The hope that B[Y Z] ∼ Rr+1 (X) is (always) the ring B[Y, Z]G of invariants under the

=

action of G is immediately disappointed: Consider B = Z, m = n = r = 1. This failure

is however caused by a notion of invariant too naive to work for commutative rings in

general; a ring like Z or a ¬nite ¬eld simply has not enough units.

Definition. An element f ∈ B[Y, Z] is called an absolute GL-invariant if for every

ring homomorphism • : B ’ S the element f is mapped to an invariant of GL(r, S)

under the natural extension B[Y, Z] ’ S[Y, Z].

We shall also consider the action of the special linear groups

SL(r, B) = {T ∈ GL(r, B) : det T = 1}

on B[Y, Z] as a subgroup of GL(r, B), and absolute SL-invariants are de¬ned analogously.

The absolute invariants are just the invariants of the “general element” of GL(r, B) and

SL(r, B) resp.:

75

B. Invariants and Absolute Invariants

(7.3) Proposition. Let U be an r — r matrix of indeterminates over B, ∆ its

determinant, S1 = B[U ][∆’1 ], S2 = B[U ]/B[U ](∆ ’ 1), and denote the matrix of residue

classes in S1 by U again. Then f ∈ B[Y, Z] is an absolute GL-invariant if and only if it

is (as an element of S1 [Y, Z]) invariant under the action of U on S1 [Y, Z]. The analogous

statement holds with GL replaced by SL and S1 by S2 .

Proof: Let S be a B-algebra, u ∈ GL(r, S). Then one has a commutative diagram

’’ S1

B

ψ

S

such that ψ sends U to u. The action of u on S[Y, Z] restricts to an action on ψ(S1 )[Y, Z],

on which it is induced by the action of U on S1 [Y, Z]. Therefore invariants of U are

mapped to invariants of u. The same argument works for SL. ”

If the ring B has enough elements (units) then every invariant is already absolutely

invariant.

(7.4) Proposition. If B is a domain with in¬nitely many elements (units), then

every invariant of SL(r, B) (GL(r, B)) in B[Y, Z] is absolutely invariant.

Proof: We take S2 as in the preceding proposition. Let L be its ¬eld of fractions.

(The veri¬cation that S2 is a domain is left to the reader.) For the contention regarding

SL, it su¬ces now to show that every invariant in B[Y, Z] is invariant under the action of

SL(r, L) on L[Y, Z]. The group SL(r, L) is generated by the elementary transformations

Eij (t), t ∈ L, i = j, where Eij (t) is the identity matrix except that its entry at position

(i, j) is t. For t ∈ B we have Eij (t) ∈ SL(r, B) (‚ SL(r, L) in a natural way). It is

more than required if we show that every element of L[Y, Z] which is invariant under the

actions of the Eij (t), t ∈ B, is an invariant of SL(r, L).

Let g ∈ L[Y, Z], g = aµ µ, µ running through the monomials in the indeterminates

of Y and Z, aµ ∈ L. Then

Eij (t)(g) = pijµg (t)µ

with polynomials pijµg in one variable over L, as is easily checked. The invariance of

g = bµ µ under Eij (t), t ∈ B, is expressed by the equations

pijµg (t) = bµ

for all t ∈ B, all i, j, µ. Since the polynomial pijµg takes the value bµ in¬nitely often, it

has to be constant on L, so g is invariant under Eij (t), t ∈ L.

In order to prove the statement about GL, we consider the ¬eld of fractions L of

S1 , S1 as in (7.3). The group GL(r, L) is generated by SL(r, L) and the matrices E1 (t),

t ∈ L \ {0}, where E1 (t) is the identity matrix except having t in its position (1,1). As

above every polynomial g de¬nes functions qµg (t), sending t to the coe¬cient of E1 (t)(g)

with respect to the monomial µ. These functions are now rational functions de¬ned on

L \ {0}, each of them taking a constant value at the points t which are units in B, if g

is an invariant of GL(r, B). Therefore qµg (t) is constant then. (Expressed very brie¬‚y,

we have used that SL(r, L) and GL(r, L) are generated by one-dimensional subgroups in

which the additive group and multiplicative group resp. of B are Zariski dense.) ”

For the computation of the absolute SL-invariants of B[Y, Z] we need to know how

they behave under the action of GL(r, B).

76 7. Generic Points and Invariant Theory

(7.5) Proposition. With the notations introduced, let f ∈ B[Y, Z] be an absolute

SL-invariant which is bihomogeneous with respect to the indeterminates in Y and Z of

partial degrees d1 and d2 resp. Then d1 ’ d2 is a multiple of r (in Z), d2 ’ d1 = tr, and

T (f ) = (det T )t f

for every B-algebra S and every r — r matrix T over S.

In invariant theory this is brie¬‚y expressed as: f is an absolute semi-invariant of

weight t (or dett ).

Proof: We consider the extension B ’ S1 as in (7.3). It is enough to prove the

contention for T = U . We further extend S1 to

S = S1 [W ]/S1 [W ](∆ ’ W r ),

W a new indeterminate. Over S the matrix U factors as

U = w(w’1 U ),

w denoting the residue class of W . Note that det w ’1 U = 1. Therefore

U (f ) = (w(w’1 U ))(f ) = w(f ) = f (Y w ’1 , wZ) = wd2 ’d1 f.

S is a free S1 -module with the basis 1, . . . , w r’1 . Since U (f ) ∈ S1 = S1 · 1 ‚ S, we

conclude d2 ’ d1 ≡ 0 (r). ”

C. The Main Theorem of Invariant Theory for GL and SL

Now we are well-prepared to state and to prove the theorem which describes the

rings of the absolute GL- and SL-invariants of B[Y, Z].

(7.6) Theorem. Let B be a commutative ring, Y an m — r matrix and Z an r — n

matrix of indeterminates, r, m, n ≥ 1.

(a) The ring of absolute GL-invariants of B[Y, Z] is B[Y Z] ∼ Rr+1 (X), X being an

=

m — n matrix of indeterminates over B.

(b) The B-subalgebra A of absolute SL-invariants of B[Y, Z] is generated by the entries

of Y Z, the r-minors of Y , and the r-minors of Z.

Conditions under which the attribute “absolute” can be omitted, are given in (7.4).

For the determinantal rings mainly the case r < min(m, n) is of interest. Under invariant-

theoretic aspects this restriction should be avoided, and so we allow arbitrary values of

m, n, r in (7.6). The B-algebra A in part (b) will be analyzed to some extent in (9.21).

As an immediate corollary we obtain G(X) as a ring of invariants:

(7.7) Corollary. Let B be a commutative ring, and X an m — n matrix of indeter-

minates over B. Then G(X) is the ring of absolute invariants under the action X ’ T X

of SL(m, B) on B[X].

In fact, it is easy to see that A © B[Z] = G(Z), and so (7.7) follows from (7.6),(b).

Nevertheless we want to give a separate proof which, relative to our preparations, is very

short. Its basic idea will be applied again in the proof of (7.6),(a).

77

C. The Main Theorem of Invariant Theory for GL and SL

Proof of (7.7): Certainly the elements of G(X) are absolutely invariant. One

¬rst observes that it is harmless to enlarge the matrix X by adding columns: If X is

the “bigger” matrix, then the action of SL on B[X] is induced by that on B[X], and

obviously G(X) = B[X] © G(X). The action of SL leaves the homogeneous components

of B[X] invariant. Therefore we may ¬rst assume n ≥ m and secondly that a given

invariant element f is homogeneous of degree d, say.

Let X consist of the ¬rst m columns of X, and put U = Cof X. Then by virtue of

(7.5) (with d1 = 0) d = tm, t ≥ 0, and

U (f ) = (det U )t f.

On the other hand the entries of U X are elements of G(X) ! Furthermore det U =

(det X)m’1 = [1, . . . , m]m’1 . Thus

[1, . . . , m]t(m’1) f = U (f ) = f (U X) ∈ G(X).

The rest is very easy for us (though it is certainly the di¬cult part of the proof from a

neutral point of view):

B[X] = G(X) • C

where C is the B-submodule generated by all standard monomials containing a factor

outside “(X). Since [1, . . . , m] is the minimal element of ∆(X), multiplication by it maps

C injectively into itself, whence f ∈ G(X). ”

In the proof of (7.6),(a) we use similar arguments. Enlarging m and n if necessary,

we may assume that m > r, n > r. In order to prove the nontrivial inclusion, it is enough

to consider invariants f which are homogeneous with respect to the variables in Y , of

degree d, say. Let Y denote the submatrix of Y consisting of the ¬rst r rows, Z the

submatrix of Z formed from the ¬rst r columns. Over B[Y, Z][(det Y Z)’1 ] the absolute

invariance of f implies

f = f (Y Y ’1 , Y Z),

so by elementary matrix algebra

f = f (Y Z(Y Z)’1 , Y Z)

= f (Y Z(det Y Z)’1 Cof(Y Z), Y Z)

= (det Y Z)’d f (Y Z Cof(Y Z), Y Z).

The entries of Y Z, Cof(Y Z), Y Z all are in B[Y Z]. Thus one has

(det Y Z)d f ∈ B[Y Z],

and it su¬ces to prove

(det Y Z)B[Y Z] = (det Y Z)B[Y, Z] © B[Y Z].

(1)

This is equivalent to the injectivity of the homomorphism

• : R/Rδ ’’ B[Y, Z]/ det(Y Z)B[Y, Z],

78 7. Generic Points and Invariant Theory

R = Rr+1 (X), δ = [1, . . . , r|1, . . . , r], • induced by the embedding R ’ B[Y, Z] as above.

By virtue of (6.6) the element δ = [1, . . . , r ’ 1, r + 1|1, . . . , r ’ 1, r + 1] is not a zero-

divisor modulo δ, since it is greater than the upper neighbours [1, . . . , r ’ 1, r + 1|1, . . . , r]

’1

and [1, . . . , r|1, . . . , r ’ 1, r + 1] of δ. Therefore the natural map R/Rδ ’ (R/Rδ)[δ ] is

’1

an injection. It can be factored through • since the image of the matrix X in (R/Rδ)[δ ]

can be factored into a product of an m — r matrix and an r — n matrix. This ¬nishes the

proof of (7.6),(a).

Before embarking on the proof of (7.6),(b), we want to point out that (7.6),(b) is

equivalent to ideal-theoretic properties of Rr+1 (X). This is already true for (7.6),(a): we

have used such a property in order to prove (7.6),(a); cf. also the remark following the

proof of (7.8). Some notations have to be introduced. Let

P = (det Y )B[Y, Z] and Q = (det Z)B[Y, Z],

P be the ideal generated by the r-minors of the ¬rst r rows of Y Z, Q the corresponding

ideal for the ¬rst r columns.

(7.8) Lemma. Let m > r and n > r. Then the following are equivalent:

(a) (7.6), (b).

(b) P j = P j © B[Y Z] and Qj = Qj © B[Y Z] for all j ≥ 1.

(c) P j and Qj are primary with radicals P and Q resp. for all j ≥ 1, provided B is an

integral domain.

(d) [1, . . . , r ’1, r +1|1, . . . , r] is not a zero-divisor modulo P j , [1, . . . , r, |1, . . . , r ’1, r +1]

is not a zero-divisor modulo Qj for all j ≥ 1.

Proof of (7.8): (a) ’ (b): All the B-submodules appearing in (b) are bihomo-

geneous in the bigraded B-module B[Y, Z], the ¬rst graduation taken with respect to Y ,

and the second one with respect to Z. Let x ∈ P j © B[Y Z] be homogeneous (thus biho-

mogeneous of partial degrees d1 = d2 ), x = pj y, p = det Y , y ∈ B[Y, Z] bihomogeneous.

Then y is an absolute SL-invariant, and

T (y) = (det T )j y

for all matrices T . Since the product of an r-minor of Y and an r-minor of Z is in B[Y Z],

(7.6),(b) implies that y can be written as a linear combination of (standard) monomials

of length j in the r-minors of Z with coe¬cients in B[Y Z]. Multiplied by pj , such a

monomial is sent into P j . The statement on the powers of Q is proved similarly.

(b) ’ (c): Obvious, since the powers of principal primes are primary.

(c) ’ (d): (c) implies that Z[Y Z]/P j and Z[Y Z]/Qj are Z-¬‚at, and (3.15) reduces

(d) to the case of a ¬eld B = K, in which (d) is a trivial consequence of (c).

(d) ’ (b): This is proved in a similar fashion as equation (1) above.

(b) ’ (a): Without restriction one may assume that a given absolute SL-invariant f

is bihomogeneous of partial degrees d1 and d2 resp. We discuss the case d2 ≥ d1 , the case

d1 ≥ d2 being analogous. By virtue of (7.5): d2 ’ d1 = tr, t ∈ Z, t ≥ 0. Let p = det Y

as above. Obviously pt f is an absolute GL-invariant, so pt f ∈ P t © B[Y Z] = P t . Write

pt f as a linear combination of (standard) monomials of length t in the r-minors of the

¬rst r rows of Y Z with coe¬cients in B[Y Z], and note that such an r-minor divided by

p gives an r-minor of Z. ”

79

C. The Main Theorem of Invariant Theory for GL and SL

It is not di¬cult to see that (7.6),(a) is equivalent to (b), (c), (d) of (7.8) with j = 1,

and via (6.6) we have derived (7.6),(a) from the fact that P and Q are prime ideals over

a domain.

We shall prove independently in Section 9 that P and Q have primary powers over a

domain, and a reference to (9.18) would be the shortest proof of (7.6),(b). A more direct

argument is in order, however.

(7.9) Lemma. (a) Let f ∈ B[Y Z] be homogeneous of degree 1 with respect to the

indeterminates in the j-th row of Y . Then the row j appears exactly once in every

standard monomial in the standard representation f = aµ µ of f as an element of

Rr+1 (X).

(b) Let t ∈ Z, t ≥ 1, and suppose m ≥ tr. Let f ∈ B[Y Z] satisfy the hypothesis of (a) for

each j, 1 ¤ j ¤ tr, and assume that f vanishes after the substitution of linearly dependent

vectors (over a B-algebra) for the rows (k ’ 1)r + 1, . . . , kr, 1 ¤ k ¤ t arbitrary. Then in

the standard representation of f the ¬rst r factors of each standard monomial have row

parts

[1, . . . , r], [r + 1, . . . , 2r], . . . , [(t ’ 1)r + 1, . . . , tr]

and none of the remaining factors contains a row j, 1 ¤ j ¤ tr.

Proof of (7.9): Part (a) is almost trivial: multiply the j-th row of Y by a new

indeterminate W , and use the linear independence of the standard monomials over B[W ].

Under the hypothesis of (b) f vanishes modulo P (as in (7.8)), which is generated by a

poset ideal of the poset underlying Rr+1 (X). Therefore every standard monomial in the

standard representation of f has a minor [1, . . . , r| . . . ] as its ¬rst factor. Splitting it o¬,

one can argue inductively because of (a). ”

Proof of (7.6),(b): Without restriction let m > r, n > r, and f ∈ B[Y, Z] be

a bihomogeneous absolute SL-invariant of partial degrees d1 and d2 resp. Suppose that

d2 ≥ d1 and let t be given by (7.5).

So far we have only repeated the ¬rst lines of the proof of (7.8),(b) ’ (a). The

essential trick now is the introduction of a new tr — r matrix of indeterminates which we

pile on top of Y such that the resulting matrix Y has Y in its rows tr + 1, . . . , tr + m.

Let yk be the determinant of the matrix consisting of the rows (k ’ 1)r + 1, . . . , kr of Y .

The element

g = f y 1 . . . yt

is an absolute GL-invariant because of (7.5), and we can apply (7.9),(b) to it. Since

[(k ’ 1)r, . . . , kr|b1 , . . . , br ] yk

is the r-minor of the columns b1 , . . . , br of Z, the result follows after division of g by

y1 . . . yt . ”

In the proof of (7.8) the hypothesis “m > r and n > r” is only needed for (d) and

the implications (c) ’ (d) and (d) ’ (b). Therefore (7.6),(b) also implies the ¬rst part

of the following corollary whose second part follows directly from (7.8):

(7.10) Corollary. Let B be an integral domain, X an m — n matrix of indetermi-

nates, m ¤ n.

(a) The prime ideal Im (X) has primary powers.

(b) Let r < min(m, n). Then the prime ideals P and Q generated by the r-minors of any

r rows and any r columns resp. of the matrix of residue classes in Rr+1 (X) have primary

powers.

80 7. Generic Points and Invariant Theory

D. Remarks on Invariant Theory

In “classical” invariant theory one considers a group G of linear transformations on

the vector space K p , K a ¬eld, preferably K = C, and wants to compute explicitely the