is not surjective, P denoting the ideal in R generated by the elements of the ¬rst row of

x. (Remember that P m’n is a canonical module of R in the case under consideration.)

Put s = m’n and denote by y1 , . . . , ym and z1 , . . . , zn the canonical bases of Rm and Rn

resp. Looking at the sequence (3) in (13.6) we see that an element of HomR (D, R/P s )

comes from an element

m

—

bi,J zJ — yi , bi,J ∈ R,

i=1 J∈S(2,n)

such that for 1 ¤ j ¤ n

m

σ(j, J\j)bi,J [i|J\j] ∈ P s ,

i=1 J∈S(2,n)

j∈J

and every such element induces an element of HomR (D, R/P s ). Now we put

xs’1 if i = 1, J = {1, 2},

11

bi,J =

0 otherwise.

Then i,J bi,J zJ — yi obviously induces an element β of HomR (D, R/P s ). β lies in the

—

—

image of h if and only if there is an element i,J ai,J zJ — yi , ai,J ∈ R, such that for

1¤j¤n

m

(6) σ(j, J\j)ai,J [i|J\j] = 0,

i=1 J∈S(2,n)

j∈J

and bi,J ’ ai,J ∈ P s , 1 ¤ i ¤ m, J ∈ S(2, n). But there is no such element: (6) in

particular implies that a1,{1,2} is contained in the ideal I of R generated by the elements

of the last m ’ 1 rows of x and x13 , . . . , x1n . Obviously xs’1 ∈ P s + I.

11 /

Since C is perfect if m ≥ n, D must be a second syzygy in this case. Now the rest

is mutatis mutandis a copy of the last part of the proof given for (13.8). ”

(13.11) Remarks. (a) Actually we proved in (13.8) and (13.10) resp. that for any

prime ideal P of R which contains Ir (x) (i) in case m < n the module (Coker x)P and (ii)

in case m > n the module (Coker ψ)P is not perfect over the corresponding localization

of B[X].

(b) Theorem (13.8) allows a generalization analogous with that of (13.4) (cf. [Br.7]):

Let A; u, r, R, u be as in (13.5),(b) and assume that grade Ir+1 (u) = (m ’ r)(n ’ r). If

m ≥ n then Coker u and hence all syzygies of Coker u in an R-free resolution of Coker u

are perfect A-modules. If m < n, Ir (u) = R and Ir (u) contains an element which is not

a zero-divisor of R, then Coker u is not a perfect A-module.

Let x be as in the proof of (13.5),(b). Since Coker u = Coker(x — A) = Coker x — A,

the ¬rst assertion follows immediately from (13.8). To prove the second, one shows

that for any P ∈ Spec R, P ⊃ Ir (u), Coker u — RP is not a perfect module over the

corresponding localization of A. In doing so one may directly assume that R and A are

171

C. Homological Properties of Generic Modules

local, P being the maximal ideal of R. The preimage Q of P in Z[X] contains Ir (X).

Put C = Coker x. From (13.4) we obtain pd CQ = grade CQ + 1 over Z[X]Q . Let

ft

F : 0 ’’ Ft ’’ Ft’1 ’’ · · · ’’ F0

be a minimal Z[X]Q -free resolution of CQ . Since Coker u = C — A has positive rank over

R, we have grade Coker u = grade Ir+1 (u) = t ’ 1. So it su¬ces to show that F — A

is a minimal A-free resolution of C — A. For every prime ideal I of Z[X] such that

I ⊃ Ir+1 (X) the complex F — Z[X]I is split-acyclic. Hence F — A is split-acyclic at all

prime ideals having grade smaller than t ’ 1. The map ft ¬nally splits at all primes

I ⊃ Ir (X) since CI is free and thus a perfect module over Z[X]I . Consequently ft — A

splits at all prime ideals of A whose grade is smaller than t. From (16.16) it follows that

F — A is acyclic. The extension Z[X]Q ’ A being local, F — A is a minimal A-free

resolution of C — A. ”

C. Homological Properties of Generic Modules

In this section we investigate some homological properties of Coker x where x is as

in the introduction. We start with a simple observation concerning projective dimension.

(13.12) Proposition. Coker x has ¬nite projective dimension as an R-module if

and only if r = 0 or r ≥ min(m, n). In case r ≥ min(m, n) one has pd R Coker x = 1 if

m < n and pdR Coker x = grade In (X) = m ’ n + 1 otherwise. If 1 ¤ r < min(m, n) and

P is a prime ideal in R then the following properties are equivalent:

(i) pdRP (Coker x)P < ∞.

(ii) (Coker x)P is a free RP -module.

(iii) P ⊃ Ir (x).

Proof: The essential part of the second statement has been proved in Section 2.

Thus the “if”-part of the ¬rst one is clear. The “only if”-part is an immediate conse-

quence of the third assertion. In case 1 ¤ r < min(m, n), P ⊃ Ir (x) if and only if

n

(Im x)P is a free direct summand of RP (cf. (16.3)), so (ii) is equivalent to (iii). Fur-

thermore pdRP (Coker x)P < ∞ implies pdRP (Coker x)P ¤ 1 because of (13.4). Thus

(Ker x)P = (Coker x— )— is free. The same is true for (Coker x— )P since Coker x— is

P

re¬‚exive. Consequently x— splits as well as xP , and (Coker x)P is free. ”

P

There is a sharp trichotomy concerning the homological properties of Coker x be-

tween the cases m = n, m < n and m > n which is not immediately apparent from the

former considerations. There is nothing to say in case r = 0. If r ≥ m and m < n then

Coker x is an (n’m)-th syzygy but not an (n’m+1)-th one, and Ext1 (Coker x, R) = 0.

R

In case r ≥ n and m ≥ n, Coker x is an R-torsion module which is perfect as an R-module

and therefore Exti (Coker x, R) = 0 for i = 1, . . . , m’n, Extm’n+1 (Coker x, R) = 0. The

R R

remaining cases are more interesting. We start with m = n:

(13.13) Theorem. Suppose that 1 ¤ r < m = n. Then Exti (Coker x, R) =

R

i —

ExtR ((Coker x) , R) = 0 for all i ≥ 1. In particular Coker x is an in¬nite syzygy module,

i.e. there is an in¬nite exact sequence

0 ’’ Coker x ’’ F1 ’’ · · · ’’ Ft ’’ Ft+1 ’’ . . .

with free R-modules Fi .

172 13. Generic Modules

Proof: In case B = Z the ring R is Gorenstein. Since C = Coker x is a maximal

Cohen-Macaulay module, Exti (C, R) = Exti (C — , R) = 0 for all i ≥ 1. Let

R R

ft f1

F : ’’ Ft ’’ · · · ’’ F1 ’’ F0

be an R-free resolution of C — . Then F — is acyclic. By the usual argumentation based

on Z-¬‚atness we obtain that F —Z B is an R —Z B-free resolution of C — —Z B and that

F — —Z B is exact for every (noetherian) ring B. This implies Exti (C — , R) = 0 for all

R

i ≥ 1 in the general case. One equally gets Exti (C, R) = 0 for all i ≥ 1. ”

R

In case m = n the homological invariants of Coker x turn out to be grade-sensitive

with respect to the ideal Ir (x).

(13.14) Theorem. Suppose that 1 ¤ r < min(m, n) and put s = grade Ir (x). Then

in case

(a) m > n:

(i) Coker x is an s-th syzygy but not an (s + 1)-th syzygy.

(ii) Exti (Coker x, R) = 0 for i = 1, . . . , s ’ 1, Exts (Coker x, R) = 0.

R R

(b) m < n:

(i) Coker x is an (s ’ 1)-th syzygy but not an s-th syzygy.

(ii) Exti (Coker x, R) = 0 for i = 1, . . . , s, Exts+1 (Coker x, R) = 0.

R R

Proof: First we use (16.32) to get that (a),(i) is equivalent to (b),(ii) and (b),(i)

equivalent to (a),(ii): This holds since D(Coker x) = Coker x— .

Part (i) of (b) is an easy consequence of (13.4) and (13.11),(a): Since Coker x is

almost perfect, depth(Coker x)P ≥ s’1 for all P ∈ Spec R, P ⊃ Ir (x), and if depth RP =

s for such a prime ideal then depth(Coker x)P = s ’ 1 because (Coker x)P is not perfect.

Finally we repeat that in any case (Coker x)P is free whenever P ∈ Spec R, P ⊃ Ir (x).

Similarly part (i) of (a) follows from (13.8) and (13.11),(a): Coker x is perfect, so

depth(Coker x)P ≥ min(s, depth RP )

and consequently Coker x is an s-th syzygy. If it would be an (s + 1)-th one, then

Exti (D(Coker x), R) = 0 for i = 1, . . . , s + 1 (cf. (16.34)). But D(Coker x) = Coker x —

R

has a free resolution

r+1 r

•x— ,r x—

n—

Rm ’ ’ (Rn )— ’’ (Rm )—

· · · ’’ (R ) — ’’

by (13.6), so (Coker x— )— = Ker x would be an (s + 3)-th and Coker ψ (cf. (3)) an s-th

syzygy. This is impossible since depth(Coker ψ)P = s ’ 1 for all P ∈ Spec R such that

P ⊃ Ir (x) and depth RP = s. ”

(13.15) Remark. (13.12), (13.13) and (13.14) have obvious generalizations to the

case considered in (13.5),(b) and (13.11),(b). Details may be found in [Br.7] or left to

the reader. ”

(13.16) Remark. We conclude the section with a few observations concerning

B[X]-free resolutions of Im x and Coker x. The complexity to construct such a resolu-

tion (which should be minimal), is certainly comparable to the complexity of the cor-

responding problem for determinantal ideals. Besides the case in which r ≥ min(m, n)

173

D. Comments and References

the maximal minor case seems to be the only one for which results are available. Let

r + 1 = min(m, n) in the following.

(i) If m ≥ n then a minimal B[X]-free resolution of Coker x is given by the complex

D1 (x) constructed in Section 2 (cf. (2.16)), since Coker x = Coker X in this case. To get

such a resolution for Im x, we use the following observation: Let A be a commutative

ring, f : F ’ G a homomorphism of free A-modules, r a non-negative integer, and

r+1 r

F — G— ’ F the homomorphism de¬ned in Subsection B. Then Ir+1 (f )F ‚

•f,r :

Im •f,r . (The easy proof is left to the reader). Applied to the special situation just

considered it yields an isomorphism

Coker •X,r ∼ Im x.

=

A candidate for a minimal B[X]-free resolution of Coker •X,r can be found in [BE.4],

1,r+1

p. 270: It is not hard to check that the complex L1 X de¬ned there is acyclic and its

map d1 is nothing but •X,r .

(ii) In case m ¤ n we observe that the kernel of the map g : B[X]m ’’ Rn induced

by X is generated by the elements

m

(’1)i+1 [I\i|J]yi , I = {1, . . . , m}, J ∈ S(r, n),

i=1

y1 , . . . , ym being the canonical basis of B[X]m (cf. the observation made in (i).) Next

we consider the isomorphism

m’1

h

(B[X]m )— ’’ B[X]m ,

m’1

where h(y — )(z — ) is the coe¬cient of y — §z — with respect to y1 §. . .§ym , y — ∈

— —

(B[X]m )— ,

m’1

z — ∈ (B[X]m )— . One readily checks that Ker g = h(Im X — ), so Im x is isomorphic

m’1

X — . Corollary 3.2 in [BE.4] provides a minimal B[X]-free resolution of

with Coker

m’1

X — . One may use the resolution of Im x just mentioned and the resolution of R

Coker

given in Section 2 (cf (2.16)) to get a resolution of Coker x by constructing the mapping

cylinder of a chain map induced by X. The resolution of Coker x thus obtained is not

minimal, not even if m < n, the case in which it has minimal length; in fact, already

the system of generators of the ¬rst syzygy module turns out to be non-minimal. The

resolution of Coker x constructed in [Av.2], Proposition 7, is for the same reason not

minimal. ”

D. Comments and References

References to the results of this section in case r ≥ min(m, n) have been given in

Section 2. The main content is taken from [Br.7]. For the perfection of the generic

module (cf. (13.8)) and its homological properties (cf. Subsection C) this applies also

to the method of proof. There is a di¬erence in demonstrating Theorem (13.4): While

we use a simple ¬ltration argument (cf. (13.2)) and the results of Section 5 concerning

wonderful posets, in [Br.7] the inductive methods of Hochster and Eagon (cf. [HE.2] and

Section 12) have been exploited to obtain the perfection of Im x. Additional literature

to the subject treated in (13.16) may be found in 2.E.

14. The Module of K¨hler Di¬erentials

a

Throughout this section X = (Xij ) is an m — n matrix of indeterminates over a

noetherian ring B, r an integer such that 1 ¤ r < min(m, n) and R = Rr+1 (X) =

B[X]/Ir+1 (X). We shall investigate the module „¦1 R/B of (K¨hler) di¬erentials of R/B.

a

(The reader who wants detailed information about this concept and its importance in

local algebra is referred to the books of Kunz [Kn] or Scheja [Sch].) We are mainly

interested in computing grade(I1 (X), „¦1 ). For this purpose the special structure of

R/B

the poset ∆(X) together with the general results on ASLs of Section 5 will be found very

useful, once more.

2

The module „¦1 R/B is closely related to Ir+1 (X)/Ir+1 (X) via the exact sequence

2

Ir+1 (X)/Ir+1 (X) ’’ „¦1 1 1

B[X]/B /Ir+1 (X)„¦B[X]/B ’’ „¦R/B ’’ 0

which can be improved to

(2)

’’ „¦1 1 1

0 ’’ Ir+1 (X)/Ir+1 (X) B[X]/B /Ir+1 (X)„¦B[X]/B ’’ „¦R/B ’’ 0

if B is a domain, so computing grade(I1 (X), „¦1 ) means to compute the grade of I1 (X)

R/B

with respect to Ir+1 (X)/Ir+1 (X)(2) in this case. Thus our investigations are connected

(2)

with Proposition (10.8) which gives a lower bound for grade(I1 (X), Ir+1 (X)/Ir+1 (X) ).

We shall see that this bound is not sharp except for the extreme cases in which m = n =

r + 1 or r = 1.

Of course the computation of grade(I1 (X), „¦1 ) is equivalent to the computation

R/B

of grade(I1 (X), M ), in general, M denoting the kernel of the projection

„¦1 1 1