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(n в€’ m в€’ r)2
j
grade(I, R/J ) в‰Ґ mr в€’ в€’1 else.
4
This result will be improved in 9.D, cf. the examples (9.27): B may be an arbitrary
noetherian ring, the п¬Ѓrst inequality is correct regardless of n в‰Ґ m + r, and if J is the
вЂњcolumn idealвЂќ, then grade(I, R/J j ) в‰Ґ nr в€’ 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 Г— r and r Г— n resp., r в‰¤ m в‰¤ n. Then

n в‰Ґ m + r,
grade I1 (Y Z)B[Y, Z] = mr if
(n в€’ m в€’ r)2
grade I1 (Y Z)B[Y, Z] = mr в€’ else.
4
[. . . ] denoting the integral part. The same equations hold for вЂњheightвЂќ.
We sketch two Proofs. The п¬Ѓrst 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 п¬Ѓelds 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 + Irв€’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 deп¬Ѓne a complex
f g
(R/P )m в€’в†’ (R/P )r в€’в†’ (R/P )n .
Since rk f + rk g в‰¤ 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 в€’ (n в€’ 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
пЈ« пЈ¶
В· В· В· Y 1rв€’1
Y 11 0
пЈ¬. .пЈ·
.
пЈ¬. . .пЈ·
. . . пЈ·.
пЈ¬
пЈ­ Y mв€’1,1 В· В· В· Y mв€’1,rв€’1 0пЈё
В·В·В·
Ym1 Ym,rв€’1 Ymr
91
H. Comments and References

в€’1
Let R = B[Y, Z][Ymr ], Y the (m в€’ 1) Г— (r в€’ 1) matrix in the left upper corner of the
matrix above, and Z the (r в€’ 1) Г— n matrix of the п¬Ѓrst r в€’ 1 rows of Z. The entries of
Y and Z are algebraically independent over
в€’1
B = B[Ym1 , . . . , Ymr , Y1r , . . . , Ymв€’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.
в€’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 вЂњgradeвЂќ by вЂњheightвЂќ throughout. вЂ”

H. Comments and References

Theorem (7.6) and Corollary (7.7) are classical for п¬Ѓelds B = K of characteristic
zero, cf. [We]. The characteristic free version of (7.7) is essentially due to Igusa [Ig], and
in their п¬Ѓnal 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 IgusaвЂ™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 deп¬Ѓnition of вЂњabsolute invariantвЂќ to concrete situations.
Our notion of вЂњgeneric pointвЂќ is inspired by Hochster and EagonвЂ™s article [HE.2]
in which the construction of generic points plays a central role, cf. Section 12. The
generic points for G(X; Оі) 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; Оґ). 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 п¬Ѓrst (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] в†’ K[Y Z] when K is a п¬Ѓeld of characteristic zero, cf. [HE.2], Intro-
duction. On the other hand we quote from [HR], p. 118: вЂњ. . . determinantal loci have
. . . ultimately motivated the conjecture of . . . the Main TheoremвЂќ 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 simpliп¬Ѓcation of its proof.
The rationality of the singularities of the Schubert varieties was п¬Ѓrst proved by
Kempf [Ke.4]. Their homogeneous coordinate rings are the G(X; Оі), and the varieties
corresponding to R(X; Оґ) 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; Оі) and the determinantal rings R(X; Оґ) (over a normal ring B of coeп¬ѓcients). 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 П‰B , G(X; Оі) and R(X; Оґ) 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; Оі) and R(X; Оґ) is NagataвЂ™s theorem which relates
Cl(S) and the class groups of its rings of quotients, cf. [Fs], В§ 7 (or [Bo.3], В§ 1, no. 10,
Prop. 17).
It has been proved in (6.3) that G(X; Оі) and R(X; Оґ) are normal domains when the
ring B of coeп¬ѓcients is a normal domain. Therefore G(X; Оі) and R(X; Оґ) have well-
deп¬Ѓned class groups then. The normality of G(X; Оі) and R(X; Оґ) has been demonstrated
by showing that the rings G(X; Оі)[Оі в€’1 ] and R(X; Оґ)[Оґ в€’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; Оі) and R(X; Оґ) and
Оµ for Оі and Оґ resp. Since R is a п¬‚at extension of B, the embedding B в†’ R induces a
homomorphism Cl(B) в€’в†’ Cl(R), and the composition

Cl(B) в€’в†’ Cl(R) в€’в†’ Cl(R[Оµв€’1 ]) в€ј Cl(B)
=

is just the natural isomorphism from Cl(B) to Cl(R[Оµв€’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) вЉ• U,

the subgroup U being generated by the classes of the minimal prime ideals of Оµ by virtue
of NagataвЂ™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/RОµ is reduced, we have
u
RОµ = 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 п¬Ѓeld of fractions of R. The
divisor div(Pi ) is contained in the kernel of the homomorphism Div(R) в†’ Div(R[Оµв€’1 ])
of groups of divisors, whence the element f is a unit in R[Оµв€’1 ]. Since R[Оµв€’1 ] arises from
a polynomial ring over B by inversion of a prime element, namely Оµ, we have

f = gОµm ,

g a unit in B, m в€€ Z. So
u u
ti div(Pi ) = div(Rf ) = m div(RОµ) = 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) вЉ• Zu ,

and every set of u of the classes of P0 , . . . , Pu generates the direct summand Zu .
Let О  = О“(X) or О  = в€†(X) resp. Then the ideal deп¬Ѓning R as a residue class ring
of G(X) or B[X] is generated by an ideal в„¦ of О , в„¦ itself being cogenerated by Оµ. The
ideal deп¬Ѓning R/RОµ is generated by в„¦ в€Є {Оµ}, and its minimal prime ideals are generated
by the ideals of О  which are cogenerated by the upper neighbours of Оµ in О . Within R
this means that the minimal prime ideals of RОµ have the form

J(x; О¶) = J(X; О¶)/J(X; Оі) or I(x; О¶) = I(X; О¶)/I(X; Оґ),

О¶ running through the upper neighbours of Оі or Оґ.
We deal with G(X; Оі) п¬Ѓrst. In Section 6 we have broken Оі = [a1 , . . . , am ] into its
blocks ОІ0 , . . . , ОІs of consecutive integers:

Оі = [ОІ0 , . . . , ОІs ], ОІi = (aki +1 , . . . , aki+1 ).

Each ОІi is followed by the gap

П‡i = (aki+1 + 1, . . . , aki+1 +1 в€’ 1),

the sequence of integers properly between the last element of ОІi and the п¬Ѓrst element of
ОІi+1 , the last gap П‡s being possibly empty. Obviously Оі has as many upper neighbours
as their are nonempty gaps П‡i , and the upper neighbours are

О¶i = [ОІ0 , . . . , ОІiв€’1 , (aki +1 , . . . , aki+1 в€’ 1), aki+1 + 1, ОІi+1 , . . . , ОІs ],

i = 0, . . . , s if am < n, i = 0, . . . , s в€’ 1 if am = n.
95
A. The Divisor Class Group

(8.1) Theorem. Let B be a noetherian normal domain, X an m Г— n matrix of
indeterminates, Оі an element of the poset О“(X) of its m-minors, Оі = [a1 , . . . , am ]. Then
the class group of G(X; Оі) is given by

Cl(B) вЉ• Zs if am < n or s = 0,
Cl(G(X; Оі)) =
Cl(B) вЉ• Zsв€’1 otherwise.

The summand Cl(B) arises naturally from the embedding B в†’ G(X; Оі), and the sum-
mand Zs or Zsв€’1 is generated by the classes of any set of s or s в€’ 1 resp. of the prime
ideals J(x; О¶i ).
Note that one can simplify the formulation of (8.1) if one п¬Ѓrst applies the reduction
to the case am < n as indicated in (6.9),(d).
(8.2) Corollary. G(X; Оі) is factorial if and only if B is factorial and there is at
most one nonempty gap in Оі.
In particular G(X) itself is factorial. The condition for Оі in (8.2) is satisп¬Ѓed exactly
in the cases in which ОЈ(X; Оі) = О“(X; Оі), cf. (6.8).
In order to determine the upper neighbours of Оґ = [a1 , . . . , ar |b1 , . . . , br ] в€€ в€†(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 О¶i from raising a row index, i = 0, . . . , u, unless ar = m, in which
case i = 0, . . . , u в€’ 1. Similary one obtains the upper neighbours О·j determined by
the column part. In case ar = m and br = n there is the further upper neighbour
П‘ = [a1 , . . . , arв€’1 |b1 , . . . , brв€’1 ], apart from the (trivial) case r = 1.
(8.3) Theorem. Let B be a noetherian normal domain, X an m Г— n matrix of
indeterminates, Оґ = [a1 , . . . , ar |b1 , . . . , br ] an element of the poset of its minors. Then
the class group of R(X; Оґ) is given by

Cl(B) вЉ• Zu+v if ar = m or br = n,
Cl(R(X; Оґ)) = u+v+1
Cl(B) вЉ• Z otherwise.

The summand Cl(B) arises naturally from the embedding B в†’ R(X; Оґ), 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 Оґ.
Evidently R(X; Оґ) is factorial if and only if B is factorial and

Оґ = [m в€’ r + 1, . . . , m|b1 , . . . , b1 + r в€’ 1] or Оґ = [a1 , . . . , a1 + r в€’ 1|n в€’ r + 1, . . . , n],

equivalently, if it is a polynomial ring over B (cf. the discussion below (6.10)). R(X; Оґ)
can be viewed as arising from a suitable ring G(X; Оґ) by dehomogenization with respect
to В±[n + 1, . . . , n + m]. As in Subsection 16.D one has a natural commutative diagram

R в€’в†’ S в€ј A[T, T в€’1 ]
=
ПЂ

A

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