Furthermore one needs of course that the powers of P and Q are divisorial ideals (by

virtue of 9.18). In order to prove the isomorphism on the left, one ¬rst observes that

∞

A is a graded subalgebra j=’∞ Aj of B[Y, Z] where Aj contains the bihomogeneous

elements of partial degrees d1 with respect to Y and d2 with respect to Z such that

d2 ’ d1 = jr. The equations

[a1 , . . . , ar ]Y [b1 , . . . , br ]Z = [a1 , . . . , ar |b1 , . . . , br ]

then su¬ce to show that the assignment given induces B-isomorphisms

Pj for j ≥ 0,

Aj ∼

=

Q’j for j < 0,

whose direct sum is an R-algebra isomorphism. In the following we identify A and its

isomorphic copy.

(b) Let S = RP (R) ‚ A. Then

SI ‚ S[(δT )’1 ],

A‚

I=(P T )S

the intersection being extended over the divisorial prime ideals I = (P T )S of S.

In fact, S[(δT )’1 ] is the intersection of all the localizations SI , I a divisorial prime,

δT ∈ I. This explains the inclusion on the right side; for the one on the left we note that

/

∞

(S : ((P T )S)j ) =

A‚ SI ,

j=0 I=(P T )S

the operation : being performed in the ¬eld of fractions of S. However, (b) is only a

preparation for (c):

117

D. The Depth of Powers of Ideals of Maximal Minors

(c) One has

∞

P jT j.

A= SI =

j=’∞

I=(P T )S

A is a normal domain. If B is factorial then A is factorial, too.

In order to prove the equality claimed, one uses the second inclusion in (b), and shows

that every element s of S[(δT )’1 ] such that s((P T )S)j ‚ S for some j, is an element of A.

Being an intersection of discrete valuation rings (and noetherian) A must be normal. For

a quick proof of the last statement one applies [HV], Theorem,(a), p. 183: The extension

R ’ S induces an isomorphism of divisor class groups. Therefore the class of P S

generates Cl(S). Now P S and (P T )S are isomorphic ideals of S, so cl(P S) = cl((P T )S),

and cl((P T )S) is in the kernel of the natural epimorphism Cl(S) ’ Cl(A), cf. [Fs], § 7.

(The last statement in (c) can be generalized: Cl(A) ∼ Cl(B).) ”

=

D. The Depth of Powers of Ideals of Maximal Minors

For a local ring R with maximal ideal P and an ideal I ‚ R the analytic spread l(I)

is de¬ned by

l(I) = dim GrI R/P GrI R,

cf. [NR], [Bh.2], and [Bd]. For a graded ASL A on Π over a noetherian ring B, and an

ideal I ‚ AΠ the corresponding quantity is

dim GrI A/ΠGrI A.

If B is a ¬eld, then dim GrI A/ΠGrI A = l(IAΠ ), since (GrI A) — AAΠ = GrIAΠ AAΠ and

(GrI A/ΠGrI A) — AAΠ = GrI A/ΠGrI A. It is easy to determine GrI A/ΠGrI A and its

dimension for our objects.

(9.22) Proposition. Let B be a noetherian ring, and R = G(X; γ) or R = R(X; δ),

I an ideal of maximal minors, „¦ the ideal in Π = “(X; γ) or Π = ∆(X; δ) generating I

and S the sub-ASL generated by „¦.

(a) Then GrI R/ΠGrI R is a homomorphic image of S and

dim GrI R/ΠGrI R ¤ dim B + rk „¦.

(b) If R = G(X; γ) or R = R(X; δ), δ = [a1 , . . . , ar |b1 , . . . , br ] and „¦ consists of r-minors

only, then GrI R/ΠGrI R ∼ S and

=

dim GrI R/ΠGrI R = dim B + rk „¦.

Proof: (a) It has been noticed in (9.15) that S can be regarded the sub-ASL of

GrI R generated by „¦— . Since the generators of the B-algebra GrI R outside „¦— are killed

in passing to GrI R/ΠGrI R, the latter ring is a homomorphic image of S which by (5.10)

has dimension dim B + rk „¦.

(b) We have to show that (ΠGrI R) © S = 0. For this it is su¬cient that every

standard monomial in the standard representation of an element in ΠGrI R contains

a factor from Π— \ „¦— . In view of the straightening procedure outlined in (4.1) this

118 9. Powers of Ideals of Maximal Minors

is equivalent to the appearance of at least one factor from Π— \ „¦— in every standard

monomial on the right hand side of a straightening relation

ξ — …— = aµ µ (in GrI R !)

with ξ — ∈ Π— \ „¦— . Since this equation is homogeneous in the graded ring GrI R, a

standard monomial µ can have at most one factor in „¦— . In case R = G(X; γ) every

such µ has automatically two factors. In the other case of (b) one argues as follows: If

… — ∈ Π— \ „¦— too, then every µ entirely consists of factors from Π— \ „¦— , and if … ∈ „¦— ,

then every µ must have two factors for reasons of degree in R. ”

The inequality in (a) can indeed be strict, as is demonstrated by the example R =

B[X], X an m — n matrix with m ≥ 2, I the ideal generated by the elements in the ¬rst

row.

As an A-module the associated graded ring represents the properties common to

all of the quotients I j /I j+1 . A quantity which can be rather comfortably computed by

means of the associated graded ring, is the minimum of their depths if A is local. The

global analogue for our objects is grade(AΠ, I j /I j+1 ), the length of a maximal (I j /I j+1 )-

sequence in AΠ. In view of a later application the following proposition is kept more

general than needed presently.

(9.23) Proposition. Let A be a noetherian ring and let F = (Ij )j≥0 , I0 = A, be

a multiplicative ¬ltration of A by ideals such that the associated graded ring Gr F A is

noetherian. Consider GrF A as an A-algebra via the natural epimorphism A ’ A/I1 ,

and let J ‚ A be an ideal. Then

min grade(J, A/Ij ) = min grade(J, Ij /Ij+1 ) = grade JGrF A.

Proof: The left equation follows from the behaviour of grade along the exact se-

quences

0 ’’ Ij /Ij+1 ’’ A/Ij+1 ’’ A/Ij ’’ 0.

If JGrF A contains an element which is not a zero-divisor of GrF A, then J is not contained

in the preimage of any of the (¬nitely many) associated prime ideals of GrF A, so contains

an element which is not a zero-divisor on any of Ij /Ij+1 . Conversely, if grade JGrF A = 0,

then J must annihilate a homogeneous element = 0 in GrF A of degree d, say, and so

grade(J, Id /Id+1 ) = 0. The rest is induction based on the equation (Ax)— = (GrF A)x—

for an element x ∈ A which is not a zero-divisor of GrF A, — again denoting “leading

form”. ”

(9.24) Corollary. Let A be a local ring, P its maximal ideal, and I ‚ A an ideal.

Then

min depth A/I j ¤ ht P GrI A ¤ dim GrI A ’ l(I).

If GrI A is a Cohen-Macaulay ring, one has equality throughout.

(9.25) Corollary. Let R = G(X; γ) or R = R(X; δ) over a noetherian ring B of

coe¬cients, I an ideal of maximal minors in R, and „¦ the ideal in Π = G(X; γ) or

Π = R(X; δ) resp. generating I. Then

min grade(RΠ, R/I j ) = min grade(RΠ, I j /I j+1 ) ≥ rk Π ’ rk „¦,

119

D. The Depth of Powers of Ideals of Maximal Minors

and if the hypothesis of part (b) of (9.22) is ful¬lled, one has equality.

Proof: Let B = Z ¬rst. Then the de¬ning ideal of GrI R as a residue class ring

of the polynomial ring Z[Tπ : π ∈ Π— ] is generically perfect as a consequence of (9.14).

Because of (9.23) and (3.14) it is enough that ht ΠGrI R ≥ rk Π ’ rk „¦ (with equality

under the hypothesis of part (b) of (9.22)) whenever B is a ¬eld, and this is guaranteed

by (9.22). ”

The best information we can give on the behaviour of grade(J, R/I j ) as a function

of j, is the following proposition.

(9.26) Proposition. Let A be a noetherian ring, I, J ideals of A such that ht I ≥ 1

and GrI A is a Cohen-Macaulay ring. If grade(J, A/I k ) = min grade(J, A/I j ), then

grade(J, A/I k ) = grade(J, A/I k+1 ).

Proof: Suppose that min grade(J, A/I j ) ≥ 1. Then there exists an x ∈ J such

that x— is not a zero-divisor of GrI A. This fact triggers a proof by induction (observe

that ht(I + Ax)/Ax ≥ 1), and one need only deal with the case grade(J, A/I k ) = 0.

Since ht I ≥ 1, dim GrI A/I — GrI A = dim A/I < dim A = dim GrI A. So I \ I 2 contains

an element y for which y — is not a zero-divisor of GrI A. Multiplication by y then induces

an embedding

A/I k ’’ A/I k+1 ,

whence grade(J, A/I k+1 ) = 0, too. ”

(9.27) Examples. In the following we assume that B = K is a ¬eld. Because

of (3.14) the grade formulas generalize to arbitrary noetherian rings. They improve

Proposition (7.24).

(a) R = B[X], X an m — n matrix, m ¤ n, I = Im (X). Then

min grade(I1 (X), R/I j ) = m2 ’ 1.

It will be shown later (cf. (14.12)) that grade(I1 (X), R/I 2 ) = 3 if m = 2, and the preced-

ing proposition then implies grade(I1 (X), R/I j ) = 3 for all j ≥ 2. Another completely

known case is n = m + 1. Since I ∼ Coker X (the linear map X : Rm ’ Rm+1 given by

=

the matrix X, cf. (16.36) for the isomorphism) and I j ∼ Sj (I) by (9.13), we conclude

=

j

from (2.19),(b)(i) that pd R/I = min(j, m) + 1, hence

grade(I1 (X), R/I j ) = m(m + 1) ’ j ’ 1 for j = 1, . . . , m,

grade(I1 (X), R/I j ) = m2 ’ 1 for j ≥ m,

because of the equation of Auslander-Buchsbaum and the equality grade(I1 (X), M ) =

depth MI1 (X) for graded K[X]-modules M . (We believe that grade(I1 (X), R/I j ) always

behaves in a regular manner; cf. the discussion below (10.8).)

(b) More generally let R = Rr+1 (X), X as in (a), I = Ir (X)/Ir+1 (X). Then

min grade(I1 (X), R/I j ) = r2 ’ 1.

(c) If R is as in (b) and Q the ideal generated by the r-minors of any r columns,

then

min grade(I1 (X), R/Qj ) = nr ’ 1.

120 9. Powers of Ideals of Maximal Minors

For the ideal P generated by the r-minors of any r rows one has

min grade(I1 (X), R/P j ) = mr ’ 1.

Since Qn’m is a canonical module and therefore a maximal Cohen-Macaulay module,

grade(I1 (X), R/Qn’m ) = dim R ’ 1 = (m + n ’ r)r ’ 1,

and the minimum can only be attained for exponents > n ’ m.

(d) The analysis of the example (c) can certainly be carried further. We content

ourselves with the case in which r + 1 = m ¤ n:

grade(I1 (X), R/Qj ) = nm ’ (n ’ m + 1) ’ 1 for j = 1, . . . , n ’ m + 1,

grade(I1 (X), R/Qj ) = nm ’ j ’ 1 for j = n ’ m + 1, . . . , n,

grade(I1 (X), R/Qj ) = nm ’ n ’ 1 for j ≥ n.

and R, Q, . . . , Qn’m+1 , P are the only Cohen-Macaulay modules of rank 1 (up to iso-

morphism). Note that the canonical module Qn’m is not the last one in the sequence of

powers of Q to be a Cohen-Macaulay module.

Every Cohen-Macaulay module of rank 1 is a divisorial ideal and therefore isomorphic

to a power of Q, cf. (8.4). In order to compute the multiplicity of R we have considered

the R-sequence

[i|j] : j ’ i < 0 or j ’ i > n ’ m ∪ [i|j] ’ [i’1|j’1] : 0 ¤ j ’ i ¤ n ’ m

y=

It generates an I1 (X)R-primary ideal, and one has

n

e(R) = »(R/Ry) = ,

m’1

cf. (2.15). Let M be a Cohen-Macaulay module of rank 1. Then dim M = dim R, and

M is a maximal Cohen-Macaulay module. Since this property localizes,

n

(—) »(M/yM ) =

m’1

by virtue of [He.2], Proposition 1.1. (For a graded R-module the property of being a

maximal Cohen-Macaulay module also globalizes, cf. (16.20): it is equivalent to being

perfect over K[X] of grade equal to grade Im (X)). Therefore the validity of (—) is su¬cient

for the modules under consideration to be Cohen-Macaulay.)

The minimal number of generators of all divisorial ideals, but the listed ones, al-

n

ready exceeds m’1 , excluding them from being Cohen-Macaulay modules. The ideal

P certainly is a Cohen-Macaulay module. For the powers of Q we use the free resolution

over K[X] constructed in (2.16) and (2.19),(b)(ii). Let f : K[X]n ’ K[X]m be given

by the matrix X — , and f = f — R. Coker f is annihilated by Im (X) (cf. (16.2)), so

Coker f ∼ Coker f . Since rk f = m ’ 1, Coker f has rank 1 as an R-module. Being a per-

=

fect K[X]-module, it is (isomorphic to) a (divisorial) ideal. Sending its i-th “canonical”

121

E. Comments and References

generator to (’1)i+1 [1, . . . , i, . . . , m|1, . . . , m ’ 1], one maps it onto Q, so Q ∼ Coker f .

=

For the formation of the symmetric powers Sj (Q), j ≥ 1, it makes no di¬erence whether

we consider Q as an R-module or a K[X]-module. In conjunction with (9.13), (2.16) and

(2.19),(b)(ii) therefore provide the projective dimension of all the powers Qj . The three

equations above now follow as those in (a).

For the general case in regard to m, n, r the preceding discussion at least implies

that the number of (isomorphism classes of) Cohen-Macaulay modules of rank 1 over

Rr+1 (X) is always ¬nite. ”

E. Comments and References

The investigation of powers of determinantal ideals was initiated by Hochster ([Ho.6])

who showed that Im (X) has primary powers if X is an m — (m + 1) matrix, cf. also

[ASV], p. 67, Beispiel 6.2. His result was generalized by Ngo ([Ng.1]) to m — n matrices;

Ngo investigated the associated graded ring by the method of principal radical systems

([HE.2]). Huneke showed in [Hu.1] that straightening-closed ideals are generated by

“weak d-sequences” ([Hu.1], Proposition 1.3). This allowed him to prove the equality

of ordinary and symbolic powers for the ideals I discussed in (9.27) and to compute

min grade(I1 (X), I j ). Example (9.27),(c) was treated in [Br.6] by an ad hoc method. A

special case of (9.27),(a) appeared in [Ro]; the special result for n = m + 1 is taken from

[AH]. The divisor class groups of the Rees algebras with respect to ideals of maximal