¬rst statement implies the second one. Furthermore I j is generated as a B-module by all

the monomials containing at least j factors from „¦. So it is su¬cient that every standard

monomial appearing in the standard representation of such a monomial contains at least

j factors from „¦, too. This standard representation is produced by repeated straightening

of pairs of incomparable elements. However, in a straightening relation ξ… = aµ µ with

one factor in „¦ every µ has to contain such a factor, too, since „¦ is an ideal in the poset

underlying G(X; γ) or R(X; δ), and if both ξ ∈ „¦ and … ∈ „¦, then µ contains (exactly)

two factors from „¦ by virtue of (9.1). ”

107

A. Ideals and Subalgebras of Maximal Minors

(9.4) Proposition. Let µ, „¦, and S be as above. Then:

(a) „¦ is a sublattice of “(X; γ) or ∆(X; δ) resp.

(b) S is a Cohen-Macaulay ring if (and only if ) B is Cohen-Macaulay.

(c) Let B be noetherian.

(i) If R = G(X; γ), then dim R ’ dim S = k(n ’ m + k ’ ak + 1).

(ii) If R = R(X; δ) and µ = [a1 , . . . , ak’1 , ak , . . . , a˜|b1 , . . . , b˜], then

r r

dim R ’ dim S = k(m + k ’ ak + 1) ’ 1.

An analogous formula holds for µ = [a1 , . . . , a˜|b1 , . . . , bl’1 , bl , . . . , b˜].

r r

(d) S is a (normal) integral domain, if (and only if ) B is a (normal) integral domain.

We outline the proof; the reader may supply the details (should there be any).

Part (a) is quite evident, and (b) follows from (5.14). For (c) one counts the number of

steps one needs to climb from the (single) maximal element of „¦ to that of “(X; γ) or

∆(X; δ), (All the maximal chains in a distributive lattice have the same length.) For (d)

one proceeds as in Section 6. Let ¬rst R = G(X; γ), and Ψ as in (6.1). Then Ψ © „¦ has

dim S elements, and one checks that

S[γ ’1 ] ∼ B[Tψ : ψ ∈ Ψ © „¦][Tγ ].

’1

=

For R = R(X; δ) one constructs R = G(X; δ), µ, „¦, S and observes that S is mapped

isomorphically onto S here; cf. the last part of the proof of (9.1). ”

The reader may try to ¬nd the class group of S and the canonical class.

(9.5) Remark. The Segre product

∞

Ak — Ak

k=0

of graded ASLs A = Ak on Π and A = Ak on Π is a graded ASL on the poset

∞

{ ξ — ξ : ξ ∈ Π © A k , ξ ∈ Π © Ak }

k=0

ordered by the decree

ξ—ξ ¤…—… ⇐’ ξ ¤ …, ξ ¤ ….

The straightforward veri¬cation of this fact can be left to the reader, and we mention

it only because some of the ASLs S considered in (9.4) can be viewed as such Segre

products. Let X be an m — n matrix of indeterminates, δ = [1, . . . , r|1, . . . , r], and

µ = [1, . . . , r ’ 1|r, . . . , r ’ 1]. Then S is the Segre product of G(Y ) and G(Z) where Y is

an r — m matrix and Z is an r — n matrix. Note that this includes the case S = R2 (X)

in which S is the Segre product of two polynomial rings in m and n variables resp. ”

108 9. Powers of Ideals of Maximal Minors

B. ASL Structures on Graded Algebras Derived from an Ideal

The algebras derived from an ideal I in a ring A we want to consider, are

∞

I j T j ‚ A[T ],

RI (A) =

the (ordinary) Rees algebra T an indeterminate,

j=0

∞

AT ’j ‚ A[T, T ’1],

RI (A) = RI (A) •

the extended Rees algebra and

j=1

∞

I j /I j+1 .

the associated graded ring GrI A =

j=0

One has the representations

GrI A = RI (A)/IRI (A),

GrI A = RI (A)/T ’1 RI (A).

∞

We suppose that j=0 I j = 0. Then every element x ∈ A has a well-de¬ned degree with

respect to the ¬ltration of A by the powers of I:

x ∈ I j \ I j+1 .

grad x = j if

The element xT grad x in RI (A) ‚ RI (A) and its residue class in GrI A will both be

denoted by x— and called the leading form of x with respect to I.

Let there be given a graded ASL A on Π over a ring B of coe¬cients, „¦ an ideal in

the poset Π, and I = A„¦. We say that I (or „¦) is straightening-closed if every standard

monomial µ appearing in the standard representation ξ… = aµ µ of incomparable

∞

elements ξ, … ∈ „¦ contains at least two factors in „¦. Note that automatically j=0 I j = 0:

(9.6) Proposition. Let A be a graded ASL on Π over B, and „¦ ‚ Π an ideal such

that I = A„¦ is straightening-closed. Then I j is the B-submodule of A generated by all

standard monomials with at least j factors in „¦.

This proposition is proved as Corollary (9.3).

(9.7) Theorem. Let A be a graded ASL on Π over B, and „¦ ‚ Π an ideal such

that I = A„¦ is straightening-closed. Then the extended Rees algebra RI (A) is a graded

ASL over B[T ’1 ] on the poset

Π— = { ξ — : ξ ∈ Π }

ordered by: ξ — ¤ … — ⇐’ ξ ¤ ….

∞

Rj T j . Each Rj is a graded B-module since I is

Proof: One has RI (A) = j=’∞

∞

Rjk , and RI (A) then is a graded B[T ’1 ]-algebra with

a homogeneous ideal: Rj = k=0

homogeneous components

∞

Rjk T j .

Rk =

j=’∞

109

B. ASL Structures on Graded Algebras Derived from an Ideal

Evidently Π— generates RI (A) as a B[T ’1 ]-algebra.

The ring A[T, T ’1 ] = A — B[T, T ’1] is obviously a graded ASL on Π over B[T, T ’1 ].

Since Π— arises from Π over B[T, T ’1] by multiplication of its elements with units of

B[T, T ’1 ], A[T, T ’1 ] is also an ASL on Π— , implying the linear independence of the

standard monomials in Π— over the smaller ring B[T ’1 ].

It follows from the preceding proposition that

ξ— = ξ ξ — = ξT

for ξ ∈ Π \ „¦ for ξ ∈ „¦.

and

Let ξ, … ∈ Π be incomparable with standard representation ξ… = aµ µ. If µ =

π1 , . . . , πk , πj ∈ Π, then µ— = π1 , . . . , µ— , and

—

k

ξ — …— = a µ T j µ µ—

is the standard representation of ξ — … — over B[T, T ’1 ]. By the hypotheses on I, jµ =

grad ξ… ’ grad µ ¤ 0 for all µ, and we have a standard representation over B[T ’1 ]. ”

(9.8) Corollary. The associated graded ring GrI A is a graded ASL on (the image

of ) Π— over B.

Proof: In passing from RI (A) to

GrI A = RI (A) —B[T ’1 ] B[T ’1 ]/T ’1 B[T ’1 ] = RI (A) —B[T ’1 ] B

we have only “extended” the ring of coe¬cients. ”

(9.9) Corollary. If moreover Π is wonderful and B is a Cohen-Macaulay ring,

then RI (A) and GrI A are Cohen-Macaulay rings, too.

Let ξ… = aµ µ be a straightening relation in A. Then

ξ — …— = a µ T j µ µ— , jµ = grad ξ… ’ grad µ,

is the corresponding straightening relation in RI (A), and in GrI A it transforms into

ξ — …— = a µ µ— ,

jµ =0

so one obtains this relation from aµ µ by dropping all the terms on the right side which

have higher degree with respect to I than ξ….

We want to make the ordinary Rees algebra RI (A) an ASL over B. Obviously Π—

does not generate RI (A) as a B-algebra: the elements ξ ∈ „¦ ‚ A ‚ RI (A) are not

representable by polynomials in Π— with coe¬cients in B, and we have to “double” „¦:

Let

Π „¦ : = Π ∪ „¦— ‚ RI (A)

where the subsets Π and „¦— are ordered naturally and every other relation is given by

ξ— < … for ξ ∈ „¦, … ∈ Π such that ξ ¤ ….

110 9. Powers of Ideals of Maximal Minors

„¦ is evidently a partially ordered subset of RI (A). For example, if Π is given by

Π

sχ

sχ

s·

d

s·

d

d ds ζ

d and „¦ = { ξ, …, ζ }, then Π „¦ is v s

ds ζ d

v

—s d s s —

s

d d ζ

d v ξ

d d

sξ

d

d —

sξ

(9.10) Theorem. Let A be a graded ASL on Π over B, „¦ ‚ Π an ideal such that

I = A„¦ is straightening-closed. Then RI (A) is a graded ASL on Π „¦ over B.

Proof: RI (A) is obviously a graded B-algebra and generated by Π „¦. The full

polynomial ring A[T ] is a graded ASL on Π ∪ {T } if we declare T to be the maximal (or

minimal) element of Π ∪ {T }. Since for a standard monomial µ = π1 . . . πk , πj ∈ Π „¦,

the factors from „¦— have to proceed the factors from Π, the standard monomials in Π „¦

correspond bijectively to those standard monomials in Π∪{T } whose degree with respect

to T does not exceed the number of factors from „¦. Therefore the standard monomials

in Π „¦ are linearly independent. In order to write down the straightening relations

we represent every standard monomial µ as µ = ±µ βµ ωµ , ±µ being the smallest factor,

βµ the second (if present), and ωµ the “tail”. There are three types of straightening

relations, always derived from the straightening relation ξ… = aµ µ in A (and, in case

ξ, … ∈ „¦, ξ > …, from the relation ξ… = …ξ):

(i) ξ, … ∈ Π : ξ… = aµ µ,

ξ—… = a µ ± — βµ ω µ ,

(ii) ξ ∈ „¦, … ∈ Π : µ

ξ — …— = a µ ± — βµ ω µ . ”

—

(iii) ξ, … ∈ „¦ : µ

For the extended Rees algebra and the associated graded algebra the poset Π has

only been replaced by an isomorphic copy. As the example above shows, Π „¦ need

not be wonderful without further hypothesis: … — and ζ — are upper neighbours of ξ — ,

but don™t have a common upper neighbour. Such an obstruction does not occur, if „¦ is

self-covering: every upper neighbour of elements …, ζ ∈ „¦ which have a common lower

neighbour ξ ∈ „¦∪{’∞}, is in „¦. For the examples of interest to us, „¦, being a sublattice

of a lattice Π then, is always self-covering. As the following example shows, even this is

not completely su¬cient:

s s

s s

, „¦ = { ξ },

Π= Π „¦= s s

ξs s

ξ— s

(9.11) Lemma. Suppose that Π is wonderful and „¦ a self-covering ideal in Π con-

taining all the minimal elements of Π. Then Π „¦ is wonderful.

Proof: The de¬nition of the partial order on Π „¦ implies: (a) ξ ∈ „¦— has a single

upper neighbour · ∈ Π, and ξ = · — . (b) If … ∈ Π and ζ ≥ … then ζ ∈ Π.

111

B. ASL Structures on Graded Algebras Derived from an Ideal

Let Π = (Π „¦) ∪ {∞, ’∞}, and suppose that …1 , …2 ∈ Π „¦, …1 = …2 , have a

common lower neighbour ξ ∈ Π and …1 , …2 ¤ ζ ∈ Π. Because of (a) and (b) we have to

consider the cases:

(i) …1 , …2 ∈ Π, ζ ∈ „¦— ; (ii) …1 ∈ Π, …2 ∈ „¦— , ζ ∈ „¦— ;

/ /

(iii) …1 , …2 ∈ „¦— , ζ ∈ „¦— ; (iv) …1 , …2 ∈ „¦— , ζ ∈ „¦— .

/

Case (i) is trivial, and in case (iii) one only needs that „¦ is an ideal in the wonderful

poset Π. In case (iv) we write …i = ωi . Since ξ ∈ „¦— ∪ {’∞}, ω1 and ω2 have a common

—

lower neighbour in Π or are minimal elements of Π. Furthermore ω1 , ω2 ¤ ζ, so they

have a common upper neighbour „ ¤ ζ, and necessarily „ ∈ „¦. Consequently „ — ¤ ζ,

and „ — is an upper neighbour of …1 and …2 . In case (ii) it is impossible that ξ = ’∞

since …1 is not minimal in Π „¦: If …1 ∈ „¦, then …1 is not even minimal in Π, and

/

otherwise …1 < …1 . The case ξ = ’∞ being excluded, necessarily ξ ∈ „¦— , and it follows

—

immediately that ξ = …1 . Now …2 = … — for a … ∈ „¦, and … is a suitable common upper

—