for all g ∈ G, x ∈ K p ,

f (x) = f (g(x))

and are therefore called invariants. The ¬rst main problem is to determine a ¬nite

set f1 , . . . , fq of “basic” invariants, i.e. invariants f1 , . . . , fq such that every invariant is

a polynomial in f1 , . . . , fq . (A paradigm for the solution of the ¬rst main problem is

Newton™s theorem on symmetric functions.) The second main problem is solved if one

has found all the relations of f1 , . . . , fq , a relation being a polynomial h in q variables

such that h(f1 , . . . , fq ) = 0.

In modern language G is a linear algebraic group over a (algebraically closed) ¬eld

K, and G operates on a ¬nite dimensional K-vector space V via a morphism or an anti-

morphism G ’ GL(V ) of linear algebraic groups (cf. [Hm], [Fo], [Kr], [MF]; the survey

[Ho.8] su¬ces for our purpose). Such a morphism is called a rational representation of G.

It makes V a G-module; more generally an arbitrary vector space W is a G-module if it is

the union of an ascending chain of ¬nite dimensional G-modules. The ring of polynomial

functions on V is the symmetric algebra S(V — ). G acts on V — via the composition of

the representation G ’ GL(V ) and the natural anti-isomorphism GL(V ) ’ GL(V — ),

sending each automorphism of V to its dual. Then S(V — ) becomes a G-module after

the natural extension of the action on V — to an action of S(V — ): every automorphism of

V — induces an algebra automorphism of S(V — ). In (7.6) and (7.7) we have let SL(r, B)

and GL(r, B) operate directly on the space of 1-forms of a symmetric algebra of a free

module over B. These theorems comprise the solutions of the ¬rst main problem for the

actions under consideration. In the situation of (7.6),(a) and (7.7) the solution of the

second main problem is also well-known to us.

From a geometric view-point V is the a¬ne n-space over K, A = S(V — ) is its

coordinate ring, G acts on the a¬ne variety V . The ring AG of invariants is the subalgebra

of functions constant on the orbits of the action of G. The ¬rst main problem has a

solution if and only if AG can be considered the coordinate ring of an a¬ne variety V .

Then the surjection V ’ V has a universal property: every morphism de¬ned on V

which is constant on the orbits, factors through V . Thus V comes as close as possible to

being the quotient of V modulo G. It is therefore called the algebraic quotient of V with

respect to G, whereas the geometric quotient may not exist: there may be nonclosed

orbits.

All this explains the signi¬cance of (7.6) and (7.7) for invariant theory. Conversely

we can use the results of (algebraic and geometric) invariant theory to gain further

knowledge about our objects. This is mainly possible in characteristic zero because

the groups GL(n, K) and SL(n, K) (and direct products of them) are linearly reductive

then, and very strong theorems hold for invariants of linearly reductive groups. Linear

reductivity can be characterized by each of the following properties:

(i) Every (¬nite dimensional) G-module is completely reducible, i.e. the direct sum of

simple G-modules (motivating the name “reductive”).

81

D. Remarks on Invariant Theory

(ii) In every (¬nite dimensional) G-module V the G-submodule V G = {x ∈ V : g(x) = x

for all g ∈ G} of invariants has a (for V G necessarily unique) G-complement. (The

G-homomorphism ρ : V ’ V G , ρ|V G = id, is called the Reynolds operator .)

(iii) For every surjective G-homomorphism V ’ W the induced map V G ’ W G is

surjective, too.

We now assume that A is a ¬nitely generated K-algebra and a G-module such that

the elements of G act as K-algebra automorphisms. Then (a) is quite evident:

(a) Let A be a domain. Then AG is the intersection of its own ¬eld of fractions with A.

In particular AG is normal if A is normal (and AG noetherian).

Suppose furthermore that G is linearly reductive. Then the ¬rst main problem always

has a solution:

(b) If A is noetherian, then AG is noetherian; if A is a ¬nitely generated K-algebra, then

AG is ¬nitely generated.

We should point out that (b) already holds under the weaker assumption that G is

reductive; cf. [Ho.8] for this notion. In characteristic 0 reductivity and linear reductivity

are equivalent, whereas in positive characteristic the groups GL(r, K) and SL(r, K) are

not linearly reductive if r ≥ 2. Property (ii) of linearly reductive groups implies:

(c) As an AG -module A splits as A = AG • C, C being the G-complement of AG ; the

Reynolds operator is an AG -homomorphism. (Cf. [Fo], p. 156, Lemma 5.4 or (7.22)

below).

The deep properties (d) and (e) of linearly reductive groups are given by the theorem

of Hochster-Roberts [HR], [Ke.5] and the even stronger and more general theorem of

Boutot [Bt] resp.:

(d) If A is regular, then AG is Cohen-Macaulay.

(e) If char K = 0 and A has rational singularities, then AG has rational singularities.

We cannot discuss the notion of rational singularity here and refer the reader to [KKMS]

and [BS]. If A has rational singularities, then it is Cohen-Macaulay. We shall see below

that G(X; γ) and R(X; δ) are invariants of groups acting on polynomial rings, the groups

being reductive in characteristic zero. Thus we conclude:

(7.11) Theorem. Let B = K be an algebraically closed ¬eld of characteristic zero.

Then the a¬ne and projective varieties corresponding to G(X; γ) and R(X; δ) have ra-

tional singularities for all γ ∈ “(X), δ ∈ ∆(X).

Even (a) above contains some new information about G(X), say. An application of

(c) is discussed in the following remark:

(7.12) Remark. In (5.21) we have given an upper bound for the arithmetical rank

of the ideals It (X) : ara It (X) ¤ mn ’ t2 + 1. Here we want to demonstrate that this

bound is sharp in case t = m ¤ n if B admits a homomorphism B ’ K to a ¬eld

of characteristic zero. Evidently we may then assume that B = K. A lower bound of

ara Im (X) is supplied by the cohomological dimension

min{i : Hi (K[X]) = 0}, I = Im (X),

I

cf. [Ha.1], p. 414, Example 2. Here Hi (. . . ) is the cohomology with support in I. Let

I

J = I © G(X). Then, for the G(X)-algebra K[X]

Hi (K[X]) = Hi (K[X]) = Hi (G(X)) • Hi (C),

I J J J

82 7. Generic Points and Invariant Theory

C being the SL(m, K)-complement of G(X) in K[X] (observe that I = JK[X]). For

i = mn ’ m2 + 1 we have Hi (G(X)) = 0, since i = dim G(X) (cf. [HK], p. 37“39).

J

Warning: Apart from trivial cases, C is not the K-vector space complement C appearing

in the proof of (7.7). It seems hopeless to compute C explicitely.

The preceding argument can neither be generalized to the case t < min(m, n)

(cf. (10.16)), nor be applied in characteristic p > 0 if t = m < n: By virtue of [PS], Propo-

sition (4.1), p. 110 one has Hi (K[X]) = 0 for all i ≥ ht I, in particular for i = mn’m2 +1,

I

and the argument based on cohomological dimension breaks down. Another consequence:

A Reynolds operator does not exist! ”

(7.13) Remark. Let K be an algebraically closed ¬eld and consider the action of

SL(m, K) on the mn-dimensional a¬ne space V of matrices as in (7.7). It follows directly

from Theorem (1.2) that the points = 0 in the a¬ne variety G with coordinate ring

n

G(X) (embedded into AN , N = m ) correspond bijectively to the orbits of SL(m, K)

containing a matrix of rank m. This fact indicates that G comes close to being the

algebraic quotient of V with respect to the action of SL(m, K), and one is justi¬ed

to ask whether Theorem (1.2) does already prove (7.7). It does so, provided one has

shown the normality of G(X), because of the following criterion (cf. [Kr], 3.4, p. 105

for the statement in characteristic 0): Let V be an irreducible a¬ne algebraic variety,

G a reductive group acting on V , and π : V ’’ W a surjective morphism from V to a

normal a¬ne variety W , which is constant on the orbits. Suppose that W contains a

dense subset U such that π ’1 (v) contains exactly one closed orbit for every v ∈ U . Then

W is the algebraic quotient of V with respect to G. (The reductivity of G guarantees

the a priori existence of a quotient, and the normality of W then allows one to conclude

that it is isomorphic to W .) It is not di¬cult to prove (7.6) by means of this criterion

(cf. [Kr], 4.1 for GL(r, K)); the normality of the algebra A in (7.6),(b) will be proved in

(9.21) independently. ”

We now proceed to give invariant-theoretic descriptions of the rings G(X; γ) and

R(X; δ) in general. The arguments needed consist of iterative applications of the ideas

underlying the proofs of (7.6) and (7.7). We start by giving the “classical” generic point

for G(X; γ).

E. The Classical Generic Point for G(X; γ)

Let • : B[X] ’ S be a B-algebra homomorphism, U the image of the matrix X in

S. Then the induced homomorphism G(X) ’ S factors through G(X; γ) if and only if

•(δ) = 0 for all δ ≥ γ = [a1 , . . . , am ]. So we can hope to ¬nd a generic point for G(X; γ)

if we choose for U a “generic” matrix for which the minors δ ≥ γ vanish. This is certainly

true, if Ik (¬rst ak ’ 1 columns of U ) = 0 for k = 1, . . . , m. Thus let Uγ be the following

matrix whose entries Uij are indeterminates over B:

«

0 ··· ··· ··· ··· ···

0 U1a1 U1a2 ’1 U1a2 U1a3 ’1 U1am U1n

··· ···

0 0 U2a2 U2a3 ’1

¬ ·

¬ .·

.

¬ .·

.

¬ .·

··· ···

0 0 .

¬. ·

. . . . .

.

. . . . .

. . . . . .

0 ··· ··· ··· ··· ···

0 0 0 0 0 Umam Umn

83

E. The Classical Generic Point for G(X; γ)

(7.14) Theorem. (a) The B-algebra homomorphism B[X] ’ B[Uγ ], X ’ Uγ ,

induces an embedding • : G(X; γ) ’ B[Uγ ], thus an isomorphism G(X; γ) ∼ G(Uγ ).

=

(b) The embedding G(X) ’ B[X] is a generic point, as is • for every γ ∈ “(X).

We give two proofs of part (a), the second one being contained in Remark (7.16).

The ¬rst proof is more “advanced”: we use that γ is not a zero-divisor of G(X; γ). It

runs like that of (7.2): we only need to factor the embedding G(X; γ) ’ G(X; γ)[γ ’1 ]

through •. In order to ¬nd such a factorization we have to construct a matrix V over

G(X; γ)[γ ’1 ] which has the same shape as Uγ and whose m-minors are the elements

δ ∈ “(X; γ). Such a problem we have faced already: the construction of a subspace (or

matrix) with given Pl¨cker coordinates is the last step in the proof of Theorem (1.2)!

u

(7.15) Lemma. Let S be a B-algebra, and ψ : G(X; γ) ’ S a B-algebra homomor-

phism. Suppose that ψ(γ) is a unit in S. Then there is a matrix V of the same shape as

Uγ such that ψ(δ) is the minor of V with the same columns as δ for all δ ∈ “(X; γ).

Proof: The key role plays the set Ψ de¬ned in (6.1):

δ = [d1 , . . . , dm ] ∈ “(X; γ): ai ∈ [d1 , . . . , dm ] for at most one i .

Ψ= /

First we let all those entries of V be zero which correspond to zero entries of U γ . The

remaining entries at positions (k, l) are de¬ned as follows: Remove ak from {a1 , . . . , am }

and replace it by l. If l = aj for some j = k, the entry is zero. Otherwise

{a1 , . . . , ak’1 , l, ak+1 , . . . , am }

de¬nes, after arrangement in ascending order, an element δ of Ψ. Then we take

σ(l, a2 , . . . , am )ψ(δ) if k = 1,

σ(a1 , . . . , ak’1 , l, ak+1 , . . . , am )ψ(δ)ψ(γ)’1 if k = 1,

as the entry of V . One checks that the minor with the same columns as δ equals ψ(δ)

for all δ ∈ Ψ.

We now have two homomorphisms G(X; γ) ’ S: ¬rst ψ, and secondly the com-

position of • : G(X; γ) ’ B[Uγ ] with the homomorphism B[Uγ ] ’ S arising from the

substitution Uγ ’ V . Since they coincide on Ψ, they are equal, cf. (6.2), and the second

homomorphism sends δ ∈ “(X; γ) to the minor of V with the same columns as δ. ”

For the proof of (7.14),(b) we ¬rst show that G(X) ’’ B[X] is a generic point.

Let ψ : G(X) ’’ L be a homomorphism to a ¬eld L. If ψ(δ) = 0 for all δ ∈ “(X), then

ψ factors through B[X] for trivial reasons. Otherwise we may assume on the grounds of

symmetry that ψ([1, . . . , m]) = 0, and then (7.15) settles the problem.

Let now γ ∈ “(X) be arbitrary, and ψ : G(X; γ) ’’ L again a homomorphism to

a ¬eld. By what has just been shown, there is a matrix V such that the minor of V

with the same columns as δ is ψ(δ) for all δ ≥ γ, and zero otherwise. Over a ¬eld such

a matrix can be transformed into one of shape Uγ by an application of elementary row

operations. ”

84 7. Generic Points and Invariant Theory

(7.16) Remark. The second proof of (7.14),(a) is given mainly because it provides

a new (and perhaps simpler) demonstration of the linear independence of the standard

monomials in G(X). We choose new notations: Let G(X) be the residue class ring of the

polynomial ring B[Tγ : γ ∈ “(X)] modulo the ideal generated by the Pl¨cker relations,

u

and G(X; γ) the residue class ring of G(X) with respect to the ideal generated by the

residue classes of the Tδ , δ ≥ γ. Then we have a homomorphism G(X; γ) ’ G(Uγ )

since the maximal minors of Uγ satisfy the de¬ning relations of G(X; γ). Furthermore it

follows as in the proof of (4.1) that G(X; γ) is generated as a B-module by the standard

monomials in the residue classes of Tδ , δ ≥ γ. In order to show that the homomorphism

G(X; γ) ’ G(Uγ ) is an isomorphism it is enough to prove that the standard monomi-

als in the maximal minors δ of Uγ , δ ≥ γ, are linearly independent! This is done by

descending induction in the partially ordered set “(X). Suppose 0 = bµ µ where µ

runs through these standard monomials, bµ ∈ B, bµ = 0 for all but a ¬nite number of

standard monomials. Let δ > γ. The matrix Uδ has nonzero entries only where Uγ

has indeterminate entries. So we have a well de¬ned substitution Uγ ’ Uδ inducing a

commutative diagram

G(X; γ) ’ ’ ’ ’ ’ G(X; δ)

’ ’ ’ ’’

¦ ¦

¦ ¦

G(Uγ ) ’ ’ ’ ’ ’ G(Uδ ).

’ ’ ’ ’’

By induction hypothesis we conclude bµ = 0 for all µ not containing γ as a factor. But

γ ∈ “(Uγ ) is a product of indeterminates, so certainly not a zero-divisor, and this implies

at once that bµ = 0 for all µ after a second application of the inductive hypothesis:

(7.14),(a) is proved again. ”

F. G(X; γ) and R(X; δ) as Rings of Invariants

Multiplication of Uγ by an element of the special linear group does not de¬ne an

automorphism of B[Uγ ] in general. In order to represent G(X; γ) as a ring of invariants

we must “symmetrize” the matrix Uγ ¬rst. Let a0 = 1, am+1 = n + 1, and

«

Z1ak · · · Z1ak+1 ’1

¬ ·

¬. ·

.

Zk = ¬ . ·,

.

¬. . ·

···

Zkak Zkak+1 ’1

k = 0, . . . , m, matrices of indeterminates (as they appear as submatrices of Uγ ). For

k = 0, . . . , m ’ 1 we choose (k + 1) — k-matrices Zk of indeterminates such that the

entries of all the matrices Zk , Zk are algebraically independent over B. Then we let

Zγ = Z m’1 . . . Z 0 Z0 Z m’1 . . . Z 1 Z1 . . . Z m’1 Zm’1 Zm

by iuxtaposing the products Z m’1 . . . Z k Zk as indicated to form the m — n matrix Zγ .

It is clear that

Ik (¬rst ak ’ 1 columns of Zγ ) = 0

(—)

85

F. G(X; γ) and R(X; δ) as Rings of Invariants

for k = 1, . . . , m. Therefore the substitution X ’ Zγ induces a homomorphism