χ : G(X; γ) ’’ G(Zγ ) ‚ B[Zγ ],

Zγ denoting the collection of all the entries of the Zk , Z k . It also induces a homomorphism

ω : R(X; [1, . . . , m|a1 , . . . , am ]) ’’ B[Zγ ] ‚ B[Zγ ].

(7.17) Proposition. The homomorphisms

χ : G(X; γ) ’’ B[Zγ ] ω : R(X; [1, . . . , m|a1 , . . . , am ]) ’’ B[Zγ ]

and

are embeddings and generic points.

Proof: Substituting the corresponding submatrix of Uγ for Zk and the (k + 1) — k

matrix «

1 0 ··· 0

.

.. ..

¬ . .·

.

¬0 .·

¬. . ·

..

¬. . 0·

..

.

¬ ·

0 ··· 0 1

0 ··· ··· 0

for Z k , one factors the embedding G(X; γ) ’ B[Uγ ] through χ to get the claim for

χ. As soon as γ is invertible, or over a ¬eld, a matrix to which X (considered over

R(X; γ)) specializes, can be “decomposed” in the same way as Zγ . (It is of course only

a problem of elementary linear algebra to ¬nd such a decomposition; for special reasons

we shall however have to outline the construction of a decomposition in Section 12, proof

of (12.3).) ”

We introduce group actions on B[Zγ ]. Let

m’1

H= GL(k, B).

k=0

The group GL(k, B) acts on B[Zγ ] by the substitution

’’ T Zk ,

Zk

’’ Z k T ’1 ,

Zk

Z k’1 ’’ T Z k’1 (k > 0),

T ∈ GL(k, B). These actions for various k commute with each other; so they de¬ne an

action of H on B[Zγ ]. Finally we let the group SL(m, B) act by

Z m’1 ’’ T Z m’1 ,

’’ T Zm ,

Zm

giving an action of H = SL(m, B) — H on B[Zγ ].

86 7. Generic Points and Invariant Theory

(7.18) Theorem. (a) B[Zγ ] ∼ R(X; [1, . . . , m|a1 , . . . , am ]) is the ring of absolute

=

H-invariants of B[Zγ ].

(b) G(Zγ ) ∼ G(X; γ) is the ring of absolute H-invariants of B[Zγ ].

=

For this theorem one of course extends the de¬nition of absolute invariants given

above. It is obvious that the Propositions (7.3), (7.4), and (7.5) hold again after the

necessary modi¬cations. For a quick proof of (7.18) in characteristic 0 see (7.21) below.

Proof: Part (a) is proved by induction. It is evident for m = 1. Let m > 1. The

indeterminates in Zm are not a¬ected by the action of H. Therefore it is enough to show

that the entries of

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

where γ = [a1 , . . . , am’1 ], generate the ring of absolute invariants after restricting the

action of H to the polynomial ring in the entries of Z k , Zk , k = 0, . . . , m ’ 1. Let

m’2

H = k=0 GL(k, B). By induction the ring of absolute invariants of H is B[Zγ ] and

the action of H can be restricted to B[Z m’1 , Zγ ]. Therefore it is now su¬cient to show

that the ring of absolute invariants of B[Z m’1 , Zγ ] under the action of GL(m’1, B) is

B[Z m’1 Zγ ].

The rest of the proof is mainly a repetition of the arguments given for (7.6),(a).

First we may enlarge Zm’1 by adding a further column of indeterminates at the right to

reach a situation in which the number of columns of Zγ exceeds am’1 . One now inverts

the minor δ = [1, . . . , m ’ 1|a1 , . . . , am’1 ] of Z m’1 Zγ and applies the substitution trick

with Y Z replaced by the product of the submatrix consisting of the ¬rst m ’ 1 rows of

Z m’1 with the submatrix consisting of columns a1 , . . . , am’1 of Zγ . Then one is left to

prove that

δ B[Z m’1 Zγ ] = δ B[Z m’1 , Zγ ] © B[Z m’1 Zγ ],

and this can also be done in analogy with (7.6), this time [1, . . . , m’1|a1 +1, . . . , am’1 +1]

being inverted instead of [1, . . . , r ’ 1, r + 1|1, . . . , r ’ 1, r + 1]. The details can be left to

the reader.

For part (b) we write B[Zγ ] in the form B[Z m’1 , Zm ], B = B[remaining variables].

Every absolute SL-invariant of B[Z m’1 , Zm ] has absolutely invariant homogeneous com-

ponents f which satisfy the equation

T (f ) = (det T )j f

for every T ∈ GL(m, S), S a B-algebra, j = (degf )/m. This implies that an invariant

f ∈ B[Zγ ] is in G(Zγ )[γ ’1 ] (γ taken as a minor of Zγ ), and the equation

γG(Zγ ) = γB[Zγ ] © G(Zγ ),

which ¬nishes the proof, is demonstrated as in the proof of (7.7): B[Zγ ] has a standard

basis inherited from R(X; [1, . . . , m|a1 , . . . , am ]). ”

It remains to consider the general case of R(X; δ), δ = [b1 , . . . , br |c1 , . . . , cr ]. Let

γ1 = [b1 , . . . , br ], γ2 = [c1 , . . . , cr ], and construct matrices Zγ1 , Zγ2 as above, Yγ1 as the

87

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

transpose of Zγ1 . The collection of indeterminates needed for Yγ1 is denoted by Yγ1 , that

r’1

for Zγ2 by Zγ2 . Let H = k=0 GL(k, B). Then H — H acts on B[Yγ1 , Zγ2 ], extending

the action of the ¬rst component on B[Yγ1 ] and the action of the second one on B[Zγ2 ].

Furthermore we let GL(r, B) operate by the substitution

Y r’1 ’’ Y r’1 T ’1 , Yr ’’ Yr T ’1 ,

Z r’1 ’’ T Z r’1 , Zr ’’ T Zr .

This action commutes with that of H —H, resulting in an action of G = H —GL(r, B)—H.

(7.19) Theorem. The substitution X ’’ Yγ1 Zγ2 induces an embedding

R(X; δ) ’’ B[Yγ1 , Zγ2 ]

which is a generic point. The image B[Yγ1 Zγ2 ] is the ring of absolute G-invariants of

B[Yγ1 , Zγ2 ].

The proof may be left to the reader. Again one should note that the attribute

“absolute” is super¬‚uous if B is a domain containing in¬nitely many units.

(7.20) Remark. The groups in (7.6) and (7.19) for δ = [1, . . . , r|1, . . . , r] are dif-

ferent, as are those appearing in (7.7) and (7.18) for γ = [1, . . . , m]. In fact one can

“minimize” the construction for γ by ¬rst applying (6.9),(d) and decomposing γ into its

blocks as in subsection 6.B:

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

Again we simultaneously consider the gaps

χ0 = (ak1 + 1, . . . , ak1 +1 ’ 1), . . . , χs = (am + 1, . . . , n).

Then one chooses matrices Zi , i = 1, . . . , s + 1, and Z i , i = 1, . . . , s, of sizes

ki — (|βi’1 | + |χi’1 |) and ki+1 — ki resp.,

and obtains an analogue of (7.17) for the substitution

X ’’ 0 Z s . . . Z 1 Z1 . . . Z s Zs Zs+1 ,

s

an analogue of (7.18),(a) for the operation of H = i=1 GL(ki , B), and an analogue of

(7.18),(b) for the operation of SL(m, B) — H . Similarly one can “minimize” (7.19). ”

(7.21) Remark. The proofs of (7.18) and (7.19) can be simpli¬ed if B = K is

a ¬eld of characteristic zero: In the inductive step of the proof of (7.18),(a) and the

proof of (b) one can directly appeal to Theorem (7.6),(a) and Corollary (7.7) resp.: Take

matrices W and Y with indeterminate entries and of the formats Z m’1 and Zγ resp.

Then the action of GL(m’1, B) on B[Z m’1 , Zγ ] is induced by that on B[W, Y ] via the

substitution W ’ Z m’1 , Y ’ Zγ , and the claim follows immediately from (7.6),(a) by

the reductivity of GL(m’1, B), cf. property (iii) of linearly reductive groups. Similarly

one concludes (7.18),(b) directly from (7.7). ”

88 7. Generic Points and Invariant Theory

G. The Depth of Modules of Invariants

Certain modules over rings of invariants arise as modules of invariants, and this fact

can be used to study some of their properties. For simplicity we assume in this subsection

(except for (7.25)) that B = K is a ¬eld.

Let G be a linear algebraic group over K which acts on a K-algebra S such that S

is a G-module. Furthermore we consider an S-G-module M , i.e. an S-module M which

is simultaneously a G-module such that

for all g ∈ G, a ∈ S, x ∈ M.

g(ax) = g(a)g(x)

In particular S itself is an S-G-module. Obviously the module

M G = {x ∈ M : g(x) = x for all g ∈ G}

of invariants is an S G -module. If G is linearly reductive (cf. D), then there is hope that

M G may be a accessible for a more detailed analysis:

(7.22) Proposition. With the notations introduced so far, suppose that S is noe-

therian, M is ¬nitely generated, and G is linearly reductive. Let ρS and ρM denote the

Reynolds operators of S and M .

(a) Ker ρM is an S G -module, so M = M G • Ker ρM is a decomposition of S G -modules,

and ρM : M ’ M G is an S G -homomorphism:

b ∈ S G , x ∈ M.

ρM (bx) = bρM (x) for all

Furthermore

a ∈ S, y ∈ M G .

ρM (ay) = ρS (a)y for all

(b) M G is a ¬nitely generated module over the noetherian ring S G .

Proof: M splits as a G-module: M = M G • C, C = Ker ρM . For the ¬rst

statement in (a) one has to prove that C is an S G -module. G being linearly reductive,

C is the sum of its irreducible G-submodules N . It is enough to show that bN ‚ C

for all b ∈ S G . Since b ∈ S G , the map N ’ bN is a G-homomorphism, hence 0 or an

isomorphism. In the ¬rst case certainly bN ‚ C, in the second bN is an irreducible

G-submodule of M on which G cannot operate trivially, for otherwise it would operate

trivially on N itself, and N ‚ M G . By construction, C is the sum of all irreducible

G-modules of M with nontrivial G-action, so bN ‚ C.

Let now a ∈ S, y ∈ M G . Write a = b + c, b = ρS (a). Then

ρM (ay) = ρM ((b + c)y) = by + ρM (cy) = ρS (a)y + ρM (cy).

So we have to verify that ρM (cy) = 0 for c ∈ Ker ρS , y ∈ M G . The argument is similar

to the one above: one takes an irreducible G-submodule T ‚ Ker ρS , and studies the

G-homomorphism T ’ T y.

For (b) it is enough to show that M G is a noetherian S G -module. Let L ‚ M G be

an S G -submodule. Then SL = L • (Ker ρS )L, (Ker ρS )L ‚ Ker ρM by virtue of (a), and

every strictly ascending chain of S G -submodules of M G gives rise to a strictly ascending

chain of S-submodules of M . ”

We are interested in the grades of ideals I ‚ S G with respect to M G . In the most

important case for us, in which the objects under consideration are graded and I is the

irrelevant maximal ideal of S G , this grade coincides with the depth of (M G )I , whence

the title of this subsection.

89

G. The Depth of Modules of Invariants

(7.23) Proposition. Under the hypotheses of the preceding proposition let I ‚ S G

be an ideal. Then

grade(I, M G ) ≥ grade(SI, M ).

Proof: M G is a direct summand of M , thus grade(I, M G ) ≥ grade(I, M ). The

proof of the equation grade(I, M ) = grade(SI, M ) is left to reader. ”

In general the estimate in (7.23) is not optimal as is demonstrated drastically by

the S-G-module S itself: then S G is a Cohen-Macaulay ring by the theorem of Hochster-

Roberts ([HR]) if S is Cohen-Macaulay, but grade SI < grade I in general. On the other

hand it is sharp sometimes, cf. the subsequent discussion of the example S = B[Y, Z],

S G = B[Y Z] ∼ Rr+1 (X).

=

Examples of S-G-modules can be constructed as follows: One chooses a ¬nite-

dimensional G-module V ; then the S-module M = V —K S becomes an S-G-module

under the G-action

g(v — a) = g(v) — g(a) for all v ∈ V, a ∈ S.

Since M is free as an S-module, the inequality in (7.23) reduces to

grade(I, M G ) ≥ grade SI.

K itself becomes a G-module via the characters χ : G ’ GL(1, K), and one can study

the G-action

for all g ∈ G, a ∈ S

gχ (a) = χ(g)g(a)

of G on S. The invariants under this action are precisely the semi-invariants of weight

χ’1 :

g(a) = χ’1 (a)a.

⇐’

gχ (a) = a

In the case of interest to us, namely S = B[Y, Z], G = GL(r, K), all the characters are

given by the powers of det, and furthermore we have already computed the module D j

of semi-invariants of weight detj :

Dj = B[Y Z]{δ1 . . . δj : δi ∈ “(Z)} if j ≥ 0,

Dj = B[Y Z]{γ1 . . . γj : γi ∈ “(Y )} if j ¤ 0,

as follows immediately from (7.6),(b). Let γ ∈ “(Y ), the rows of γ being a1 , . . . , ar .

Then for all j ≥ 0

Dj ∼ P j ,

γ j Dj = P j , so =

P being the ideal generated by the r-minors of the rows a1 , . . . , ar of Y Z in B[Y Z] ∼

=

∼ Qj , Q being the ideal generated by the r-minors of

Rr+1 (X). Similarly one has D’j =

any r columns. We formulate the ¬nal result in terms of Rr+1 (X).

90 7. Generic Points and Invariant Theory

(7.24) Proposition. Let B = K be a ¬eld of characteristic 0, X an m—n matrix of

indeterminates over K, m ¤ n. Let J and r be given as follows: (i) r = m, J = Im (X),

or (ii) r < m, J the ideal generated by the r-minors of any r rows or any r columns resp.

of the matrix x of residue classes in R = Rr+1 (X). Let furthermore I = I1 (X)R. Then

grade(I, R/J j ) ≥ mr ’ 1 n ≥ m + r,

if