is regular if and only if BQ is regular and P ⊃ It’1 (X)/It (X).

RP

Proof: The statement is obvious if t = 1. Suppose that t > 1.

Abbreviating Is = Is (X), we claim that RP cannot be regular if P ⊃ I1 /It . Oth-

erwise we may assume that P is a minimal prime of I1 /It and B = BQ . Since B is an

integral domain, I1 /It = P . Therefore Q ‚ I1 /It © B = 0, and B is a ¬eld, say K. Now

2

it su¬ces to note that (K[X]/It )I1 /It = K[X]I1 / It K[X]I1 is not regular since It ‚ I1 .

For the rest of the proof we may therefore assume that the residue class xmn of Xmn

is not contained in P . According to (2.4) we have an isomorphism

R[x’1 ] ∼ Rt’1 (Y )[Xm1 , . . . , Xm’1,n ][Xmn ].

’1

mn =

Let P be the contraction to Rt’1 (Y ) of the image of P R[x’1 ] under this isomorphism.

mn

Then P contains It’1 /It if and only if P contains It’2 (Y )/It’1 (Y ). And RP is regular

if and only if Rt’1 (Y )P is regular. The inductive hypothesis now immediately yields the

˜

required result. ”

13

B. The Perfection of Im (X) and Some Consequences

B. The Perfection of Im (X) and Some Consequences

From now on we shall restrict our attention to the ideal Im (X) in B[X], generated

by the maximal minors of X. It will be shown that Im (X) is a perfect ideal. Because

of (2.5) this means that pdB[X] Rm (X) = n ’ m + 1. In Subsection C we shall prove

this equation by constructing a free resolution of Rm (X) over B[X]. On the other hand

there is a simple proof, which does not use a concrete free resolution. We formulate the

following theorem for arbitrary matrices U . Apart from giving a more general result,

this formulation is better adapted to the method of proof being used below.

(2.7) Theorem. Let A be a noetherian ring and U an m — n matrix, m ¤ n, with

entries in A. Suppose grade Im (U ) = n ’ m + 1. Then Im (U ) is a perfect ideal.

Using (2.5) and (16.19) we obtain:

(2.8) Corollary. The ideal Im (X) is perfect. In particular Rm (X) is a Cohen-

Macaulay ring if this holds for B.

As we shall see in Section 3, it would be equally justi¬ed to call (2.7) a corollary of

(2.8). (2.7) will follow from Proposition (2.9). In the proof of (2.9) we will frequently

use arguments from 16.A, and we assume that the reader is familiar with the material of

that subsection.

(2.9) Proposition. Let A be a noetherian ring, F and G free A-modules of ranks

m and n, resp. Further, let f : F ’ G be a homomorphism such that the ideal I m (f ) has

p’1

grade at least p ≥ 1. Then f is injective, and, M denoting the cokernel of f , M is

p

M ¤ p.

torsionfree and pd

First we will derive (2.7) from (2.9). Let f : Am ’ An given by U , and r = n ’ m.

Denote by u1 , . . . , um the rows of U , and consider the map

r n

n

An ,

A ’’ ν(x) = x § u1 § · · · § um ,

ν:

n

An ∼ A. Put M = Coker f . Then

Obviously Im ν = Im (U ) by an identi¬cation =

we have a presentation

r’1 r r

m n n

A— A ’’ A ’’ M ’’ 0,

x—y ’’f (x) § y,

r r r

so ν factors through M . Since rk M = rk Im ν = 1, and M is torsionfree by (2.9),

r

we conclude pd Im ν = pd M = r.

proof of (2.9): By induction on m. The proposition is trivial for m = 0 (Im (f ) =

A in this case). Let m > 0. Since p ≥ 1, Im f has rank m, and f is injective for trivial

reasons. Furthermore there is nothing to prove if p = 1, and we can proceed by induction

14 2. Ideals of Maximal Minors

p’1

M ¤ p ’ 1. Since MP is free

on p. By the inductive hypothesis with respect to p, pd

for all prime ideals P in A with depth AP < p, we get

p’1

M — AP ≥ min(1, depth AP )

depth

p’1

for all prime ideals P . Consequently M is torsionfree.

Write F = F • A, F free of rank m ’ 1, let f : F ’ G be the restriction of f ,

and put M = Coker f . Since Im’1 (f ) ⊃ Im (f ), the inductive hypothesis on m can be

applied to M . We claim that there exists an exact A-sequence

p’1 p p

0 ’’ M ’’ M ’’ M ’’ 0.

p

This immediately yields pd M ¤ p.

Let π : M ’ M be the natural projection, and y a generator of Ker π. Then we

have canonical presentations

p

p’1 p p

π

σ

M ’’ M ’’ M ’’ 0,

p’2 p’1 p’1

M ’’ M ’’ M ’’ 0,

x ’’ x § y.

The second of these presentations shows that the map σ introduced in the ¬rst one

p’1 p’1 p’1 p

factors through M . Since M is torsionfree and rk M = rk Ker π, we obtain

p p’1

Ker π ∼ M , as desired. ”

=

As a consequence of (2.8) one can answer questions about the ideals Im (X) which,

from a naive point of view, concern their prime [sic] properties.

(2.10) Theorem. If B is an integral domain, then Im (X) is a prime ideal.

Proof: One may assume B to be noetherian, for the general statement is easily

reduced to this case. Then we use induction on m. If m = 1, the theorem is obvious.

We assume that m > 1. Since Im (X) is perfect of grade n ’ m + 1 and grade I1 (X) =

mn > n ’ m + 1, the ideal I1 (X) is not contained in any associated prime ideal of Im (X),

cf. (16.17).

Denote by xij the residue class of Xij in R = Rm (X). Since R[x’1 ] is a domain

mn

by (2.4) and the inductive hypothesis, there is exactly one associated prime ideal P of

R such that xmn ∈ P . If P is the single associated prime ideal, then xmn is not a

/

zero-divisor in R, and R is a domain, too. Suppose there is a second associated prime

ideal Q = P . By what we have stated above and since xmn ∈ Q, there is some xij ∈ Q. /

Arguing inductively again, we get xij ∈ P . Now P R[x’1 ] = 0, but the image of xij in

mn

P R[x’1 ] is di¬erent from 0, cf. (2.4). Contradiction! ”

mn

15

B. The Perfection of Im (X) and Some Consequences

(2.11) Theorem. If B is reduced (a normal domain), then Rm (X) is reduced (a

normal domain), too.

Proof: Suppose that B is a domain. Then R = Rm (X) is a domain by (2.10).

In order to show that B is reduced or normal resp. we apply criteria based on Serre™s

conditions.

The statements are obvious if m = 1. Let m > 1 and suppose that B is reduced

(a normal domain). Consider a prime ideal P in R such that depth RP = 0 (¤ 1).

Then grade P = 0 (¤ 1). Because of grade I1 (X) = mn > n ’ m + 2 there is an

indeterminate Xij which has residue class xij not contained in P . Clearly we may

assume xij = xmn . Then by (2.4) and the inductive hypothesis R[x’1 ] is reduced (a

mn

normal domain). Consequently RP is reduced (a normal domain), too. ”

(2.12) Remark. In Section 5 we shall prove that It (X) is a perfect ideal for every

t, 1 ¤ t ¤ m. Since grade I1 (X) > grade It (X)+1 if t > 1, the arguments in the proofs of

(2.10) and (2.11) demonstrate that (2.10) and (2.11) hold for arbitrary t (the case t = 1

being trivial).

(2.13) Remark. As to converse statements of (2.8) and (2.12), applying (2.4)

one easily deduces that B is reduced (a normal domain, a Cohen-Macaulay ring) if

B[X]/It (X) is reduced (a normal domain, a Cohen-Macaulay ring). ”

So far we have used Corollary (2.8) only, and it seems adequate to discuss an appli-

cation of (2.7) which is independent of (2.8). Let y1 , . . . , yk be elements of a commutative

ring A, J the ideal generated by them, and Y the m — (m + k ’ 1)-matrix

«y 0

··· ··· ···

y2 y3 yk 0

1

.·

.. .. ..

¬ .·

. . .

¬0 y1 y2 .

·.

¬.

.. .. .. .. ..

.

. . . . .

. 0

··· ··· ···

0 0 y1 y2 y3 yk

For trivial reasons Im (Y ) ‚ J m . We claim Im (Y ) = J m . It is of course enough to

prove this for the case in which A = Z[y1 , . . . , yk ], the yi being indeterminates. Arguing

inductively we conclude y1 J m’1 ‚ Im (Y ) and Ay1 + Im (Y ) = Ay1 + J m . Next it follows

that Ay1 © Im (Y ) = y1 J m’1 = Ay1 © J m , and altogether this yields the desired equality.

Letting n = m + k ’ 1 we have

n ’ m + 1 = k,

and (2.7) implies that J m is perfect (of grade k) if grade J = k:

(2.14) Proposition. Let A be a noetherian ring, and y1 , . . . , yk an A-sequence.

Then all the ideals J m , m ≥ 1, are perfect (of grade k).

The matrix Y above helps us to get more information on the rings Rm (X). Given

an m — n matrix X of indeterminates, we put k = n ’ m + 1 and choose Y1 , . . . , Yk

as indeterminates over B. Let S = B[Y1 , . . . , Yk ]/Im (Y ). Then the substitution which

assigns each entry of X the corresponding entry of Y (formed from Y1 , . . . , Yk ), induces

surjections

ψ : B[X] ’’ S • : Rm (X) ’’ S.

and

16 2. Ideals of Maximal Minors

The kernel of ψ is generated by the linear polynomials

j ’ i < 0 or j ’ i > k ’ 1,

Xij ,

Xij ’ Xi’1,j’1 , i = 2, . . . , m, 0 ¤ j ’ i ¤ k ’ 1,

and the ideal Im (X) . The residue classes of the polynomials listed generate the kernel

of •. Their number is exactly

nm ’ (n ’ m + 1) = grade Ker ψ ’ grade Im (X)

= grade Ker •

by virtue of (16.18): both Ker ψ and Im (X) are perfect. Here we assume B to be

noetherian, of course. Since the generators of Ker • are homogeneous (of degree 1), one

concludes easily that they form an Rm (X)-sequence (in any order). This fact makes it

possible to transfer information from Rm (X) to S and vice versa. After all, S can be

considered a well-understood B-algebra.

We use the connection between Rm (X) and S to compute the multiplicity of Rm (X)

in case B = K is a ¬eld. The graded K-algebra Rm (X) then has a well-de¬ned multi-

plicity (given by the multiplicity of its localization with respect to the irrelevant maximal

ideal). We refer the reader to [Na] for multiplicity theory.

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

and y the Rm (X)-sequence generating Ker •, as speci¬ed above. Then the multiplicity of

Rm (X) is given by

n

e(Rm (X)) = »(Rm (X)/yRm (X)) = .

m’1

Proof: Since the sequence y is a “super¬cial sequence” (de¬ned to be a sequence

of super¬cial elements in the same way as an A-sequence is a sequence of elements

not dividing zero), the multiplicities of Rm (X) and Rm (X)/yRm (X) coincide. The

multiplicity of the latter ring is just its length. ”

One could further exploit the relationship between Rm (X) and S in order to deter-

mine the Gorenstein rings among the rings Rm (X). We shall do this in (2.21), based on

a di¬erent argument.

C. The Eagon-Northcott Complex

In the preceding subsection we have investigated the ideal Im (X) by considering X

as the matrix of a linear map f : F ’ G. In this subsection it is better to start from the

dual map f — : G— ’ F — . To avoid notational complications we replace G— and F — by G

and F and f — by a map g : G ’ F . Instead the map f will be treated as the dual of g,

and the ideal Im (f ) of Subsection B is Im (g) below. While the perfection of Im (g) has

been proved already, cf. (2.8), we will construct a free resolution of the corresponding

residue class ring and some related modules. The approach taken in the following may

be rather abstract, but it is certainly very e¬ective.

17

C. The Eagon-Northcott Complex

Let A be an arbitrary ring, and suppose that F and G are ¬nitely generated free

A-modules of rank m and n, resp. Since the natural homomorphism

G— —A F ’ HomA (G, F )

is an isomorphism in this situation, one may view every A-homomorphism g : G ’ F an

element of G— —F . The free module F is the degree 1 homogeneous part of the symmetric

algebra S(F ), so we can consider g even an element of

G— — S(F ) ∼ HomS(F ) (G — S(F ), S(F )).

=

Viewed as an S(F )-linear form on

G = G — S(F ),

g gives rise to a Koszul complex (cf. [Bo.4], § 9)

n n’1

‚ ‚ ‚ ‚

C(g) : 0 ’’ G ’’ G ’’ . . . ’’ G ’’ S(F ) ’’ 0,

i+1 i

G ’’