Fitting Ideals

17 Pages·2009·0.195 MB·English

by  Filippo A. E. Nuccio

Fitting Ideals
Filippo A. E. Nuccio
Guwahati, September 22nd − 29th
This document is a summary of two talks given at the School on Arith-
metic Geometry held in Guwahati, India, from September 22nd − 29th on
Fitting ideals and their applications to Wiles’ proof of Iwasawa Conjecture
for totally real fields as in [Wil90]. The main references for the theory of Fit-
ting ideals are [Nor76] and the appendix of [MW84]. Nothing in this notes
is new, and every fact is taken - either directly or slightly adapted - from
[Nor76]or[MW84]. Observe that aringwith onlyonemaximal idealis called
semi-local in [Nor76] and it is called local if it is noetherian with only one
maximal ideal.
1 General results on Fitting ideals
By “ring” we mean a commutative ring with 1. We fix such a ring and we
call it R. If M is a finitely generated R-module and x is a set of generator
of M (we will assume all such sets to be ordered), we indicate by |x| the
number of elements it contains; an element r = (r ,...,r ) ∈ R|x| is called
1 |x|
a relation among the set of generators x if
|x|
X
r x = 0 ∈ M .
i i
i=1
Definition 1. Let M be a finitely generated R-module and let x be a set of
generators of M. A matrix X is said to be a matrix of relations for x if
1. X ∈ M (R) with q ≥ |x|;
q×|x|
2. X ·x = 0, i. e. every row is a relation among the x ’s.
i
1
We are now ready to define the main object of our study. If X is a n×m-
matrix (with n ≥ m), we define its determinantal ideal DetId(X) ⊆ R to be
the ideal of R generated by all m×m-minors of X.
Definition 2. Let M be a finitely generated R-module. We define the R-
Fitting ideal of M to be
X
Fitt (M) = DetId(X)
R
X
where X runs through all matrices of relations for all sets of generators of
M.
Remark. Observe that a priori the sum in the definition is extended to
infinitely many ideals. Moreover, we want to stress that our convention that
a matrix of relations has always more rows than columns (these being as
many as the generators) is crucial for the definition of the determinantal
ideal. Indeed, suppose that we allow every matrix whose rows are relations
among a set of generators to be a matrix of relations and that we define its
determinantal ideal to be the ideal generated by all minors of largest possible
rank. Then if x is a set of generators, let x0 = {x ,x ,x ,...,x } be the old
1 1 2 |x|
set of generators with one entry doubled - say the first. Then
(−1,1,0,...,0)
would be a matrix of relations, and DetId(X) = R.
The definition of Fitting ideal that we gave is practically useless for any
application: butitmaybeofsomeuseincertainproofs. Thus, beforepassing
to concrete examples, we want to show that in the definition it would have
beenenoughtosumoverallmatricesofrelationsofonefixed setofgenerators.
Indeed we have:
Proposition 1. Let x be a fixed, finite set of generators of the finitely gen-
erated module M. Then
X
Fitt (M) = DetId(X)
R
X
where X runs through all matrices of relations for x.
2
Proof. Only for the proof, for every set of generators y of M, set
X
A(y) := DetId(Y)
Y
for Y running, as above, through all matrices of relations for y. We need
to show that for every set of generators y, the equality A(x) = A(y) holds:
P
indeed, by definition, Fitt (M) = A(y) and if they all coincide this sum
R y
reduces to A(x).
Let then y be any set of generators and set xy := {x ,...,x ,y ,...,y }:
1 |x| 1 |y|
clearly |xy| = |x|+|y|. Since the x ’s generate M, we can find a relation
i
c x +c x +···c x +y = 0 ,
k1 1 k2 2 k|x| |x| k
for suitable c ∈ R, for every 1 ≤ k ≤ |y|. Let now X = (a ) be any
ki ij
p×|x|-matrix appearing in the sum A(x) and define a new matrix, having
|x|+|y| columns,
 c ... c 1 ... 0 
11 1|x|
 ... ... ... ... ... ... 
 
 c ... c 0 ... 1 
Xe =  |y|1 |y||x|  .
 a ... a 0 ... 0 
 11 1|x| 
 ... ... ... ... ... ... 
a ... a 0 ... 0
p1 p|x|
It is clearly a matrix of relations for xy (it has p+|y| ≥ |x|+|y| rows) and
every |x| × |x|-minor of X also appears as |xy| × |xy|-minor of Xe, showing
the inclusion A(x) ⊆ A(xy).
Let now Y be a matrix of relations for xy; define a new matrix
 
c c ... c
11 12 1|x|
 ... ... ... ... I 
Ye =  |y| 
 
c c ... c
 |y|1 |y|2 |y||x| 
Y
where I is the |y|×|y| identity matrix. Then DetId(Ye) ⊇ DetId(Y) (just
|y|
observethatY andYe havethesamenumberofcolumnsandevery|xy|×|xy|-
3
minor of Y is also a |xy|×|xy|-minor of Ye) and Ye is similar to
 
c c ... c
11 12 1|x|
 ... ... ... ... I 
Yf0 =  |y| 
 
c c ... c
 |y|1 |y|2 |y||x| 
X 0
(for this, use the identity matrix in the upper right corner of Y to make every
element of the lower right corner trivial by means of elementary row oper-
ations); since the determinantal ideals of similar matrices clearly coincide,
we find that DetId(Y) ⊆ DetId(Ye) = DetId(Ye0) = DetId(X)DetId(I ) =
|y|
DetId(X). Moreover, since the rows of Ye are relations among the elements
of xy, the same holds for the rows of Ye0, so X is a matrix of relations for x
(since Y was a matrix of relations, X has more rows than columns). This
shows the inclusion A(xy) ⊆ A(x), and repeating the same argument with y
at the place of x shows that A(y) = A(xy) = A(x), as we needed.
Remark. Observe that it is enough to restrict the computation to square
|x| × |x|-matrices. Indeed, substituting a matrix of relations X with all its
|x|×|x|-submatrices in the sum defining DetId(X) gives the same elements
in the determinantal ideal.
Example. LetM = Ra forsomea ≥ 1. AsetofgeneratorsforM ise, where
e is the vector such that (e ) = δ . For every relation (r ,...,r ) ∈ Ra,
i i j i,j 1 a
a
X
r e = 0 =⇒ r = 0 ∀ i .
i i i
i=1
Thereforetherearenonon-trivialrelationsand,inparticular,theonlymatrix
of relations for e is the 0 matrix. Accordingly, Fitt (Ra) = 0.
R
Example. Let p be a prime number, R = Z and let M = Z/pZ × Z/p2.
Then clearly Ann (M) = p2Z. We want to compute Fitt (M). For this, let
Z Z
{(a,0),(0,b)} a set of generators of M, where p - ab (we are going of course
to use Proposition 1). Then a matrix of relations for (a,b) is of the form
 
α β
1 1
 α β 
2 2
X =  . . 
 . . 
. .
 
α β
q q
4
with v (α ) ≥ 1 and v (β ) ≥ 2. Therefore, any 2 × 2-minor of X is of the
p i p i
form
(cid:12) (cid:12)
det(X ) = (cid:12)(cid:12) αi βi (cid:12)(cid:12)
ij (cid:12) α β (cid:12)
j j
and v (det(X ) = v (α β −α β ) ≥ 3, showing that Fitt (M) = p3Z. If we
p ij p i j i i Z
allowed “matrices of relations” having only one row, (p,p2) would have been
one such a matrix, generating the ideal p2Z. In this sense, this invariant is
“finer” than the usual annihilator.
Remark. Inhisbook[Nor76]Northcottdefinesthei-thFittingidealforevery
natural number i ≥ 0. What we have defined is then the 0-th Fitting ideal,
and is enough for the purposes of Wiles’ proof.
We now gather the main elementary results on Fitting ideals.
Lemma 1. Let I ⊆ R be an ideal. Then Fitt (R/I) = I.
R
Proof. A generator for R/I is 1 and the general matrix of relations for it is
 
α
1
 ... 
α
q
such that α ·1 = 0 ∈ R/I, i. e. such that α ∈ I for all i’s.
i i
In the following, M will always denote a finitely generated R-module.
Lemma 2. Let M0 be a quotient of M and let π : M (cid:16) M0 be a surjection:
then Fitt (M) ⊆ Fitt (M0).
R R
Proof. Let x be a set of generators for M; then π(x) = {π(x) ,...,π(x) }
1 |x|
is a set of generators for M0. If {r ,...,r } is a relation for x, it is also one
1 |x|
for π(x). Therefore any matrix of relations for x is also a matrix of relations
for π(x), showing the desired inclusion.
∼
Lemma 3. If M = M ×M , then Fitt (M) = Fitt (M )·Fitt (M ).
1 2 R R 1 R 2
Proof. On one side, any two matrices X,Y of relations for M and M ,
1 2
respectively, give a matrix of relations
(cid:18) (cid:19)
X 0
0 Y
5
for M ×M . On the other side, choosing as set of generators for M ×M
1 2 1 2
one of the form xy = {x ,...,x ,y ,...,y } where x is a set of genera-
1 |x| 1 |y|
tors for M and y is one for M , we can form from any matrix of relations
1 2
for xy two matrices of relations for x and y. These matrices verify clearly
DetId(X)DetId(Y) = DetId(XY), and our statement follows.
Combining Lemma 3 and Lemma 1 we get
Corollary 1. If M = R/a ×···×R/a , then Fitt (M) = a ···a .
1 n R 1 n
(cid:3)
Lemma 4. If M can be generated by n elements, then
Ann (M)n ⊆ Fitt (M) ⊆ Ann (M) .
R R R
Proof. First of all, let x be a set of generators with |x| = n. Then, if c ∈
i
Ann (M) for 1 ≤ i ≤ n, the matrix
R
 
c ... 0
1
C =  ... ... ... 
 
0 ... c
n
is a matrix of relations for x and c ···c = DetId(C) ∈ Fitt (M), so
1 n R
Ann (M)n ⊆ Fitt (M).
R R
On the other hand, let X be a square n×n-matrix of relations for x: let X◦
be its adjugate matrix1. Then multiplying the equation
X ·x = 0
by X◦ shows that det(X) ∈ Ann (M). Since, as we observed, Fitt (M) is
R R
generated by these determinants, we find Fitt (M) ⊆ Ann (M).
R R
Remark. This Lemma gives another proof that Fitt (R/I) = I, since in that
R
case n = 1 and Ann (M) = I. During the school, Otmar Venjakob proposed
R
1The adjugate matrix X◦ of a square n×n-matrix X with coefficients in a ring R is a
matrix such that XX◦ =det(X)I . It may be defined as the matrix whose (i,j)-entry is
n
the (j,i)-cofactor of X.
6
the following more elegant proof of the first inclusion: since M admits n
generators, there is a surjection
(cid:16) (cid:17)n
R/Ann (M) (cid:16) M
R
and applying Lemma 2 to it we find the desired inclusion.
Recall that a R-module M is said to be faithful if Ann (M) = 0. Thus
R
the Lemma above gives
Corollary 2. If M is a faithful R-module, then Fitt (M) = 0.
R (cid:3)
Lemma 5. If I ⊆ R is any ideal of R, then
Fitt (M/IM) = Fitt (M)+I ⊆ R/I
R/I R
Proof. It is clear that if x is a set of generators for M, than its image under
the canonical projection is a set of generators for M/I, and that any relation
for the first set is also a relation for the second, giving Fitt (M) + I ⊆
R
Fitt (M/IM).
R/I
If now [X] = ([ξ ]) is a matrix of relation for the set of generators [x] (we
ij
denote by [·] the class ·+I), we can find elements α ∈ I such that for all
ij
1 ≤ i ≤ |x| we have
|x| |x|
X X
ξ x = α x
ij j ij j
j=1 j=1
because every element of IM can be written as a finite sum of the generators
of M with coefficients in I. The matrix Xe whose i-th row is
(ξ −α ,...,ξ −α )
i1 i1 i|x| i|x|
is a matrix of relations for M and X ≡ Xe (mod I). Therefore DetId([X]) =
DetId([Xe]) = DetId(Xe)+I ∈ Fitt (M)+I. This gives the other inclusion.
R
Lemma 6. If 0 −→ M −→ M −→ M −→ 0 is an exact sequence of
1 2 3
finitely generated R-modules, then
Fitt (M )·Fitt (M ) ⊆ Fitt (M ) .
R 1 R 3 R 2
7
Proof. Without loss of generality, we may and will assume that M ⊆ M
1 2
and that the first map above is the natural inclusion. Let now [x] be a set
of generators for M and let x be a lifting of it to M . Let y be a set of
3 2
generators for M ⊆ M . Then xy is a set of generators for M .
1 2 2
Let X ∈ M (R),Y ∈ M (R) be matrices of relations for [x] and y,
p×|x| q×|y|
respectively, and let C = (−c ) ∈ M (R), where
ij p×|y|
 
P|y|
c y
i=1 1i i
 P|y| 
X ·x =  i=1.c2iyi  .
 .. 
 
P|y|
c y
i=1 pi i
Then the matrix
(cid:18) (cid:19)
X C
Z =
0 Y
is a (p + q) × |xy|-matrix of relations for M and DetId(X) · DetId(Y) ⊆
2
DetId(Z) since every minor of Z is of the form
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)(cid:12) MX MC (cid:12)(cid:12) = (cid:12)(cid:12) MX 0 (cid:12)(cid:12)
(cid:12) 0 M (cid:12) (cid:12) 0 M (cid:12)
Y Y
where M is a submatrix of X and M is a submatrix of Y.
X Y
2 Applications to Wiles’ proof
In this section we focus on the applications of the theory of Fitting ideals
developed up to now to Wiles’ proof. In a first part we will start by assuming
that not only M is finitely generated, but that is in fact finitely presented;
and we eventually specialize to the case R = O[[T]] where O is the ring of
integers of a p-adic field.
2.1 Finite presentations
First of all, recall that an R-module M is said to be finitely presented if there
is an exact sequence of R-modules
Ra →h Rb →ϕ M → 0
8
for suitable a,b ∈ N. Clearly, every finitely presented module is finitely
generated, sinceitcanbegeneratedbyϕ(e )for1 ≤ i ≤ b. IfR isnoetherian,
i
the converse is also true: indeed, if M is finitely generated, it is a quotient
of a finitely generated free module and there is an exact sequence
0 → S →ψ Rb →ϕ M → 0
for some S and for some b ∈ N. Using that R is noetherian, S should also
be finitely generated, say a quotient of Ra by a module T: we find
0 //T //RaA_ _ _ _h _ _ _//>>Ra ϕ //M //0
AAAAkAAA }ψ}}}}}}
}
>>S A
} A
} A
} A
} A
} A
} A
} A
}
0 0
where h := ψ ◦ k. We can therefore extract from the above diagram the
sequence
Ra →h Rb →ϕ M → 0
that is exact since Ker(ϕ) = Im(ψ) = Im(h).
Now we present some results that are true for general finitely presented R-
modules. Thus, assuming noetherianity reduces this hypothesis to M being
finitely generated.
From now on we assume that M is a finitely presented R-module.
Proposition 2. Let Ra →h Rb →ϕ M → 0 be a finite presentation of M and
let H ∈ M (R) be a matrix attached to h. If a ≥ b, then Fitt (M) =
b×a R
DetId(Htr) and if a < b then Fitt (M) = 0.
R
Proof. The finite presentation
Ra →h Rb →ϕ M → 0
shows that a set of generators of M is {ϕ(e ),...,ϕ(e )} and that the rela-
1 b
tions among these elements are precisely the kernel of ϕ, namely Im(h), that
9
is spanned by the columns of H. By Proposition 1, to compute Fitt (M) it
R
sufficestoconsidermatriceswhoserowsarerelationsamongthesegenerators.
If a ≥ b, then Ht is a matrix of relations; moreover, if X is any other
matrix of relations, then all its sumbatrices are linear combinations of rows
of Ht and their determinant coincide with the determinant of those rows.
Therefore DetId(X) ⊆ DetId(Ht), showing the desired equality.
If a < b, then every matrix of relations X ∈ M (R) has at least two linearly
b×b
dependent rows, since they belong to Im(h) and this module has R-rank a <
b. Therefore DetId(X) = 0 for all such matrices, showing Fitt (M) = 0.
R
Lemma 7. If I is a finitely generated ideal, then
Fitt (M) ⊆ Fitt (M/IM) ⊆ (Fitt (M),I) ⊆ R .
R R R
Proof. The first inclusion is Lemma 2. For the second, if
Ra →h Rb → M → 0
is a presentation of M and if I = hα ,...,α i, then a presentation of M/IM
1 n
is
Ra+nb →h∗ Rb → M/IM → 0
where
h∗(w ,...,w ,v ,...,v ,...,v ,...,v ) =h(w ,...,w )
1 a 1,1 1,b n,1 n,b 1 a
n
X
+ (α v ,...,α v ) .
j j,1 j j,b
j=1
A matrix associated to h∗ is
 
H 0 ... 0
 0 A ... 0 
H∗ =  ... ... ... ... 
 
0 0 ... A
where H is a matrix associated to h and A is the diagonal matrix
 
α 0 ... 0
1
 0 α ... 0 
2
A =  ... ... ... ...  .
 
0 0 ... α
n
10