In (9.5) it has been pointed out that certain rings appearing in (9.4) can be viewed

as Segre products. Conditions under which Segre products of Cohen-Macaulay rings are

Cohen-Macaulay again are investigated by Chow ([Ch]). Chow™s results in particular

imply that R2 (X) is a Cohen-Macaulay ring.

The material of Subsection B is taken from [Ei.1], [DEP.2], Section 2, and [EiH].

The extended Rees algebra and the associated graded ring can be treated in greater

generality; one only needs that the ¬ltrations on which they are based satisfy a certain

condition, cf. [DEP.2].

There are more ideals satisfying the hypotheses of (9.12), say, than just the ideals of

maximal minors considered above. It is quite obvious that some of their subideals share

their characteristic properties; cf. [AS.1], [AS.2], [BrS], [BNS] for a detailed analysis of

certain ideals of this type.

In [Hu.3] Huneke has determined all the values of m, n, t such that RIt (X) (B[X]) ∼ =

S(It (X)), X an m — n matrix of indeterminates.

Proposition (9.23) and its corollary generalize Burch™s inequality ([Bh.2]), cf. also

[Bd].

10. Primary Decomposition

Let X be an m — n matrix of indeterminates over a domain B, m ¤ n. Contrary

to the ideal Im (X) (and I1 (X), of course) the ideals It (X), 2 ¤ t ¤ m ’ 1, have non-

primary powers. In this section we shall determine the symbolic powers of the It (X),

discuss the “symbolic graded ring” as the proper analogue for It (X) of the ordinary asso-

ciated graded ring for Im (X), and ¬nally compute a primary decomposition for products

It1 (X) . . . Its (X), essentially under the condition that B contains a ¬eld of characteristic

zero. It is a remarkable fact that the primary decomposition depends on characteristic.

A. Symbolic Powers of Determinantal Ideals

In (2.4) we have established an isomorphism which will prove very useful here: Let

Y be an (m ’ 1) — (n ’ 1) matrix of indeterminates; then the substitution

’1

Xij ’’ Yij + Xmj Xin Xmn , 1 ¤ i ¤ m ’ 1, 1 ¤ j ¤ n ’ 1,

Xmj ’’ Xmj , Xin ’’ Xin

induces an isomorphism

’1 ’1

• : B[X][Xmn ] ’’ B[Y ][Xm1 , . . . , Xmn , X1n , . . . , Xm’1,n ][Xmn ]

’1

whose inverse is given by Yij ’’ Xij ’ Xmj Xin Xmn , Xmj ’’ Xmj , Xin ’’ Xin . For

simplicity we identify the two rings by putting

’1

Yij = Xij ’ Xmj Xin Xmn ,

remembering of course that the Yij are algebraically independent over B. In order to

distinguish minors of X and Y we write [. . . | . . . ]X and [. . . | . . . ]Y .

(10.1) Lemma. (a) For all minors [a1 , . . . , as |b1 , . . . , bs ]Y one has

’1

[a1 , . . . , as |b1 , . . . , bs ]Y = Xmn [a1 , . . . , as , m|b1 , . . . , bs , n]X .

’1

(b) Let B be an integral domain, R = B[X], S = B[X][Xmn ]. Then

(k) (k)

S) © R,

It (X) = (It (X)

(k) (k)

It (X) S = (It’1 (Y ) )S

for all t, 2 ¤ t ¤ m, and all k ∈ N.

The equation in (a) is proved using the invariance of determinants under elementary

transformations. The ¬rst equation in (b) follows from R ‚ S ‚ RIt (X) , and for the

second it is important that the extensions R ’ S and B[Y ] ’ S commute with the

formation of symbolic powers.

The symbolic powers of I1 (X) coincide with the ordinary powers for trivial reasons;

(k) (k+1)

one has Ik (X) ‚ I1 (X) for all k and δ ∈ I1 (X) for a k-minor δ, k ≥ 1. Starting

/

with t = 1 and applying (10.1) inductively one gets:

123

A. Symbolic Powers of Determinantal Ideals

(10.2) Proposition. Let B be an integral domain. Then

(k)

It+k’1 (X) ‚ It (X)

for all k, and δ ∈ It (X)(k+1) for a (t + k ’ 1)-minor δ if 1 ¤ k ¤ m ’ t + 1.

/

The symbolic powers of a prime ideal P form a multiplicative ¬ltration,

P (k) P (l) ‚ P (k+l) ,

(k)

and this fact together with (10.2) determines the symbolic powers It (X) completely

as will be seen below. Because of (10.2) the degree of a (t + k ’ 1)-minor with respect to

this ¬ltration is k. Therefore we de¬ne the function γt (for arbitrary t) by

0 if δ is an s-minor, s < t,

γt (δ) =

s’t+1 if δ is an s-minor, s ≥ t,

and extend this de¬nition to the set of all (formal) monomials of minors by

p

γt (δ1 . . . δp ) = γt (δi ).

i=1

Let J(t, k) be the ideal generated by all the monomials π such that γt (π) ≥ k. Then

(k)

J(t, k) ‚ It (X) , and since we have obviously equality for t = 1, we could prove equality

for all t by induction via (10.1) if we knew that J(t, k)S © R = J(t, k), equivalently, that

Xmn is not a zero-divisor modulo J(t, k).

(10.3) Lemma. J(t, k) is generated as a B-module by the standard monomials µ

such that γt (µ) ≥ k. In particular, Xmn is not a zero-divisor modulo J(t, k) if t ≥ 2.

Proof: The proof of Proposition (4.1) details the “straightening procedure” by

which repeated applications of the straightening relations transform an arbitrary mono-

mial into its standard representation. It therefore su¬ces that in a straightening relation

ξ… = aµ µ one has γt (µ) ≥ γt (ξ) + γt (…) for all µ. This is easily seen to be true if one

takes into account that µ and ξ… have the same degree as polynomials in the entries of

X and that µ has at most two factors.

The second statement is obvious now: For every standard monomial µ the product

µXmn is a standard monomial again, and γt (µXmn ) = γt (µ) for t ≥ 2. ”

(10.4) Theorem. Let B be an integral domain. Then for all t, 1 ¤ t ¤ m, and all

k the k-th symbolic power of It (X) is generated by the (standard) monomials µ such that

γt (µ) ≥ k. Equivalently,

(k)

It (X) = It+κ1 ’1 (X) . . . It+κs ’1 (X),

the sum being extended over all κ1 , . . . , κs ≥ 1, s ¤ k, such that κ1 + · · · + κs ≥ k.

(k)

Furthermore µ ∈ It (X) if and only if γt (µ) ≥ k.

Proof: Only the last statement for non-standard monomials still needs a proof.

In the next subsection we will introduce the associated graded ring with respect to the

124 10. Primary Decomposition

¬ltration given by the symbolic powers of It (X). This ring is a domain, cf. (10.7). So the

leading forms of the minors of X are not zero-divisors, and for µ = δ1 . . . δp , δi ∈ ∆(X),

one therefore has

µ— = δ 1 . . . δ p ,

— —

—

denoting leading form. Thus the degree of µ— is the sum of the degrees of its factors

—

δi , whence it coincides with γt (µ). ”

(10.5) Remark. Without essential changes the ideals It (X)/Iu (X) ‚ Ru (X), 1 ¤

t ¤ u can be considered. (10.4) remains true modulo Iu (X), in other words:

(k)

(It (X)/Iu (X))(k) = (It (X) + Iu (X))/Iu (X).

The generalization to R(X; δ) is not immediate, because the induction argument breaks

(k)

down. Nevertheless we expect that (It (X)R(X; δ))(k) = It (X) R(X; δ) throughout. ”

B. The Symbolic Graded Ring

For a prime ideal P in a ring A the ring

()

P (j) /P (j+1)

GrP A =

j≥0

should properly be called the graded ring associated with the ¬ltration by symbolic

powers. In general, one cannot say much about it; it may even be non-noetherian though

A is noetherian (cf. [Rb.4]). As in the case of ordinary powers we denote the leading

()

form of x ∈ A by x— , in GrP A as well as in the “extended symbolic Rees ring”

∞ ∞

() (j) j

AT ’j ‚ A[T, T ’1 ].

RP (A) T•

= P

j=0 j=1

() ()

In order to make GrP A and RP (A) well-de¬ned objects over every ring B, we consider

(k)

It (X) to be given by the description in (10.4) if B is not a domain.

(10.6) Theorem. Let B be a commutative ring, X an m — n matrix of indeter-

minates, m ¤ n, and ∆ its poset of minors. Let 1 ¤ t ¤ m and P = It (X). Then

() ()

RP (B[X]) is a graded ASL on ∆— over B[T ’1 ], and GrP B[X] is a graded ASL on ∆—

over B, ∆— inheriting its partial order from ∆ as in (9.7).

Proof: This theorem is proved in the same fashion as (9.7) and (9.8). One needs

of course that ∆— generates the extended symbolic Rees ring as follows from (10.4), and

aµ µ the inequality γt (ξ…) ¤ γt (µ) holds for all

that in a straightening relation ξ… =

µ. ”

(10.7) Corollary. (a) If B is a Cohen-Macaulay ring, then the rings considered in

(10.6) are Cohen-Macaulay rings, too.

(b) If B is reduced (a (normal) domain), then the rings considered in (10.6) are reduced

((normal) domains).

Part (a) and the assertion on being reduced are immediate. For (b) one applies

(16.24) after the inversion of the maximal element Xmn of ∆— (modulo which the rings

—

under consideration are again ASLs) together with induction on t.

()

In the following proposition we consider GrP B[X] a B[X]-algebra via the natural

()

epimorphism B[X] ’ B[X]/P ‚ GrP B[X].

125

B. The Symbolic Graded Ring

(10.8) Proposition. Let B be a noetherian domain. Then with the notations of

(10.6), one has

()

min grade(I1 (X), P (j) /P (j+1) ) = grade I1 (X)GrP B[X] = t2 ’ 1.

()

Proof: The ¬rst equality follows from (9.23). Let now R = B[X], S = GrP R, and

J = I1 (X)S. The ideal J is generated by the subset

∆ = { δ — : δ an s-minor, 1 ¤ s < t }

of ∆— . We want to show that S/J is a graded ASL on „¦ = ∆— \ ∆. By Proposition (5.1),(a)

it is enough to show that as a B-module J is generated by the standard monomials

containing a factor from ∆, and, by reference to the straightening procedure, one only

has to show that in a straightening relation

ξ — …— = aµ µ (in S !)

every standard monomial µ contains a factor from ∆ if ξ — ∈ ∆. If additionally … — ∈ ∆,

then this is the straightening relation in R/P , and every µ consists entirely of factors

from ∆. Let … — ∈ ∆. If µ = ζ — · — , ζ, · ∈ ∆, ζ ¤ ·, then · — ≥ ξ — and · — ∈ ∆. (Remember

/

that, after all, the straightening relations are inherited from “(X).) If µ = ν — , ν ∈ ∆,

then, as polynomials in B[X],

deg ν = deg ξ + deg … > deg …

and

γt (ξ…) = γt (…) < γt (ν).

This is impossible, since the straightening relations are homogeneous equations in the

graded ring S.

So S/J is a graded ASL over the wonderful poset „¦. As in (9.25) one ¬rst reduces

to the case in which B is a ¬eld (via (3.14)). Since S is Cohen-Macaulay then, one has

grade I1 (X)S = rk ∆ ’ rk „¦

= mn ’ (mn ’ t2 + 1) = t2 ’ 1. ”

It will be shown in (14.12) that for all t ≥ 2 one has

(2)

grade(I1 (X), It (X)/It (X) ) = grade(I1 (X), Rt’1 (X)) + 3

= (m + n ’ t + 2)(t ’ 2) + 3.

Therefore

(2)

d = grade(I1 (X), Rt (X)) ’ grade(I1 (X), It (X)/It (X) )

= m + n ’ 2t

divides

(j) (j+1)

grade(I1 (X), Rt (X)) ’ min grade(I1 (X), It (X) /It (X) )

= (m + n ’ 2t)(t ’ 1).

(j) j+1

We believe that grade(I1 (X), It (X) /It (X) ) goes down by d if j is increased by 1

until it reaches its minimal value (and stays constant then). Admittedly there is not

much support for this claim, cf. (9.27),(a).

126 10. Primary Decomposition

C. Primary Decomposition of Products of Determinantal Ideals

None of the results proved so far depends on the characteristic of the ring B of

coe¬cients. Quite surprisingly, the primary decomposition of products It1 (X) . . . Its (X),

in particular of powers It (X)k , cannot be given without reference to the characteristic of

B, and we shall succeed in complete generality only for the rings containing the rational

numbers.

Let B be an integral domain, X an m — n matrix of indeterminates. The smallest

(e(j,t))

symbolic power of Ij (X) containing It (X) is Ij (X) , where

t’j +1 1 ¤ j ¤ t,

if

e(j, t) =

0 if t < j,

t arbitrary. This implies immediately the inclusion “‚” in:

(10.9) Theorem. Let B be an integral domain, X an m — n matrix of indetermi-

nates, and t1 , . . . , ts integers, 1 ¤ ti ¤ min(m, n). Let w = max ti , and suppose that

(min(ti , m ’ ti , n ’ ti ))! is invertible in B for i = 1, . . . , s. Then

w s

(ej )

It1 (X) . . . Its (X) = Ij (X) , ej = e(j, ti ),

j=1 i=1

is a (possibly redundant) primary decomposition.

It will be indicated in (10.12) how to re¬ne this decomposition to an irredundant

one.

As a speci¬c example we take n ≥ m ≥ 3, s = 2, t1 = t2 = 2. Then (10.9) in

conjunction with (10.4) says

I2 (X)2 = I1 (X)4 © (I3 (X) + I2 (X)2 ).

2

In particular the product of a 1-minor and a 3-minor must be in I2 (X) , the ¬rst nontrivial

case of the following lemma which is the crucial argument in the proof of (10.9).

(10.10) Lemma. Let B = Z, and F(i, j) be the Z-submodule of Z[X] generated