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