ńņš. 9 |

ht(Ann M )S = grade(Ann M )S ā¤ g.

On the other hand ((Ann M )S + I)/I = (Ann M )A; so (Ann M )S + I is primary with

respect to the maximal ideal of S. By Serreā™s intersection theorem ([Se], ThĀ“or`me 3,

ee

p. V-18)

ht(Ann M )S + ht I ā„ ht((Ann M )S + I),

and

ht((Ann M )S + I) = dim S = ht(Ann M )A + ht I,

so

g ā„ ht(Ann M )S ā„ ht(Ann M )A.ā”

For the determinantal ideals (3.16) has been known to us already; it was proved in

this special case by more direct methods in (2.1). Taking M = Z[X]/( Xi Z[X]) we see

that (3.16) is a generalization of Krullā™s principal ideal theorem.

E. Comments and References

The notion āgenerically perfectā was introduced by Eagon and Northcott in [EN.2]

using the description in (3.2) as a deļ¬nition. As their main results one may consider

(3.5) ([EN.2], Corollary 1, p. 158) and (3.16) ([EN.2], Theorem 3). The non-obvious

implication (a) ā’ (b) of (3.3) was proved by Hochster [Ho.1]. In Hochsterā™s terminology

(3.3) states that every generically perfect module is strongly generically perfect. Our

proof of (3.3) is a substantial simpliļ¬cation for the case considered by us, namely Z as

the base ring. The theory of generic perfection can be extended in diļ¬erent directions:

(i) One can work relative to a base ring Ī and consider Ī-algebras throughout. (ii) The

ānoetherianā hypothesis can be dropped after the introduction of an adequate deļ¬nition

of grade for general commutative rings. (iii) As a minor modiļ¬cation, one can weaken

the hypothesis āfaithfully ļ¬‚atā into āļ¬‚atā allowing in (3.3), say, that āM ā— B = 0 or

M ā— B is perfect . . . ā. We refer to [Ba], [Ho.5], [No.3], [No.4], [No.5] and [No.6] for more

information.

37

E. Comments and References

There is however one generalization the reader can perform without substantial

changes in the proofs: Z as the base can be replaced by any Dedekind domain or ļ¬eld

D, since the properties of Z are used only for the equivalence of ātorsionfreeā and āļ¬‚atā.

Then of course only D-algebras may be considered and (3.3),(c) has to be modiļ¬ed in

an obvious way.

With the same generalization, Theorem (3.9) was proved by Eagon and Hochster

in [EH], where the replacement of indeterminates by elements in a regular sequence

was investigated in a more general situation. The method of proof employed in [EH] is

indicated in (3.10),(d). Our proof of (3.9) and those of (3.11) and (3.13) are patterned

after [No.2], where Northcott considered ideals of maximal minors. Part of (3.13) can

also be found in [Ng.2]. Proposition (3.12) is taken from [Ho.3] where B is not supposed

to be noetherian, an assumption which simpliļ¬es the proof for ānormalā. The example

(3.10),(c) was given in [EH].

Since (3.5) indicates how one could derive results for ānon-genericā determinantal

ideals from those on the ideals It (X), this may be an appropriate place to list some

articles in which such determinantal ideals have been investigated. Deļ¬nitive results

have been obtained by Eisenbud in [Ei.2] on ideals It (X) where X denotes the matrix of

residue classes of the entries of X in a ring B[X]/J, J being generated by linear forms in

the indeterminates. The case t = min(m, n) had previously been treated by Giusti and

Merle ([GM]).

The ultimate generalization of (3.16) would be the āhomological height conjectureā:

Let R ā’ S be a homomorphism of noetherian rings, and M an R-module; then ht P ā¤

pd M for every minimal prime ideal P ā (Ann M )S. It is known to hold if S contains a

ļ¬eld, cf. [Ho.9].

4. Algebras with Straightening Law on Posets of Minors

Among the residue class rings B[X]/I the most easily accessible ones are those for

which the ideal I is generated by a set of monomials, since one can use the structure of

B[X] as a free B-module very favourably: I itself is generated as a B-module by a subset

of the monomial B-basis of B[X]. The multiplication table with respect to this basis is

very simple, a property inherited by B[X]/I.

With respect to the monomial basis of B[X], a minor of X is a very complicated

expression. Therefore it is desirable to ļ¬nd a new basis of B[X] which contains the

minors and as many of their products as possible. The construction of such a basis is

the main object of this section. This basis will consist of monomials whose factors are

minors of X, and whether such a monomial is an element of the basis can be decided by

a simple combinatorial criterion.

The set of maximal minors of a matrix has a combinatorially simpler structure than

the set of all minors: one needs only one set of indices to specify a maximal minor, and

all the maximal minors have the same size. Therefore it is simpler to treat the rings

G(X) ļ¬rst and to derive the structure sought for B[X] afterwards (from G(X) for an

extended matrix X).

A. Algebras with Straightening Law

When all the minors of X appear in a B-basis of B[X], then, apart from trivial

cases, it is impossible that a product of two elements of the basis is in the basis always;

nevertheless one has suļ¬cient control over the multiplication table. This situation is met

often enough to justify the introduction of a special class of algebras:

Definition. Let A be a B-algebra and Ī ā‚ A a ļ¬nite subset with a partial order ā¤,

called a poset for short. A is a graded algebra with straightening law (on Ī , over B) if

the following conditions hold:

(H0 ) A = iā„0 Ai is a graded B-algebra such that A0 = B, Ī consists of homogeneous

elements of positive degree and generates A as a B-algebra.

(H1 ) The products Ī¾1 . . . Ī¾m , m ā N, Ī¾i ā Ī , such that Ī¾1 ā¤ Ā· Ā· Ā· ā¤ Ī¾m are linearly inde-

pendent. They are called standard monomials.

(H2 ) (Straightening law) For all incomparable Ī¾, Ļ… ā Ī the product Ī¾Ļ… has a representa-

tion

aĀµ ā B, aĀµ = 0, Āµ standard monomial,

Ī¾Ļ… = aĀµ Āµ,

satisfying the following condition: every Āµ contains a factor Ī¶ ā Ī such that Ī¶ ā¤ Ī¾, Ī¶ ā¤ Ļ…

(it is of course allowed that Ī¾Ļ… = 0, the sum aĀµ Āµ being empty).

The rather long notation āalgebra with straightening lawā will be abbreviated by

ASL.

We shall see in Proposition (4.1) that the standard monomials form in fact a basis

of A as a B-module, the standard basis of A. The representation of an element x ā A as

39

A. Algebras with Straightening Law

a linear combination of standard monomials is called its standard representation. The

relations in (H2 ) will be referred to as the straightening relations.

To be formally precise one would better consider a partially ordered set Ī outside

A and an injection Ī ā’ A. We have preferred to avoid this notational complication and

warn the reader that Ī (or a subset of it) may be treated as a subset of diļ¬erent rings,

in particular when A and a residue class ring of A occur simultaneously. Similarly we

do not distinguish between a formal monomial in Ī and the corresponding ring element.

Condition (H1 ) of course says that the family of ring elements parametrized by the

formal standard monomials is linearly independent. Whenever a function is deļ¬ned on

(a subset of) the set of monomials by reference to the factors of the monomials, then

such a deļ¬nition properly applies to the formal monomials.

Before we discuss an example, one simple observation: If A is a graded ASL over B

on Ī and C a B-algebra, then A ā— C is a graded ASL over C on Ī in a natural way.

The polynomial ring B[T1 , . . . , Tu ] is a graded ASL in a trivial fashion: one orders

T1 , . . . , Tu linearly. For a less trivial example we let X be a 2Ć—2-matrix, Ī“ its determinant.

We order the set Ī of minors of X according to the diagram

sX22

Ā d

Ā d

Ā ds X

X12 s

d Ā 21

dĀ

dĀ

sX

11

sĪ“.

The conditions (H0 ) and (H2 ) are obviously satisļ¬ed: Only X12 and X21 are incompa-

rable, and the straightening law consists of the single relation X12 X21 = X11 X22 ā’ Ī“.

Replacing every occurence of the product X12 X21 in a monomial by X11 X22 ā’ Ī“ one

obtains a representation as a linear combination of standard monomials. Furthermore

one has a bijective degree-preserving correspondence between the ordinary monomials

and the standard monomials:

X11 X12 Ī“ k X22

i jā’k l

if j ā„ k,

j

i k l

āā’

X11 X12 X21 X22

X11 X21 Ī“ j X22

i kā’j l

if k > j.

Therefore the standard monomials must be linearly independent, and B[X] is an ASL

on Ī . It is much more diļ¬cult to establish the analogous result for bigger matrices.

(4.1) Proposition. Let A be a graded ASL over B on Ī . Then:

(a) The standard monomials generate A as a B-module, thus forming a B-basis of A.

(b) Furthermore every monomial Āµ = Ī¾1 . . . Ī¾m , Ī¾i ā Ī , has a standard representation in

which every standard monomial contains a factor Ī¾ ā¤ Ī¾1 , . . . , Ī¾m .

Proof: For Ī¾ ā Ī let u(Ī¾) = |{Ī“ ā Ī : Ī¾ ā¤ Ī“}| and w(Ī¾) = 3u(Ī¾) ; for Āµ = Ī¾1 . . . Ī¾m ,

m

Ī¾i ā Ī , we put w(Āµ) = i=1 w(Ī¾i ). (This is an example of a deļ¬nition properly applying

to the formal monomials.) Obviously w(Ī¾Ļ…) < w(Āµ) for all the monomials Āµ appearing

on the right side of the standard representation Ī¾Ļ… = aĀµ Āµ.

Because of (H1 ) it is enough for part (a) to show that every monomial is a linear

combination of standard monomials. If all the factors Ī¾1 , . . . , Ī¾m of Āµ are comparable, Āµ

is a standard monomial. Otherwise two of the factors are incomparable. Replacing their

40 4. Algebras with Straightening Law on Posets of Minors

product by the right side of the corresponding straightening relation produces a linear

combination of monomials which, if diļ¬erent from 0, have a greater value than Āµ under

the function w. On the other hand their values are bounded above since they have the

same degree as homogeneous elements of A: w(Āµ) ā¤ d Ā· 3|Ī | for monomials Āµ of degree d.

Thus we are through by descending induction. The easy proof of the second assertion, a

similar induction, is left to the reader. ā”

The preceding proof shows that the standard representation of an element of A

can be obtained by successive applications of the straightening relations, regardless of

the order in which the steps of āstraighteningā are performed. As a consequence the

straightening relations generate the deļ¬ning ideal of A:

(4.2) Proposition. Let A be a graded ASL over B on Ī , and TĪ¾ , Ī¾ ā Ī a family

of indeterminates over B. For each monomial Āµ = Ī¾1 . . . Ī¾m , Ī¾i ā Ī , let TĀµ = TĪ¾1 . . . TĪ¾m .

Then the kernel of the epimorphism

Ļ• : B[TĪ¾ : Ī¾ ā Ī ] ā’ā’ A, TĪ¾ ā’ā’ Ī¾,

is generated by the elements TĪ¾ TĻ… ā’ aĀµ TĀµ representing the straightening relations.

Proof: Let f ā Ker Ļ•, f = bĀµ TĀµ , bĀµ ā B. If all the monomials Āµ are standard

monomials, bĀµ = 0 for all Āµ. Otherwise we apply the straightening procedure indicated

above: we subtract successively multiples of the elements representing the straightening

relations. Thus we create a sequence f = f1 , f2 , . . . , fn of polynomials in Ker Ļ•, whose

successive terms diļ¬er by a multiple of such an element and for which fn , representing a

linear combination of standard monomials, is zero. ā”

B. G(X) as an ASL

Let B be a commutative ring and X an m Ć— n-matrix of indeterminates over B,

m ā¤ n. As a B-algebra G(X) is generated by the set

Ī“(X)

of maximal minors of X, cf. 1.D. Ī“(X) is ordered partially in the following way:

[i1 , . . . , im ] ā¤ [j1 , . . . , jm ] āā’ i 1 ā¤ j 1 , . . . , im ā¤ j m .

Only in the cases n = m and n = m+1 the set Ī“(X) is linearly ordered. For m = 2, n = 4

and m = 3, n = 5 the partial orders have the diagrams

s [3 4 5]

s [2 4 5]

Ā d

Ā d

Ā ds [2 3 5]

s [3 4] [1 4 5] s

d Ā d

dĀ d

dĀ ds[2 3 4]

s [2 4] [1 3 5] s

Ā d Ā d Ā

Ā d Ā dĀ

[1 4] Ā ds [2 3] [1 2 5] Ā ds [1 3 4]

Ā

s s

d Ā d Ā

dĀ dĀ

dĀ [1 3] dĀ [1 2 4]

s s

s [1 2] s [1 2 3]

and

41

B. G(X) as an ASL

Ī“(X) can be considered as a subset of the poset Nm in a natural way, and it inherits

from Nm the structure of a distributive lattice, the lattice operations and given by

[i1 , . . . , im ] [j1 , . . . , jm ] = [min(i1 , j1 ), . . . , min(im , jm )],

[i1 , . . . , im ] [j1 , . . . , jm ] = [max(i1 , j1 ), . . . , max(im , jm )];

one has Ī“1 ā¤ Ī“2 if and only if Ī“1 Ī“2 = Ī“ 1 .

(4.3) Theorem. G(X) is a graded ASL on Ī“(X).

Condition (H0 ) is obviously satisļ¬ed. The linear independence of the standard mono-

mials will be proved in Subsection C below. In the ļ¬rst part of the proof we want to show

that condition (H2 ) holds, assuming linear independence of the standard monomials. By

virtue of Proposition (4.1) this implies that the standard monomials generate G(X) as a

B-module. We shall not describe the straightening relations themselves explicitely; they

will result from the PlĀØcker relations.

u

(4.4) Lemma. (PlĀØcker relations) For every m Ć— n-matrix, m ā¤ n, with elements

u

in a commutative ring and all indices a1 , . . . , ak , bl , . . . , bm , c1 , . . . , cs ā {1, . . . , n} such

that s = m ā’ k + l ā’ 1 > m, t = m ā’ k > 0 one has

Ļ(i1 , . . . , is )[a1 , . . . , ak , ci1 , . . . , cit ][cit+1 , . . . , cis , bl , . . . , bm ] = 0.

i1 <Ā·Ā·Ā·<it

it+1 <Ā·Ā·Ā·<is

{1,...,s}={i1 ,...,is }

Proof: It suļ¬ces to prove this for a matrix X of indeterminates over Z. We consider

the Z[X]-module C generated by the columns of X. As a Z[X]-module it has rank m.

Let Ī± : C s ā’ Z[X] be given by

Ī±(y1 , . . . , ys ) = Ļ(Ļ) det(Xa1 , . . . , Xak , yĻ(1) , . . . , yĻ(t) )

ĻāSym(1,...,s)

Ā· det(yĻ(t+1) , . . . , yĻ(s) , Xbl , . . . , Xbm ),

Xj denoting the j-th column of X, Sym(1, . . . , s) the group of permutations of {1, . . . , s}.

It is straightforward to check that Ī± is a multilinear form on C s . When two of the vectors

yi coincide, every term in the expansion of Ī±, which does not vanish anyway, is cancelled

by a term of the opposite sign: Ī± is alternating. Since s > rk C, Ī± = 0.

We ļ¬x a subset {i1 , . . . , it }, i1 < Ā· Ā· Ā· < it , of {1, . . . , s}. Then, for all Ļ such that

Ļ({1, . . . , t}) = {i1 , . . . , it } the summand corresponding to Ļ in the expansion of Ī± equals

Ļ(i1 , . . . , is ) det(Xa1 , . . . , Xak , yi1 , . . . , yit ) det(yit+1 , . . . , yis , Xbl , . . . , Xbm ),

it+1 , . . . , is chosen as above. Therefore each of these terms occurs t! (s ā’ t)! times in the

expansion of Ī±. In Z[X] the factor t! (s ā’ t)! may be cancelled. ā”

ńņš. 9 |