , What can the height h of the
is local and that
R
Now suppose that
he ight
R, with coefficients
of (p
has
height
for every
of the ide al of l
l
minors
is
again the largest possible value.
In
particular,
To
see
minor of
cp
is
0 .
that the
cessary for
hypothesis about coefficients being in the maximal ideal are
this, consider the following matrix over F[x ,
when F is a
field :
Here the
first
height of the ideal of
2 x 2
minor is
0 e
2 x 2
minors is
2(
=
(3 -
2 +
1) ,
~ut the
ne-
20-04
3. A remark
on
intersection theory.
speculative : Suppose
The remark is easy and
ranks
and
m
that (p
over a
n
and ~
E
Then the ideal X
at most
form
m + n .
cp(M) .
rem
([7],
e
E
JN* ,
Y
can
+
This
gives
be written
some
hold
(c~ , ~)(M
as
on
ch. V,
c~(M~
theorem 3).
are
field,
modules of
say). Suppose
=
P , then
8
N) , so
X
+
Y
has
height
theory" of ideals of the
every prime ideal P of a
the "intersection
>
with
a
N
and write
example, if one could prove that
ring R ~ J’ there exists a module
JM*
and
(R, J) (containing
For
regular local
element cp
local ring
M
that
one
for
M
with rank
M
=
ht P
and
an
could deduce Serre’s intersection theo-
REFERENCES
[1]
BUCHSBAUM (D. A.) and RIM (D. S.). - A generalized Koszul complex, III., Proc.
Amer. math. Soc., t. 16, 1965, p. 555-558.
[2]
EAGON
(J. A.). -
University
[3]
of
Ideals
generated by
Chicago, 1961.
the subdeterminants of
a
matrix, Thesis,
EAGON (J. A.) and NORTHCOTT (D. G.). - Ideals defined by matrices and a certain complex associated with them, Proc. Royal Soc. London, Series A, t. 269,
1962, p. 188-204.
[4]
[5]
[6]
(M.). - Deep local rings (to appear).
(W.). - Über eine Hauptsatz der allgemeinen
HOCHSTER
KRULL
richte
Idealtheorie, Sitzungsbe-
Heidelberg Akad. Wiss., 1929, p. 11-16.
(F. S.). - The algebraic theory of modular systems. - Cambridge, at
University Press, 1916 (Cambridge Tracts in Mathematics and mathematical
Physics, 19).
MACAULAY
the
[7]
SERRE
1965
(J.-P.). - Algèbre
(Lecture Notes in
locale -
Multiplicités. - Berlin, Springer-Verlag,
Mathematics,
11).
(Texte
David E ISENBUD
Department of Mathematics
Brandeis University
WALTHAM, Mass. 02154
(Etats-Unis)
reçu Ie 20 mai
1975)