using the ¬rst inclusion of (a). So

aµ ∈ R, a ∈ µ · I(x; ζj ).

a= aµ µ + a,

µ∈Ψ

µ∈Ψ\{„ }

Applying the inductive hypothesis we get the result for s = 1.

Let s > 1. From (12) and the inductive hypothesis on s we obtain

Qs’2 [I|H] + Qs’1 I(x; ζj )

a„ ∈

I≥J

(substituting J for K and {[I|H] : I ∈ S(r, m)} for Ψ). We claim that

Qs’1 [I|H] + „ · Qs’1 I(x; ζj ).

a„ „ ∈

(13)

I<J

To show this we must only look at terms of the form b[L|H]„, b ∈ Qs’2 , in which [L|H]

and „ = [J|H] are incomparable. According to (9.1) every standard monomial in the

standard representation of [L|1, . . . , r][J|1, . . . , r] is a product [U |1, . . . , r][V |1, . . . , r],

U, V ∈ S(r, m), U ¤ V . So [L|H]„ has a standard representation whose monomials are

of the form [U |H][V |H], U, V as above. Thus we get

Qs’1 [I|H].

b[L|H]„ ∈

I<J

So (13) holds. With that we obtain a representation

aµ ∈ Qs’1 , a ∈ µQs’1 I(x; ζj ),

a= aµ µ + a,

µ∈Ψ

µ∈Ψ\{„ }

and the proof of (11) can be ¬nished as for s = 1. ”

197

C. Syzygetic Behaviour and Rigidity

C. Syzygetic Behaviour and Rigidity

We now investigate some homological properties of „¦— as we did for the generic

modules in Section 13. Abbreviating

(= m + n ’ 2r + 1)

s = grade Ir (x)

we state the following

(15.10) Theorem. (a) Let m = n. Then „¦— is an (s’1)-th syzygy or, equivalently,

Exti („¦, R) = 0, 1 ¤ i ¤ s ’ 3.

R

(b) „¦— is an s-th syzygy in case m = n. Equivalently Exti („¦, R) = 0, 1 ¤ i ¤ s ’ 2.

R

Proof: Since „¦ = „¦—— (cf. (14.10)) and „¦— is free for all prime ideals P ‚ R

P

such that depth RP < s, the assertions given in (a) and (b) are in fact both equivalent

to depth „¦— ≥ min(s ’ 1, depth RP ) and depth „¦— ≥ min(s, depth RP ), resp., for all

P P

P ∈ Spec R (cf. (16.33)). From (15.7) we get depth „¦— ≥ depth RP ’ 1 in case m = n

P

and depth „¦— = depth RP in case m = n. This proves the theorem. ”

P

To derive some supplementary results on the syzygetic behaviour of „¦— we need the

map •1 de¬ned in (14.8),(a), the cokernel of which coincides with the ¬rst syzygy M of

„¦.

(15.11) Supplement to (15.10).

(a) Let m = n. Then „¦— is not an s-th syzygy and consequently Exts’2 („¦, R) = 0.

R

(b) Let m < n’1. Then „¦— is not an (s+1)-th syzygy and consequently Exts’1 („¦, R) = 0.

R

(c) Let m = n ’ 1. Then:

(c1 ) Exti („¦, R) = 0, 1 ¤ i ¤ s, and Exts+1 („¦, R) = 0 in case r + 1 < m (and

R R

consequently „¦— is at least an (s + 2)-th syzygy).

(c2 ) Exti („¦, R) = 0, 1 ¤ i ¤ s + 1 = 5, and Ext6 („¦, R) = 0 in case r + 1 = m (and

R R

therefore „¦— is at least a seventh syzygy).

Proof: (a) Because of (15.8) depth „¦— = s ’ 1 for all minimal prime ideals P of

P

Ir (x), so „¦— is not an s-th syzygy.

(b) If „¦— were an (s + 1)-th syzygy, then it would be (s + 1)-torsionless (cf. 16.34)).

So M — were an s-th syzygy. We claim however that depth MP = depth RP ’ 1 for all

—

prime ideals P ⊃ Ir (x). To prove this, we ¬rst reduce to the case in which B = Z. We

may then argue as in the last part of the proof of (13.8).

—

Let B = Z. Clearly depth MP ≥ depth RP ’ 1 for all prime ideals P ⊃ Ir (x) since

„¦— is perfect. So it is enough to show that depth MI— (x) < depth RIr (x) . Localizing as

r

in the proof of (14.9) we can reduce to the case in which r = 1. Furthermore we may

replace Z by Q since Ir (x) © Z = 0. Then we need only to prove that M — is not perfect

(cf. (16.20)) or equivalently that Ext1 (M — , Qn’m ) = 0, Q being the ideal in R generated

R

by the entries of the ¬rst column of x. Consider the map

h : HomR (M — , R) ’’ HomR (M — , R/Qn’m )

induced by the residue class projection R ’ R/Qn’m . It is easy to see, that

[2|1]n’m’1 y1 — z1 ’ [1|1][2|1]n’m’2y2 — z1

— —

198 15. Derivations and Rigidity

represents an element of HomR (M — , R/Qn’m ) which is not in Im h (y1 , . . . , ym and

z1 , . . . , zn denoting the canonical bases of Rm and Rn , resp.). So h is not surjective

and consequently Ext1 (M — , R/Qn’m ) = 0.

R

(c) In case r + 1 = m the kernel of the map • (cf. Subsection A) is obviously

isomorphic with Im x. The assertion of (c2 ) follows therefore from (15.10) and (13.14),(b).

As to (c1 ) we consider the three R-modules Ker •— (= M — ), Im •— and Coker •— .

1 1 1

The assertion is an easy consequence of the following claim:

(14) Ker •— and Im •— are perfect B[X]-modules whereas

1 1

depth(Coker •— )P = depth RP ’ 1

1

for all prime ideals P ⊃ Ir (x).

We outline the proof of (14); the computational details are very similar to those

used up to now and may be left to the reader: Since Ker •— and Im •— are Z-free in

1 1

case B = Z, the proof of the perfection of Ker •— can be reduced to this case (cf. (3.3)).

1

1 —

Then it is enough to show that ExtR (M , Q) = 0 where Q is the ideal in R generated

by the r-minors of the ¬rst r columns of x. The same way leads to the perfection

of Im •— . (Observe that depth(Coker •— )P ≥ depth MP ’ 2 for all prime ideals P in

—

1 1

R, so Coker •— is R-torsionfree and therefore Z-¬‚at in case B = Z.) It follows that

1

depth(Coker •— )P ≥ depth RP ’ 1 for all prime ideals P in R. To get equality when

1

P ⊃ Ir (x), one reduces to the case in which r = 1 and B = Q as one did in the proof of

(b). An easy computation yields Ext1 (Coker •— , Q) = 0. ”

R 1

(15.12) Remark. The proof shows that the assertions of (15.11) remain true if we

localize at some prime ideal containing Ir (x).”

Finally we shall derive some results concerning the rigidity of determinantal rings,

the base ring B presumed to be a ¬eld K from now. Some concepts and results in a more

general situation are needed.

Let S be a ¬nitely generated K-algebra, S = K[X1 , , . . . , , Xu ]/I, X1 , . . . , Xu being

indeterminates and I an ideal in K[X1 , . . . , Xu ]. “The” Auslander-Bridger dual of the

S-module I/I 2 (cf. 16.E) will be called an Auslander-module of S and is denoted by DS ;

up to projective direct summands it does not depend on the special presentation taken

for S.

An S-algebra T will be called a complete intersection over S if T is a factor ring of a

polynomial ring S[Y1 , , . . . , , Yv ] with respect to an ideal generated by an S[Y1 , , . . . , , Yv ]-

sequence.

We will not discuss here what it means that S is rigid. The reader may ¬nd detailed

information about this concept in the literature (cf. [Ar], [J¨], [KL] or [Sl] for instance).

a

The only fact we notice is that in case K is perfect and S is reduced, S is a rigid K-algebra

if and only if Ext1 („¦1 , S) = 0.

S S/K

Definition. Let S be as above and k a natural number. Assume S to be rigid. S

is k-rigid if the following condition holds: If T is a complete intersection over S whose

¬‚at dimension as an S-module is at most k, then Tors (T, DS ) = 0 for all i > 0. If S is

i

k-rigid for all k, then S is very rigid .

199

C. Syzygetic Behaviour and Rigidity

(15.13) Proposition. Let S be as above.

(a) If S is k-rigid and a1 , . . . , aj an S-sequence, j ¤ k, then TorS (S/(a1 , . . . , aj )S, DS )

i

= 0 for all i > 0.

(b) If DS satis¬es the condition (Sk ) (cf. 16.E), then S is k-rigid.

Proof: (a) Assume a1 , . . . , aj , 1 ¤ j ¤ k, to be an S-sequence. The S-module

S = S/(a1 , . . . , aj )S is a complete intersection over S which has ¬‚at dimension at most

k, so TorS (S, DS ) = 0 for all i > 0.

i

(b) Let T be a complete intersection over S, S = S[Y1 , . . . , Yv ], T = S/(f1 , . . . , fl )S,

where f1 , . . . , fl is an S-sequence. Assume that the ¬‚at dimension of T over S is at most

k. We claim that

depth(DS —S S)Q ≥ l

(15) ˜

for all Q ∈ Spec S containing f1 , . . . , fl . This implies

˜

S

TorS (T, DS ) = Tori (T, DS —S S) = 0

i

for all i > 0: Let l ≥ 1 and

•l •2 •1

F : 0 ’’ Fl ’’ · · · ’’ F1 ’’ F0

be the Koszul-complex over S derived from f1 , , . . . , , fl . F is an S-free resolution of T .

To see that F —S (DS —S S) remains exact, we use Theorem (16.15), the numbers ri

˜

and the ideals Ji being de¬ned as there. It is well known that Rad Ji = Rad l Sfj ,j=1

i = 1, . . . , l. So the exactness we want to prove follows immediately from (15).

To verify (15) let Q be the image of Q in T and P the preimage of Q in S with

respect to the canonical maps. From the ¬‚atness of the extension SP ’ S Q we obtain

˜

depth(DS —S S)Q = depth(DS )P + depth S Q ’ depth SP .

˜ ˜

Since depth(DS )P ≥ min(k, depth SP ) by assumption, we are done in case k ≥ depth SP .

Let k < depth SP and consider elements a1 , . . . , aj ∈ P which form a maximal SP -

sequence. Denote by K = K(a1 , . . . , aj ) the Koszul-complex (over S) derived from

a1 , . . . , aj . Since the ¬‚at dimension of T over S is at most k, we get

Hi (K —S SP —SP TQ ) = TorSP (SP /(a1 , . . . , aj )SP , TQ )

i

= TorS (S/(a1 , . . . , aj )S, TQ ) —S SP

i

= TorS (S/(a1 , . . . , aj )S, T ) —T TQ

i

=0

for i > k. Thus the ideal (a1 , . . . , aj )TQ has grade at least j ’ k (cf. Theorem (16.15) for

example). Consequently depth TQ ≥ depth SP ’ k. Since depth S Q ≥ l + depth TQ , the

˜

proof of (15) is complete now. ”

200 15. Derivations and Rigidity

(15.14) Corollary. Let S be Cohen-Macaulay. Then S is k-rigid if and only if D S

satis¬es the condition (Sk ).

Proof: It is easy to see that in the case we consider, a ¬nitely generated S-module

M satis¬es the condition (Sk ) if and only if TorS (S/(a1 , . . . , aj )S, M ) = 0 for every

i

S-sequence a1 , . . . , aj , 1 ¤ j ¤ k, and all i > 0. ”

Now we specialize to the determinantal ring R = Rr+1 (X) with base ring B = K.

As above put s = grade Ir (x). From (15.14) and the syzygetic behaviour of „¦— we derive:

(15.15) Theorem. Assume that K is a perfect ¬eld. Then R is rigid except for

the case in which r + 1 = m = n. Furthermore:

(a) If r + 1 < m = n then R is (s ’ 4)-rigid but not (s ’ 3)-rigid.

(b) If m < n ’ 1 then R is (s ’ 3)-rigid but not (s ’ 2)-rigid.

(c) Let m = n ’ 1.

(c1 ) If r + 1 < m then R is (s ’ 1)-rigid but not s-rigid.

(c2 ) If r + 1 = m then R is very rigid.

Proof: First we observe that there is a commutative diagram

r+1 r+1 •

m

(Rn )— Rm — (Rn )—

R— ’’

˜

•

2

Ir+1 (X)/Ir+1 (X)

where • = •x,r and

2

—

•(yI — zJ ) = [I|J] mod Ir+1 (X) ,

y1 , . . . , ym and z1 , . . . , zn being the canonical bases of Rm and Rn resp. (cf. Section 14). It

2

is a well known fact that Ir+1 (X)/Ir+1 (X) and Im • have the same rank as R-modules.

So their R-duals coincide, and consequently „¦— is a third syzygy of an Auslander-module

DR of R. Clearly (DR )P is free for all prime ideals P ‚ R such that depth RP < s.