j

grade(I, R/J ) ≥ mr ’ ’1 else.

4

This result will be improved in 9.D, cf. the examples (9.27): B may be an arbitrary

noetherian ring, the ¬rst inequality is correct regardless of n ≥ m + r, and if J is the

“column ideal”, then grade(I, R/J j ) ≥ nr ’ 1. These estimates, on the other hand, are

optimal: one has equality for j large.

It remains to compute grade I1 (Y Z)B[Y, Z]. We restrict ourselves to the case of

interest to us.

(7.25) Proposition. Let B be a noetherian ring, Y and Z matrices of indetermi-

nates of sizes m — r and r — n resp., r ¤ m ¤ n. Then

n ≥ m + r,

grade I1 (Y Z)B[Y, Z] = mr if

(n ’ m ’ r)2

grade I1 (Y Z)B[Y, Z] = mr ’ else.

4

[. . . ] denoting the integral part. The same equations hold for “height”.

We sketch two Proofs. The ¬rst one uses the theory of varieties of complexes

([DS]). Let I = I1 (Y Z). Then R = B[Y, Z]/I is a Hodge algebra in the sense of [DEP.2],

in particular it is a free B-module. It is enough to consider ¬elds B (cf. (3.14)). R is

reduced now (actually, over any reduced B), and the minimal prime ideals of I in B[Y, Z]

are given by

Pi = I + Ir’i+1 (Y ) + Ii+1 (Z), i = 0, . . . , r.

In fact, let P be any minimal prime ideal of R. The residue classes of the matrices Y

and Z over R/P de¬ne a complex

f g

(R/P )m ’’ (R/P )r ’’ (R/P )n .

Since rk f + rk g ¤ r, the preimage of P in B[Y, Z] has to contain one of the ideals Pi ,

namely Prk g . On the other hand the Pi are prime ideals ([DS], Theorem 2.11) and

grade Pi = rn + i2 ’ (n ’ m + r)i

by virtue of [DS], Lemma 2.3. Now one takes the minimum of these grades.

The second proof is elementary. It goes by induction on r. If r = 1, then I =

I1 (Y )I1 (Z), and the formula for grade I is obviously correct. We invert Ymr and perform

elementary row transformations on Y to obtain

«

· · · Y 1r’1

Y 11 0

¬. .·

.

¬. . .·

. . . ·.

¬

Y m’1,1 · · · Y m’1,r’1 0

···

Ym1 Ym,r’1 Ymr

91

H. Comments and References

’1

Let R = B[Y, Z][Ymr ], Y the (m ’ 1) — (r ’ 1) matrix in the left upper corner of the

matrix above, and Z the (r ’ 1) — n matrix of the ¬rst r ’ 1 rows of Z. The entries of

Y and Z are algebraically independent over

’1

B = B[Ym1 , . . . , Ymr , Y1r , . . . , Ym’1,r ][Ymr ],

as are the elements of the product Ym Z of the last row Ym of Y and Z over B[Y Z].

Furthermore

R = (B[Y Z])[Ym Z] and I R = I1 (Y Z)R + I1 (Ym Z)R,

hence

grade I R = grade I1 (Y Z) + n.

’1

Letting R = B[Y, Z][Zrn ], one concludes similarly that

grade I R = grade I1 (Y Z) + m,

Y , Z being constructed analogously. Since obviously

grade I = min(grade I R, grade I R),

the claim follows by the inductive hypothesis. In order to obtain the equations for height,

one replaces “grade” by “height” throughout. ”

H. Comments and References

Theorem (7.6) and Corollary (7.7) are classical for ¬elds B = K of characteristic

zero, cf. [We]. The characteristic free version of (7.7) is essentially due to Igusa [Ig], and

in their ¬nal form presented here, (7.6) and (7.7) were given by de Concini and Procesi

[DP]; cf. [BB] and [Ri] for possibly simpler or more elementary proofs. The proofs of (7.7)

and (7.6),(a) result from an attempt to understand Igusa™s arguments. Our treatment is

certainly close to [DP], from which we copied the proofs of (7.7),(b) and (7.5). At least

for (7.6),(a), however, the standard monomial theory is not essential; it can be derived

from the result of Hochster and Eagon ([HE.2]) already, cf. Section 12. In order to avoid

the intricacies of the notion of algebraic group over general commutative rings we have

restricted the de¬nition of “absolute invariant” to concrete situations.

Our notion of “generic point” is inspired by Hochster and Eagon™s article [HE.2]

in which the construction of generic points plays a central role, cf. Section 12. The

generic points for G(X; γ) in (7.14) were given by Hodge [Hd], and the proof of the

linear independence of the standard monomials in (7.16) is taken from Musili [Mu]. The

construction of the generic points in (7.17) and the invariant theoretic description (7.18)

are borrowed from [HE.2], Sections 7 and 8, and [Ho.3], Section 5. Hochster proves (7.18)

in characteristic zero by the reductivity argument indicated in Remark (7.21). We have

freed his constructions from the assumption of characteristic zero and generalized to all

the rings R(X; δ). The determination of the semi-invariants of the group H in (7.19) and

a generalization of (7.10) are left to the reader. The ideal-theoretic consequences to be

expected will be proved in Section 9 by methods perhaps more convenient.

92 7. Generic Points and Invariant Theory

The ¬rst (unpublished) proof of Hochster and Eagon for the perfection of determi-

nantal ideals was based on invariant theory, in particular the existence of a Reynolds

operator K[Y, Z] ’ K[Y Z] when K is a ¬eld of characteristic zero, cf. [HE.2], Intro-

duction. On the other hand we quote from [HR], p. 118: “. . . determinantal loci have

. . . ultimately motivated the conjecture of . . . the Main Theorem” of [HR] mentioned

above. Further examples for which the theorem of Hochster-Roberts implies the Cohen-

Macaulay property are listed in [HR]. Cf. [Ke.5] for a generalization of the theorem of

Hochster-Roberts and a simpli¬cation of its proof.

The rationality of the singularities of the Schubert varieties was ¬rst proved by

Kempf [Ke.4]. Their homogeneous coordinate rings are the G(X; γ), and the varieties

corresponding to R(X; δ) are open subvarieties of the Schubert varieties, so they have

rational singularities, too.

Remark (7.12) was communicated to us by M. Hochster, and Subsection G owes its

existence to discussions with J. Herzog.

References for (7.10) and (7.24) will be given in Section 9 where results of the same

kind will be derived in a more general context. We do not know of an invariant-theoretic

approach in the literature, however.

8. The Divisor Class Group and the Canonical Class

This section is devoted to the study of the divisor class groups of the Schubert cycles

G(X; γ) and the determinantal rings R(X; δ) (over a normal ring B of coe¬cients). Their

computation has been prepared in Section 6, and will turn out rather easy. If B is a

Cohen-Macaulay ring with a canonical module ωB , G(X; γ) and R(X; δ) have canonical

modules, too, which, under the assumption of normality, are completely determined by

their divisor class, called the canonical class. The crucial case in the computation of the

canonical class is R2 (X), to which the general case can be reduced by surprisingly simple

localization arguments. As an application we determine the Gorenstein rings among

the rings under consideration. In Section 9 we shall give a complete description of the

canonical module in terms of the standard monomial basis.

A. The Divisor Class Group

For the theory of divisorial ideals and the (divisor) class group Cl(S) of a normal

domain S we refer the reader to [Fs] (or [Bo.3]). The main tool for the computation

of the class groups of the rings G(X; γ) and R(X; δ) is Nagata™s theorem which relates

Cl(S) and the class groups of its rings of quotients, cf. [Fs], § 7 (or [Bo.3], § 1, no. 10,

Prop. 17).

It has been proved in (6.3) that G(X; γ) and R(X; δ) are normal domains when the

ring B of coe¬cients is a normal domain. Therefore G(X; γ) and R(X; δ) have well-

de¬ned class groups then. The normality of G(X; γ) and R(X; δ) has been demonstrated

by showing that the rings G(X; γ)[γ ’1 ] and R(X; δ)[δ ’1 ] arise from a polynomial ring

over B after the inversion of a prime element, rendering their class groups naturally

isomorphic with Cl(B), cf. (6.1) and (6.4). Let us write R for G(X; γ) and R(X; δ) and

µ for γ and δ resp. Since R is a ¬‚at extension of B, the embedding B ’ R induces a

homomorphism Cl(B) ’’ Cl(R), and the composition

Cl(B) ’’ Cl(R) ’’ Cl(R[µ’1 ]) ∼ Cl(B)

=

is just the natural isomorphism from Cl(B) to Cl(R[µ’1 ]).

Naturality here means: These maps are induced by homomorphisms of the groups

of divisors which send a divisorial ideal to its extension. It follows at once that

Cl(R) = Cl(B) • U,

the subgroup U being generated by the classes of the minimal prime ideals of µ by virtue

of Nagata™s theorem. Corollary (6.5) names these prime ideals, and we will specify them

below. Let they be denoted by P0 , . . . , Pu here. Since R/Rµ is reduced, we have

u

Rµ = Pi

i=0

94 8. The Divisor Class Group and the Canonical Class

u

and thus the relation i=0 cl(Pi ) = 0. We claim: This is the only relation between the

classes cl(Pi ), and every subset of u of them is linearly independent. Suppose that

u

ti cl(Pi ) = 0.

i=0

u

Then i=0 ti div(Pi ) is a principal divisor div(Rf ), f in the ¬eld of fractions of R. The

divisor div(Pi ) is contained in the kernel of the homomorphism Div(R) ’ Div(R[µ’1 ])

of groups of divisors, whence the element f is a unit in R[µ’1 ]. Since R[µ’1 ] arises from

a polynomial ring over B by inversion of a prime element, namely µ, we have

f = gµm ,

g a unit in B, m ∈ Z. So

u u

ti div(Pi ) = div(Rf ) = m div(Rµ) = m div(Pi ).

i=0 i=0

Since the divisors div(Pi ), i = 0, . . . , u, are linearly independent, we conclude that ti = m

for i = 0, . . . , u as desired. Therefore

Cl(R) = Cl(B) • Zu ,

and every set of u of the classes of P0 , . . . , Pu generates the direct summand Zu .

Let Π = “(X) or Π = ∆(X) resp. Then the ideal de¬ning R as a residue class ring

of G(X) or B[X] is generated by an ideal „¦ of Π, „¦ itself being cogenerated by µ. The

ideal de¬ning R/Rµ is generated by „¦ ∪ {µ}, and its minimal prime ideals are generated

by the ideals of Π which are cogenerated by the upper neighbours of µ in Π. Within R

this means that the minimal prime ideals of Rµ have the form

J(x; ζ) = J(X; ζ)/J(X; γ) or I(x; ζ) = I(X; ζ)/I(X; δ),

ζ running through the upper neighbours of γ or δ.

We deal with G(X; γ) ¬rst. In Section 6 we have broken γ = [a1 , . . . , am ] into its

blocks β0 , . . . , βs of consecutive integers:

γ = [β0 , . . . , βs ], βi = (aki +1 , . . . , aki+1 ).

Each βi is followed by the gap

χi = (aki+1 + 1, . . . , aki+1 +1 ’ 1),

the sequence of integers properly between the last element of βi and the ¬rst element of

βi+1 , the last gap χs being possibly empty. Obviously γ has as many upper neighbours

as their are nonempty gaps χi , and the upper neighbours are

ζi = [β0 , . . . , βi’1 , (aki +1 , . . . , aki+1 ’ 1), aki+1 + 1, βi+1 , . . . , βs ],

i = 0, . . . , s if am < n, i = 0, . . . , s ’ 1 if am = n.

95

A. The Divisor Class Group

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

indeterminates, γ an element of the poset “(X) of its m-minors, γ = [a1 , . . . , am ]. Then

the class group of G(X; γ) is given by

Cl(B) • Zs if am < n or s = 0,

Cl(G(X; γ)) =

Cl(B) • Zs’1 otherwise.

The summand Cl(B) arises naturally from the embedding B ’ G(X; γ), and the sum-

mand Zs or Zs’1 is generated by the classes of any set of s or s ’ 1 resp. of the prime

ideals J(x; ζi ).

Note that one can simplify the formulation of (8.1) if one ¬rst applies the reduction

to the case am < n as indicated in (6.9),(d).

(8.2) Corollary. G(X; γ) is factorial if and only if B is factorial and there is at

most one nonempty gap in γ.

In particular G(X) itself is factorial. The condition for γ in (8.2) is satis¬ed exactly

in the cases in which Σ(X; γ) = “(X; γ), cf. (6.8).

In order to determine the upper neighbours of δ = [a1 , . . . , ar |b1 , . . . , br ] ∈ ∆(X) we

have to decompose the row part [a1 , . . . , ar ] and the column part [b1 , . . . , br ] similarly,

obtaining u + 1 blocks for [a1 , . . . , ar ] and v + 1 blocks for [b1 , . . . , br ]. There arise

upper neighbours ζi from raising a row index, i = 0, . . . , u, unless ar = m, in which

case i = 0, . . . , u ’ 1. Similary one obtains the upper neighbours ·j determined by

the column part. In case ar = m and br = n there is the further upper neighbour

‘ = [a1 , . . . , ar’1 |b1 , . . . , br’1 ], apart from the (trivial) case r = 1.

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

indeterminates, δ = [a1 , . . . , ar |b1 , . . . , br ] an element of the poset of its minors. Then

the class group of R(X; δ) is given by

Cl(B) • Zu+v if ar = m or br = n,

Cl(R(X; δ)) = u+v+1

Cl(B) • Z otherwise.

The summand Cl(B) arises naturally from the embedding B ’ R(X; δ), and the direct

summand Zu+v or Zu+v+1 is generated by the classes of any u + v or u + v + 1 resp.

prime ideals corresponding to the upper neighbours of δ.

Evidently R(X; δ) is factorial if and only if B is factorial and

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

equivalently, if it is a polynomial ring over B (cf. the discussion below (6.10)). R(X; δ)

can be viewed as arising from a suitable ring G(X; δ) by dehomogenization with respect

to ±[n + 1, . . . , n + m]. As in Subsection 16.D one has a natural commutative diagram

R ’’ S ∼ A[T, T ’1 ]

=

π

A

R = G(X; δ), A = R(X; δ), T an indeterminate, S = R[[n + 1, . . . , n + m]’1 ]. There