regular sequence (S(F ) is the polynomial ring R[d1 , . . . , dm ]). Therefore the Koszul

complex C(X) is acyclic.

It has been shown in [Av.2] that in general g(e1 ), . . . , g(en ) is an S(F )-regular se-

quence if and only if grade In’i (g) ≥ m ’ n + i + 1 for i = 0, . . . , n ’ 1. Thus (i) holds

under this more general condition.

(ii) If n ≥ m, then the conclusion of (2.18) holds for C(X). In particular C i (X)

resolves Si (Coker X) and pd Si (Coker X) = min(i, n) for all i ≥ n ’ m + 1 (whereas, by

(2.16), pd Si (Coker X) = n ’ m + 1 for i = 1, . . . , n ’ m).

This is proved by the same induction as (2.18) starting with the case n = m covered

by (i). In fact, the exactness of the sequence of complexes used in the proof of (2.18) does

not depend on the special hypotheses there nor on the choice of e such that G = H • Ae,

H free. (The reader may investigate whether (ii) can be generalized in the same way as

(i).) ”

The resolution of Rm (X) obtained from (2.16) carries much more information about

Rm (X) than just its perfection. For example, one can compute its canonical module

(cf. 16.C) and decide whether it is a Gorenstein ring.

22 2. Ideals of Maximal Minors

(2.20) Theorem. Let B be a Cohen-Macaulay ring having a canonical module

ωB and X an m — n matrix of indeterminates over B, m ¤ n. Furthermore let C

denote the cokernel of the map B[X]n ’ B[X]m given by the transpose X — of X. Then

Sn’m (C) —B ωB is a canonical module of Rm (X).

Proof: Let g denote the map given by X — , r = n ’ m, and R = Rm (X). Then

ωR = Extr+1 (R, ωB[X] ) = Hr+1 (HomB[X] (C0 (g), B[X]))

B[X]

= Hr+1 (C0 (g) —B[X] ωB ) = Hr+1 (C0 (g) —B[X] (B[X] — ωB ))

— —

= Hr+1 (C0 (g)) —B[X] (B[X] —B ωB )

—

= Sr (C) — ωB . ”

(2.21) Corollary. Let X be an m—n matrix of indeterminates over the noetherian

ring B. Then Rm (X) is a Gorenstein ring if and only if (i) B is a Gorenstein ring and

(ii) m = 1 or m = n.

Proof: The “if”-part is obvious (without (2.20)). Assume that R = Rm (X) is a

Gorenstein ring. As in the case of the Cohen-Macaulay property (cf. (2.13)) we deduce

that B is a Gorenstein ring (using a suitable argument stated in [Wt]). Let P be a prime

ideal in A containing the entries of X and r, g, C as in (2.20). Sr (C)P is the canonical

module of RP . By the de¬nition of C0 (g) the minimal number of generators of Sr (C)P

is rk Sr (B[X]m ). It has to be 1 if RP is a Gorenstein ring. ”

Later we will determine the canonical module for each of the rings Rt (X) , cf. Sec-

tions 8 and 9. Then the canonical module will be described as an ideal of Rt (X) (provided

B is Gorenstein). The reader may try to derive such a description from (2.20).

D. The Complex of Gulliksen and Neg˚

ard

We shall now construct a ¬nite free resolution of It (X) for the case in which m = n,

t = n ’ 1, n ≥ 2. Let A be an arbitrary commutative ring. By Mn (A) we denote the

ring of n — n matrices with entries in A. We also use the structure of Mn (A) as a free

A-module of rank n2 . Let U ∈ Mn (A). Then the complex of A-modules

d d d d

4 3 2 1

G(U ) : 0 ’’ G(U )4 ’’ G(U )3 ’’ G(U )2 ’’ G(U )1 ’’ G(U )0 ’’ 0

is given as follows: Put G(U )0 = G(U )4 = A, G(U )1 = G(U )3 = Mn (A). To get G(U )2

we consider the zero-sequence

ι π

A ’’ Mn (A) • Mn (A) ’’ A

(2)

where ι(a) = (aE, aE), E being the unit matrix of Mn (A), and π(V, W ) = trace(V ’W ).

Let Eij , 1 ¤ i, j ¤ n, be the canonical basis of Mn (A). Then Ker π is generated by the

elements (Eij , 0), i = j, (0, Euv ), u = v, (Eii , E11 ), 1 ¤ i ¤ n, and (0, Euu ’ E11 ), 2 ¤

n n n

u ¤ n. Since Im ι is generated by i=1 (Eii , Eii ) = i=1 (Eii , E11 )+ u=2 (0, Euu ’E11 ),

23

D. The Complex of Gulliksen and Neg˚

ard

G(U )2 = Ker π/ Im ι is a free A-module. Now let U be the matrix of cofactors of U . Then

we put

d1 (V ) = trace(U V ), d4 (a) = aU .

To de¬ne d2 , d3 we consider the zero-sequence

ψ •

Mn (A) ’’ Mn (A) • Mn (A) ’’ Mn (A)

(3)

where ψ(V ) = (U V, V U ), •(V, W ) = V U ’ U W . Clearly Im ι ‚ Ker • and Im ψ ‚ Ker π

so that we may de¬ne d2 , d3 as the maps induced by • and ψ, resp.

A trivial calculation shows that di —¦ di+1 = 0, i = 1, 2, 3, whence G(U ) is in fact

a complex. Furthermore Im d1 = In’1 (U ). We make another trivial observation: If

h : A ’ A is a homomorphism of commutative rings and if h(U ) denotes the matrix

obtained from U by applying h to the entries of U , then one has a natural isomorphism

of A -complexes

C(U ) —A A ∼ C(h(U )).

=

(2.22) Proposition. The complex G(U ) is self-dual.

Proof: We have to de¬ne isomorphisms νi : G(U )i ’ [G(U )4’i ]— , 0 ¤ i ¤ 4, such

that νi’1 —¦ di = d—4’i+1 —¦ νi for i = 1, . . . , 4. Let ν0 = ν4 be the canonical isomorphism

— —

A ’ A . Next we take the canonical basis Eij of Mn (A), 1 ¤ i, j ¤ n, and its dual Eij

to de¬ne ν : Mn (A) ’ Mn (A)— by ν(Eij ) = Eij . Put ν1 = ν3 = ν. Let ι, π be the maps

—

from the sequence (2) above and denote by χ : Ker π ’ Mn (A) • Mn (A) the canonical

injection. Then χ— —¦ (ν, ’ν) as well as the elements of Im(χ— —¦ (ν, ’ν)) vanish on Im ι.

Consequently χ— —¦ (ν, ’ν) induces a homomorphism ν2 : G(U )2 ’ G(U )— which is easily

2

seen to be bijective. The equations νi’1 —¦ di = d— —¦ νi may be veri¬ed directly. ”

4’i+1

(2.23) Proposition. If U is invertible, then G(U ) is split-exact.

Proof: It is no problem to see by direct computation that H(G(U )) = 0 in the case

under consideration. On the other hand the proposition will follow from the next one

once we have shown that H2 (G(U )) = 0. For this purpose let V, W ∈ Mn (A) and suppose

V U ’ U W = 0. Let U be the matrix of cofactors of U and put Z = (det U )’1 U V . Then

U Z = V and ZU = (det U )’1 UV U = (det U )’1 U U W = W . ”

(2.24) Proposition. Let N be any A-module. Then the ideal In’1 (U ) annihilates

Hi (G(U ) —A N ) for i = 2.

Proof: Let Eij , 1 ¤ i, j ¤ n, be the canonical basis of M = Mn (A). We consider

the Koszul complex

2

‚ ‚

2 1

K: . . . M ’’ M ’’ A ’’ 0

derived from the linear form ‚1 = d1 : M ’ A. We claim that Im ‚2 ‚ Im d2 . In

24 2. Ideals of Maximal Minors

connection with (2.22) this yields a commutative diagram

2

‚ ‚

’2 ’1

M ’’’

’ ’’’

’

M A

¦

¦f

d d d d

’4 ’3 ’2 ’1

G(U )4 ’ ’ ’ G(U )3 ’ ’ ’ G(U )2 ’ ’ ’ G(U )1 ’ ’ ’ G(U )0

’ ’ ’ ’

¦ ¦ ¦

¦∼ ¦∼ ¦∼

= = =

d— d—

G(U )— G(U )— ’ ’ ’ G(U )—

1 2

’’’

’’ ’’

0 1 2

¦

¦—

f

2

‚— ‚—

A— M— M —.

’1 ’2

’’’

’ ’’’

’

Since Im ‚1 = In’1 (U ) annihilates the homology of K —A N as well as that of K— — N

([Bo.4], § 9, no. 1, Cor. 2, p. 148), tensoring of the diagram by N then proves the statement

of the proposition.

As to the proof of Im ‚2 ‚ Im d2 , let π, • the maps from (2) and (3) above and

I = {1, . . . , n}. An easy computation shows that

‚2 (Eiu § Eiv ) = ± σ(j, I\i)[1, . . . , i, . . . , j, . . . , n|1, . . . , u, . . . , v, . . . , n]•(Eij , 0)

j=i

if u = v, so that ‚2 (Eiu §Eiv ) ∈ Im d2 . In the same way one obtains ‚2 (Eiu §Eju ) ∈ Im d2 .

Finally let i = j, u = v. Then

‚2 (Eiu § Ejv ) =

± σ(u, I\v)σ(k, I\i)[1, . . . , i, . . . , k, . . . , n|1, . . . , u, . . . , v, . . . , n]•(Ejk , 0)

k=i,j

+ σ(u, I\v)σ(j, I\i)[1, . . . , i, . . . , j, . . . , n|1, . . . , u, . . . , v, . . . , n]•(Ejj , Euu )

+ σ(j, I\i)σ(w, I\v)[1, . . . , i, . . . , j, . . . , n|1, . . . , v, . . . , w, . . . , n]•(0, Ewu )

w=u,v

so that ‚2 (Eiu § Ejv ) ∈ Im d2 . ”

(2.25) Proposition. Let N be an A-module. Then the ideal (In’1 (U ))2 annihilates

H2 (G(U ) —A N ).

Proof: Consider A as an algebra over the ring A = A[Xij : 1 ¤ i, j ¤ n] via the

substitution Xij ’ uij where U = (uij ). Let

0 ’’ K ’’ F ’’ N ’’ 0

be an exact sequence of A -modules, F being free. Then one obtains an exact sequence

H2 (G(X) —A F ) ’’ H2 (G(U ) —A N ) ’’ H1 (G(X) —A K)

(4)

25

E. Comments and References

where X = (Xij ), as usual. Put d = det X and let L denote the cokernel of the canon-

ical embedding A ’ A [d’1 ]. By (2.23) the homology of G(X) —A A [d’1 ] vanishes.

Therefore H2 (G(X)) H3 (G(X) —A L). Now In’1 (X) annihilates H3 (G(X) —A L) and

H1 (G(X) —A K) by (2.24). Since (4) is an exact sequence H2 (G(U ) —A N ) is annihilated

by (In’1 (U ))2 . ”

From (2.24), (2.25) and the acyclicity criterion (16.16) we get

(2.26) Theorem. Let A be a noetherian ring, U an n — n matrix with entries in

A. Assume that grade In’1 (U ) ≥ 4. Then G(U ) is acyclic.

In view of (2.5) we obtain, in particular, that G(X) yields a free resolution of In’1 (X)

and Rn’1 (X) is a perfect B[X]-module, so Rn’1 (X) is a Cohen-Macaulay ring if this

holds for B. Because of (2.22) the Gorenstein property is also preserved in passing over

from B to Rn’1 (X). This (and (2.21), of course) is a special case of Corollary (8.9) below

which says that Rt (X) is a Gorenstein ring if and only if (i) B is a Gorenstein ring and

(ii) m = 1 or m = n.

E. Comments and References

The history of determinantal ideals in case t > 1 seems to begin with Macaulay

[Ma]. He stated (2.1) when A is a polynomial ring over a ¬eld and t = m ([Ma], Section

53). See also [Gb], pp. 199“204, for a simple proof). After a slight generalization of this

result, due to Northcott ([No.1], Theorem 9), the general case has been treated by Eagon

([Ea.1], Corollary 4.1). Our proof together with (2.3) is drawn form [EN.1] (Theorem 3).

The localization argument of (2.4) was used, perhaps not for the ¬rst time, by

Northcott in proving (2.10) ([No.2], Proposition 2). Our version can be found in [Ea.2]

(Proof of Theorem 2). (2.5) goes back to Northcott in case t = m ([No.2], Proposition 1),

to Mount ([Mo]) in case B is a ¬eld (of characteristic zero), and to Eagon in the general

case ([Ea.2], Theorem 2).

(2.7) is exactly Corollary 5.2 in [Ea.1]; our proof (i.e. (2.9)) is taken from [Ve.2]. The

Cohen-Macaulay property of Rm (X) stated in (2.8), was already proved by Northcott

([No.1], Theorems 10 and 11). More precisely he showed that for a matrix U with entries

in a Cohen-Macaulay ring A the residue class ring A/Im (U ) is Cohen-Macaulay, too, if

Im (U ) has the maximally possible grade. This assertion as well as the idea of the proof,

which goes by an inductive argument using the knowledge of the ¬rst syzygy module

of Im (U ), is a generalization of corresponding considerations in [Ma], Section 53. The

generalizations of (2.7), (2.8) and (2.10) to It (X) for arbitrary t were proved by Hochster

and Eagon ([HE.2], Theorem 1). That of (2.10) has a precursor due to Mount ([Mo]) in

case B is a ¬eld (of characteristic zero). For t = 2 the results corresponding to (2.8) and

(2.10) had already been proved by Sharpe ([Sh.1], Theorem 3 and Corollary to Theorem

1, resp.) who had also shown the Cohen-Macaulay property of A/I2 (U ) in case the entries

of U belong to a Cohen-Macaulay ring A and I2 (U ) has maximal grade ([Sh.2], Theorem).

In proving these statements Sharpe followed the idea of proof Macaulay and Northcott

had applied already: He concurrently computed the ¬rst syzygy module of I2 (U ). (2.11)

and (2.12) together with their proofs are drawn from [HE.2] (Corollary 3).

A rather remarkable proof of the perfection of the ideals Im (U ) has been given by

Huneke in [Hu.2]. Huneke concludes the perfection of Im (U ), m ¤ n, from the fact that

the ideals Im (U ), n = m + 1, are even “strongly Cohen-Macaulay”.

26 2. Ideals of Maximal Minors

The representation of a power of an ideal as a determinantal ideal may be an old

idea (though it appears in [BR.1], p. 215 without further reference), and in [Ka], p. 107 it

is said that (2.14) goes back to Macaulay. The multiplicity of Rm (X) has been calculated

in [EN.3] as part of an investigation of the Hilbert functions of rings of type A/Im (U )

based on the Eagon-Northcott complex.

The Eagon-Northcott complex has a long and extensive history. It begins with

Hilbert who computed explicitely what we call the Koszul complex derived from a ¬nite