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.

129

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).