in particular, for δ = [1, . . . , t ’ 1|1, . . . , t ’ 1]:

dim Rt (X) = dim B + (m + n ’ t + 1)(t ’ 1).

(b) Let γ = [a1 , . . . , am ] ∈ “(X). Then

m

m(m + 1)

dim G(X; γ) = dim B + m(n ’ m) + ’ ai + 1,

2 i=1

in particular, for γ = [1, . . . , m]:

dim G(X) = dim B + m(n ’ m) + 1.

58 5. The Structure of an ASL

D. Wonderful Posets and the Cohen-Macaulay Property

As noticed in (5.3) and (5.4) the posets ∆(X; δ) and “(X; γ) are distributive lattices.

We shall see below that this implies the Cohen-Macaulay property for the corresponding

rings (provided B is Cohen-Macaulay). However, a weaker condition will turn out to be

su¬cient already, a condition which can be controlled rather easily in an inductive proof:

Definition. A partially ordered set Π is called wonderful (in systematic combi-

natorial language: locally upper semi-modular) if the following holds after a smallest

and a greatest element ’∞ and ∞ resp. have been added to Π: If …1 and …2 are upper

neighbours of ξ ∈ Π ∪ {’∞, ∞} and …1 , …2 < ζ ∈ Π ∪ {’∞, ∞} then there is an upper

neighbour … of …1 , …2 such that … ¤ ζ, pictorially

sζ

sv

d

d

v1 ds v2

s

d

d

d

sξ

the existence of … being required.

In a lattice Π there is of course only one choice for …: … = …1 …2 . In general …1 …2

need not to be an upper neighbour of …1 and …2 , as the following example indicates:

s v = v 1 v2

t

t t

s

ts v 1

v2

s

d

d

ds ξ

A distributive lattice is always wonderful: Suppose there is an · ∈ Π, …2 < · < …. We

put ω = · …1 . Then

ξ = … 1 …2 ¤ … 1 · = ω ¤ … 1 ,

leaving the cases ω = ξ or ω = …1 . In the last case …1 ¤ ·, so … = …1 …2 ¤ · < …, a

contradiction, whereas in the ¬rst (and critical, cf. the example above) case

(· …1 ) …2 = ξ … 2 = …2 , but also

(· …1 ) …2 = (· …2 ) (…1 …2 ) = · … = ·,

again a contradiction.

For a lattice Π one could obviously weaken the condition for being wonderful: A

lattice is already wonderful if it is upper semi-modular, i.e. if elements …1 and …2 with a

common lower neighbour ξ also have a common upper neighbour. For posets in general

this weaker property does not imply that an ASL is Cohen-Macaulay; a counterexample

will be discussed below.

The next lemma collects some combinatorial properties of wonderful posets.

59

D. Wonderful Posets and the Cohen-Macaulay Property

(5.13) Lemma. Let Π be a wonderful poset.

(a) If „¦ ‚ Π is an ideal and if for all minimal elements …1 , …2 of Π \ „¦ and all ζ ∈ Π \ „¦

such that …1 , …2 < ζ there is a common upper neighbour … ¤ ζ of …1 , …2 , then Π \ „¦ is

wonderful.

(b) If Π has a single minimal element ξ, then Π \ {ξ} is wonderful.

(c) Let „¦ be the ideal cogenerated by a subset of Min Π. Then Π \ „¦ is wonderful.

(d) Every maximal chain in Π has length rk Π.

(e) Suppose that Π has minimal elements ξ1 , . . . , ξk , k ≥ 2, let „¦ be the ideal cogenerated

by ξ1 , Ψ the ideal cogenerated by {ξ2 , . . . , ξk }. Then:

(i) Π \ „¦, Π \ Ψ and Π \ (Ψ ∪ „¦) are wonderful.

(ii) rk(Π \ „¦) = rk(Π \ Ψ) = rk Π, whereas rk(Π \ (Ψ ∪ „¦)) = rk Π ’ 1.

(iii) „¦ © Ψ = ….

Proof: Part (a) is rather trivial, and parts (b) and (c) follow immediately from

(a), whereas (d) is proved by induction on |Π|: Let ξ1 < · · · < ξk and …1 < · · · < …l be

maximal chains in Π. If ξ1 = …1 , one passes to

(Π \ {ideal cogenerated by ξ1 }) \ {ξ1 }

which is wonderful by virtue of (c) and (b). Otherwise ξ1 and …1 have a common upper

neighbour ζ2 (in Π ∪ {’∞, ∞} they both are upper neighbours of ’∞). There is a

maximal chain ξ1 < ζ2 < · · · < ζm . Applying the argument of the case ξ1 = …1 twice, we

see that the chains ξ1 < · · · < ξk , ξ1 < ζ2 < · · · < ζm , …1 < ζ2 < · · · < ζm , …1 < · · · < …l

all have the same length.

In (e) the assertions concerning Π \ „¦, Π \ Ψ follow directly from (c) and (d).

Furthermore Π \ (Ψ ∪ „¦) does not contain any minimal element of Π, but it contains a

common upper neighbour of ξ1 and ξ2 , unless it is empty; therefore rk(Π \ („¦ ∪ Ψ)) =

rk Π ’ 1. „¦ © Ψ = … is trivial. It only remains to prove that Π \ („¦ ∪ Ψ) is wonderful, and

here we need the full strength of the property “wonderful”! We want to apply (a) and

consider minimal elements …1 , …2 of Π \ („¦ ∪ Ψ). Then …1 , …2 ≥ ξ1 . The crucial point is to

show that …1 and …2 are both upper neighbours of ξ1 ; then (a) can be applied. Suppose

…1 is not an upper neighbour of ξ1 . Since …1 > ξ1 and …1 > ξi for some i ∈ {2, . . . , k}, ξ1

and ξi have an upper neighbour ζ < …1 . This is a contradiction: ζ ∈ Π \ („¦ ∪ Ψ), too. ”

(5.14) Theorem. Let B be a Cohen-Macaulay ring, Π a wonderful poset, and A a

graded ASL over B on Π. Then A is a Cohen-Macaulay ring, too.

The proof of the theorem is by induction on |Π|, and Lemma (5.13) contains the

combinatorial arguments. The algebraic arguments will be the Cohen-Macaulay criterion

for ¬‚at extensions and the following lemma which is also crucial in the proof of Hochster-

Eagon for the perfection of determinantal ideals (cf. Section 12).

(5.15) Lemma. Let K be a ¬eld, A = i≥0 Ai a graded K-algebra with A0 = K.

(a) Let x ∈ A be homogeneous of positive degree such that x is not a zero-divisor. Then

A is Cohen-Macaulay if and only if A/Ax is Cohen-Macaulay.

(b) Let I, J be homogeneous ideals such that

dim A/(I + J) = dim A ’ 1, I © J = 0.

dim A/I = dim A/J = dim A, and

Suppose that A/I and A/J are Cohen-Macaulay. Then A is Cohen-Macaulay if and only

if A/(I + J) is Cohen-Macaulay.

60 5. The Structure of an ASL

Proof: By virtue of (16.20) we may ¬rst localize with respect to the irrelevant

maximal ideal. The local analogues of (a) and (b) are easy to prove. For (a) one observes

dim A/Ax = dim A ’ 1 and depth A/Ax = depth A ’ 1, whereas for (b) it is crucial that

in the exact sequence

0 ’’ A/(I © J) ’’ A/I • A/J ’’ A/(I + J) ’’ 0

A/(I © J) can be replaced by A:

0 ’’ A ’’ A/I • A/J ’’ A/(I + J) ’’ 0

is exact. The middle term has depth equal to dim A. If depth A = dim A, then

depth A/(I + J) ≥ dim A ’ 1 = dim A/(I + J). Conversely, if A/(I + J) is Cohen-

Macaulay, then depth A/(I + J) = dim A ’ 1, and depth A ≥ dim A. ”

Let us prove (5.14) now. For a prime ideal P of A the localization AP is Cohen-

Macaulay if and only if for Q = P © B the rings BQ and (BQ /QBQ ) — AP are Cohen-

Macaulay. The last ring is a localization of (BQ /QBQ ) — A, a graded ASL over a ¬eld.

Hence we may assume that B = K is a ¬eld. Now one applies induction on |Π|. If

A has a single minimal element ξ, it follows from (5.13),(b) and (5.15),(a) that A is

Cohen-Macaulay. Otherwise there are minimal elements ξ1 , . . . , ξk , k ≥ 2. Let „¦ and

Ψ be chosen as in (5.13),(e) and I = A„¦, J = AΨ. Then, by virtue of (5.13),(e) and

induction, the hypothesis of (5.15),(b) is satis¬ed and A/(I + J) is Cohen-Macaulay. We

conclude that A is Cohen-Macaulay itself. ”

In the same manner as Theorem (5.14) one proves the following generalization:

(5.16) Proposition. Let B be a noetherian ring, and A a graded ASL on a won-

derful poset over B. Then A satis¬es Serre™s condition (Sn ) if (and only if ) B satis¬es

(Sn ).

The following example for Π may show that the condition “wonderful” cannot be

weakened in an obvious way:

rr ¨ ζ2

ζ1 s s

rr¨

¨¨r

s ¨ s v 2 rs v 3

„¦ = {ξ2 , …3 }

¨

v1

d Ψ = {ξ1 , …1 }

d

ds ξ2

ξ1 s

Though every pair of elements of the same rank has a common upper neighbour, an ASL

over Π cannot be Cohen-Macaulay. Since Π \ „¦ and Π \ Ψ are wonderful, A/A„¦ and

A/AΨ are Cohen-Macaulay. Moreover A/(A„¦ + AΨ) has dimension one less than A and

is not Cohen-Macaulay, hence A cannot be Cohen-Macaulay by virtue of (5.15),(b).

It remains to specialize (5.14) for determinantal rings and Schubert cycles. Their

underlying posets are distributive lattices, hence wonderful, as remarked above.

(5.17) Corollary. Let B be a Cohen-Macaulay ring. Then all the rings R(X; δ)

and G(X; γ) are Cohen-Macaulay rings, too.

Using the theory of generic perfection one can strengthen and generalize (5.17):

61

E. The Arithmetical Rank of Certain Ideals

(5.18) Corollary. Let X be an m — n matrix of indeterminates.

(a) For B = Z the ideal I(X; δ) is generically perfect. Hence I(X; δ) is perfect over an

arbitrary noetherian ring, and, with δ = [a1 , . . . , ar |b1 , . . . , br ],

r

grade I(X; δ) = mn ’ (m + n + 1)r + (ai + bi ).

i=1

(b) Let J(X; γ) denote the kernel of the epimorphism B[Yρ : ρ ∈ “(X)] ’’ G(X; γ)

induced by the substitution Yρ ’’ ρ. For B = Z the ideal J(X; γ) is generically perfect.

Hence J(X; γ) is perfect over an arbitrary noetherian ring, and, with γ = [a 1 , . . . , am ],

m

n m(m + 1)

’ m(n ’ m) ’ ai ’ 1.

grade J(X; γ) = +

m 2 i=1

Proof: By virtue of (3.3) it is enough to prove the corollary as far as it applies

to ¬elds B, for which perfection follows from (5.17) via (16.20). The formulas for grade

result immediately from those for dimension in (5.12) when B is a ¬eld. ”

As we observed in Section 2, the Cohen-Macaulay property of the rings Rt (X) implies

that they are (normal) domains whenever the ring B of coe¬cients is a (normal) domain.

We shall see in the following section that in this case all the rings R(X; δ) and G(X; γ)

are (normal) domains. For later application we note a generalization of (5.17) and (5.18):

(5.19) Proposition. Let „¦ be an ideal in ∆(X) or “(X) such that the minimal

elements of its complement have a common lower neighbour if there are at least two

minimal elements. Then (5.17) and (5.18) hold mutatis mutandis for the ideal generated

by „¦ in B[X] or “(X) resp. and the residue class ring de¬ned by it.

Proof: It follows readily from (5.13),(a) that the complement of „¦ is a wonderful

poset. ”

Needless to say, the theory of generic perfection applies to all the ideals in (5.19),

in particular to the ideals I(X; δ) and J(X; γ); the speci¬c consequences are left to the

reader.

E. The Arithmetical Rank of Certain Ideals

One of the main problems of algebraic geometry is the determination of the minimal

number of equations de¬ning a given variety. As a by-product of the theory of ASLs

we can obtain an upper bound for this number in the case of the determinantal rings.

The corresponding algebraic problem is to ¬nd the minimal number of elements which

generate a given ideal up to radical. For an ideal I in a commutative noetherian ring S

let therefore the arithmetical rank of I be given by

k

ara I = min k : there exist x1 , . . . , xk ∈ I such that Rad I = Rad Sxi .

i=1

The following proposition is a direct consequence of Lemma (5.9).

62 5. The Structure of an ASL

(5.20) Proposition. Let A be a graded ASL over B on Π, and „¦ ‚ Π an ideal.

Then

ara A„¦ ¤ rk „¦.

Of course Lemma (5.9) does not only supply this bound; it also shows how to ¬nd

a sequence x1 , . . . , xk , k = rk „¦, such that rad A„¦ = rad Axi . The case in which

A = B[X], X an m — n matrix, I = It (X) is particularly simple, since the corresponding

ideal has a single maximal element, namely [m ’ t + 1, . . . , m|n ’ t + 1, . . . , n].

(5.21) Corollary. Let X be an m — n matrix. Then

ara It (X) ¤ mn ’ t2 + 1.

A generalization to arbitrary ideals I(X; δ), J(X; γ) is left to the reader.

Unfortunately we do not know how to derive a lower bound from ASL theory in

general. The problem one is faced can already be illustrated by means of the example

A = B[X], X a 2 — 2 matrix, with the poset

sX22

d

d

ds X

X12 s

d 21

d

ds X

11

sδ.

and „¦ = {δ, X11 , X12 }: rk „¦ even exceeds the minimal number of generators of A„¦.

However, one can reverse (5.20) if A is a symmetric ASL. As stated in (4.6), G(X) is

a symmetric ASL, and therefore all the G(X; γ) are symmetric, too. Another class of

symmetric ASLs is given by the discrete ASLs, in which every straightening relation has

the form ξ… = 0. (In the general theory of ASLs the discrete ones play a central role,

cf. [DEP.2].) Discrete ASLs are graded in a natural way: assign the degree 1 to every

element of Π.

(5.22) Proposition. Let A be a symmetric graded ASL on Π. Then for every ideal

„¦ ‚ Π one has

ara A„¦ = rk „¦.

Proof: The complement of „¦ is an ideal in Π equipped with the reverse order.

Since A is symmetric, S = A/(Π\„¦)A is again an ASL, the underlying poset being „¦

with its order reversed. Now obviously ara A„¦ ≥ ara S„¦, and by Krull™s Principal Ideal

Theorem ara S„¦ ≥ ht S„¦ = rk „¦ (cf. (5.10) for the last equation). ”