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B[X]/B /Ir+1 (X)„¦B[X]/B ’’ „¦R/B .

For technical reasons we shall primarily deal with M .
To get a lower bound for grade(I1 (X), M ) we shall construct a (¬nite) ¬ltration of
M , the quotients of which are isomorphic to direct sums of certain good-natured ideals
in the rings R(X; δ), δ = [1, . . . , r ’ 1, s|1, . . . , r ’ 1, t]. These ideals are investigated in
the ¬rst subsection. The second deals with the ¬ltration. It is not hard to see that the
lower bound obtained for grade(I1 (X), „¦1 ) is an upper bound, too. Finally we shall
R/B
1
discuss the syzygetic behaviour of „¦R/B .
175
A. Perfection and Syzygies of Some Determinantal Ideals

A. Perfection and Syzygies of Some Determinantal Ideals
Let s, t denote integers such that r ¤ s ¤ m, r ¤ t ¤ n. We put

δst = [1, . . . , r ’ 1, s|1, . . . , r ’ 1, t].

The ideal in R(X; δst ), we are interested in, is generated by the residue classes of all
r-minors [a1 , . . . , ar |b1 , . . . , br ] of X such that ar = s, br = t. More formally one can give
the following description: Consider the ideal

± ∈ ∆(X) : ± ≥ δs+1,t and ± ≥ δs,t+1

in ∆(X) cogenerated by δs+1,t , δs,t+1 (cf. Section 5; of course we allow the extreme cases
δm+1,t = δs,n+1 = [1, . . . , r ’ 1|1, . . . , r ’ 1].) This generates an ideal I(X; δs+1,t , δs,t+1 )
in B[X]. The quotient

I(x; δs+1,t , δs,t+1 ) = I(X; δs+1,t , δs,t+1 )/I(X; δst )

is the ideal in R(X; δst ) we have in mind. The special case in which s = r, t = r + 1 has
already been treated in Section 13. In accordance with the notation just introduced we
put
∆(X; δs+1,t , δs,t+1 ) = γ ∈ ∆(X) : γ ≥ δs+1,t or γ ≥ δs,t+1 .
(14.1) Proposition. Choose notations as above. Then I(X; δs+1,t , δs,t+1 ) is a per-
fect ideal of B[X]. Furthermore

rk ∆(X; δs+1,t , δs,t+1 ) = rk ∆(X; δst ) ’ 1.

Proof: The minimal elements in ∆(X; δs+1,t , δs,t+1 ) are exactly

δs+1,t , δs,t+1 if s < m, t < n,
s ¤ m, t = n,
δs+1,t if
s = m, t ¤ n.
δs,t+1 if

In any case these elements are upper neighbours of δst , which is the only minimal element
of ∆(X; δst ). Hence I(X; δs+1,t , δs,t+1 ) is perfect in view of (5.19), and the rank formula
is obvious. ”
For an application in the next subsection we need a description of the ¬rst syzygy of
I(x; δs+1,t , δs,t+1 ) as an R(X; δst )-module. The following proposition generalizes (13.3):
(14.2) Proposition. Let s, t be integers such that r ¤ s ¤ m, r ¤ t ¤ n. We
put δst = [1, . . . , r ’ 1, s|1, . . . , r ’ 1, t] as above and R = R(X; δst ). Let y1 , . . . , ys and
z1 , . . . , zt be the canonical bases of Rs , Rt , and consider Rs’1 , (Rt’1 )— as submodules
of Rs , (Rt )— generated by y1 , . . . , ys’1 and z1 , . . . , zt’1 resp. Denote by x : Rs ’ Rt the
— —

map given by the s — t matrix which arises from X by cancelling the last m ’ s rows and
the last n ’ t columns. Let
r’1 r’1
s’1
(Rt’1 )— ’’ R,

•: R
176 14. The Module of K¨hler Di¬erentials
a

be the composition of the map
r’1 r’1 r r
s’1 t’1 — s
(Rt )— , — —
— ) ’’ R— yI — zJ ’’ yI∪{s} — zJ∪{t}
R (R

and •x,r . Then the kernel of • is generated by the elements

σ(i, I\i)[i|j] yI\i — zJ , I ∈ S(r, s’1), j ∈ S(1, n), J ∈ S(r’1, t’1),
i∈I

and

σ(j, J\j)[i|j] yI — zJ\j , J ∈ S(r, t’1), i ∈ S(1, m), I ∈ S(r’1, s’1).
j∈J

Proof: Let N be the submodule generated by these elements. Obviously

•(yI — zJ ) = [I, s|J, t],

I ∈ S(r’1, s’1), J ∈ S(r’1, t’1), so N ‚ Ker •. Proposition (5.6),(b) provides the
proof of the opposite inclusion: Put

[I, s|J, t] : I ∈ S(r’1, s’1), J ∈ S(r’1, t’1) .
Ψ=

Ψ is an ideal in ∆(X; δst ). Let [I, s|J, t] ∈ Ψ, [K|L] ∈ ∆(X; δst ) such that [K|L] ≥
[I, s|J, t]. We claim that

— —
[K|L] yI — zJ ∈ N + R yU — zV .
[U |V ]<[I|J]

To show this we write [K|L] = [k1 , . . . , ku |l1 , . . . , lu ], [I|J] = [i1 , . . . , ir’1 |j1 , . . . , jr’1 ].
By assumption there is a ρ ¤ r ’ 1 such that kρ < iρ or lρ < jρ . Suppose that kρ < iρ
for some ρ. Denote by σ the smallest such index.
If σ = u, then we put K = {i1 , . . . , iu’1 }, I = {k1 , . . . , ku , iu , . . . , ir’1 }. From the
lemma below (with F = Rs’1 , G = Rt’1 , f the matrix x decreased by its last row and
its last column, v = r ’ 1) we obtain that

— — —
•u (yK — y˜ — zL ) — zJ = σ(U, I\U )[U |L] yK § y˜\U — zJ ∈ N .
˜ ˜
I I
U ∈S(u,˜)
I

Since [K, I\U |J] < [I|J] for all U ∈ S(u, I) such that U = K and [K, I\U |J] = 0, the
claim follows at once.
If σ < u, then the inductive hypothesis on u yields

— —
[K\ku |L\lρ ] yI — zJ ∈ N + R yU — zV
[U |V ]<[I|J]

for ρ = 1, . . . , u. But [K|L] yI — zJ is a linear combination of the elements on the left
hand side, so we are done in this case, too.
Clearly the proof runs analogously if lρ < jρ for some ρ. ”
177
B. The Lower Bound for the Depth of the Di¬erential Module

(14.3) Lemma. Let A be an arbitrary ring, f : F ’’ G a homomorphism of A-
modules, and u, v integers such that 1 ¤ u ¤ v + 1. Consider the map
u’1 v+1 u v

F— F— G ’’
•u : F

given by
— —
•u (wK — yI — zL ) = wK § •f,u (yI — zL ),
wK = wk1 § . . . § wku’1 , yI = yi1 § . . . § yiv+1 , zL = zl1 § . . . § zlu , wkρ , yiσ ∈ F , zl„ ∈ G— .
— — — —

Then Im •u ‚ Im •1 .
Proof: We use induction on u. There is nothing to prove for u = 1. Let u ≥ 1,
wK = wk1 § . . . § wku , yI = yi1 § . . . § yiv+1 , zL = zl1 § . . . § zlu+1 , wkρ , yiσ ∈ F , zl„ ∈ G— .
— — — —

Then

•u+1 (wK — yI — zL )
u

f (yU )) wK § yI\U
= σ(U, I\U ) zL (
U ∈S(u+1,I)
u’1
— —
σ(U, I\U )σ(i, U \i) zl1 (f (yi )) zL\l1 ( f (yU \i )) wK § yI\U
=
U ∈S(u+1,I) i∈U
u’1
u — —
σ(U \i, I\(U \i))σ(i, I\U ) zl1 (f (yi )) zL\l1 ( f (yU \i )) wK § yI\U
= (’1)
U ∈S(u+1,I) i∈U
u’1
— —
=± f (yV )) wK § y(I\V )\i
σ(V, I\V )σ(i, (I\V )\i) zl1 (f (yi )) zL\l1 (
V ∈S(u,I) i∈I\V
u’1
— —
=± σ(i, (I\V )\i) zl1 (f (yi )) wK § y(I\V )\i
σ(V, I\V ) zL\l1 ( f (yV ))
V ∈S(u,I) i∈I\V
u’1
— —
≡± σ(i, K\i) zl1 (f (wi )) wK\i § yI\V
σ(V, I\V ) zL\l1 ( f (yV )) mod Im •1
i∈K
V ∈S(u,I)
— —
≡± σ(i, K\i) zl1 (f (wi )) •u (wK\i — yI — zL\l1 ) mod Im •1
i∈K
≡ 0 mod Im •1 by the inductive hypothesis. ”

B. The Lower Bound for the Depth of the Di¬erential Module
The R-module M considered in the introduction is generated by the residue classes
d± modulo Ir+1 (X)„¦1 B[X]/B of the elements d± where d is the universal B-derivation of
B[X] and ± runs through the (r + 1)-minors of X. To have a simpler notation we shall
write d± instead of d±. If we identify „¦1 1 m n—
B[X]/B /Ir+1 (X)„¦B[X]/B with R — (R ) via
the map

dXij ’’ yi — zj ,
178 14. The Module of K¨hler Di¬erentials
a

y1 , . . . , ym and z1 , . . . , zn being the canonical bases of Rm , Rn resp., then clearly

σ(i, I\i)σ(j, J\j)[I\i|J\j] yi — zj
d([I|J]) =
i∈I
j∈J

= •x,r (yI — zJ )

for all I ∈ S(r+1, m), J ∈ S(r+1, n), the map x : Rm ’ Rn given by the matrix X
modulo Ir+1 (X).
We start with a simple observation concerning the free locus of „¦. It may also be
derived from (2.6).
(14.4) Proposition. Let P be a prime ideal of R. Then MP is a free direct sum-
mand of RP — (RP )— (of rank (m ’ r)(n ’ r)) if and only if P ⊃ Ir (X)/Ir+1 (X).
m n

Proof: The less trivial “if” part is an immediate consequence of the following
general fact: Let r be a nonnegative integer, F , G modules over an arbitrary ring A
and f : F ’ G an A-homomorphism such that Im f is a free direct summand of G and
rk f = r. Then the image of the map

r+1 r+1
G— ’’ F — G—
F—
•f,r :

is the direct summand Ker f — Ker f — of F — G— . ”
Let r ¤ s < m, r ¤ t < n, and put

M (s, t) = submodule of M generated by all d±,
± = [a1 , . . . , ar+1 |b1 , . . . , br+1 ], (ar , br ) (s, t)

(“ ” means “lexicographically ¤”). Clearly {M (s, t) : r ¤ s < m, r ¤ t < n} gives
an increasing ¬ltration of M if the pairs (s, t) are ordered lexicographically. Next we
consider the quotients of this ¬ltration:
±
 M (r, r) if s = t = r

M (s, t) = M (s, t)/M (s, t’1) if t > r

M (s, r)/M (s’1, n’1) if s < r, t = r.

(14.5) Proposition. Put δst = [1, . . . , r ’ 1, s|1, . . . , r ’ 1, t] whenever r ¤ s < m,
r ¤ t < n, and choose notations as above. Then
(a) AnnR M(s, t) = I(X; δst )/Ir+1 (X).
(b) As an R(X; δst )-module M (s, t) is isomorphic to the (m ’ s)(n ’ t)-fold direct sum
of the ideal I(x; δs+1,t , δs,t+1 ).
Proof: Let (s, t) be such that r ¤ s < m, r ¤ t < n. Looking at R s’1 , Rt’1 as
submodules of Rm , Rn generated by y1 , . . . , ys’1 and z1 , . . . , zt’1 resp., we consider the
map
r’1 r’1
s’1
(Rt’1 )— ’’ M(s, t)

• = •(s, i; t, j) : R
179
B. The Lower Bound for the Depth of the Di¬erential Module

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