<< Предыдущая стр. 34(из 47 стр.)ОГЛАВЛЕНИЕ Следующая >>
170 13. Generic Modules

is not surjective, P denoting the ideal in R generated by the elements of the п¬Ѓrst row of
x. (Remember that P mв€’n is a canonical module of R in the case under consideration.)
Put s = mв€’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
в€—
bi,J zJ вЉ— yi , bi,J в€€ R,
i=1 Jв€€S(2,n)

such that for 1 в‰¤ j в‰¤ n
m
Пѓ(j, J\j)bi,J [i|J\j] в€€ P s ,
i=1 Jв€€S(2,n)
jв€€J

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

xsв€’1 if i = 1, J = {1, 2},
11
bi,J =
0 otherwise.

Then i,J bi,J zJ вЉ— yi obviously induces an element ОІ of HomR (D, R/P s ). ОІ lies in the
в€—
в€—
image of h if and only if there is an element i,J ai,J zJ вЉ— yi , ai,J в€€ R, such that for
1в‰¤jв‰¤n
m
(6) Пѓ(j, J\j)ai,J [i|J\j] = 0,
i=1 Jв€€S(2,n)
jв€€J

and bi,J в€’ ai,J в€€ P s , 1 в‰¤ i в‰¤ m, J в€€ 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 в€’ 1 rows of x and x13 , . . . , x1n . Obviously xsв€’1 в€€ P s + I.
11 /
Since C is perfect if m в‰Ґ 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). вЂ”
(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 П€)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 в€’ r)(n в€’ r). If
m в‰Ґ 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 вЉ— A) = Coker x вЉ— A,
the п¬Ѓrst assertion follows immediately from (13.8). To prove the second, one shows
that for any P в€€ Spec R, P вЉѓ Ir (u), Coker u вЉ— 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 в€’в†’ Ft в€’в†’ Ftв€’1 в€’в†’ В· В· В· в€’в†’ F0
be a minimal Z[X]Q -free resolution of CQ . Since Coker u = C вЉ— A has positive rank over
R, we have grade Coker u = grade Ir+1 (u) = t в€’ 1. So it suп¬ѓces to show that F вЉ— A
is a minimal A-free resolution of C вЉ— A. For every prime ideal I of Z[X] such that
I вЉѓ Ir+1 (X) the complex F вЉ— Z[X]I is split-acyclic. Hence F вЉ— A is split-acyclic at all
prime ideals having grade smaller than t в€’ 1. The map ft п¬Ѓnally splits at all primes
I вЉѓ Ir (X) since CI is free and thus a perfect module over Z[X]I . Consequently ft вЉ— A
splits at all prime ideals of A whose grade is smaller than t. From (16.16) it follows that
F вЉ— A is acyclic. The extension Z[X]Q в†’ A being local, F вЉ— A is a minimal A-free
resolution of C вЉ— A. вЂ”

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 п¬Ѓnite projective dimension as an R-module if
and only if r = 0 or r в‰Ґ min(m, n). In case r в‰Ґ min(m, n) one has pd R Coker x = 1 if
m < n and pdR Coker x = grade In (X) = m в€’ n + 1 otherwise. If 1 в‰¤ r < min(m, n) and
P is a prime ideal in R then the following properties are equivalent:
(i) pdRP (Coker x)P < в€ћ.
(ii) (Coker x)P is a free RP -module.
(iii) P вЉѓ Ir (x).
Proof: The essential part of the second statement has been proved in Section 2.
Thus the вЂњifвЂќ-part of the п¬Ѓrst one is clear. The вЂњonly ifвЂќ-part is an immediate conse-
quence of the third assertion. In case 1 в‰¤ r < min(m, n), P вЉѓ 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 < в€ћ implies pdRP (Coker x)P в‰¤ 1 because of (13.4). Thus
(Ker x)P = (Coker xв€— )в€— is free. The same is true for (Coker xв€— )P since Coker xв€— is
P
reп¬‚exive. Consequently xв€— splits as well as xP , and (Coker x)P is free. вЂ”
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 в‰Ґ m and m < n then
Coker x is an (nв€’m)-th syzygy but not an (nв€’m+1)-th one, and Ext1 (Coker x, R) = 0.
R
In case r в‰Ґ n and m в‰Ґ 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, . . . , mв€’n, Extmв€’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 в‰¤ r < m = n. Then Exti (Coker x, R) =
R
i в€—
ExtR ((Coker x) , R) = 0 for all i в‰Ґ 1. In particular Coker x is an inп¬Ѓnite syzygy module,
i.e. there is an inп¬Ѓnite exact sequence
0 в€’в†’ Coker x в€’в†’ F1 в€’в†’ В· В· В· в€’в†’ Ft в€’в†’ Ft+1 в€’в†’ . . .
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 в€— , R) = 0 for all i в‰Ґ 1. Let
R R

ft f1
F : в€’в†’ Ft в€’в†’ В· В· В· в€’в†’ F1 в€’в†’ F0

be an R-free resolution of C в€— . Then F в€— is acyclic. By the usual argumentation based
on Z-п¬‚atness we obtain that F вЉ—Z B is an R вЉ—Z B-free resolution of C в€— вЉ—Z B and that
F в€— вЉ—Z B is exact for every (noetherian) ring B. This implies Exti (C в€— , R) = 0 for all
R
i в‰Ґ 1 in the general case. One equally gets Exti (C, R) = 0 for all i в‰Ґ 1. вЂ”
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 в‰¤ 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 в€’ 1, Exts (Coker x, R) = 0.
R R
(b) m < n:
(i) Coker x is an (s в€’ 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 xв€— .
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 в‰Ґ sв€’1 for all P в€€ Spec R, P вЉѓ Ir (x), and if depth RP =
s for such a prime ideal then depth(Coker x)P = s в€’ 1 because (Coker x)P is not perfect.
Finally we repeat that in any case (Coker x)P is free whenever P в€€ Spec R, P вЉѓ Ir (x).
Similarly part (i) of (a) follows from (13.8) and (13.11),(a): Coker x is perfect, so

depth(Coker x)P в‰Ґ 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 в€—
R
has a free resolution
r+1 r
П•xв€— ,r xв€—
nв€—
Rm в€’ в€’ (Rn )в€— в€’в†’ (Rm )в€—
В· В· В· в€’в†’ (R ) вЉ— в€’в†’

by (13.6), so (Coker xв€— )в€— = Ker x would be an (s + 3)-th and Coker П€ (cf. (3)) an s-th
syzygy. This is impossible since depth(Coker П€)P = s в€’ 1 for all P в€€ Spec R such that
P вЉѓ Ir (x) and depth RP = s. вЂ”
(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. вЂ”
(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 в‰Ґ 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 в‰Ґ 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 в†’ G a homomorphism of free A-modules, r a non-negative integer, and
r+1 r
F вЉ— Gв€— в†’ F the homomorphism deп¬Ѓned in Subsection B. Then Ir+1 (f )F вЉ‚
П•f,r :
Im П•f,r . (The easy proof is left to the reader). Applied to the special situation just
considered it yields an isomorphism
Coker П•X,r в€ј Im x.
=
A candidate for a minimal B[X]-free resolution of Coker П•X,r can be found in [BE.4],
1,r+1
p. 270: It is not hard to check that the complex L1 X deп¬Ѓned there is acyclic and its
map d1 is nothing but П•X,r .
(ii) In case m в‰¤ n we observe that the kernel of the map g : B[X]m в€’в†’ Rn induced
by X is generated by the elements
m
(в€’1)i+1 [I\i|J]yi , I = {1, . . . , m}, J в€€ 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
mв€’1
h
(B[X]m )в€— в€’в†’ B[X]m ,
mв€’1
where h(y в€— )(z в€— ) is the coeп¬ѓcient of y в€— в€§z в€— with respect to y1 в€§. . .в€§ym , y в€— в€€
в€— в€—
(B[X]m )в€— ,
mв€’1
z в€— в€€ (B[X]m )в€— . One readily checks that Ker g = h(Im X в€— ), so Im x is isomorphic
mв€’1
X в€— . Corollary 3.2 in [BE.4] provides a minimal B[X]-free resolution of
with Coker
mв€’1
X в€— . 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 п¬Ѓrst 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. вЂ”

D. Comments and References
References to the results of this section in case r в‰Ґ 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 diп¬Ђerence in demonstrating Theorem (13.4): While
we use a simple п¬Ѓltration 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 KВЁhler Diп¬Ђerentials
a

Throughout this section X = (Xij ) is an m Г— n matrix of indeterminates over a
noetherian ring B, r an integer such that 1 в‰¤ r < min(m, n) and R = Rr+1 (X) =
B[X]/Ir+1 (X). We shall investigate the module в„¦1 R/B of (KВЁhler) diп¬Ђerentials 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), в„¦1 ). For this purpose the special structure of
R/B
the poset в€†(X) together with the general results on ASLs of Section 5 will be found very
useful, once more.
2
The module в„¦1 R/B is closely related to Ir+1 (X)/Ir+1 (X) via the exact sequence

2
Ir+1 (X)/Ir+1 (X) в€’в†’ в„¦1 1 1
B[X]/B /Ir+1 (X)в„¦B[X]/B в€’в†’ в„¦R/B в€’в†’ 0

which can be improved to

(2)
в€’в†’ в„¦1 1 1
0 в€’в†’ Ir+1 (X)/Ir+1 (X) B[X]/B /Ir+1 (X)в„¦B[X]/B в€’в†’ в„¦R/B в€’в†’ 0

if B is a domain, so computing grade(I1 (X), в„¦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), в„¦1 ) is equivalent to the computation
R/B
of grade(I1 (X), M ), in general, M denoting the kernel of the projection

в„¦1 1 1
 << Предыдущая стр. 34(из 47 стр.)ОГЛАВЛЕНИЕ Следующая >>