be de¬ned as a subvariety of the ambient Grassmann variety by rk(“(X) \ “(X; γ))

equations, but not by a smaller number of equations.

In general the arithmetical rank may go down when one passes from G(X) to B[X],

as the example above shows. Without further or completely di¬erent arguments one

can therefore not conclude that the bound in (5.21) is sharp. Hochster has given an

invariant-theoretic argument for the case of maximal minors, B containing a ¬eld of

63

F. Comments and References

characteristic 0. We shall discuss Hochster™s argument in Section 7. Newstead uses topo-

logical arguments in order to show that the bound in (5.21) is an equality for t = 2,

B again containing a ¬eld of characteristic zero, cf. [Ne], p. 180, Example (i), (a). As

Cowsik told us, Newstead™s argument goes through for every t and can be transferred to

characteristic p > 0 via the use of ´tale cohomology. (There is of course no restriction in

e

assuming that B is a ¬eld; otherwise one factors by a maximal ideal of B ¬rst.)

F. Comments and References

Our representation of ASL theory follows Eisenbud [Ei.1]. However we avoid the

passage to the discrete ASL in proving (5.7), (5.9) and (5.10), and in the proof of (5.14)

we have replaced an argument of Musili ([Mu], Proposition 1.3) by the closely related

(5.15), drawn from [HE.2], section 4. (5.20), (5.21), (5.22) seem to be new, at least in

regard to the method of proof.

Since all our examples are graded, we have made “graded” a standard assumption.

This allows us to weaken the ASL axioms slightly (relative to [Ei.1]) as indicated in

[DEP.2], Proposition 1.1. Proposition (4.2) is the only result for which the assumption

“graded” seems to be unavoidable after one has made the conclusion of (4.1) an ax-

iom. The reader may check that the assumption “graded” is not essential for (5.7) and

(5.10)(cf. also [DEP.2], Prop 6.1). Without the assumption “graded” Theorem (5.14) is

to be replaced by the statement that the sequence x1 , . . . , xk , k = rk Π, constructed for

(5.9) is an A-regular sequence, cf. [Ei.1].

The Cohen-Macaulay property of the determinantal rings was ¬rst proved by Hoch-

ster and Eagon in [HE.2] without a standard monomial theory, cf. Section 12. Shortly

later Laksov and Hochster proved that the homogeneous coordinate rings of the Schubert

subvarieties of the Grassmannians are Cohen-Macaulay, cf. [La.1] and [Ho.3]. Their

rather similar proofs were then followed by a proof of Musili [Mu], which di¬ers in

the technicalities of the induction step only, and the proof of Theorem (5.14) may be

considered an abstract version of it. The proof of Hochster and Laksov has also been

reproduced in [ACGH].

The theory of ASLs is a connection between combinatorics and commutative algebra.

For a development of ASL theory from a more combinatorial view-point we refer the

reader to [Bc].

6. Integrity and Normality. The Singular Locus

As we have noticed in (2.12) already, it follows from a localization argument and

the Cohen-Macaulay property that the rings Rt (X) are (normal) domains whenever the

ring B of coe¬cients is a (normal) domain. In this section we want to extend this result

to all the Schubert cycles and determinantal rings. Furthermore their singular locus will

be computed.

A. Integrity and Normality

As a normality criterion we shall use Lemma (16.24): Let S be a noetherian ring,

and x ∈ S such that x is not a zero-divisor, S/Sx is reduced, and S[x’1 ] is normal.

Then S is normal.

In a graded ASL A, whose underlying poset has a single minimal element, this

element is a natural candidate for x: A/Ax is a graded ASL again and therefore reduced

if B is reduced (cf. (5.1), (5.7)). Then it has “only” to be checked, whether A[x’1 ] is

normal. In the cases of interest to us the ring A[x’1 ] has a particularly simple structure:

(6.1) Lemma. Let X be an m — n matrix of indeterminates over B, m ¤ n,

γ = [a1 , . . . , am ] ∈ “(X) and

[d1 , . . . , dm ] ∈ “(X; γ) : ai ∈ [d1 , . . . , dm ] for at most one index i .

Ψ= /

Then

G(X; γ)[γ ’1 ] = B[Ψ][γ ’1 ],

the extensions being formed within the total ring of quotients of G(X; γ), and notably,

the set Ψ is algebraically independent over B. Therefore G(X; γ)[γ ’1 ] is isomorphic to

’1

B[T1 , . . . , Td ][T1 ], d = dim G(X; γ) ’ dim B, T1 , . . . , Td indeterminates.

Proof: The inclusion “ ⊃ ” is clear. We show that [b1 , . . . , bm ] ∈ B[Ψ][γ ’1 ] for

all [b1 , . . . , bm ] ∈ “(X; γ) by induction on the number k of indices i such that bi ∈ /

[a1 , . . . , am ]. For k = 0 and k = 1, [b1 , . . . , bm ] ∈ Ψ by de¬nition. Let k > 1 and choose

an index j such that bj ∈ [a1 , . . . , am ]. We use the Pl¨cker relation (4.4), the data “. . . ”

/ u

of (4.4) corresponding to the present ones in the following manner:

“k” = 0, “(b2 , . . . , bm )” = (b1 , . . . , bj’1 , bj+1 , . . . , bm ),

“l” = 2, “(c1 , . . . , cs )” = (a1 , . . . , am , bj ),

“s” = m + 1.

In this relation all the terms di¬erent from

[a1 , . . . , am ][bj , b1 , . . . , bj’1 , bj+1 , . . . , bm ] = (’1)j’1 [a1 , . . . , am ][b1 , . . . , bm ]

65

A. Integrity and Normality

and = 0 in G(X; γ) have the form δµ such that δ ∈ Ψ and µ has only k ’ 1 in-

dices not occuring in γ. Solving for [a1 , . . . , am ][b1 , . . . , bm ] and dividing by γ, one gets

[b1 , . . . , bm ] ∈ B[Ψ][γ ’1 ].

In proving the algebraic independence of Ψ we ¬rst consider a ¬eld B of coe¬cients.

If x is not a zero-divisor in a ¬nitely generated algebra A over a ¬eld, one has dim A =

dim A[x’1 ]. An easy count yields |Ψ| = rk “(X; γ): there are n ’ ai ’ (m ’ i) elements

in Ψ which do not contain ai . (The rank of “(X; γ) has been computed above (5.12)).

So

|Ψ| = rk “(X; γ) = dim G(X; γ)

= dim G(X; γ)[γ ’1 ] = dim B[Ψ][γ ’1 ] = dim B[Ψ],

and Ψ is algebraically independent.

Let now B = Z. Since Ψ is algebraically independent over Q, it is algebraically

independent over Z. In order to derive the general case one needs that G(X; γ)/Z[Ψ] is

Z-¬‚at. This is equivalent to

TorZ (G(X; γ)/Z[Ψ], Z/pZ) = 0

1

for all prime numbers p, and this again follows from the case of a ¬eld of coe¬cients

considered already. ”

The following lemma will be needed in Section 7, in particular for the proof of

Theorem (1.2) given there:

(6.2) Lemma. Let S be a B-algebra, and suppose that •, ψ : G(X; γ) ’ S are B-

algebra homomorphisms. If •(γ) is not a zero-divisor and •(δ) = ψ(δ) for all δ ∈ Ψ, Ψ

as in (6.1), then • = ψ.

Proof: Consider the commutative diagram in which the vertical arrows are injec-

tions:

• (ψ)

’’’’’’

’’’’’’

G(X; γ) S

¦ ¦

¦ ¦

•[γ ’1 ] (ψ[γ ’1 ])

’1

] ’ ’ ’ ’ ’ ’ S[•(γ)’1 ]

’’’’’’

G(X; γ)[γ

By virtue of (6.1) and hypothesis: •[γ ’1 ] = ψ[γ ’1 ]. ”

If B is an integral domain, G(X; γ), a subring of the domain B[Ψ][γ ’1 ], is a do-

main, too, and for normal B the ring B[Ψ][γ ’1 ] is even normal, so normality of G(X; γ)

then follows from the criterion cited above. The ring R(X; δ) arises from G(X; δ) by

dehomogenization with respect to ±[n + 1, . . . , n + m] as stated in (5.5). So R(X; δ) is a

(normal) domain, too, by virtue of (16.23).

(6.3) Theorem. Let B be a (normal) domain, X an m — n matrix, m ¤ n, of

indeterminates, and γ ∈ “(X), δ ∈ ∆(X). Then G(X; γ) and R(X; δ) are (normal)

domains.

Though a determinantal analogue of (6.1) has not been needed for the proof of (6.3),

it will be useful later.

66 6. Integrity and Normality. The Singular Locus

(6.4) Lemma. Let δ = [a1 , . . . , ar |b1 , . . . , br ] ∈ ∆(X), and

[ai |bj ] : i, j = 1, . . . , t ∪ δ ∈ ∆(X; δ) : δ di¬ers from δ in exactly one index .

Ψ=

Then R(X; δ)[δ ’1 ] = B[Ψ][δ ’1 ], and Ψ is algebraically independent over B. Thus

R(X; δ)[δ ’1 ] is isomorphic to

B[T1 , . . . , Td ][ζ ’1 ], ζ ∈ B[T1 , . . . , Td ], d = dim R(X; δ) ’ dim B,

T1 , . . . , Td indeterminates. If B is an integral domain, ζ is a prime element.

Proof: For R(X; δ) ‚ B[Ψ][δ ’1 ] it is enough that [u|v] ∈ B[Ψ][δ ’1 ] for all [u|v] ∈

∆(X; δ). Suppose ¬rst, that u = ai . Then (in B[X] already)

[u, a1 , . . . , ar |v, b1 , . . . , br ] = 0.

Expansion of this minor along row u shows that [u|v] can be expressed (over Z) by the

[ai |bj ] ∈ Ψ, [a1 , . . . , ar |v, b1 , . . . , bi , . . . , br ] ∈ Ψ and δ ’1 . Let u be arbitrary now. In

R(X; δ) one has

[u, a1 , . . . , ar |v, b1 , . . . , br ] = 0,

and now one expands along column v, expressing [u|v] by the [ai |v] ∈ B[Ψ][δ ’1 ] (“∈”

has been shown already), [u, a1 , . . . , ai , . . . , ar |b1 , . . . , br ] ∈ Ψ and δ ’1 . In proving the

algebraic independence of Ψ one proceeds as in the proof of (6.1). At this point one can

derive the contention of (6.3) with respect to R(X; δ) or use (6.3) directly in order to

conclude that ζ, being the determinant of a matrix of indeterminates, is a prime element

over a domain B. ”

The representation of R(X; δ) as a dehomogenization of G(X; δ) renders R(X; δ)[δ ’1 ]

a residue class ring of B[Ψ][δ ’1 ], Ψ constructed for δ according to (6.1). The reader may

¬nd the resulting representation of R(X; δ)[δ ’1 ].

Theorem (6.3) has consequences for a more general class of rings.

(6.5) Corollary. Let B be an integral domain, „¦ ‚ “(X) an ideal. Then the

minimal prime ideals of „¦G(X) are the ideals J(X; γ), γ a minimal element of “(X) \ „¦,

and „¦G(X) is their intersection. The analogous statement holds for ideals „¦ ‚ ∆(X).

In fact, the ideals J(X; γ) are prime, and „¦G(X) = J(X; γ) follows from „¦ =

(“(X) \ “(X; γ)) by virtue of (5.2). We leave it to the reader to ¬nd the most general

version (in regard to B) of (6.5) and to prove the following corollary (as an application

of (3.15), say):

(6.6) Corollary. Let B be an arbitrary ring, „¦ ‚ “(X) an ideal. An element

γ ∈ “(X) \ „¦ is not a zero-divisor modulo „¦G(X) if and only if it is comparable to every

minimal element of “(X) \ „¦. The analogous statement holds for ideals „¦ ‚ ∆(X).

67

B. The Singular Locus

B. The Singular Locus

Let B be a ¬eld momentarily. Then every localization of G(X; γ) with respect

to a prime ideal not containing γ is a localization of a polynomial ring over B, and

therefore regular. The element γ is distinguished only in the combinatorial structure of

“(X; γ). For the purpose of (6.1) every element of G(X; γ) which can be mapped to γ

by an automorphism of G(X; γ), is as good as γ itself. In the extreme case in which

γ = [1, . . . , m], G(X; γ) = G(X), every element of “(X) can be mapped (up to sign) to γ

by a suitable permutation of the columns of X which of course induces an automorphism

of G(X). In general we can only use the permutations which leave J(X; γ) invariant.

Every permutation π of {1, . . . , n} induces a permutation of “(X) (which up to sign has

the same e¬ect as the corresponding automorphism). Let γ = [a1 , . . . , am ]. If

π({ai , . . . , n}) = {ai , . . . , n} for i = 1, . . . , m

then certainly π(δ) ∈ J(X; γ) for all δ ∈ “(X; γ), this being equivalent to the invariance

of J(X; γ) under (the automorphism induced by) π. The example γ = [1, . . . , m] however

shows that the condition above is too coarse: an appropriate condition must take care

of how [a1 , . . . , am ] breaks into blocks of consecutive integers

β0 = (a1 , . . . , ak1 ), β1 = (ak1 +1 , . . . , ak2 ), . . . , βs = (aks +1 , . . . , am ).

For systematic reasons we let k0 = 0, ks+1 = m, am+1 = n + 1. Similarly we decompose

the complement of γ with respect to the interval {a1 , . . . , n} to obtain the gaps of γ:

χ0 = (ak1 + 1, . . . , ak1 +1 ’ 1), . . . , χs = (am + 1, . . . , n).

Here χs is empty if am = n. If a permutation π satis¬es the condition

π(βi ∪ χi ) = βi ∪ χi ,

(1) i = 0, . . . , s,

then π certainly leaves “(X; γ) invariant as a set, thus maps “(X) \ “(X; γ) onto itself,

and induces an automorphism of G(X; γ). An element δ = [b1 , . . . , bm ] ∈ “(X; γ) can be

mapped to γ by such a permutation if and only if

bki ∈ βi’1 ∪ χi’1 ,

(2) i = 1, . . . , s + 1.

Let Σ(X; γ) be the set of elements δ ∈ “(X; γ) which satisfy (2). It is an ideal in the

partially ordered set “(X; γ) !

To give an example: Let m = 4, n = 7, γ = [1 3 4 6]. Then the blocks and gaps of

γ are

β0 = (1), β1 = (3 4), β2 = (6) and χ0 = (2), χ1 = (5), χ2 = (7).

Σ(X; γ) consists of all δ = [b1 , . . . , b4 ] ∈ “(X; γ) such that b1 ¤ 2, b2 ¤ 5.

68 6. Integrity and Normality. The Singular Locus

(6.7) Theorem. Let B be a noetherian ring, X an m—n matrix of indeterminates,

γ ∈ “(X), P a prime ideal of G(X; γ), and Q = B © P . Suppose that “(X; γ) is not a

chain. Then G(X; γ)P is regular, if and only if BQ is regular and P does not contain

the ideal Σ(X; γ)G(X; γ).

The case in which “(X; γ) is a chain, is trivial: G(X; γ) is a polynomial ring over B

then. It is easy to check that “(X; γ) is a chain if and only if γ = [a1 , n ’ m + 2, . . . , n] or

γ = [n’m, . . . , aj , aj +2, . . . , n]. Before we prove (6.7) we supplement it by a description

of the minimal elements in the complement of Σ(X; γ) and a slight strengthening.

(6.8) Supplement to (6.7). One has Σ(X; γ) = “(X; γ) if and only if (i) s = 0

or (ii) s = 1 and am = n (the latter implying χ1 = …). If (iii) s ≥ 1 and χ1 = …, then the

minimal elements in the complement of Σ(X; γ) with respect to “(X; γ) are

σ1 = [(a1 , . . . , ak1 ’1 ), (ak1 +1 , . . . , ak2 , ak2 + 1), β2 , . . . , βs ],

.

.

.

σt = [β0 , . . . , βt’2 , (akt’1 +1 , . . . , akt ’1 ), (akt +1 , . . . , akt+1 , akt+1 + 1), βt+1 ],

where t = s if χs = …, and t = s ’ 1 if χs = … (in the ¬rst of these cases we let βt+1 = …).

In case (iii) the localizations of G(X; γ) with respect to a prime ideal P ⊃ Σ(X; γ)G(X; γ)

are not even factorial domains.

In our example t = 2, σ1 = [3 4 5 6], σ2 = [1 3 6 7].

If the singular locus of Spec B is closed, then the singular locus of G(X; γ) is