Proof: By virtue of (12.7) and (3.14) we may assume that B = K is a ¬eld. The

chain

I(s0 , . . . , sr ; sw’1 ) ‚ I(s0 , . . . , sr ; v) ‚ I(s0 , . . . , sr ; sw )

=

of inclusions reduces the second equation to the special case in which v = sw . Since

I(s0 , . . . , sr ; sw ) is a minimal prime ideal of

I(s0 , . . . , sr ; sw’1 ) + X1sw’1 +1 K[X],

the second equation can be derived inductively from the ¬rst one.

If s0 > 0, one can write

s0

m

I(s0 , . . . , sr ) = I(0, s1 ’ s0 , . . . , sr ’ s0 ) + Xij K[X],

i=1 j=1

the ideal I(. . . ) being taken from a smaller matrix of indeterminates in an obvious way.

Thus we are left with the case s0 = 0, for which we remind the reader of the inductive

argument (2.4). Here it is of course convenient to invert X11 and to perform elementary

transformations with respect to the ¬rst row and column. A glance at the generating set

’1

of I(s0 , . . . , sr ) shows that the extension of I(s0 , . . . , sr ) in K[X][X11 ] can be identi¬ed

with the extension of an ideal

ti = si+1 ’ 1,

I(t0 , . . . , tr’1 ),

taken from an (m ’ 1) — (n ’ 1) matrix of indeterminates. ”

(12.9) Theorem. Let B be a noetherian ring, X an m—n matrix of indeterminates.

Then the ideals I(s0 , . . . , sr ; sw ), 0 ¤ w ¤ r, and I(s0 , . . . , sr ; sw + 1), 0 ¤ w < r, are

perfect. In particular the ideals It (X) are perfect.

Proof: Again it is harmless to work with a ¬eld B, cf. (3.3), thus rendering Lemma

(5.15) applicable. By part (a) of (5.15) (and (16.20)) the perfection of I(s 0 , . . . , sr ; sw )

follows from that of I(s0 , . . . , sr ; sw + 1), a larger ideal, unless sw = n, for which case we

invoke induction on m. If sw + 1 < sw+1 , one writes

I(s0 , . . . , sr ; sw + 1) = I(s0 , . . . , sw’1 , sw + 1, sw+1 , . . . , sr ; sw + 1) © I(s0 , . . . , sr ; sw+1 ).

Since the sum of the ideals on the right hand side is

I(s0 , . . . , sw’1 , sw + 1, sw+1 , . . . , sr ; sw+1 ),

and all ideals involved have the “correct” grade according to (12.8), a reference to

(5.15),(b) ¬nishes the proof. ”

160 12. Principal Radical Systems

(12.10) Remarks. (a) It seems worthwhile to look back to the sections 5 “ 11

and to check which of the results in these sections, as far as they apply to the ideals

I(s0 , . . . , sr ), can be derived from (12.5) “ (12.9). The properties of being a radical or a

prime ideal and the grade formula are covered explicitely, as well as perfection, of course.

Section 6: Lemma (6.4) only builds on the dimension (or grade) formula, and one

concludes normality of the residue class ring. The computation of the singular locus is

as easy as in (2.6) if one uses the inductive device sketched in the proof of (12.8).

Section 7: The construction of generic points (at least over domains B) is a main

argument in that section. The proof of (7.6),(a), relies on the fact that the minor

[1, . . . , r ’ 1, r + 1|1, . . . , r ’ 1, r + 1] is not a zero-divisor modulo [1, . . . , r|1, . . . , r] in

Rr+1 (X). This can be derived from (12.9) and (12.7), since the minimal prime ideals

of Ir+1 (X) + [1, . . . , r|1, . . . , r]B[X] (over a domain) belong to the class of ideals consid-

ered. This is no longer true for I(s0 , . . . , sr ) + [1, . . . , r|s0 + 1, . . . , sr’1 + 1]B[X], and

it is doubtful whether one can derive the result of (7.19) for I(s0 , . . . , sr ). At least in

characteristic zero there is a loop-hole, however, cf. (7.21).

Section 8: For the reason just mentioned the computation of the divisor class group

of B[X]/I(s0 , . . . , sr ) is certainly not immediate in the general case, though it is in the

case Rr+1 (X), for which one can then compute the canonical class.

Section 9 “ 11: Here the full strength of the ASL structure on B[X] is used, and

there seems to be no chance to obtain the main results without considerable e¬ort. This

does not exclude the possibility of constructing principal radical systems containing the

ideals of interest, and at least in one case this has been successful, cf. [Ng.1].

(b) The choice of a principal radical system embracing the ideals It (X) is by no

means unique ! In fact, the ideals J(X; γ) ‚ G(X) have been investigated in [Ho.3] by a

blend of methods based on a principal radical system and standard monomial theory. It

doesn™t seem a very bold speculation to believe that a principal radical system containing

the ideals I(X; δ) can be constructed even without standard monomial theory (though

one will certainly need the partial order on ∆(X) as a systemizing tool).

(c) It should be possible to explore the rings B[X]/I(s0 , . . . , sr ; sw ) in regard to their

divisor class group, canonical class etc. At least they are normal over a normal domain

B, cf. [HE.2], p. 1024, Corollary 3.

(d) A modi¬cation of the scheme of proof developped in this section has been sug-

gested in [KlL.1]. It avoids generic points and exploits dimension-theoretic arguments in

order to prove that “x is not a zero-divisor modulo Rad I” or “. . . Rad J” resp. In fact,

if all the minimal prime ideals P of I have height ¤ h, but ht I + Ax ≥ h + 1, then x is

not a zero-divisor modulo Rad I. ”

D. Comments and References

The main source for this section is Hochster and Eagon™s fundamental article [HE.2]

whose line of reasoning is followed very closely. The construction of a generic point for

I(s0 , . . . , sr ) is their Proposition 25, (12.3) corresponds to Proposition 29, part 1), (12.4)

combines the Propositions 21 and 22. (12.8) reproduces Proposition 32 of [HE.2], with a

di¬erent proof however, and the derivation of (12.9) is exactly as given in [HE.2], section

11.

The notion of principal radical system as de¬ned in (12.1) has been suggested by Ngo,

cf. [Ng.2], Proposition 3. It certainly simpli¬es the de¬nition of [HE.2]. Ngo discusses a

161

D. Comments and References

generalization called a principal system of ideals. A module-theoretic version as indicated

in (12.2),(b) has been used in [Br.7].

In [KlL.1] Kleppe and Laksov give a detailed account of their modi¬cation to the

proof of Hochster and Eagon, as pointed out in (12.10),(d). Originally they had devel-

opped it for their investigation of ideals generated by pfa¬ans ([KlL.2]), which have been

treated by Marinov ([Mr.1],[Mr.2],[Mr.3]) in complete analogy to [HE.2]. Other applica-

tions of the method of principal radical systems are to be found in [Ho.3] (cf. (12.10),(b)),

Kutz™s work [Ku] on ideals generated by minors of symmetric matrices, and [Ng.2]

(cf. 9.E). It is interesting to note that the classes of ideals studied in the papers mentioned

have later been explored by standard monomial methods, too, cf. [DEP.2] for a survey.

Nonetheless there seem to be situations in which the method of principal radical systems

solves a problem which cannot be tackled by standard monomial methods, cf. [HL].

13. Generic Modules

Once more let X = (Xij ) be an m—n matrix of indeterminates over a noetherian ring

B and r a nonnegative integer. Put R = Rr+1 (X). In this section we shall investigate the

image and the cokernel of the map x : Rm ’ Rn given by the matrix of the residue classes

of the indeterminates Xij . The map x and its cokernel C have the following universal

properties: Let S be a (noetherian) B-algebra. If f : S m ’ S n is a homomorphism of

rank r represented by a matrix (uij ), then f = x — S, S made an R-algebra via the

substitution Xij ’ uij . If M is an S-module given by n generators and m relations and

of rank ≥ n ’ r, then M = C — S (since M is represented by a map S m ’ S n of rank

¤ r). The universal properties of x and C justify the notions generic map and generic

module.

The main results of the section will be that Im x is a perfect B[X]-module (with

one trivial exception) and that Coker x is perfect (provided r ≥ 1) if and only if m ≥ n.

Some special cases have been treated already: For r ≥ min(m, n) we refer to (2.16). This

result also implies that Coker x is perfect in case r + 1 = n ¤ m since Ir+1 (X) annihilates

Coker X, then.

A. The Perfection of the Image of a Generic Map

We start with a simple lemma which will be used several times. Its proof consists

in a repeated application of (16.14),(b).

(13.1) Lemma. Let S be a noetherian ring, I an ideal in S, and M a ¬nitely

generated S-module, M = M0 ⊃ . . . ⊃ Ms = 0 a ¬ltration of M . Then

grade(I, M ) ≥ min{ grade(I, Mi /Mi+1 ) : 0 ¤ i ¤ s’1 }.

The crucial step in proving the perfection of Im x is contained in:

(13.2) Proposition. Let r < min(m, n), C = Coker x.

(a) C is a re¬‚exive R-module.

(b) There exists a free submodule F of C such that C/F is annihilated by I(X; δ), δ =

[1, . . . , r|1, . . . , r ’ 1, r + 1], and as an R(X; δ)-module is isomorphic with the ideal in

R(X; δ) generated by the residue classes of the r-minors [1, . . . , r|1, . . . , k, . . . , r + 1], 1 ¤

k ¤ r.

Proof: Let z1 , . . . , zn denote the canonical basis of Rn . We put

n

0 ¤ i ¤ r,

Fi = Rzj mod Im x,

j=i+1

Fr+1 = 0, and

1 ¤ i ¤ r + 1.

δi = [1, . . . , r|1, . . . , i, . . . , r + 1],

163

A. The Perfection of the Image of a Generic Map

I(X; δi ) is the ideal generated by the (r + 1)-minors of X and the i-minors of its ¬rst i

columns. Then for i = 1, . . . , r + 1:

(i) C/Fi is annihilated by I(X; δi ).

(ii) Fi’1 /Fi is an R(X; δi )-free submodule of C/Fi .

(iii) The map x induces an exact sequence

x(i’1)

R(X; δi )m ’’ R(X; δi )i’1 ’’ C/Fi’1 ’’ 0,

the m — j matrix x(j) consisting of (the residues of) the ¬rst j columns of x.

We shall ¬nish the proof of (13.2) ¬rst before demonstrating the assertions (i)“(iii).

Obviously CP is RP -free for all prime ideals P ‚ R which do not contain Ir (x). From

(ii), (5.18), and (16.18) we obtain

grade(Ir (x), Fi’1 /Fi ) = grade(Ir (x), R(X; δi ))

= grade I(X; [1, . . . , r ’ 1|1, . . . , i, . . . , r])/I(X; δi )

= grade I(X; [1, . . . , r ’ 1|1, . . . , i, . . . , r]) ’ grade I(X; δi )

= m + n ’ 2r

for i = 1, . . . , r, and grade(Ir (x), Fr ) = m + n ’ 2r + 1. Now (a) is an immediate

consequence of (13.1) (cf. (16.33)). As to (b) we put F = Fr and let J be the ideal in

R(X; δ) generated by the r-minors δk , 1 ¤ k ¤ r. From (iii) one gets a presentation

(r)

mx r

’’ R(X; δ) ’’ C/F ’’ 0

R(X; δ)

by tensoring with R(X; δ). On the other hand there is a zero-sequence

x(r) h

R(X; δ)m ’’ R(X; δ)r ’’ J ’’ 0

(1)

where the surjective map h is de¬ned by

h(z k ) = (’1)k+1 δk , 1 ¤ k ¤ r,

r

z 1 , . . . , z r being the canonical basis of R(X; δ) . So we obtain a surjection C/F ’’ J.

Furthermore C/F has rank 1 and is torsionfree (as an R(X; δ)-module): It is free of

rank 1 at all prime ideals P ‚ R(X; δ) which do not contain the ideal J. Using (13.1)

and (ii) once more, we get

grade(J, C/F ) ≥ min{ grade(J, Fi /Fi+1 ) : 0 ¤ i ¤ r ’ 1 } ≥ 1.

Consequently C/F ∼ J (and (1) is exact).

=

(i)“(iii) will be proved by descending induction on i. Since the ¬rst r columns of x

are linearly independent over R, Fr is a free submodule of C. Obviously x induces an

exact sequence

x(r)

Rm ’’ Rr ’’ C/Fr ’’ 0.

164 13. Generic Modules

So the assertions hold for i = r + 1. Let 1 ¤ i ¤ r. By the inductive hypothesis we have

an exact sequence

x(i)

R(X; δi+1 )m ’’ R(X; δi+1 )i ’’ C/Fi ’’ 0,

so C/Fi is annihilated by I(X; δi ) (cf. (16.2)). Tensoring with R(X; δi ) yields an exact

sequence

x(i)

R(X; δi )m ’’ R(X; δi )i ’’ C/Fi ’’ 0.

Since the ¬rst i ’ 1 columns of x(i) are linearly independent over R(X; δi ), Fi’1 /Fi is an

R(X; δi )-free submodule of C/Fi of rank 1, and (iii) is an immediate consequence. ”

(13.3) Remark. By the way, the proof of (13.2) shows that the ¬rst syzygy module

of the ideal in (13.2),(b) is as one expects at ¬rst sight: cf. the exact sequence (1). The

exactness of (1) can also be derived from (5.6) or even checked directly. ”

Taking into account the special structure of the ideal described in (13.2),(b) we are

now able to prove the main result of this subsection.

(13.4) Theorem. Choose notations as at the beginning of the section. Then Im x

is a perfect B[X]-module except for the case in which r ≥ n and m > n, and Coker x is

an almost perfect B[X]-module, i.e.

grade Coker x ≥ pd Coker x ’ 1.

Proof: Assume ¬rst that r ≥ min(m, n), the case in which Ir+1 (X) = 0. If m ¤ n,

then Im x is free and pd Coker x = 1. In case m > n we obtain the (almost) perfection

of Coker x from (2.16).

Suppose now that r < min(m, n). By Proposition (13.2) Coker x is a torsionfree R-

module, so grade Coker x = grade R (over B[X]). Therefore the second assertion follows

from the ¬rst one via the exact sequence 0 ’ Im x ’ R n ’ Coker x ’ 0.

Because of (3.3) we only have to prove that Im x is generically perfect and that

Coker x is Z-¬‚at in case B = Z. Since Im x is a graded torsionfree R-module, it is a free

Z-module. The same is true for Coker x because of (13.2). It remains to show that Im x

is perfect if B = Z. This is equivalent with the fact that (Im x)P is a Cohen-Macaulay

module over RP for all prime ideals P ‚ R (cf. 16.19), or that

depth CP ≥ dim RP ’ 1,

C = Coker x. In view of (13.2),(b) it will be enough that

depth(R(X; δ)/J)P ≥ dim RP ’ 2

for all prime ideals P ‚ R, P ∈ Supp(R(X; δ)/J), where J is the ideal in R(X; δ)

generated by the residue classes of the elements [1, . . . , r|1, . . . , k, . . . , r + 1], 1 ¤ k ¤ r.

It is easy to check that R(X; δ)/J = B[X]/„¦B[X], „¦ being the ideal in ∆(X) cogenerated

by the elements

δ1 = [1, . . . , r ’ 1, r + 1|1, . . . , r ’ 1, r + 1], δ2 = [1, . . . , r|1, . . . , r ’ 1, r + 2].

By (5.19) R(X; δ)/J is a Cohen-Macaulay ring, and obviously dim R(X; δ)/J = dim R’2.

The proof of (13.4) is complete now. ”

165

B. The Perfection of a Generic Module

(13.5) Remarks. (a) It is a simple but noteworthy fact that under the assumptions

of (13.4) all R-syzygies of Coker x are perfect B[X]-modules along with Im x.

(b) In [Br.7] the following generalized version of Theorem (13.4) has been stated: Let

A be a noetherian ring and u = (uij ) be an m — n matrix of elements in A. Suppose that

0 ¤ r ¤ min(m, n’1) and that Ir+1 (u) has (the maximally possible) grade (m’r)(n’r).

Denote by R the residue class ring A/Ir+1 (u). Let u : Rm ’ Rn be the map given by

the matrix of the residue classes of the elements uij . Assume further that Ir (u) contains

an element which is not a zero-divisor of R. Then Im u and hence all higher syzygies of

Coker u are perfect A-modules.

To prove this, one has only to change the arguments which reduce the general case

to the generic one: Let X = (Xij ) be an m — n matrix of indeterminates over Z. Then

A is a Z[X]-algebra via the substitution Xij ’ uij . Put S = Z[X]/Ir+1 (X) and denote

by x : S m ’ S n the map given by the matrix of the residue classes of the elements Xij .

Consider the natural surjection h : Im x — R ’’ Im(x — R). By the assumption on Ir (u)

one obtains that h — RP is an isomorphism of free RP -modules of positive rank whenever

P is an associated prime ideal of R. By (3.5) we can derive that Im x — R = Im x — A

is a perfect A-module. But then Im x — R is necessarily a torsionfree R-module and

consequently h is an isomorphism. ”

B. The Perfection of a Generic Module

To get further information on the generic module, we consider the following (more