At this point we have to determine the structure of S. Analogously with (6.1) we let

δ ∈ “(X; γ): δ di¬ers from σi in at most one index

Ψ=

and subdivide Ψ in

Ψ1 = {δ ∈ Ψ : δ ≥ σi } and Ψ2 = {δ ∈ Ψ : δ ≥ σi }.

Evidently, Ψ2 contains those δ ∈ Ψ which arise from σi by replacement of an element of

the block

β i = (aki +1 , . . . , aki+1 + 1)

by an element of the gap

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

Let β i’1 = (aki’1 +1 , . . . , aki ’1 ), p = |χi’1 |, q = |β i |. We choose a p — q matrix T and

an independent family {Tψ : ψ ∈ Ψ1 } of indeterminates over B.

(8.11) Lemma. The substitution Tψ ’’ ψ, ψ ∈ Ψ1 , and

Tjk ’’ [β0 , . . . , βi’2 , β i’1 , aki ’1 + j, β i \{aki+1 + 2 ’ k}, βi+1 , . . . , βs ]

induces an isomorphism

’1

’1

(B[T ]/I2 (T ))[Tψ : ψ ∈ Ψ1 ][Tσi ] ’ R[σi ].

Furthermore the prime ideal P generated in B[T ]/I2 (T ) by the elements of the ¬rst row

’1

of T extends to J(x; ζi’1 )R[σi ], the prime ideal Q generated by the elements of the ¬rst

’1

column extends to J(x; ζi )R[σi ].

This lemma ¬nishes the computation of cl(ω). It follows immediately from (8.8)

that

κi’1 ’ κi = |χi’1 | ’ |β i |

= (|χi’1 | + 1) ’ (|βi | + 1)

= |χi’1 | ’ |βi |.

Before we state the main result, (8.11) should be proved.

102 8. The Divisor Class Group and the Canonical Class

Proof of 8.11: The substitution induces a surjective map

’1

’1

B[T ][Tψ : ψ ∈ Ψ1 ][Tσi ] ’’ R[σi ].

This is proved as in (6.1). To see that I2 (T ) is sent to zero, we look at the Pl¨cker

u

relation, for which, with the notations of (4.4),

“[a1 , . . . , ak ]” = [β0 , . . . , βi’2 , β i’1 , β i \{aki+1 + 2 ’ k}, βi+1 , . . . , βs ],

“s” = m + 1,

“[c1 , . . . , cs ]” = [aki ’1 +j, β0 , . . . , βi’2 , β i’1 , aki ’1 +u, β i \{aki+1 +2’v}, βi+1 , . . . , βs ].

At most three indices in “[c1 , . . . , cs ]” do not occur in “[a1 , . . . , ak ]”, so at most three

products can appear in this relation, one of which drops out in G(X; γ): the “second”

factor which contains both aki ’1 + j and aki ’1 + u is ≥ γ. This leaves the desired

relation. In order to show injectivity it is enough to prove that the ring on the left

’1

side has the same dimension as R[σi ]. This is easily checked if one remembers that

|Ψ1 | = dim G(X; σi ) ’ dim B (cf. (6.1)).

’1

It is obvious that the extension of P is contained in J(x; ζi’1 )R[σi ]; since the latter

is divisorial, inclusion implies equality. For Q one argues similarly. (It is of course also

possible to prove equality directly.) ”

(8.12) Theorem. Let B be a normal Cohen-Macaulay domain having a canonical

module ωB , X an m — n matrix of indeterminates, and γ ∈ G(X). Then a canonical

module of R = G(X; γ) is given by a divisorial ideal with class

t

cl(ωB R) + κi cl(J(x; ζi ))

i=0

such that

κi’1 ’ κi = |χi’1 | ’ |βi |, i = 1, . . . , t,

where t = s if am < n, t = s ’ 1 if am = n, β0 , . . . , βs are the blocks of γ, and χi’1 is

the gap between βi’1 and βi . Every canonical module of R has this representation, and

up to the choice of ωB it is unique.

(8.13) Corollary. Let B be an (arbitrary) noetherian ring. Then G(X; γ) is a

Gorenstein ring if and only if B is Gorenstein and |χi’1 | = |βi | for i = 1, . . . , t.

(8.12) has been completely proved already, and (8.13) follows from it as (8.9) followed

from (8.8). In particular we have |χi’1 | = |βi | for i = 1, . . . , t if t = 0, in which case

G(X; γ) is factorial (over a factorial B).

The easiest way to deal with R = R(X; δ) is to relate it to R = G(X; δ) as usual

(cf. (5.5)):

y = [n + 1, . . . , n + m] ± 1.

R = R/Ry,

(κi )

Assuming that B is a ¬eld one writes the canonical module ωR of R as Pi , P i running

˜

through the prime ideals corresponding to the upper neighbours of δ, and κi ≥ 0. Then

(—) cl(ωR ) = κi cl(Pi )

103

C. The General Case

where the sum is now extended over the prime ideals corresponding to the upper neigh-

bours of δ, and Pi is the image of P i in R. The equation (—) can be derived from (8.10),

but it is easier to use the properties of dehomogenization. Since y is not a zero-divisor,

ωR = ωR /yωR ,

˜ ˜

and since y is not a zero-divisor modulo ω (as an ideal) one has yωR = ωR © Ry, hence

˜ ˜

ωR = (ωR + Ry)/Ry.

˜

Now (—) follows from (16.27): dehomogenization preserves primary decomposition. We

remind the reader that the upper neighbours of δ = [a1 , . . . , ar |b1 , . . . , br ] have been

named above (8.3): ζi and ·i arising from raising a row and a column index resp., and,

in case ar = m, br = n, ‘ = [a1 , . . . , ar’1 |b1 , . . . , br’1 ]. The blocks of [a1 , . . . , ar ] are

β0 , . . . , β u with gaps χ 0 , . . . , χu ,

those of [b1 , . . . , br ] are denoted

— —

χ — , . . . , χ— .

β0 , . . . , β v with gaps 0 v

Furthermore let w = u if ar < m, w = u ’ 1 if ar = m, z = v if br < n, z = v ’ 1 if

br = n.

Relating the blocks and gaps of [a1 , . . . , ar ] and [b1 , . . . , br ] to those of δ, the reader

will easily derive the following theorem:

(8.14) Theorem. Let B be a normal Cohen-Macaulay domain having a canonical

module ωB , X an m — n matrix of indeterminates, and δ ∈ ∆(X). Then a canonical

module of R = R(X; δ) is given by a divisorial ideal with class

±w z

κi cl(I(x; ζi )) + »i cl(I(x; ·i )) if ar < m or br < n,

i=0 i=0

cl(ωB R) +

w z

κi cl(I(x; ζi )) + »i cl(I(x; ·i )) + µ cl(I(x; ‘)) if ar = m and br = n,

i=0 i=0

where

κi’1 ’ κi = |χi’1 | ’ |βi |, i = 1, . . . , w,

»i’1 ’ »i = |χ— | ’ |βi |,

—

i = 1, . . . , z,

i’1

±—

|χz | ’ |χw |(= n ’ br ’ (m ’ ar )) if ar < m, br < n,

»z ’ κw = (|χ— | + |βw+1 |) ’ |χw | if ar = m, br < n,

—z

—

|χz | ’ (|βz+1 | + |χw |) if ar < m, br = n,

µ ’ κw = |βw+1 | ’ |χw | if ar = m, br = n,

µ ’ »z = |βz+1 | ’ |χ— |

—

if ar = m, br = n.

z

Every canonical module of R has this representation, and up to the choice of ω B it is

unique.

104 8. The Divisor Class Group and the Canonical Class

Again, R is Gorenstein if and only if B is Gorenstein, and all the di¬erences listed

above vanish (as far as they apply to a speci¬c R).

Within divisor theory the preceding theorems are completely satisfactory. Neverthe-

less they su¬er from an ideal-theoretic de¬ciency: We don™t have a concrete description

of the symbolic powers of the prime ideals generating the class group. As we shall see

in the next section, they coincide with the ordinary powers. Only for Rr+1 (X) this has

been proved already, cf. (7.10).

D. Comments and References

For the simplest case R2 (X), X a 2 — 2 matrix, the class group is computed in

Fossum™s book ([Fs], § 14), and Theorems (8.1) and (8.3) may be viewed as natural

generalizations, the intermediate case Rr+1 (X) being covered by [Br.3]. The factoriality

of G(X) was proved by Samuel ([Sa], p. 38), cf. also [Ho.3], Corollary 3.15.

The computation of the canonical class was initiated in [Br.6] for Rr+1 (X). Accord-

ing to [Hu.1], p. 500 this case was also solved by Hochster. Yoshino [Yo.1] computed the

canonical module of Rm (X) directly from the Eagon-Northcott resolution (cf. Section 2).

Svanes determined the Gorenstein rings among the homogeneous coordinate rings of the

Schubert varieties and derived (8.9), cf. [Sv.1], pp. 451,452. The ¬rst attempt towards

(8.9) was made by Eagon [Ea.2] who obtained the result for ideals of maximal minors.

Goto [Go.1] proved the necessity of m = n in (8.9) in general and the su¬ciency for the

case r = 1.

Stanley showed that the Gorenstein property is re¬‚ected in the Hilbert function of a

graded Cohen-Macaulay domain. This fact can also be used to determine the Gorenstein

rings among the rings R(X; δ) and G(X; γ), cf. [St].

9. Powers of Ideals of Maximal Minors

In Section 7 we have derived results on the powers of certain ideals in the rings

Rr+1 (X) by invariant-theoretic methods (cf. (7.10) and (7.24)). The ideals considered

there are Im (X), the ideal generated by the m-minors of an m — n matrix of indeter-

minates, and the ideals P and Q appearing in the description of the class group and

the canonical class of Rr+1 (X) in Section 8. In this section we want to investigate more

generally the powers of ideals in G(X; γ) and R(X; δ) which can justi¬ably be called

ideals of maximal minors. They share a remarkable property: their generators in the

poset underlying R(X; δ) or G(X; γ) generate a sub-ASL in a natural way. The graded

algebras related to the powers of such ideals, the ordinary and extended Rees algebra,

and the associated graded ring, are again ASLs over wonderful posets and (normal) do-

mains if the ring B of coe¬cients is a (normal) domain. In particular, the ideals have

primary powers (over an integral B), and one obtains a lower bound of their depths (in

suitable localizations).

A. Ideals and Subalgebras of Maximal Minors

Let U be a matrix over a commutative ring. If Ir+1 (U ) = 0 and Ir (U ) = 0, then an

r-minor = 0 is called a maximal minor of U . In R(X; δ) (considered over an arbitrary

commutative ring B) an ideal I is said to be an ideal of maximal minors if it is generated

by the maximal minors of a submatrix U of the matrix of residue classes of X which

consists of the ¬rst u rows or ¬rst v columns, 1 ¤ u ¤ m, 1 ¤ v ¤ n. More formally, the

ideals of maximal minors are the ideals

I(X; µ)/I(X; δ),

δ = [a1 , . . . , ar |b1 , . . . , br ], µ = [a1 , . . . , ak’1 , ak , ak+1 , . . . , a˜|b1 , . . . , b˜], where ak is a

r r

given integer such that

ak < ak ¤ ak+1

and µ is the smallest element in ∆(X; δ) whose row part starts as a1 , . . . , ak’1 , ak , or

µ = [a1 , . . . , a˜|b1 , . . . , bl’1 , bl , . . . , b˜] with a similar condition. We allow the extreme

r r

cases k = r and ak = m + 1, so µ = [a1 , . . . , ar’1 |b1 , . . . , br’1 ] and I being generated

by all the r-minors then. The ideal indicated is generated by the k-minors of the rows

1, . . . , ak ’ 1, and the condition ak ¤ ak+1 guarantees that the (k + 1)-minors of these

rows are zero. For simplicity we call the corresponding ideals

J(X; µ)/J(X; γ),

γ = [a1 , . . . , am ], µ = [a1 , . . . , ak’1 , ak , . . . , am ], ak ¤ ak+1 , and µ as small as possible,

ideals of maximal minors, too, and say that µ de¬nes an ideal of maximal minors. Note

that all the elements in ∆(X; δ) or “(X; γ) which have been important for the structure

of R(X; δ) or G(X; γ), de¬ne ideals of maximal minors: the upper neighbours of γ or δ

as well as the elements describing the singular locus.

The crucial property of ideals of maximal minors is given by the following lemma:

106 9. Powers of Ideals of Maximal Minors

(9.1) Lemma. Let µ de¬ne an ideal of maximal minors in G(X; γ) or R(X; δ) and

„¦ = “(X; γ) \ “(X; µ) or „¦ = ∆(X; δ) \ ∆(X; µ). Let ξ, … ∈ „¦ be incomparable. Then

every standard monomial appearing in the standard representation

ξ… = aµ µ, aµ = 0,

is the product of two factors in „¦.

Proof: Consider the case G(X; γ) ¬rst, γ = [a1 , . . . , am ]. Then every standard

monomial appearing on the right side of the straightening relation has exactly two factors,

and the union of their indices coincides with the union of the incides of ξ, …. Now

ζ ∈ “(X; γ) is in „¦ if and only if it has k indices < ak . On the other hand ζ cannot

have k + 1 indices < ak , for ζ ≥ γ then. Since ξ and … together contain 2k indices < ak

(counted with multiplicities), and both factors of µ can have at most k such indices, both

of them have exactly k of them, and so are in „¦.

Again it is useful to consider R(X; δ) arising from G(X; δ) in the usual way. For

every element ζ ∈ ∆(X; δ) let ζ denote the corresponding element in “(X; δ). Then

„¦ = “(X; δ) \ “(X; µ) = { ζ : ζ ∈ „¦ },

and one checks immediately that µ de¬nes an ideal of maximal minors in G(X; δ). Let

ξ… = a˜ µ be the standard representation of ξ…. From the ¬rst part of the proof we

µ

know that each of the µ has both its factors in „¦, so does not contain [n + 1, . . . , n + m].

In passing from G(X; δ) to R(X; δ) one only replaces [n + 1, . . . , n + m] by (’1)m(m’1)/2 .

(It is of course as easy to argue directly for R(X; δ).) ”

(9.2) Corollary. Let S be the B-submodule generated by the standard monomials

which have all their factors in „¦. Then S is a subalgebra of G(X; γ) or R(X; δ) resp.,

and therefore automatically a graded ASL on „¦.

In fact, the argument that proved (4.1), shows that S is a subalgebra, and the rest

is obvious. The properties of S will be noted below: they are as good as one could

reasonably hope for.

(9.3) Corollary. Let I be the ideal of maximal minors generated by the ideal „¦ in

“(X; γ) or ∆(X; δ). Then I j is the submodule of G(X; γ) or R(X; δ) resp. generated by

the standard monomials containing at least j factors in „¦, so as an ideal it is generated

by the standard monomials of length j in „¦.