J(X; σi ))G(X; γ), where Q runs through the minimal singular prime ideals of B and

i = 1, . . . , t. (Similar statements hold in the remaining cases.)

Proof of (6.7) and (6.8): The “if” part of (6.7) has been indicated already: Let

S = G(X; γ). The ring SP is a localization of (S — BQ )[δ ’1 ] for some δ ∈ Σ = Σ(X; γ).

Since δ can be mapped to γ by an automorphism of S — BQ , SP is regular by (6.1).

For the converse we ¬rst note that regularity of SP implies regularity of BQ through

¬‚atness, and factoriality implies that BQ is a domain (at least). Next we may assume

that P is a minimal prime ideal of SΣ, in order to derive a contradiction. Then, after

having replaced B by BQ , we conclude Q = 0 from (6.6), and B = K is a ¬eld.

In case (i) of (6.8) condition (2) holds for every δ ≥ γ, so Σ = “(X; γ) and P = SΣ is

the irrelevant maximal ideal of a graded K-algebra generated by its 1-forms. If “(X; γ)

is not a chain, the dimension of the K-vector space of 1-forms di¬ers from the Krull

dimension of S : SP is not regular. In case (ii) one has aj = n ’ (m ’ j), and therefore,

letting δ = [b1 , . . . , bm ], bj = n ’ (m ’ j) for j ≥ k1 + 1. This implies bk1 < ak1 +1 , and

we are through by the same argument.

In case (iii) one certainly has Σ = “(X; γ), since σ1 , . . . , σt ∈ Σ and t ≥ 1. It is easy

/

to see that σ1 , . . . , σt are the minimal elements of the complement of Σ ‚ “(X; γ). By

our assumption on P being minimal over SΣ, P = J(x; σj )(= J(X; σj )/J(X; γ), cf. 5.A)

for an index j (cf. (6.5)). Since S/SΣ is reduced, Σ generates P SP , in particular contains

a minimal system of generators of P SP ; its elements are irreducible. The permutations

π which satisfy (1), have the property corresponding to (1) for σj , too: The sets βi ∪ χi

for γ coincide with the corresponding sets for σj ! Therefore these permutations leave

69

B. The Singular Locus

“(X; σj ) and “(X; γ) \ “(X; σj ) invariant: they induce automorphisms of SP . Since γ

can be moved to every element of Σ by such a permutation, γ is an irreducible element

of SP . On the other hand it cannot be prime: Let

„1 = [β0 , . . . , βj’2 , (akj’1 +1 , . . . , akj ’1 ), akj + 1, βj , . . . , βs ]

and

„2 = [β0 , . . . , βj’1 , (akj +1 , . . . , akj+1 ’1 ), akj+1 + 1, βj+1 , . . . , βs ].

Then „1 , „2 are upper neighbours of γ, and „1 , „2 ¤ σj , so P = J(x; σj ) contains J(x; „1 )

and J(x; „2 ), two di¬erent minimal primes of Sγ, excluding that γ is prime in SP . ”

(6.9) Remarks. (a) As stated already, G(X; γ) is a polynomial ring over B if

“(X; γ) is a chain. (6.7) shows that the converse is likewise true: Suppose that G(X; γ)

is a polynomial ring over B. Then G(X; γ) — (B/Q) is a polynomial ring over the ¬eld

B/Q, Q a maximal ideal of B. So all the localizations of G(X; γ) — (B/Q) are regular,

and “(X; γ) must be a chain by (6.7).

(b) Since the cue “factorial” has been given already, we should point out that in

the exceptional cases (i) and (ii) of (6.8) the ring G(X; γ) is indeed factorial, provided

B is factorial: γ has only a single upper neighbour then, so is prime by (6.5), and the

factoriality of G(X; γ)[γ ’1 ] implies the factoriality of G(X; γ) itself; cf. Section 8 for a

detailed discussion.

(c) We have started the proof of (6.8) by trying to ¬nd as many elements of “(X; γ)

which are conjugate to γ under an automorphism of G(X; γ), and have found the set

Σ(X; γ) of such elements. After (6.8) it is clear that elements σ outside Σ(X; γ) are not

conjugate to γ under a B-automorphism: the B-algebras G(X; γ)[γ ’1 ] and G(X; γ)[σ ’1 ]

’1

are not isomorphic. The structure of G(X; γ)[σi ] will be revealed in (8.11).

(d) Without restriction one can exclude the case am = n ¬rst, and thus reduce the

number of cases to be considered in (6.8). In fact, if γ = [a1 , . . . , ap , n’(m’p)+1, . . . , n],

then

G(X; γ) ∼ G(X ; γ )

=

where X is a p — (n ’ (m ’ p)) matrix of indeterminates and γ = [a1 , . . . , ap ]. We leave

it to the reader to check that the map which sends [b1 , . . . , bp , n ’ (m ’ p) + 1, . . . , n] ∈

“(X; γ) to [b1 , . . . , bp ] ∈ “(X ; γ ), is well-de¬ned and an isomorphism. ”

The most convenient way to ¬nd the singular locus of R(X; δ) is again the method

of dehomogenization. Though very suggestive, the automorphism argument (now in

conjunction with (6.4)) does not produce the correct result in all cases, as will be demon-

strated below.

We write R(X; δ) as the dehomogenization of G(X; δ) again. It is immediate from

(16.26) that the ideal I of R(X; δ) generated by an ideal „¦ ‚ ∆(X; δ) is the dehomog-

enization of the ideal J of G(X; δ) generated by the corresponding ideal „¦ in “(X; δ):

J is homogeneous, ±[n + 1, . . . , n + m] is not a zero-divisor modulo J (since it is the

maximal element of the poset underlying the ASL G(X; δ)/J), and the generating set „¦

is mapped (up to sign) onto „¦. Let Ξ(X; δ) be the subset of ∆(X; δ) corresponding to

Σ(X; δ) ‚ “(X; δ). Then, by virtue of (16.28) and (6.7), a localization R(X; δ)P is regu-

lar if and only if BP ©B is regular and P ⊃ Ξ(X; δ). It only remains to give a description

70 6. Integrity and Normality. The Singular Locus

of Ξ(X; δ) in terms of δ. We state the result, leaving the translation back and forth to

the reader.

Let δ = [a1 , . . . , ar |b1 , . . . , br ]. We decompose the row part [a1 , . . . , ar ] into its blocks:

[a1 , . . . , ar ] = [β0 , . . . , βu ], βi = (aki +1 , . . . , aki+1 ).

Then we let

ξi = [β0 , . . . , βi’2 , (aki’1 +1 , . . . , aki ’1 ), (aki +1 , . . . , aki+1 , aki+1 +1),

βi+1 , . . . , βu |b1 , . . . , br ],

i = 1, . . . , u ’ 1, and i = u if ar < m and u ≥ 1. Analogously one constructs elements …j ,

j = 1, . . . , w ’ 1, and j = w if br < n and w ≥ 1, for the column part. In the exceptional

case ar = m the element ξu is given by

ξu = [β0 , . . . , βu’2 , (aku’1 +1 , . . . , aku ’1 ), (aku +1 , . . . , ar )|b1 , . . . , br’1 ],

and if br = n the element …w is choosen analogously. Finally,

ζ = [a1 , . . . , ar’1 |b1 , . . . , br’1 ].

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

and δ = [a1 , . . . , ar |b1 , . . . , br ] ∈ ∆(X). Assume that δ = [m ’ r + 1, . . . , m|b1 , . . . ,

b1 + r ’ 1] and δ = [a1 , . . . , a1 + r ’ 1|n ’ r + 1, . . . , n]. Let P be a prime ideal of R(X; δ)

and Q = B © P . Then the localization RP is regular if and only if BQ is regular and

P ⊃ Ξ(X; δ), where Ξ(X; δ) is given as follows:

(i) If r = 1, then Ξ(X; δ) = ∆(X; δ).

(ii) If r > 1, ar < m and br < n, Ξ(X; δ) is the ideal in ∆(X; δ) cogenerated by

ξ1 , . . . , ξu , …1 , . . . , …w , ζ.

(iii) If r > 1, ar = m or br = n, Ξ(X; δ) is the ideal in ∆(X; δ) cogenerated by

ξ 1 , . . . , ξ u , …1 , . . . , … w .

The singular locus of Rr+1 (X) = R(X; [1, . . . , r|1, . . . , r]) has been computed in

(2.6) already. This case is recovered in (ii): u = w = 0 then, and the singular locus

is determined by ζ = [1, . . . , r ’ 1|1, . . . , r ’ 1]. Again one of the exceptional cases δ =

[m ’ r + 1, . . . , m|b1 , . . . , b1 + r ’ 1] and δ = [a1 , . . . , a1 + r ’ 1|n ’ r + 1, . . . , n] occurs

if and only if R(X; δ) is a polynomial ring over B. (The “if” part is obvious, and for the

“only if” part one argues as in (6.9),(a).)

The reader may check that only in the cases (i) and (ii) Ξ(X; δ) is the set of elements

of ∆(X; δ) which are conjugates of δ (up to sign) under row and column permutations of

X. That the set of conjugates fails to give the singular locus in general can also be seen

from the following example : B = K a ¬eld, m = 2, n = 3, δ = [1 2|1 3]. The prime ideal

P = I(x; [1|1]) has height 1, since [1|1] is an upper neighbour of δ. By (6.3) the local ring

R(X; δ)P is regular, though P contains all the conjugates of δ. The exceptional nature

of case (iii) is easily explained: Let δ = [a1 , . . . , am ]. Then ar < m and br < n if and

71

B. The Singular Locus

only if ar+1 = n + 1 > br + 1 = ar + 1. Therefore in cases (i) and (ii) every permutation

π satisfying condition (1) above induces an automorphism of R(X; δ).

We can combine the di¬erent cases of (6.10) to a single statement if we choose to

describe determinantal ideals by their generators. It has been noted in 5.A already that

the ideal I = I(X; δ) has a system of generators consisting of the

(ri + 1)-minors of the rows 1, . . . , ui , i = 1, . . . , p,

and the

(sj + 1)-minors of the columns 1, . . . , vj , j = 1, . . . , q,

where the ri , ui , sj , qj are suitably chosen integers satisfying the conditions

0 ¤ r1 < · · · < rp < m, 0 ¤ s1 < · · · < sq < n,

ui+1 > ui + (ri+1 ’ ri ), vj+1 > vj + (sj+1 ’ sj ), i = 1, . . . , p ’ 1, j = 1, . . . , q ’ 1,

and

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

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

Suppose that the ideal I is generated as just speci¬ed. Then for a prime ideal P of

R = B[X]/I the localization RP is regular if and only if BQ is regular for Q = B © P

and P does not contain the ideal

P1 © · · · © P p © Q 1 © · · · © Q q ,

where Pi is generated by the ri -minors of rows 1, . . . , ui , and the Qj are de¬ned analo-

gously for the columns.

The derivation of (6.11) from (6.10) can be left to the reader.

After one has explicitely described the singular locus of the rings G(X; γ) and

R(X; δ) one can compute its codimension. The best possible general estimate is given in

the following proposition:

(6.12) Proposition. Let B be a noetherian ring which satis¬es Serre™s condition

(R2 ). Then all the rings G(X; γ) and R(X; δ) satisfy (R2 ), too.

Proof: Because of (16.28) it is enough to consider the rings R = G(X; γ). Let P be

a singular prime ideal of R and Q = B ©P . If BQ is singular, then dim RP ≥ dim BQ ≥ 3.

Thus we may assume that B = BQ is a regular local ring and P is minimal among the

singular prime ideals of R. In the cases (i) and (ii) of (6.10) one has P = “(X; γ)R,

hence ht P = rk “(X; γ) ≥ 3 (if rk “(X; γ) ¤ 2, R is a polynomial ring over B). In case

(iii) of (6.10) P = J(X; σi )/J(X; γ) for a suitable i, and there are at least two elements

π < ρ of “(X; γ) strictly between γ and σi , and therefore ht P ≥ 3 because of (6.3). ”

It is easy to see that (R2 ) is the best we can expect in general; take γ = [1 3 5] for

example or δ = [a1 , . . . , ar |b1 , . . . , br ] such that ar = m ’ 1, br = n ’ 1.

72 6. Integrity and Normality. The Singular Locus

C. Comments and References

The key lemma (6.1) is essentially Lemma 3.11 from Hochster™s article [Ho.3],

whereas a variant of (6.4) seems to appear ¬rst in [Br.3] (for Rr+1 (X)). “Classical-

ly” the integrity of the Schubert cycles G(X; γ) is proved by the construction of generic

points, cf. Section 7. Hochster shows the “if” part of (6.7) using the automorphism argu-

ment and concludes the normality of G(X; γ) from the Cohen-Macaulay property and the

Serre condition (R1 ); as we have seen, even (R2 ) follows from (6.7). The singular locus

of G(X; γ) is given (in the language of Schubert varieties) by Svanes in [Sv.1], p. 451,

(5.5.2).

References for the integrity of the rings Rr+1 (X) were given in Section 2. Their

normality was ¬rst proved by Hochster and Eagon [HE.2] as a consequence of the Cohen-

Macaulay property and (R1 ), the latter resulting from a demonstration of the “if” part

of (6.11) (as far as the rings R(X; δ) are treated in [HE.2]). (6.10) and (6.11) may be

considered a natural generalization of their results.

7. Generic Points and Invariant Theory

The main objective of this section is to describe the rings Rr+1 (X) and G(X), more

generally R(X; δ) and G(X; γ), as the rings of invariants of actions of linear groups on

polynomial rings, thereby solving classical problems of invariant theory. This requires

the construction of suitable embeddings into polynomial rings, and the embeddings con-

structed below are generic points. Furthermore we illustrate the connection between

invariant theory and the ideal theory of Rr+1 (X).

A. A Generic Point for Rr+1 (X)

Definition. Let B be a commutative ring, A a B-algebra. A homomorphism •

from A into a polynomial ring B[W ] is called a generic point if every homomorphism

from A to a ¬eld L factors through •:

•

’’ B[W ]

A

L

Let us consider A = Rr+1 (X) as a simple example. The image U of the matrix X

with respect to a homomorphism from A into a ¬eld L satis¬es the condition rk U ¤ r.

The homomorphism Lm ’ Ln given by U can therefore be factored through Lr , and the

matrix U may be written

U = V 1 V2

where V1 is an m — r matrix, V2 an r — n matrix. So we take an m — r matrix Y and an

r — n matrix Z of (independent) indeterminates over B and factor the homomorphism

A ’’ L through

• : A ’’ B[Y, Z], X ’’ Y Z,

by substituting V1 for Y , V2 for Z. Thus • is a generic point. The existence of a

generic point has consequences which are known to us for the rings under considera-

tion. When we shall discuss a di¬erent approach to the theory of determinantal rings in

Section 12 starting from scratch, part (c) of the following proposition will be extremely

useful though. The reader should note that the construction of generic points for the

rings R(X; δ) below only relies on elementary matrix algebra !

(7.1) Proposition. Let • : A ’ B[W ] be a generic point for the B-algebra A.

(a) The kernel of • is contained in the nilradical of A.

(b) If B is reduced, then Ker • is the nilradical of A.

(c) If B is an integral domain, then the nilradical of A is prime.

(d) If B is a domain and A is reduced, then • is injective and A a domain itself.

All this is evident. If B is a domain, then it follows from (d) and (5.7) that the

generic point constructed for Rr+1 (X) is an embedding. But, all we need to prove this

in general, is the fact that [1, . . . , r|1, . . . , r] is not a zero-divisor in Rr+1 (X):

74 7. Generic Points and Invariant Theory

(7.2) Theorem. For every ring B the homomorphism • : Rr+1 (X) ’ B[Y, Z],

X ’ Y Z, is a generic point and an embedding.

Proof: Over an arbitrary commutative ring S a matrix U which has Ir+1 (U ) = 0

and an r-minor which is a unit in S, can be factored U = V1 V2 as above. So we only

need an embedding Rr+1 (X) ’ S such that this condition is true for the image U of

X; then the embedding factors through •. A suitable S is supplied by Rr+1 (X)[δ ’1 ],

δ = [1, . . . , r|1, . . . , r]. ”

The argument just given is typical for many proofs below: After the inversion of a

suitable minor the matrix under consideration can be manipulated like a matrix over a

¬eld.

The ring G(X) is de¬ned as a subring of B[X]. Let ψ : G(X) ’ L be a homomor-

phism into a ¬eld. Then the “vector” (ψ(γ) : γ ∈ “(X)) satis¬es the Pl¨cker relations,

u

and one can factor ψ through B[X] if and only if it is possible to construct a matrix U

over L such that its set of Pl¨cker coordinates is (ψ(γ) : γ ∈ “(X)). This is guaranteed

u

by Theorem (1.2) which, however, still waits for the completion of its proof. We shall

complete its proof within the proof of (7.14) below where it will also be stated that the

embedding G(X) ’ B[X] is a generic point.

B. Invariants and Absolute Invariants

In the situation of (7.2) let T be an element of GL(r, B), i.e. an invertible r — r

matrix over B. Then

Y Z = Y T ’1 T Z,

so the entries of Y Z are invariant under the substitution Y ’ Y T ’1 , Z ’ T Z considered

as an automorphism of B[Y, Z]. As T runs through G = GL(r, B), this de¬nes an

action of G on B[Y, Z] as a group of B-automorphisms. For T ∈ G and a polynomial

f (Y, Z) ∈ B[Y, Z] one puts

T (f ) = f (Y T ’1 , T Z).