γ = [b1 , . . . , bm ], δ = [d1 , . . . , dm ],

•(γ) = [a1 , . . . , au |b1 , . . . , bu ], •(δ) = [c1 , . . . , cv |d1 , . . . , dv ].

Suppose ¬rst that γ ¤ δ. Then obviously v ¤ u and b1 ¤ d1 , . . . , bv ¤ dv . We assume

a1 ¤ c1 , . . . , aw ¤ cw and aw+1 > cw+1 in order to derive a contradiction. Since cw+1 ∈

/

{a1 , . . . , au }, there is a t such that cw+1 = (n + m + 1) ’ bt . Then

(n + m + 1) ’ dt ¤ (n + m + 1) ’ bt = cw+1 < aw+1 ,

47

D. B[X] as an ASL

even

(n + m + 1) ’ dt < cw+1 ,

equality being excluded. The indices which are smaller than (n + m + 1) ’ bt , are

a1 , . . . , aw , (n + m + 1) ’ bm , . . . , (n + m + 1) ’ bt+1 ,

hence (n + m + 1) ’ bt = w + m ’ t + 1. On the other hand the following indices are

smaller than cw+1 = (n + m + 1) ’ bt :

c1 , . . . , cw , (n + m + 1) ’ dm , . . . , (n + m + 1) ’ dt .

So (n + m + 1) ’ bt ≥ w + m ’ t + 2, a contradiction.

Let now •(γ) ¤ •(δ). This implies v ¤ u and b1 ¤ d1 , . . . , bv ¤ dv . Again we want

to reach a contradiction and suppose that b1 ¤ d1 , . . . , bw ¤ dw , bw+1 > dw+1 . Then

bw+1 > dw+1 ≥ dv+1 > n. Consequently there exists a t such that

at = (m + n + 1) ’ dw+1 .

There are at least m ’ w + t ’ 1 indices smaller than (m + n + 1) ’ dw+1 :

a1 , . . . , at’1 , (m + n + 1) ’ bm , . . . , (m + n + 1) ’ bw+1 ,

in particular (m + n + 1) ’ dw+1 ≥ m ’ w + t. On the other hand

(m + n + 1) ’ dw+1 ¤ (m + n + 1) ’ dw+1 , . . . , (m + n + 1) ’ dv+1 .

Hence m ’ w + t ’ 1 + (w + 1) ’ (v + 1) + 1 ¤ m, so t ¤ v. Since a1 ¤ c1 , . . . , av ¤ cv ,

all the indices smaller than (m + n + 1) ’ dw+1 occur among

c1 , . . . , ct’1 , (m + n + 1) ’ dm , . . . , (m + n + 1) ’ dw+2 ,

again a contradiction. ”

Lemma (4.9) shows that as a poset ∆(X) is isomorphic to “(X)\{[n+1, . . . , n+m]}.

Since the top of “(X) looks like

s [n + 1, . . . , n + m]

s [n, n + 2, . . . , n + m]

d

d

ds

s

“(X) \ {[n + 1, . . . , n + m]} and ∆(X) are distributive lattices, too.

(4.10) Lemma. (a) G(X)([n + 1, . . . , n + m] ’ µ) is a prime ideal if B is an integral

domain.

(b) Ker • = G(X)([n + 1, . . . , n + m] ’ µ).

Proof: (a) Consider the commutative diagram

ψ

B[Tγ : γ ∈ “(X)] ’ ’ ’ B[Tγ : γ ∈ “(X) \ {[n + 1, . . . , n + m]}]

’’

¦ ¦

¦ ¦χ

π

’’’

’’ G(X)/G(X)([n + 1, . . . , n + m] ’ µ)

G(X)

48 4. Algebras with Straightening Law on Posets of Minors

of epimorphisms, where ψ(T[n+1,...,n+m] ) = µ. Since Ker χ = ψ(Ker π) it is enough to

know that ψ maps homogeneous prime ideals P not containing T[n+1,...,n+m] onto prime

ideals. The map ψ is just the “dehomogenization” with respect to T = µT[n+1,...,n+m] ,

and therefore has the desired property, cf. (16.26).

(b) Since both G(X) and B[X] as well as the map • arise from the corresponding

objects over Z by tensoring with B, it is su¬cient to prove (b) in the case B = Z. Since

dim G(X) = mn + 1 + dim B,

as will be shown in Section 5,

dim G(X)/G(X)([n + 1, . . . , n + m] ’ µ) = dim B[X].

By virtue of (a) both of them are integral domains, and the epimorphism induced by •

is an isomorphism. ”

Now all the arguments for the proof of the main result have been collected:

(4.11) Theorem. B[X] is a graded ASL on ∆(X).

Proof: It follows directly from (4.9) that the standard monomials in “(X) \ [m + 1,

. . . , m + n] are mapped to standard monomials in ∆(X) (up to sign). Property (H2 )

cannot be destroyed, since the maximal element of “(X) is replaced by µ: any monomial

appearing on the right side of a straightening relation in G(X) contains a factor di¬erent

from [n + 1, . . . , n + m]. The only critical point is whether the standard monomials in

∆(X) are linearly independent. Suppose we have a relation aµ •(µ) = 0, µ representing

a standard monomial in “(X) not containing [n + 1, . . . , n + m]. Then, by virtue of (4.10)

aµ µ = (µ ’ [n + 1, . . . , n + m]) bν ν,

ν representing a standard monomial, too. It is obvious that such an equation can only

hold if all the coe¬cients aµ , bν are zero. ”

For a generalization in the next section we record:

(4.12) Proposition. B[X] is the dehomogenization of G(X) with respect to µ[n+1,

. . . , n + m].

The geometric analogue of (4.12) has been observed above Theorem (1.3): The a¬ne

mn-space is the open subvariety of the projective variety Gm (K n+m ) complementary to

the hyperplane de¬ned by [n + 1, . . . , n + m] (or any of the coordinate hyperplanes).

E. Comments and References

The ¬rst standard monomial theory was established by Hodge [Hd] for the homo-

geneous coordinate rings of the Grassmannians and their Schubert subvarieties. Having

found an explicite basis, he could derive the “postulation formula” for the Schubert sub-

varieties (previously conjectured by him and proved by Littlewood) in an elementary

manner. (In algebraic language the “postulation formula” is an explicit formula for the

dimension of the i-th homogeneous component of the homogeneous coordinate ring of a

49

E. Comments and References

projective variety.) A complete treatment was given by Hodge and Pedoe in their classical

monograph [HP]; the tacit assumption that the ring of coe¬cients contains the rational

numbers is only used there in proving that the relations in (4.4) are linear combinations

of the relations

m+1

(’1)k [a1 , . . . , aj’1 , bk , aj+1 , . . . , am ][b1 , . . . , bk’1 , bk+1 , . . . , bm+1 ]

k=1

instead of establishing them directly. (In positive characteristic the just-mentioned rela-

tions are not su¬cient in general to generate the ideal of Pl¨cker relations, cf. [Ab.2].)

u

More recent accounts of this standard monomial theory were given by Laksov [La.1]

and Musili [Mu]. Musili™s article is fairly selfcontained; his proof for the linear indepen-

dence of the standard monomials will be indicated in Section 6. It is actually simpler

than the one given whose merits will however become apparent in Section 11.

Like all the other authors we essentially follow Hodge™s “canonical” way in proving

that the standard monomials generate the B-module G(X). The proof of the linear

independence is borrowed from DeConcini™s, Procesi™s and Eisenbud™s article [DEP.1].

The only place however, where we could ¬nd a proof for the validity of (H2 ), is Lemma

2.1 of Hochster™s paper [Ho.3]; Hochster also observed that G(X) has the property dual

to (H2 ).

Our derivation of (4.11) from (4.3) is taken from [DEP.1] again, where priority

for Theorem (4.11) is attributed to Doubilet, Rota and Stein [DRS]. The geometric

relationship between the Grassmann variety and the a¬ne space is classical, however;

an algebraic argument involving standard products of arbitrary minors can be found in

[Mo] already; cf. also [HE.2], p. 1045.

The notion “algebra with straightening law” is drawn from Eisenbud™s introductory

survey [Ei.1] of the more voluminous monograph [DEP.2], in which the name “Hodge

algebra” is used for the members of a more general class and ASLs ¬gure as “ordinal

Hodge algebras”.

5. The Structure of an ASL

In this section we want to derive the properties of determinantal rings and Schubert

cycles which follow from the general theory of ASLs and the particular nature of the

partially ordered sets “(X) and ∆(X) introduced in the preceding section. Determinantal

rings and Schubert cycles inherit their structure as an ASL from B[X] and G(X), simply

because their de¬ning ideals are generated by an ideal in ∆(X) and “(X) resp.

We shall see that ASLs are reduced over reduced rings B and that ASLs on posets

of a certain class (containing the distributive lattices) are Cohen-Macaulay rings over

Cohen-Macaulay rings B. Furthermore there is a simple combinatorial formula for the

dimension of an ASL, for the proof of which one needs “natural” regular elements of an

ASL. One of the lemmas on which the formula for dimension is based, is general enough

to supply an upper bound for the number of elements needed to generate certain ideals up

to radical. This has consequences for the number of equations de¬ning a determinantal

or Schubert variety.

A. ASL Structures on Residue Class Rings

In order to apply ASL theory to determinantal rings and Schubert cycles one ¬rst

has to show that these rings are ASLs. This will follow readily from the fact that

their de¬ning ideals have a system of generators which is distinguished in regard of the

underlying poset.

(5.1) Proposition. Suppose A is a graded ASL on Π over B.

(a) Let Ψ ‚ Π, I = AΨ. If I is generated as a B-module by all the standard monomials

containing a factor ξ ∈ Ψ, then A/I is again a graded ASL on Π \ Ψ (in a natural way).

(b) In particular A/A„¦ is a graded ASL on Π \ „¦ if „¦ is an ideal in Π (i.e. ξ ∈ „¦ and

… ¤ ξ implies … ∈ „¦).

Proof: Part (a) is obvious. In (b) the ideal A„¦ is generated by all the monomials

containing a factor ξ ∈ „¦. Thus (b) follows directly from (a) and Proposition (4.1),

(b). ”

If Π has a single maximal element π, then Ψ = {π} satis¬es the hypothesis of (5.1),(a)

(though it is not an ideal, provided Π = Ψ). This is a trivial but useful example.

En passant we note:

(5.2) Proposition. Let „¦ and Ψ be ideals in Π. Then A„¦ © AΨ is generated by

the ideal „¦ © Ψ in Π.

Proof: Every standard monomial in the standard representation of an element of

A„¦ © AΨ has to contain a factor ω ∈ „¦ and a factor ψ ∈ Ψ. At least one of them lies in

„¦ © Ψ. ”

51

A. ASL Structures on Residue Class Rings

Together with the trivial statement that A„¦+AΨ is generated by „¦∪Ψ, Proposition

(5.2) shows that the ideals A„¦, „¦ an ideal in Π, form a distributive lattice with respect

to intersections and sums (which is isomorphic to the lattice of ideals „¦ ‚ Π).

In order to have a compact description of our examples and for systematic reasons

we introduce one more piece of notation:

Definition. Let Σ ‚ Π. The ideal generated by Σ in Π is the smallest ideal in Π

containing Σ:

{ξ ∈ Π : ξ ¤ σ for a σ ∈ Σ},

whereas the ideal cogenerated by Σ in Π is the greatest ideal disjoint from Σ:

{ξ ∈ Π : ξ ≥ σ for every σ ∈ Σ}.

As usual let X be an m — n matrix of indeterminates over B, ∆(X) its set of minors,

partially ordered as introduced in 4.D. The ideal It (X) is generated by the t-minors and

contains every u-minor such that u ≥ t: it contains all the minors γ ¤ δ for a t-minor δ.

One has

It (X) = B[X]Σ,

Σ being the ideal in ∆(X) generated by [m’t+1, . . . , m|n’t+1, . . . , n], equivalently: the

ideal cogenerated by [1, . . . , t ’ 1|1, . . . , t ’ 1]. The last description is the most convenient

one, and as will be seen shortly, the de¬ning ideals of all the determinantal ideals can be

described in this way. For δ ∈ ∆(X) we let

I(X; δ) = B[X]{π ∈ ∆(X) : π ≥ δ},

R(X; δ) = B[X]/I(X; δ), and

∆(X; δ) = {π ∈ ∆(X) : π ≥ δ}.

In exploring R(X; δ), δ ¬xed, we shall have to consider ideals of the form I(X; µ)/I(X; δ).

It is therefore convenient to write

I(x; µ) = I(X; µ)/I(X; δ)

then. (There is of course no need for the notations R(x; µ) and ∆(x; µ).) Let δ =

[a1 , . . . , ar |b1 , . . . , br ]. Then I(X; δ) is generated by the

s-minors of the rows 1, . . . , as ’ 1

s = 1, . . . , r, and the

s-minors of the columns 1, . . . , bs ’ 1

(r + 1)-minors of X.

So the rings R(X; δ) are determinantal rings in the sense of 1.C, and conversely the

determinantal rings B[X]/I are of type R(X; δ). Let

1 ¤ u1 < · · · < up ¤ m, 0 ¤ r 1 < · · · < rp < m

and

1 ¤ v1 < · · · < vq ¤ n, 0 ¤ s1 < · · · < sq < n,

52 5. The Structure of an ASL

such that I is generated by the

(ri + 1)-minors of the ¬rst ui rows

and the

(sj + 1)-minors of the ¬rst vj columns,

i = 1, . . . , p, j = 1, . . . , q. In general this system of generators is far from being minimal:

If ui+1 ¤ ui + ri+1 ’ ri , then all the (ri+1 + 1)-minors of the rows 1, . . . , ui+1 are linear

combinations of the (ri + 1)-minors of the rows 1, . . . , ui . Furthermore all the (r + 1)-

minors are in I if r + 1 is given as

r + 1 = min(rp + 1 + m ’ up , sq + 1 + n ’ vq )

In case rp + 1 ≥ r + 1 we can discard the (rp + 1)-minors of the rows 1, . . . , up , since

they are contained in the ideal generated by the (sq + 1)-minors of the columns 1, . . . , vq .

Similar observations apply to the “column-de¬ned” generators, and therefore it is no

restriction to assume that

ui+1 > ui + ri+1 ’ ri , vj+1 > v j + sj+1 ’ sj ,

i = 1, . . . , p ’ 1, j = 1, . . . , q ’ 1,

(—)

rp + 1 < s q + 1 + n ’ v q , sq + 1 < r p + 1 + m ’ u p .

Now we can describe δ such that I = I(X; δ):

δ = [(1, . . . , r1 ), (u1 + 1, . . . , u1 + (r2 ’ r1 )), . . . , (up + 1, . . . , up + (rp+1 ’ rp ))|

(1, . . . , s1 ), (v1 + 1, . . . , v1 + (s2 ’ s1 )), . . . , (vq + 1, . . . , vq + (sq+1 ’ sq ))],

where of course rp+1 = sq+1 = r + 1 and the blocks of consecutive integers in the row

and column parts of δ have been enclosed in parentheses.

(5.3) Theorem. (a) The determinantal rings B[X]/I are given exactly by the rings

R(X; δ), δ ∈ ∆(X).

(b) R(X; δ) is a graded ASL on ∆(X; δ).

(c) ∆(X; δ) is a distributive lattice.

The analogues of I(X; δ), R(X; δ), ∆(X; δ) with respect to G(X) are

J(X; γ) = G(X){δ ∈ “ : δ ≥ γ},

G(X; γ) = G(X)/J(X; γ), and

“(X; γ) = {δ ∈ “ : δ ≥ γ}.