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vP (δ) = vQ (δ) = 1, vI (δ) = 0 for I = P, Q.

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

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