i+1

(’1)j+1 g(xj )x1 § · · · § xj § · · · § xi+1 .

‚(x1 § · · · § xi+1 ) =

j=1

As a complex of A-modules C(g) splits into direct summands

i i’1

Ci (g) : · · · ’’ 0 ’’ G — S0 (F ) ’’ G — S1 (F ) ’’ . . .

1 0

’’ G — Si’1 (F ) ’’ G — Si (F ) ’’ 0.

m

We ¬x orientations γ on F — and δ on G— , i.e. F — ’’ A and

isomorphisms γ :

n

G— ’’ A. Let

δ:

r = n ’ m.

—

Then for i = 0, . . . , r we can splice the A-dual Cr’i (g) of Cr’i (g) and Ci (g) to a sequence

0 r’i i

—

‚— νi

—‚ ‚

—

Di (g) : 0 ’’ ( G — Sr’i (F )) ’’ . . . ’’ ( G — S0 (F )) ’’ G — S0 (F ) ’’ . . .

0

‚

’’ G — Si (F ) ’’ 0,

where νi is described as follows: First one de¬nes νi as a map

r’i i

—

G — )—

G ’’ (

18 2. Ideals of Maximal Minors

by

m r’i i

— —

G— , z = γ ’1 (1),

(νi (x))(y) = δ(x § y § x∈ y∈

g (z)), G,

r’i i

G — S0 (F ))— ’’ G — S0 (F ) via the natural

and then one regards νi as a map (

r’i r’i r’i i i i

G ∼ ( G)— ∼ ( G — S0 (F ))— and ( G— )— ∼ G∼

—

G — S0 (F ).

isomorphisms = = = =

An easy calculation shows that

νi —¦ ‚ — = 0, ‚ —¦ νi = 0.

Furthermore γ, δ, and, hence, νi are unique up to a unit factor. So Di (g) is a complex

whose homology depends only on g. In order to specify homology modules we consider

0 0

G — Si (F ) to be in position 0 and ( G — Sr’i (F ))— in position r + 1. Then

H0 (D0 (g)) = A/Im (g),

H0 (Di (g)) = Si (Coker g), i > 0.

The second of these equations is quite obvious whereas one has to analyze ν 0 to

observe that Im ν0 = Im (g).

Our purpose will be achieved when the following theorem has been proved:

(2.16) Theorem. Let A be a noetherian ring, g : G ’ F a homomorphism of

¬nitely generated free A-modules. Put n = rk G, m = rk F and choose orientations γ, δ

of F — and G— , resp. Suppose m ¤ n and grade Im (g) = n ’ m + 1. Then the following

holds:

(a) The complexes Di (g), 0 ¤ i ¤ n ’ m, are acyclic.

(b) D0 (g) resolves A/Im (g), Di (g), i = 1, . . . , r, resolves Si (Coker g).

(c) A/Im (g) and Si (Coker g) , i = 1, . . . , r, are perfect A-modules.

If we look at the next to the last homomorphism of D0 (g), we see that in the situation

of (2.16) the ¬rst syzygy module of Im (g) is generated by the “expected” relations: U

being a matrix representing g, they are obtained by Laplace column expansion of the

(m + 1)-minors of all matrices which result from U by doubling a row.

Of course (2.16) can be applied to the case in which g is given by an n — m matrix

X of indeterminates over a noetherian ring B, and part of it has already been proved

(cf. (2.8)). In Section 13 we shall again take up the problem concerning the perfection

of Coker g. More generally the map x : Rn ’ Rm will be investigated where R = Rt (X)

and x is given by the residue classes of the entries of X. Coker x will turn out to be a

perfect B[X]-module if and only if n ≥ m.

Only part (a) of (2.16) needs a proof; (b) and (c) then follow easily from what

has been said above. The complexes Di (g) are complexes of free A-modules of length

r + 1 = n ’ m + 1. By virtue of the exactness criterion (16.16) it is enough to show

that their localizations Di (g)P , P ⊃ Im (g), are split-exact. For these prime ideals P the

localization gP is surjective, so we have reduced (2.16) to the following proposition.

(2.17) Proposition. Let A be a noetherian ring, g : G ’ F a homomorphism of

¬nitely generated free A-modules. If g is surjective, then the complexes D i (g) are split-

exact.

19

C. The Eagon-Northcott Complex

Proof: By the de¬nition of Di (g) one has

±

H0 (Ci (g)) if j = 0 and i > 0,

Hj (Ci (g)),

j = 1, . . . , i ’ 1,

Ker ‚/ Im νi , j = i,

Hj (Di (g)) =

Ker νi / Im ‚ — , j = i + 1,

r+1’j —

H

(Cr’i (g)), j = i + 2, . . . , r,

0—

H (Cr’i (g)) if j = r + 1 and i < r.

We may assume that A has exactly one maximal ideal. Then Ker g is a free direct

summand of G. As stated above, Ker ‚/ Im νi and Ker νi / Im ‚ — do not depend on the

orientations γ and δ. Therefore one may take a basis x1 , . . . , xm of F and a basis

y1 , . . . , yn of G such that g(yk ) = xk , k = 1, . . . , m, g(yk ) = 0, k = m + 1, . . . , n, to de¬ne

γ and δ by

γ(x— § · · · § x— ) = 1 and δ(y1 § · · · § yn ) = 1

— —

1 m

(x— , . . . , x— being the basis dual to x1 , . . . , xm etc.). With these data it is very easy to

1 m

calculate that Im ‚ — = Ker νi , Ker ‚ = Im νi . The rest essentially follows from:

(2.18) Proposition. Let A be a commutative ring, g : G ’ F a surjective homo-

morphism of ¬nitely generated free A-modules. If g is surjective, then

j > rk G ’ rk F

Hj (C(g)) = 0 for

and

Hj (Ci (g)) = 0 for j = i.

Let us ¬rst ¬nish the proof of (2.17). Proposition (2.18) shows that

i 0

‚ ‚

0 ’’ Hi (Ci (g)) ’’ G — S0 (F ) ’’ . . . ’’ G — Si (F ) ’’ 0

is a split-exact sequence of A-modules. Therefore its dual is split-exact, too. Taking into

account that this holds for i = 0, . . . , r, (2.17) follows immediately. ”

As just seen, the important part of (2.18) is the second equation. The ¬rst one

can be viewed a special case of the general theorem concerning the vanishing of Koszul

homology ([No.6], Theorem 4, p. 262): the image of the linear form g : G ’ S(F ) is the

ideal Si (F ). After the choice of a basis for F one can identify S(F ) with a polynomial

i≥1

ring over A within which Im g is just the ideal generated by the indeterminates, an ideal

of grade rank F (with the suitable de¬nition of grade if A is non-noetherian).

Proof of (2.18): As in the proof of (2.17) it is useful (and harmless) to assume

that A has exactly one maximal ideal.

One proceeds by induction on rk G ’ rk F . In case rk G = rk F , the argument just

explained shows that Hj = 0 for j > 0 (without any reference to the notion “grade”): the

Koszul complex associated with (the linear form given by) a sequence of indeterminates

is acyclic in positive degrees (cf. [Bo.4], § 9, no. 6, Prop. 5). Furthermore Hj (C0 (g)) = 0

for j > 0 by de¬nition of C0 (g).

20 2. Ideals of Maximal Minors

Let now rk G > rk F . Then one splits G into a direct sum G = H • Ae, H free,

rk H = rk G ’ 1, e ∈ Ker g. Let h = g|H. The decomposition induces split-exact

sequences

i+1 i+1 i

0 ’’ H ’’ G ’’ H ’’ 0,

(1)

the map on the left being the natural embedding, the map on the right sending x § e,

i i+1

x∈ H, to x and vanishing on H.

Passing to S(F ) one obtains a diagram

0 0 0 0

¦ ¦ ¦ ¦

¦ ¦ ¦ ¦

n’1 1 0

C(h) : 0 ’’’

’’ ’’’

’’ H ’’’ ··· ’’’

’’ ’’ H ’’’

’’ H ’’’ 0

’’

0

¦ ¦ ¦ ¦

¦ ¦ ¦ ¦

n n’1 1 0

C(g) : 0 ’’’

’’ G ’’’

’’ G ’’’ ··· ’’’

’’ ’’ G ’’’

’’ G ’’’ 0

’’

¦ ¦ ¦ ¦

¦ ¦ ¦ ¦

n’1 n’2 0

C(h)[’1] : 0 ’ ’ ’

’’ H ’’’

’’ H ’’’ ··· ’’’

’’ ’’ H ’’’

’’ ’’’ 0

’’

0

¦ ¦ ¦ ¦

¦ ¦ ¦ ¦

0 0 0 0

whose split-exact columns are induced by (1). It is easy to check that this diagram is

commutative, whence we have an exact sequence

· · · ’’ Hj (C(h)) ’’ Hj (C(g)) ’’ Hj’1 (C(h)) ’’ . . .

of homology modules. The ¬rst equation follows immediately.

For the demonstration of the second we regard the diagram above as a diagram of

0

graded S(F )-modules. For convenience one chooses the graduation of G = S(F ) as the

natural one, and then shifts all the other graduations such that every homomorphism is

of degree zero. The i-th homogeneous part of Hj (C(g)) is then given by

j j’1

G — Si’j (F ) ’’ G — Si’j+1 (F )

Ker

Hj (C(g))i = = Hj (Ci (g)).

j+1 j

G — Si’j’1 (F ) ’’ G — Si’j (F )

Im

Analogously

Hj (C(h))i = Hj (Ci (h)),

whereas

Hj (C(h)[’1])i = Hj’1 (Ci’1 (h)).

The decomposition of the exact homology sequence above makes the second equation

evident now. ”

21

C. The Eagon-Northcott Complex

(2.19) Remarks. (a) It is not di¬cult to identify the homology modules Hi (Ci (g)),

i = 0, . . . , r, in the situation of (2.16) or (2.18). The reader may check that

i

M )— , M = Coker g — .

Hi (Ci (g)) = (

Furthermore

r’i

r’i —

H (Cr’i (g)) = M,

r’i r+1

— —

so Cr’i (g) resolves M , i = 0, . . . , r (and Cr+1 (g) resolves M ). The map νi can be

interpreted (or constructed) as an isomorphism

r’i i

M ∼( M )—

=

r

M ’ A.

derived from the linear form ν0 :

(b) The complexes discussed so far do not exhaust the class of resolutions which can

be extracted from the complexes C(g). We prove the following results only for the case

R = B[X], B noetherian, g : G ’ F given by the n — m matrix X of indeterminates

with respect to bases e1 , . . . , en and d1 , . . . , dm of G and F . To indicate this clearly we

use X in place of g.

(i) If n ¤ m, then the complex C(X) is acyclic. It resolves S(Coker X) over S(F );

its homogeneous component Ci (X) resolves Si (Coker X) for all i ≥ 0. In particular

pd Si (Coker X) = min(i, n).