is always a natural epimorphism Cl(R) ’ Cl(S), and a natural isomorphism Cl(A) ’

Cl(S). Here the resulting epimorphism Cl(R) ’ Cl(A) is an isomorphism. This follows

from (8.1) and (8.3) since the upper neighbours of δ and δ are in one-one correspondence,

and the ideals in A and R resp. “cogenerated” by them correspond to each other under

π as in (16.26).

A by-product: The maximal element µ of “(X; γ) is always a prime element (over

an integral domain B). It is enough to show this for a ¬eld B since G(X; γ)/µG(X; γ)

is a graded ASL (cf. (3.12)). Then we write G(X; γ) as G(X; δ). Since Cl(R) ’ Cl(S)

is an isomorphism, the minimal prime ideal of µ must be principal. Being irreducible, µ

is prime itself.

We single out the most important case of (8.3):

(8.4) Corollary. The class group of Rr+1 (X), 0 < r < min(m, n), is

Cl(Rr+1 (X)) = Cl(B) • Z,

the summand Z generated by the class of the prime ideal P generated by the r-minors of

r arbitrary rows or its negative, the class of the prime ideal Q generated by the r-minors

of r arbitrary columns.

In fact, the generators speci¬ed in (8.3) correspond to the ¬rst r rows or ¬rst r

columns. Since an automorphism exchanging the rows leaves each of the prime ideals

Q in (8.4) invariant, the induced automorphism of Cl(Rr+1 (X)) is the identity, and the

same holds for a permutation of the columns. (The reader may describe the isomorphisms

between the prime ideals P or Q resp. directly; cf. also the discussion above (7.24).)

(8.5) Remarks. (a) Let B = K be a ¬eld. The completion R of R = G(X; γ) or

R = R(X; δ) with respect to its irrelevant maximal ideal is again normal (since the asso-

ciated graded ring of R with respect to its maximal ideal, namely R, is normal; cf. also

(3.13)). In general one has only an injection Cl(R) ’ Cl(R). The ring under consid-

eration satis¬es the Serre condition (R2 ) (cf. (6.12)) and is Cohen-Macaulay; therefore

Cl(R) ’ Cl(R) is even an isomorphism here ([Fl], (1.5)).

(b) In the preceding section we have described the rings G(X; γ) and R(X; δ) as rings

of (absolute) invariants of linear algebraic groups acting on polynomial rings A over B.

Assume that B = K is an algebraically closed ¬eld. Then it is well-known that (under

hypotheses satis¬ed for our objects) R = AG is a factorial domain if G is connected

and has a trivial character group G— (cf. [Kr], p. 100, Satz 2 and p. 101, Bemerkungen).

It follows that G(X) and the K-algebra of SL(r, K)-invariants of K[Y, Z] are factorial

domains (cf. (7.6) and (7.7)).

The main result of [Mg] connects G— and Cl(AG ) under much more general circum-

stances: G is supposed to be a connected algebraic group acting rationally on a normal

a¬ne K-algebra A. Suppose for simplicity that A is factorial. Then, by [Mg], Theorem 6,

Cl(AG ) is a homomorphic image of G— . The reader may investigate G(X; γ) and R(X; δ)

from this point of view. It is clear that one can only expect an isomorphism Cl(AG ) ∼ G—

=

if G is taken as “small” as possible; in this regard (7.20) may be useful. ”

With the hypotheses and notations of the preceding corollary, the ideal generated

by an r-minor (of the matrix of residue classes) is the intersection of the prime ideals

97

B. The Canonical Class of Rr+1 (X)

P and Q generated by the r-minors of its rows and columns resp. On the contrary the

lower order minors are prime elements.

(8.6) Proposition. Let B be an (arbitrary) integral domain. Then for s < r an

s-minor of the matrix of residue classes is a prime element of Rr+1 (X).

We outline the proof, leaving the details to the reader™s scrutiny. It is enough

to consider the cases in which B = Z or B is a ¬eld: The case B = Z provides the

¬‚atness argument needed for (3.12). For an inductive reasoning let s = 1 ¬rst, R =

Rr+1 (X). Because of (2.4) and (8.4) the natural epimorphism Cl(R) ’ Cl(R[x’1 ]) is

mn

an isomorphism, whence xmn , being irreducible, is prime. Let s > 1 now. Since xmn is

a prime element and R is a domain,

δ = [m ’ s + 1, . . . , m|n ’ s + 1, . . . , n], xmn

is an R-sequence. In order to prove that R = R/δR is a domain, it is now enough to

show that R[(xmn )’1 ] is a domain, and this follows from (2.4) in conjunction with the

inductive hypothesis.

B. The Canonical Class of Rr+1 (X)

A canonical module (cf. 16.C) of a normal Cohen-Macaulay domain S is a re¬‚exive S-

module of rank 1, therefore (isomorphic to) a divisorial ideal and completely determined

by its class which is called a canonical class (cf. [Fs], § 12 or [HK], 7. Vortrag). We want

to compute the canonical classes of G(X; γ) and R(X; δ), and to decide which of these

rings are Gorenstein rings: S is Gorenstein if and only if S is a canonical module of itself.

Let B be a Cohen-Macaulay ring possessing a canonical module ωB , R a generically

perfect residue class ring of a polynomial ring Z[X], and S = R —Z B. From (3.6) we

know that

ωS = ω R — Z ωB

is a canonical module of S. For the rings under consideration this formula can be re¬ned.

(8.7) Proposition. Let R be one of the rings G(X; γ) or R(X; δ) de¬ned over

Z. Let B be a normal Cohen-Macaulay domain having a canonical module ω B , and

S = R —Z B.

(a) The modules ωB —S and ωR —S are divisorial ideals. The class of ωR —S is in the free

direct summand F of Cl(S) appearing in (8.1) and (8.3) resp. Under the isomorphism

F ’ Cl(S — L), L the ¬eld of fractions of B, it is mapped to cl(ωR — L).

(b) An element of Cl(S) represents a canonical module of S if and only if it has the form

cl(ωB — S) + cl(ωR — S) for a canonical module ωB of B, so is unique up to the choice

of ωB .

Proof: (a) ωB — S is a divisorial ideal, since the extension B ’ S is ¬‚at. Let

A be a polynomial ring over Z of which R is a residue class ring. By virtue of (3.6)

ωR is generically perfect of the same grade as R. So ωR —R S = ωR —Z B is a perfect

A —Z B-module, and one has depth ωR — SP = depth SP for every prime ideal P of S. It

has rank 1, as can be seen by passing from R to S[µ’1 ] through R[µ’1 ], where µ is the

98 8. The Divisor Class Group and the Canonical Class

minimal element of the poset de¬ning R:

’’ ’ ’ ’

’ ’ ’ ’’

R S

¦ ¦

¦ ¦

R[µ’1 ] ’ ’ ’ ’ ’ S[µ’1 ].

’ ’ ’ ’’

Being locally a maximal Cohen-Macaulay module, it is torsionfree, so isomorphic with

an ideal I of S, and S/I, if = 0, is a Cohen-Macaulay ring of dimension dim S ’ 1. Being

equal to S or unmixed of height 1, I is divisorial. Since ωR — S[µ’1 ] is free of rank 1, the

class of ωR — S is in the kernel of Cl(S) ’ Cl(S[µ’1 ]), thus in F . The last statement is

obvious.

(b) We have learnt that ωR —Z ωB = (ωR — S) —S (ωB — S) is a canonical module of

S, and the class of the tensor product is the sum of the classes. Thus a class cl(ωB — S) +

cl(ωR — S) represents a canonical module of S. Conversely let a class c = c1 + c2 , c1 ∈

Cl(B), c2 in the free direct summand, represent a canonical module. The class c1 contains

the extension of a divisorial ideal I of B, whose extension to S[µ’1 ] becomes a canonical

module of S[µ’1 ]. Using the characterization in [HK], Satz 6.1,d) (for example) and the

properties of the extension B ’ S[µ’1 ], it is easy to show that I must be a canonical

module of B. In order to isolate c2 , we consider the extension S ’ S —B L = R —Z L.

An extension of a divisorial ideal in the class c2 then is a canonical module of R —Z L,

and so is ωR —Z L. The passage from R —Z L to its localization with respect to the

irrelevant maximal ideal induces an isomorphism of class groups ([Fs], Corollary 10.3).

Since the canonical module of a local ring is uniquely determined, we ¬nally conclude

c2 = cl(ωR — S). ”

As we shall see, the general case R = G(X; γ) or R = R(X; δ) can be reduced in a

strikingly simple manner to the case R = R2 (X). We start by noting the result for the

case R = Rr+1 (X):

(8.8) Theorem. Let B be a normal Cohen-Macaulay domain with a canonical mod-

ule ωB , X an m — n matrix of indeterminates, 0 < r < min(m, n), R = Rr+1 (X). Then

a divisorial ideal ω with class

cl(ω) = cl(ωB R) + m cl(P ) + n cl(Q)

is a canonical module of R. (As in (8.4), P is the prime ideal generated by the r-minors

of any r rows, Q the prime ideal generated by the r-minors of any r columns.) Every

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

Since cl(P ) = ’ cl(Q), the di¬erence of m and n determines the class of ω already.

(8.9) Corollary. Let B be an (arbitrary) noetherian ring. Then Rr+1 (X), 0 < r <

min(m, n), is Gorenstein if and only if B is a Gorenstein ring and m = n.

Proof: Along ¬‚at local extensions the Gorenstein property behaves like the Cohen-

Macaulay property: For a prime ideal I of R = Rr+1 (X) and J = B © I the localization

RI is Gorenstein if and only if both BJ and (BJ /JBJ )—RI have this property (cf. [Wt]).

As usual, this argument reduces the general case to the one in which B is a ¬eld, and for

which the corollary is a direct consequence of the theorem. ”

99

B. The Canonical Class of Rr+1 (X)

The proof of (8.8) in the crucial case r = 1 is an induction on m + n. Because

of (8.7) it is enough to treat the case in which B = K is a ¬eld. For the minimal choice

m = n = 2 of m and n, (8.8) is true: R is the residue class ring of K[X] modulo a

principal ideal, so Gorenstein. In the inductive step we want to descend from R to the

residue class ring modulo the elements in the last row or column of the matrix, thereby

passing to a “smaller” ring.

(8.10) Lemma. Let A be a normal Cohen-Macaulay domain, and I a prime ideal of

height 1 in A such that A/I is again a normal Cohen-Macaulay domain. Let P 1 , . . . , Pu

be prime ideals of height 1 in A and suppose that the class of I and the class of a canonical

module ω of A have representations

u u

cl(I) = si cl(Pi ) and cl(ω) = ri cl(Pi ).

i=1 i=1

Assume further that:

(i) ri ’ si ≥ 0 for i = 1, . . . , u.

(r ’s )

(ii) Ann(Pi i i /Piri ’si ) ‚ Pi + I for i = 1, . . . , u.

(iii) The ideals (Pi + I)/I are distinct prime ideals of height 1 in A/I. Then A/I has a

canonical module ωA/I with

u

(ri ’ si ) cl((Pi + I)/I).

cl(ωA/I ) =

i=1

We ¬rst ¬nish the proof of (8.8). Without restriction we may assume that m ≥ n,

so m ≥ 3. Let P be the prime ideal generated by the elements in the ¬rst row, Q the

prime ideal corresponding to the ¬rst column, and I being generated by the elements

of the last row. Whatever ω is, its class can be written cl(ω) = p cl(P ) + q cl(Q) with

p, q > 0, since cl(P ) and cl(Q) generate Cl(R) and cl(P ) + cl(Q) = 0. Now cl(I) = cl(P ).

The lemma gives

cl(ωA/I ) = (p ’ 1) cl((P + I)/I) + q cl((Q + I)/I)

= (m ’ 1) cl((P + I)/I) + n cl((Q + I)/I)

by induction, and p ’ q = m ’ n as desired. The hypotheses of the lemma are indeed

satis¬ed: Except for (ii), everything is trivial (for (iii) note that m ≥ 3). Condition (ii)

holds, since P and Q become principal when a matrix element not occuring in P , Q, or

I is inverted (one may take [2|n]).

Let now r > 1. The reader may argue inductively, using the isomorphism in (2.4)

which allows one to pass from the data (m, n, r) to the data (m ’ 1, n ’ 1, r ’ 1) after

the inversion of [m|n]. (8.8) is again covered by (8.14). ”

Proof of (8.10): A canonical module ωA/I is given by Ext1 (A/I, ω). So we have

A

an exact sequence

0 ’’ HomA (A, ω) ’’ HomA (I, ω) ’’ ωA/I ’’ 0.

100 8. The Divisor Class Group and the Canonical Class

HomA (I, ω) is the quotient ω : I within the ¬eld of fractions of A, and

u

cl(ω : I) = cl(ω) ’ cl(I) = (ri ’ si ) cl(Pi ).

i=1

(t ) (t )

Let ti = ri ’ si . Then HomA (I, ω) is isomorphic to J = P1 1 © · · · © Pu u . Since all the

exponents are non-negative, J ‚ A, and we have an exact sequence

0 ’’ J ’’ J ’’ ωA/I ’’ 0.

The ideal J is isomorphic to ω and contains IJ. Since I is a prime ideal di¬erent from

P1 , . . . , Pu , the smallest divisorial ideal containing IJ is I © J ‚ J. On the other hand

no proper quotient of J/I © J can be (isomorphic to) a nonzero ideal in A/I. We have

J = I © J, and must prove that the equality in

(t1 ) (tu )

J/(I © J) ∼ (J + I)/I = P 1 © · · · © P u

=

holds, where P i = (Pi + I)/I. Hypothesis (ii) implies that

(ti )

(P (ti ) + I)/I ‚ P .

Therefore one has a chain of inclusions

t1 tu (t1 ) (tu )

P 1 . . . P u ‚ (J + I)/I ‚ P 1 © ··· © Pu ,

t1 tu

the last ideal being the smallest divisorial ideal containing P 1 . . . P u , and the desired

equality holds since (J + I)/I ∼ ωA/I is likewise divisorial. ”

=

C. The General Case

Next we treat the case R = G(X; γ) for which we may again assume that B is a

¬eld (cf. (8.7)). The class of the canonical module has a representation

t

cl(ω) = κi cl(J(x; ζi ))

i=0

where t = s if am < n, t = s ’ 1 if am = n, γ having s + 1 blocks, the ζi being the upper

neighbours of γ. Since cl(J(x; ζi )) = 0, the di¬erences

κi’1 ’ κi , i = 1, . . . , t,

determine cl(ω) uniquely. In (6.8) we introduced the elements

σi = [β0 , . . . , βi’2 , (aki’1 +1 , . . . , aki ’1 ), (aki +1 , . . . , aki+1 , aki+1 + 1), βi+1 , . . . , βs ]

101

C. The General Case

and we noted that the localizations of R with respect to the prime ideals J(x; σi ) are not

’1

factorial. All the more, R[σi ] is not factorial, and from the minimal primes of γ only

J(x; ζi’1 ) and J(x; ζi )

’1

survive in R[σi ], since σi is contained in all the other ones (σi ≥ ζj for j = i ’ 1, i).

’1

Let S = R[σi ]. Then