We now make the crucial assumption:

(2) The ideals J1 , . . . , J4 are radical ideals.

154 12. Principal Radical Systems

As pointed out in 7.A, elementary linear algebra provides us with a generic point for

A/J1 (and analogously for A/J3 and A/J4 ). Furthermore J3 J4 ‚ J2 , and one concludes

immediately:

(3) J1 , J3 , J4 are prime ideals. In addition (a) X11 is not a zero-divisor mod J1 , and

(b) J2 = J3 © J4 .

At this point it is clear that Z[X]/J1 is Z-free. In proving perfection one can

therefore restrict the ring of coe¬cients to be a ¬eld.

m

The grade of J1 has been computed in (2.5). Writing J3 = I2 (X) + i=1 AXi1 , X

consisting of the last n ’ 1 columns of X, and representing J4 and J3 + J4 in a similar

way, one has:

(4) grade J2 = grade J3 = grade J4 = grade J1 +1, and grade(J3 +J4 ) = grade J1 +2.

The auxiliary arguments are complete now. Inductively one may suppose that J3 ,

J4 , and J3 + J4 are perfect ideals. Then it follows from (3),(b) and (4) that J2 is perfect

by virtue of Lemma (5.15),(b), and part (a) of this lemma, in conjunction with (3),(a),

implies the desired perfection of J1 .

Statement (2) above which has only been an assumption so far, is proved by induc-

tion, too. At least we know from the existence of generic points:

(5) Rad J1 is prime. In particular X11 is not a zero-divisor modulo Rad J1 .

Assume for the ¬rst part of the inductive proof of (2) that J2 = J1 + AX11 is a

radical ideal. Let y ∈ A be nilpotent modulo J1 . Then

y1 ∈ J1 , z1 ∈ A.

y = y1 + z1 X11 ,

Since X11 is not a zero-divisor modulo Rad J1 , one has iteratively

u

y u ∈ J1 , z u ∈ A

y = yu + zu X11 ,

for all u ≥ 1. Arguing in the graded ring A/J1 , we immediately conclude that y ∈ J1 , as

desired.

Unfortunately there seems to be no way to derive the radical property of J2 from

that of J3 , J4 , and J3 + J4 which we could safely assume to be radical. We are forced to

enlarge the class of ideals:

v

Gv = J1 + AX1j ,

j=1

Hv = J1 + I1 (Xij : 1 ¤ i ¤ m, 1 ¤ j ¤ v).

Observe that G1 = J2 , Gn = J4 , H1 = J3 . By descending induction on v one now sees

that all the ideals Gv are radical. Gn and H1 , . . . , Hn may be assumed to be prime. Let

1 ¤ v ¤ n. Then:

(6) X1v Hv’1 ‚ Gv’1 ‚ Hv’1 and X1v is not a zero-divisor mod Rad Hv’1 .

155

B. A Principal Radical System for the Determinantal Ideals

Let y ∈ A be nilpotent modulo Gv’1 . Then y ∈ Gv (by induction !),

y1 ∈ Gv’1 , z1 ∈ A.

y = y1 + z1 X1v ,

Next z1 X1v ∈ Rad Hv’1 , so z1 ∈ Rad Hv’1 = Hv’1 , and y ∈ Gv’1 .

This scheme of reasoning can be cast in abstract form:

(12.1) Theorem. Let A be a noetherian ring, and F a family of ideals in A,

partially ordered by inclusion. Suppose that for every member I ∈ F one of the following

assumptions is ful¬lled:

(a) I is a radical ideal.

(b) There exists an element x ∈ A such that I + Ax ∈ F and

(i) x is not a zero-divisor modulo Rad I and ∞ (I + Axi )/I = 0, or

i=0

(ii) there exists an ideal J ∈ F, J ⊃ I, J = I, such that xJ ‚ I and x is not a

zero-divisor modulo Rad J.

Then all the ideals I ∈ F are radical ideals.

A family of ideals satisfying the hypothesis of (12.1) is called a principal radical

system. The attribute “principal” refers to the fact that one ascends in the system by

adding a principal ideal to a given ideal.

In our example above the family F consists of the ideals J1 = I2 (X), G1 , . . . , Gn ,

H1 , . . . , Hn . For H1 , . . . , Hn and Gn the assumption (a) is ful¬lled by induction on the

size of the matrix, (b),(i) holds for J1 , and (b),(ii) is valid for G1 , . . . , Gn’1 .

The theorem is proved by noetherian induction with the same arguments as in the

example above.

(12.2) Remarks. (a) We may replace the hypothesis that A be noetherian by

the weaker assumption that every subfamily of F has a maximal element. In most

applications F will even be ¬nite of course.

(b) The family of ideals can be replaced by a family of submodules of a (¬nitely

generated) A-module, the role of the radical then played by a certain “envelope” E(. . . )

such that M ‚ E(M ), and M ‚ N implies E(M ) ‚ E(N ). The conclusion is that

M = E(M ) for all M ∈ F. ”

B. A Principal Radical System for the Determinantal Ideals

In the following it will be inconvenient to stick too much to the notations used for

the exploration of determinantal rings from the ASL point of view. We introduce a new

description. Let X be an m — n matrix of indeterminates, and s0 , . . . , sr integers such

that 0 ¤ s0 < · · · < sr = n. Then

I(s0 , . . . , sr )

denotes the ideal generated by the collection of t-minors, 1 ¤ t ¤ r + 1, of the submatrix

formed by the columns 1, . . . , st’1 of X. Obviously

I(s0 , . . . , sr ) = I(X; [1, . . . , r|s0 + 1, . . . , sr’1 + 1])

and

It (X) = I(0, . . . , t ’ 2, n).

156 12. Principal Radical Systems

The ideals corresponding to G1 , . . . , Gn in the example above are given by

v

I(s0 , . . . , sr ; v) = I(s0 , . . . , sr ) + X1j B[X],

j=1

B as usual denoting the ring of coe¬cients.

(12.3) Lemma. Let B be an integral domain and v = sw for some w, 0 ¤ w ¤ r.

Then the radical of I(s0 , . . . , sr ; v) is a prime ideal.

Proof: Let ¬rst w = 0, that is I(s0 , . . . , sr ; v) = I(s0 , . . . , sr ). Denote this ideal by

I. Based on completely elementary linear algebra we have already constructed a generic

point for R = B[X]/I in (7.19), and thus proved the lemma for these ideals, cf. (7.1).

Since we have to refer to the details of this construction in order to obtain the claim for

w > 0, we repeat it here. Let s’1 = 0 and choose a matrix

«

···

Z1sk’1 +1 Z1sk

¬ .·

.

. .

Zk = . .

···

Zksk’1 +1 Zksk

of indeterminates, k = 0, . . . , r. Then one puts

Z = Z r’1 · · · Z 0 Z0 | · · · |Z r’1 Zr’1 |Zr

where Z j is a (j + 1) — j matrix of (new) indeterminates. In the (relative to (7.19))

special case considered here, one simply takes an m — r matrix Y of indeterminates, and

then the substitution

X ’’ Y Z

induces the generic point R ’ B[Y , Z]: If L is a ¬eld and R ’ L a B-homomorphism,

then the matrix to which the matrix X (modulo I) specializes can be decomposed in the

same way as Y Z; this gives rise to a commutative diagram

’’ B[Y , Z]

R

L

as desired. There are of course various ways to construct such a decomposition, and

below we shall outline a speci¬c one in order to guarantee an extra condition.

v

If w = r, then I(s0 , . . . , sr ; v) is of the form I(s0 , . . . , s˜)B[X] + j=1 X1j B[X],

r

I(s0 , . . . , s˜) taken with respect to the rows 2, . . . , m of X. So one may assume 0 < w < r.

r

We write

I(s0 , . . . , sr ; v) = I + J,

v

I = I(s0 , . . . , sr ), J = j=1 X1j B[X]. Let R = B[X]/I be as above, and J = JR. Then

for every B-homomorphism R/J ’ L, L a ¬eld, the composition R ’ R/J ’ L can be

157

B. A Principal Radical System for the Determinantal Ideals

factored through B[Y , Z]. It is enough that at least one such a factorization gives rise

to a commutative diagram

R/J ’’ B[Y , Z]/P,

L

P a ¬xed prime ideal in B[Y , Z]. Here is the only point in this section where we have to

work a little. We choose P as the ideal generated by the coe¬cients in the ¬rst row of

Y Z r’1 · · · Z w .

Let x be the image of the matrix X under B[X] ’ R ’ L. In order to reach the desired

factorization R/J ’ B[Y , Z]/P ’ L we now specify how to decompose x. The matrices

appearing in the decomposition of x are denoted by small letters. For systematic reasons

it is convenient to write Z r = Y .

First we represent x as

x = x0 | . . . |xr

where the separators | are placed after the columns s0 , . . . , sr’1 as in Z above. Since

rk x ¤ r there is an m — r matrix z r such that its column space equals the column space

of x = x0 | . . . |xr . Next we ¬nd an r — (r ’ 1) matrix z r’1 for which the column space

of z r z r’1 coincides with the column space of x0 | . . . |xr’1 . Continuing this procedure

we have eventually chosen matrices z r , . . . , z 0 of formats m — r, r — (r ’ 1), . . . , 1 — 0 such

that z r . . . z j has the same column space as x0 | . . . |xj . The choice of z0 , . . . , zr is the

last step (and no problem, of course). The matrix x0 | . . . |xw has only zeros in its ¬rst

row, and since its column space equals that of z r . . . z w , the latter matrix has zeros in its

¬rst row, too. This is exactly the condition to be satis¬ed in order to factor R/J ’ L

through B[Y , Z]/P .

It remains to show that P is a prime ideal. The generators of P are the entries of

the 1 — sw matrix

Y1 Z r’1 · · · Z w ,

Y1 denoting the ¬rst row of Y . The number of columns of Y1 , Z r’1 , · · · , Z w is decreasing

from left to right, and the claim therefore follows inductively from the lemma below:

(12.4) Lemma. Let A be a noetherian ring, and f1 , . . . , fu elements of A generating

an ideal I of grade g. Let U be an u — v matrix of indeterminates Xij over A.

u

(a) If v ¤ g, then the elements i=1 fi Xij , j = 1, . . . , v, form an A[U ]-sequence.

(b) If A is a domain and v < g, then the ideal J generated by them is a prime ideal.

Proof: Inductive reasoning reduces both (a) and (b) immediately to the case v = 1.

u

Since every zero-divisor in A[U ] is annihilated by an element of A, i=1 fi Xi1 cannot

be a zero-divisor. This proves (a) already.

For (b) we have g ≥ 2, and grade IA[U ]/J ≥ 1 because of (a). There is no harm in

assuming that f1 , . . . , fu = 0. We ¬rst show that A[U ]/J is reduced. To be reduced is a

local property, and it certainly su¬ces that the rings (A[U ]/J)[fi’1 ] are domains. This

is easy to see:

(A[U ]/J)[fi’1 ] ∼ ((A[fi’1 ])[U ])/(extension of J),

=

158 12. Principal Radical Systems

and over A[fi’1 ] the generator of J becomes an indeterminate.

Since (A[U ]/J)[fi’1 ] is a domain, fi must be contained in all the minimal primes of

J but one. Since fi fj ∈ J for all i, j, the “excluded” minimal prime must be the same

/

for all i. Since, on the other hand, grade IA[U ]/J ≥ 1, there cannot be a second minimal

prime: it would contain f1 , . . . , fu . ”

Now it is easy to show:

(12.5) Proposition. Let B be a noetherian domain, X an m — n matrix of inde-

terminates over B. Then the ideals I(s0 , . . . , sr ; v), 0 ¤ r ¤ min(m, n), 0 ¤ v ¤ n, form

a principal radical system. Hence all these ideals are radical.

Proof: First we invoke induction on the size of the matrix to conclude that all the

ideals I(s0 , . . . , sr ; n) are radical ideals. For the other ideals I = I(s0 , . . . , sr ; v) one may

suppose that v ≥ s0 . In order to show that assumption (b) of (12.1) is ful¬lled for all of

them we take x = X1v+1 . Then I + xB[X] = I(s0 , . . . , sr ; v + 1). Case (i): v = sw for

some w. Then x is not a zero-divisor mod Rad I. Otherwise it would be nilpotent modulo

I by (12.3); this is impossible because the B-homomorphism B[X] ’ B, X1v+1 ’ 1,

Xij ’ 0 for all other indeterminates, factors through B[X]/I. Since B[X]/I is graded

and the residue class of x has positive degree, ∞ (I + xi B[X])/I = 0. Case (ii):

i=0

sw < v < sw+1 for some w. Let

J = I(s0 , . . . , sw’1 , v, sw+1 , . . . , sr ; v).

and y = [a1 , . . . , aw+1 |b1 , . . . , bw+1 ] be a generator of J not already in I. Then Laplace

expansion of [1, a1 , . . . , aw+1 |b1 , . . . , bw+1 , v + 1] ∈ I along its ¬rst row shows xy ∈ I. It

is seen as in case (i) that x is not a zero-divisor modulo Rad J. ”

(12.6) Corollary. Let B be a noetherian domain, X an m — n matrix of indeter-

minates over B.

(a) If v = sw , then I(s0 , . . . , sr ; v) is a prime ideal, and X1v+1 is not a zero-divisor mod-

ulo I(s0 , . . . , sr ; v).

(b) Let sw < v < sw+1 . Then

I(s0 , . . . , sr ; v) = I(s0 , . . . , sr ; sw+1 ) © I(s0 , . . . , sw’1 , v, sw+1 , . . . , sr ; v)

is the prime decomposition of I(s0 , . . . , sr ; v).

(12.7) Remark. As a consequence of (12.6) the Z-algebras Z[X]/I(s0 , . . . , sr ; v)

are Z-free. Therefore one may use (3.12) in order to relax the hypotheses of (12.5) and

(12.6). As far as the property “radical” is concerned, it is enough to assume that B is

reduced; for “prime” one only needs that B is an integral domain.

C. The Perfection of Determinantal Ideals

Looking back to the example in Subsection A one notes that the only missing step

in the proof of perfection is an analogue of (4), a formula for the grade of the ideals

I(s0 , . . . , sr ; v).

159

C. The Perfection of Determinantal Ideals

(12.8) Proposition. Let B be a noetherian ring, X an m — n matrix of indeter-

minates. Then

r’1

r(r + 1)

grade I(s0 , . . . , sr ) = mn ’ (m + n)r + + si ,

2 i=0

and if sw’1 < v ¤ sw , 1 ¤ w ¤ r,