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170 13. Generic Modules

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

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