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by the products δ1 δ2 of the i-minors δ1 and the j-minors δ2 . Then for π = [a1 , . . . , au |
b1 , . . . , bu ], ρ = [c1 , . . . , cv |d1 , . . . , dv ], u ¤ v ’ 2, and

u = max |{a1 , . . . , au } © {c1 , . . . , cv }|, |{b1 , . . . , bu } © {d1 , . . . , dv }|

one has
(u + 1 ’ u)! πρ ∈ F(u+1, v’1).
(We include the case u = 0 in which π = 1.)
Proof of (10.9): The inclusion “‚” has been noticed already. The converse is
proved by induction on s, the case s = 1 being trivial. Consider a (standard) monomial
µ = δ1 . . . δp of minors δi such that γj (µ) ≥ ej for j = 1, . . . , w. If one of the minors δi
127
C. Primary Decomposition of Products of Determinantal Ideals

has size w, one is through by induction. Otherwise we arrange the factors δ1 , . . . , δp in
ascending order relative to their sizes, and split µ into the product
µ1 = δ 1 . . . δ q
of minors of size < w, and
µ2 = δq+1 . . . δp
of minors of size > w. Let u and v be the sizes of δq and δq+1 resp. Applying (10.10) to
δq δq+1 we get a representation

ai ∈ B,
µ= ai νi , νi = δ1 . . . δq’1 ζi ·i δq+2 . . . δp ,
in which ζi has size u + 1 and ·i has size v ’ 1. Evidently

γj (νi ) = γj (µ), j = 1, . . . , u + 1,
γj (νi ) = γj (µ) ’ 1, j = u + 2, . . . , w
By induction on v ’ u or by reference to the case in which a w-minor is present, we are
through if γj (µ) > ej for j = u + 2, . . . , w. It remains the case in which γr (µ) = er for
some r, u + 2 ¤ r ¤ w.
One may assume that t1 , . . . , tk < r ’ 1 and tk+1 , . . . , ts ≥ r ’ 1. Let
k s
J1 = Iti (X) , J2 = Iti (X),
i=1 i=k+1
k s
e1 e2
= e(j, ti ), = e(j, ti ).
j j
i=1 i=k+1

Then γj (µ) = γj (µ2 ) for j ≥ r ’ 1. Furthermore γr’1 (µ) ’ γr (µ) = p ’ q and er’1 ’ er =
s ’ k. Since γr’1 (µ) ≥ er’1 it follows that p ’ q ≥ s ’ k, whence µ2 ∈ J2 for trivial
reasons. We claim: γj (µ1 ) ≥ e1 for j ¤ r ’ 2 and ¬nish the proof by induction on s.
j
Relating γr+1 and γr , er+1 and er one gets
γr (µ) ’ γr+1 (µ) = p ’ q,
er ’ er+1 ¤ s ’ k.
Therefore p ’ q ¤ s ’ k, too, and p ’ q = s ’ k. Since
γj (µ2 ) = γr (µ2 ) + (p ’ q)(r ’ j) = er + (s ’ k)(r ’ j)
= e2
j

for all j ¤ r, γj (µ1 ) ≥ e1 for all j ¤ r ’ 2 as claimed. ”
j
Proof of (10.10): In case u = 0 the contention is a trivial consequence of Laplace
expansion. Let u > 0 and suppose that
u = |{a1 , . . . , au } © {c1 , . . . , cv }|,
transposing if necessary. We use descending induction on u, starting with the maximal
value u = u. The fundamental relation which is crucial for this case as well as for the
inductive step, is supplied by the following lemma. Its very easy proof is left to the
reader:
128 10. Primary Decomposition

(10.11) Lemma. One has
u+1
(’1)i’1 [a1 , . . . , ai , . . . , au+1 |b1 , . . . , bu ][ai , c2 , . . . , cv |d1 , . . . , dv ] ∈ F(u+1, v’1).
i=1


In proving (10.10) for u = u we may assume that c1 ∈ {a1 , . . . , au }. Then, with
/
au+1 = c1 , all the terms of the sum in (10.11) except

(’1)u [a1 , . . . , au |b1 , . . . , bu ][c1 , . . . , cv |d1 , . . . , dv ]

are zero, and πρ ∈ F(u+1, v’1) as desired.
Let u < u now and again c1 ∈ {a1 , . . . , au }. We put β = (b1 , . . . , bu ), δ = (d1 , . . . , dv ),
/
au+1 = c1 . The terms [a1 , . . . , ai , . . . , au+1 |β][ai , c2 , . . . , cv |δ] with ai ∈ {c2 , . . . , cv } drop
out, and

(’1)i’1 [a1 , . . . , ai , . . . , au+1 |β][ai , c2 , . . . , cv |δ] ∈ F(u+1, v’1).
(1)
i
ai ∈{c2 ,...,cv }
/

We claim: If ai ∈ {c2 , . . . , cv }, then
/

(2) (u ’ u)! (’1)i’1 [a1 , . . . , ai , . . . , au+1 |β][ai , c2 , . . . , cv |δ]
≡ (u ’ u)! (’1)u [a1 , . . . , au |β][c1 , . . . , cv |δ] mod F(u+1, v’1).

Multiplying the sum in (1) by (u ’ u)! and applying the preceding congruence we get

(u + 1 ’ u)! [a1 , . . . , au |β][c1 , . . . , cv |δ] ∈ F(u+1, v’1)

as desired.
In order to prove (2) one replaces the rows ai and c1 of X both by the sum of these
rows, creating a matrix X. Let π and ρ be the minors of X arising from π and ρ under the
substitution X ’ X. The minors π and ρ both can be interpreted as minors of a matrix
with m ’ 1 rows, and then have u + 1 rows in common. The Z-module F(u+1, v’1)
relative to the new matrix is contained in F(u+1, v’1) relative to X, and by induction,

(u + 1 ’ (u + 1))! πρ ∈ F(u+1, v’1).

On the other hand

π ρ = [a1 , . . . , ai , . . . , au |β][ai , c2 , . . . , cv |δ] + [a1 , . . . , c1 , . . . , au |β][ai , c2 , . . . , cv |δ]
+ [a1 , . . . , ai , . . . , au |β][c1 , c2 , . . . , cv |δ] + [a1 , . . . , c1 , . . . , au |β][c1 , c2 , . . . , cv |δ].

The inductive hypothesis applies to the ¬rst and the fourth term on the right side of this
equation, whence

(u ’ u)! πρ + [a1 , . . . , c1 , . . . , au |β][ai , c2 , . . . , cv |δ] ∈ F(u+1, v’1). ”

The intersection in (10.9) is obviously redundant if s = 1, t1 > 1 or t1 = · · · =
ts = min(m, n) > 1. In the latter case It1 (X) . . . Its (X) is primary itself, cf. (9.18). The
following proposition shows how to single out the essential primary ideals.
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C. Primary Decomposition of Products of Determinantal Ideals

(10.12) Proposition. Let B be an integral domain, X an m — n matrix of indeter-
minates such that m ¤ n (for notational simplicity). Furthermore let e1 , . . . , ew be given
as in (10.9). Then
w
(ek ) (ej )
⊃ ⇐’ ek+1 ¤ (m ’ k)(gk ’ 1),
Ik (X) Ij (X)
j=1
j=k

gk denoting the number of indices i, 1 ¤ i ¤ s, such that ti ≥ k.
Proof: Passing to a ring of quotients does not a¬ect the question whether a primary
ideal Q is irredundant in a given decomposition, provided Q stays a proper ideal under
extension. Therefore we may invert a 1-minor if k > 1, and eventually reduce the
proposition to the case in which k = 1. So one has to show:
w
(e1 )
Ij (X)(ej )
⊃ ⇐’ e2 ¤ (m ’ 1)(s ’ 1),
I1 (X)
j=2

s as in (10.9) denoting the number of factors of the product to be decomposed. Observe
that e1 = e2 + s, so
e2 ¤ (m ’ 1)(s ’ 1) ⇐’ e1 ’ 1 ¤ m(s ’ 1).
(3)
We write e1 ’ 1 = qm + r, q, r ∈ Z, 0 ¤ r < m and choose an m-minor δ and an r-minor
µ (µ = 1 if r = 0). Then it is easy to see that
γj (µ) ¤ γj (δ q µ), j = 1, . . . , m
for all (standard) monomials µ such that γ1 (µ) < e1 . Therefore
w
(e1 )
Ij (X)(ej ) γj (δ q µ) ≥ ej
⊃ ⇐’
(4) I1 (X) for j = 2, . . . , w.
j=2

We now show that the right sides of (3) and (4) are equivalent. Suppose ¬rst that
e1 ’ 1 > m(s ’ 1). Since, on the other hand, e1 ¤ ms, one has q = s ’ 1 and r > 0, so
δ q µ has exactly s factors (= 1), and
γ2 (δ q µ) = e1 ’ 1 ’ s < e1 ’ s = e2 .
Suppose now that e1 ’ 1 ¤ m(s ’ 1). Then
γ2 (δ q µ) ≥ e1 ’ 1 ’ (s ’ 1) = e2 .
Observe in the following that the di¬erences ei ’ ei+1 form a non-increasing sequence
(i.e. ei+2 ’ ei+1 ¤ ei+1 ’ ei for all i) and that γj (δ q µ) ’ γj+1 (δ q µ) can only take the values
q and q + 1. In order to obtain a contradiction we assume that there exists a v such that
γv (δ q µ) ≥ ev , but γv+1 (δ q µ) < ev+1 . Then for all j ≥ v one has
γj (δ q µ) ’ γj+1 (δ q µ) ≥ γv (δ q µ) ’ γv+1 (δ q µ) ’ 1
≥ ev ’ ev+1
≥ ej ’ ej+1 .
Summing up these di¬erences for j = v + 1 to j = m + 1 one obtains the desired
contradiction since γm+1 (δ q µ) = em+1 = 0. ”
An immediate consequence of (10.9) and (10.12) is the following irredundant primary
decomposition of the powers of the ideals It (X):
130 10. Primary Decomposition

(10.13) Corollary. Let B be an integral domain, X an m — n matrix, m ¤ n.
Suppose that (min(t, m ’ t))! is invertible in B. Then
t
s
Ij (X)((t’j+1)s) , r = max(1, m ’ s(m ’ t)),
It (X) =
j=r

is an irredundant primary decomposition.
(k)
(10.14) Remarks. (a) If one de¬nes It (X) by means of the description given in
(k)
(10.4), i.e. It (X) is the B-submodule generated by the standard monomials µ such
that γt (µ) ≥ k, then the intersection formulas of (10.9) and (10.13) hold for every ring
in which the elements (min(ti , m ’ ti , n ’ ti ))!, i = 1, . . . , s, are units.
(b) The proof of (10.9) shows that for B = Z the ideal w Ij (X)(ej ) is the Z-torsion
j=1
of Z[X] modulo It1 (X) . . . Its (X).
(c) If in addition to the hypotheses of (10.13) B is noetherian, then the associated
s
prime ideals of It (X) are precisely the ideals Ij (X), j = r, . . . , t. If m is large compared
to t, then I1 (X) is associated with It (X)k for all k ≥ 2, and if t ¤ m ’ 1, then I1 (X) is
k
associated with It (X) for k ≥ t, as is easily seen. We will show below that the latter
fact holds over every noetherian domain.
(d) The example given below shows that it is not possible to remove the assumption
on the characteristic of B in (10.9) or (10.13). It should be noted however that in (10.13)
it becomes void not only in the cases t = 1 or t = m, but also when t = m ’ 1, the case of
submaximal minors. On the other hand these are the only exceptional cases for (10.13),
cf. the end of (g) below.
(e) Without any change in their statements or proofs, the results (10.9) “ (10.13)
carry over from the polynomial ring B[X] to the residue class ring Rr+1 (X), provided of
course that one considers products and powers of the ideals It (X), t ¤ r, only. Cf. (10.5)
for the corresponding remark in regard to the symbolic powers.
(f) With the notations of (10.10) the order of πρ modulo F(u+1, v’1) may be smaller
than (u + 1 ’ u)! . For example let u = 1 and v > 2 be an even number. Then πρ ∈
F(2, v’1). The general case follows from the case π = [1|1], ρ = [2, . . . , v + 1|2, . . . , v + 1]
by specialization. One has
v+1
(’1)i’1 [i|1][1, . . . , i, . . . , v+1|2, . . . , v+1] = [1, . . . , v+1|1, . . . , v+1] ∈ F(2, v’1)
(5)
i=1

by Laplace expansion, and

(’1)i’1 [i|1][1, . . . , i, . . . , v + 1|2, . . . , v + 1]
+ (’1)j’1 [j|1][1, . . . , j, . . . , v + 1|2, . . . , v + 1] ∈ F(2, v’1)

by virtue of (10.11), and the sum in (5) has an odd number of terms.
(g) On the other hand the order of πρ modulo F(u’1, v+1) is greater than 1 in
general. We claim: [1|1][2 3 4|2 3 4] ∈ F(2, 2) and (F(1, 3) + F(2, 2))/F(2, 2) ∼ Z/2Z if
/ =
m = n = 4.
131
C. Primary Decomposition of Products of Determinantal Ideals

A simple observation helps to reduce the amount of computation needed to prove
this. Let µ = δ1 . . . δk , δi ∈ ∆(X). Then the support of µ is the smallest submatrix of
X from which all the minors δi can be taken. It is fairly obvious that B[X] and all the
modules F(u, v) (and F(u1 , . . . , ut ) as a self-suggesting generalization) decompose into
the direct sum of their submodules generated by monomials with a ¬xed support. In
order to show [1|1][2 3 4|2 3 4] ∈ F(2, 2) it is therefore enough to consider the submodule
/
N of F(2, 2) which is generated by the products δ1 δ2 , δi a 2-minor, whose support is the
entire 4 — 4 matrix X.
In order to write 2[1|1][2 3 4|2 3 4] as an element of F(2, 2) we follow the proof of
(10.10). The reader who has proved (10.11) knows that
[1|1][1 2 3|2 3 4] = ’[1 2|1 2][1 3|3 4] + [1 2|1 3][1 3|2 4] ’ [1 2|1 4][1 3|2 3]
and
[1|1][2 3 4|2 3 4] ’ [2|1][1 3 4|2 3 4]
(6)
= [1 2|1 2][3 4|3 4] ’ [1 2|1 3][3 4|2 4] + [1 2|1 4][3 4|2 3].
Disregarding all terms of support smaller than X, one gets from the ¬rst of these equa-
tions:
[1|1][2 3 4|2 3 4]+[2|1][1 3 4|2 3 4]
= ’ [1 3|1 2][2 4|3 4] + [1 3|1 3][2 4|2 4] ’ [1 3|1 4][2 4|2 3]
(7)
’ [2 3|1 2][1 4|3 4] + [2 3|1 3][1 4|2 4] ’ [2 3|1 4][1 4|2 3].
Addition of (6) and (7) yields the desired representation of 2[1|1][2 3 4|2 3 4], and it is
enough to prove that the nine products appearing in it are part of a Z-basis of N . N has
the same rank as the submodule generated by the standard monomials with support X
in F(0, 4), F(1, 3), and F(2, 2). An easy count yields rk N = 14 whereas 18 products δ 1 δ2
of 2-minors δi have support X. Relations of these products are produced by equating two
expansions of [1 2 3 4|1 2 3 4] along two rows or two columns. Let Rij be the expansion
along rows i, j and Cij the expansion along columns i, j. It is not di¬cult to see that
the relations
C12 ’ C13 = 0, C12 ’ C14 = 0, C12 ’ R12 = 0, C12 ’ R13 = 0
can be solved for four products none of which appears in the representation of 2[1|1][2 3 4|
2 3 4] derived above.
The second claim follows very easily now: The generators of F(1, 3) with support
smaller than X are in F(2, 2), and those with support X all have order 2 modulo F(2, 2).

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