(9.12) Corollary. Let A be a graded ASL on Π over B. Suppose that B is a

Cohen-Macaulay ring, and Π a wonderful poset. Let „¦ be an ideal in Π such that „¦ is

self-covering, contains all the minimal elements of Π, and I = A„¦ is straightening-closed.

Then RI (A) is a Cohen-Macaulay ring, too.

(9.13) Remark. Besides the Rees algebra(s) and the associated graded ring there

is another commutative algebra de¬ned by “powers” of I: the symmetric algebra

∞

S(I) = Sj (I).

j=0

The natural epimorphisms Sj (I) ’ I j , sending a product of j elements of I in Sj (I) to

their product in I j , de¬nes a natural epimorphism

S(I) ’’ RI (A).

It would be unreasonable to expect that this epimorphism is an isomorphism for our

rings and ideals, except under very rare circumstances. One has a commutative diagram

•

A[Tω : ω ∈ „¦] ’’ S(I)

ψ

RI (A),

the indeterminate Tω being sent to ω ∈ S1 (I) and ω — resp. Being a graded ASL, RI (A)

is represented over B by its generators Π „¦ and the straightening relations. Therefore

the kernel of ψ is generated by the elements representing the straightening relations of

types (ii) and (iii) in the proof of (9.10). The elements of type (ii) are in the kernel of •,

too. So S(I) and RI (A) are isomorphic if „¦ is linearly ordered and no relations of type

(iii) are present. ”

112 9. Powers of Ideals of Maximal Minors

C. Graded Algebras with Respect to Ideals of Maximal Minors

Without further ado we draw the consequences of the results in Subsections A and B:

(9.14) Theorem. Let R = G(X; γ) or R = R(X; δ) over a ring B of coe¬cients,

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

in Π generating I.

(a) Then GrI R and RI (R) are graded ASLs over B on Π— ‚ GrI R and Π „¦ resp.

(b) If B is a Cohen-Macaulay ring, then GrI R, RI (R), and RI (R) are Cohen-Macaulay

rings, too.

Next we want to prove that GrI R, RI (R), and RI (R) are (normal) domains over

a (normal) domain B. First we observe that the sub-ASL generated by „¦ is present in

GrI R (and RI (R)), too:

(9.15) Lemma. Under the hypothesis of the preceding theorem „¦— generates a sub-

ASL of GrI R. It is isomorphic to the sub-ASL generated by „¦ in R.

This is obvious: the straightening relation for incomparable ξ — and … — in „¦— is

produced from that for ξ and … in „¦ by “starring” all the factors ζ ∈ Π occuring.

(Remember that a graded ASL is completely determined by its straightening relations !)

Di¬erent from our usual procedure we start with the case R = R(X; δ) for the

investigation of integrity and normality, mainly because we regard expansions of deter-

minants more “visible” then Pl¨cker relations. Let A = GrI R, and assume that B is

u

—

normal. The element δ is minimal in the poset underlying A, and one would like to

show that A[(δ — )’1 ] is normal in order to apply (16.24) then. Let

δ = [a1 , . . . , ar |b1 , . . . , br ],

and the element µ de¬ning I be given by

µ = [a1 , . . . , ak’1 , ak , . . . , a˜|b1 , . . . , b˜].

r r

In R we expand the minor δ along its ¬rst k rows:

±[a1 , . . . , ak |C][ak+1 , . . . , ar |{b1 , . . . , br }\C],

(1) δ=

C

C running through the subsets of cardinality k of {b1 , . . . , br }. Every term in this equation

has exactly one factor in „¦: δ on the left and [a1 , . . . , ak |C] on the right side. Because

of (9.3) none of these factors is in I 2 , and [ak+1 , . . . , ar |{b1 , . . . , br }\C] ∈ I. Therefore

/

δ— = ±[a1 , . . . , ak |C]— [ak+1 , . . . , ar |{b1 , . . . , br }\C]—

(2)

in A = GrI R. Let A = A[(δ — )’1 ]. It su¬ces to show that AP is normal for the prime

ideals P of A. Since δ — is a unit in A, one of the elements [a1 , . . . , ak |C]— has to be a

unit in AP , too. Eventually it is enough to prove normality for the extensions

A ([a1 , . . . , ak |C]— )’1 .

113

C. Graded Algebras with Respect to Ideals of Maximal Minors

(9.16) Lemma. With the hypotheses of (9.14) and the notations just introduced, let

ζ = [a1 , . . . , ak |C]— , C = [c1 , . . . , ck ]. Furthermore Let S be the sub-ASL of A generated

by „¦— , U denote a k — k matrix, and V an (m ’ ak + 1) — k matrix of indeterminates

over B. Then the homomorphism

•

S[ζ ’1 , U, V ]/Ik (U ) ’’ A[ζ ’1 ]

which is the identity on S ‚ A, sends Uij to the residue class of [ai |cj ] in R/I and Vuv

to the residue class of [ak ’ 1 + u|cv ], is an isomorphism.

Proof: Let [u|v] be the residue class of [u|v] in R/I. Since [a1 , . . . , ak |b1 , . . . , bk ] ∈

I, the determinant of the matrix formed by the elements [ai |cj ] (which is di¬erent from

[a1 , . . . , ak |c1 , . . . , ck ]— !) is zero, and the homomorphism is well-de¬ned.

The rings and the homomorphism • under consideration are constructed from the

corresponding objects over Z by tensoring with B, since both rings are, roughly spoken,

de¬ned by their straightening relations. So we may assume that B is a noetherian integral

domain. A glance at (9.4),(c) shows that the dimensions are equal (note that dim A =

dim R), and it is enough that the homomorphism • is surjective. As an S-algebra, A

is generated by the elements [i|j] . Let ¬rst i < ak and j = cv . If i ∈ {a1 , . . . , ak },

[i|j] ∈ Im • by de¬nition. Otherwise we look at the equation

[i|cv ][a1 , . . . , ak |C] = ±[au |cv ][{a1 , . . . , ak , i}\{au }|C]

u

in R which simply results from the Laplace expansion of a minor with two equal columns.

If k = 1, then [i, cv ] = 0, and [i, cv ] ∈ Im • trivially. If k > 1, the k-minors = 0 in this

equation all lie in I \ I 2 , and the 1-minors are in R \ I. Therefore in A one has

[au |cv ] = [au |cv ]— ,

[i|cv ] = [i|cv ]— = ζ ’1 ±[au |cv ]— [{a1 , . . . , ak , i}\{au }|C]— ,

u

and [i|cv ] ∈ Im •. Combined with the de¬nition of •, we conclude [i|cv ] ∈ Im • for all

i = 1, . . . , m and all v = 1, . . . , k.

In order to “cover” [i|j] with i ∈ {a1 , . . . , ak }, j ∈ {c1 , . . . , ck } one works with the

/

relation

[au |j][a1 , . . . , ak |C] = ±[au |cv ][a1 , . . . , ak |{c1 , . . . , ck , j}\{cv }],

v

and ¬nally for [i|j] with i ∈ {a1 , . . . , ak }, j ∈ {c1 , . . . , ck } the equation

/ /

[i|j][a1 , . . . , ak |C] = [a1 , . . . , ak , i|c1 , . . . , ck , j]

±[i|cv ][a1 , . . . , ak |{c1 , . . . , ck , j}\{cv }]

+

v

implies a suitable equation in A: the k-minors and the (k + 1)-minor appearing all are in

114 9. Powers of Ideals of Maximal Minors

I \ I 2 , unless they are zero or k = 1, i < ak , in which case [i, j] = 0 ∈ Im • anyway. ”

(9.17) Theorem. Let R = G(X; γ) or R = R(X; δ) over a ring B of coe¬cients,

and I be an ideal of maximal minors in R. If B is a (normal) domain, then Gr I R,

RI (R), and RI (R) are (normal) domains, too.

Proof: Let R = R(X; δ), A = GrI R. Since δ — is not a zero-divisor of A, A is

a subring of A[(δ — )’1 ]. Every localization of A[(δ — )’1 ] is a localization of one of the

rings A[ζ ’1 ] as in the preceding lemma. A[ζ ’1 ] is a (normal) domain by virtue of the

lemma, (9.4), and (6.3). A little exercise shows that A[(δ — )’1 ] cannot contain a nontrivial

idempotent (if B has none), and therefore A[(δ — )’1 ] is a (normal) domain together with

all its localizations. So A itself is a domain, and normal, when A[(δ — )’1 ] is normal,

cf. (16.24).

In case R = G(X; γ), A = GrI R, we view R as the homogenization of R = R(X; δ),

R = R/R(y ± 1), y = [n ’ m + 1, . . . , n]. Let I be the dehomogenization of I. Since y —

is the maximal element in the poset underlying the ASL A, y — ± 1 = (y ± 1)— is not a

zero-divisor, consequently

GrI R = A/A(y — ± 1)

(cf. (3.7)). Obviously A can be viewed as a graded B-algebra in which y — is an element

of degree 1. Since A/Ay — is an ASL and therefore reduced, we can apply (16.24) and

conclude integrity and normality of A from the corresponding properties of GrI R.

For the Rees algebras integrity is not an issue. Since

RI (R)[(· — )’1 ] = RI (R)[· ’1 ],

· = δ or · = γ resp., it is enough to prove normality for RI (R), one more application of

(16.24). Now

RI (R)[(T ’1 )’1 ] = R[T, T ’1]

is normal and

RI (R)/T ’1 RI (R) = GrI R

is certainly reduced. (So the normality of RI (R) and RI (R) results from (9.8) already.) ”

Generalizing (7.10) we obtain:

(9.18) Corollary. If B is an integral domain, then an ideal I of maximal minors

in R (is prime and) has primary powers: I j = I (j) for all j ≥ 0.

Proof: Suppose that the contention is false, and let k be the smallest exponent

for which I k = I (k) . For x ∈ I (k) \ I k there exists an y ∈ R \ I such that yx ∈ I k .

By assumption on k, x ∈ I k’1 , and y — x— = 0 in GrI R, contradicting the integrity of

GrI R. ”

This corollary allows us to complete the description of the canonical module whose

class was computed in the preceding section:

115

C. Graded Algebras with Respect to Ideals of Maximal Minors

(9.19) Corollary. Let B be an Cohen-Macaulay ring having a canonical module

ωB . If the integers κi ≥ 0 satisfy the condition in (8.12) or the analogous condition in

(8.14) resp., then a canonical module of R is given (as a B-module) by the direct sum

ωR = ωB µ,

µ ranging over the standard monomials which have at least κi factors in “(X; γ)\“(X; ζi )

or ∆(X; δ) \ ∆(X; ζi ) resp., the elements ζi being the upper neighbours of γ or δ resp.

(The assumption κi ≥ 0 can always be satis¬ed.)

Proof: Let R0 be the ring G(X; γ) or R(X; δ) over the integers Z. Then

ω R = ω B — Z ω R0

reducing everything to R0 . We have ωR0 = Piκi now, Pi = J(x; ζi ) or Pi = I(x; ζi ),

and Piκi has a basis consisting of standard monomials as given by (9.6). ”

(9.20) Remark. In principle (9.19) allows the computation of the Cohen-Macaulay

type of G(X; γ) and R(X; δ) over a ¬eld, say. (The Cohen-Macaulay type is the minimal

number of generators of the canonical module.) A relatively simple case is R = R r+1 (X).

Assume that m ¤ n, k = n ’ m. Then ωR ∼ Qk , Q generated by the r-minors of the

=

¬rst r columns of the matrix of residue classes, and a minimal system of generators of Q k

(in R as well as in the localization with respect to the irrelevant maximal ideal) is given

by the standard monomials of length k in the r-minors of the ¬rst r columns. Therefore

it coincides with the minimal number of generators of the k-th power of the irrelevant

maximal ideal of G(Y ), Y an r — m matrix, and the type of Rr+1 (X) can be read o¬

from the Hilbert series of G(Y ). The latter has been computed explicitely in [HP], p.

387, Theorem III. J. Brennan communicated the following expression for the type of

Rr+1 (X):

n’m+r

· · · n’1

r r

.

r m’1

··· r

r

In the cases in which the generators of Q are linearly ordered (i.e. r + 1 = m or r = 1)

n’1

this simpli¬es to n’m , a result which also follows directly from (9.19). ”

(9.21) Remark. As in 7.C let R = Rr+1 (X) ∼ B[Y Z], Y be an m — r matrix

=

and Z an r — n matrix of indeterminates over B. In the following we want to analyze

the algebra A generated by the entries of the product matrix Y Z, the r-minors of Y

and the r-minors of Z. It has been demonstrated (cf. (7.6),(b)) that A is the ring of

absolute SL(r, B)-invariants of B[Y, Z]. In view of Remark (7.13) it is desirable to prove

the normality of A independently from invariant theory. For the rest of this remark we

assume that B is a normal domain. We sketch the arguments, leaving some details to

the reader. As usual, let P (Q) be the ideal in R ∼ B[Y Z] generated by the r-minors

=

of the ¬rst r rows (columns), and T an independent indeterminate over R. Furthermore

[a1 , . . . , ar ]Y is the r-minor of the rows a1 , . . . , ar of Y , whereas [b1 , . . . , br ]Z denotes the

r-minor of the columns b1 , . . . , br of Z. Finally, δ = [1, . . . , r|1, . . . , r] (as a minor of

X = Y Z in R).

116 9. Powers of Ideals of Maximal Minors

(a) The assignment

[a1 , . . . , ar ]Y ’’ [a1 , . . . , ar |1, . . . , r]δ ’1 T ’1 ,

[b1 , . . . , br ]Z ’’ [1, . . . , r|b1 , . . . , br ]T,

induces an isomorphism of R-algebras

∞ ∞ ∞

A∼ j ’j ’j j j

P jT j.

•R•

Qδ T PT =

=

j=1 j=1 j=’∞

For the object on the right side, the powers P j , j < 0, are of course to be considered

fractionary ideals of the domain R. The equality in the preceding formula is easily

checked if one applies the valuations associated with the divisorial prime ideals of R: