<<

. 9
( 47 .)



>>


ht(Ann M )S = grade(Ann M )S ¤ 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 Serre™s intersection theorem ([Se], Th´or`me 3,
ee
p. V-18)
ht(Ann M )S + ht I ≥ ht((Ann M )S + I),
and
ht((Ann M )S + I) = dim S = ht(Ann M )A + ht I,
so
g ≥ ht(Ann M )S ≥ ht(Ann M )A.”

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 Krull™s principal ideal theorem.

E. Comments and References

The notion “generically perfect” was introduced by Eagon and Northcott in [EN.2]
using the description in (3.2) as a de¬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) ’ (b) of (3.3) was proved by Hochster [Ho.1]. In Hochster™s terminology
(3.3) states that every generically perfect module is strongly generically perfect. Our
proof of (3.3) is a substantial simpli¬cation for the case considered by us, namely Z as
the base ring. The theory of generic perfection can be extended in di¬erent directions:
(i) One can work relative to a base ring Λ and consider Λ-algebras throughout. (ii) The
“noetherian” hypothesis can be dropped after the introduction of an adequate de¬nition
of grade for general commutative rings. (iii) As a minor modi¬cation, one can weaken
the hypothesis “faithfully ¬‚at” into “¬‚at” allowing in (3.3), say, that “M — B = 0 or
M — B is perfect . . . ”. 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 ¬eld
D, since the properties of Z are used only for the equivalence of “torsionfree” and “¬‚at”.
Then of course only D-algebras may be considered and (3.3),(c) has to be modi¬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 simpli¬es the proof for “normal”. The example
(3.10),(c) was given in [EH].
Since (3.5) indicates how one could derive results for “non-generic” 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. De¬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 “homological height conjecture”:
Let R ’ S be a homomorphism of noetherian rings, and M an R-module; then ht P ¤
pd M for every minimal prime ideal P ⊃ (Ann M )S. It is known to hold if S contains a
¬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 ¬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) ¬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 su¬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 Π ‚ A a ¬nite subset with a partial order ¤,
called a poset for short. A is a graded algebra with straightening law (on Π, over B) if
the following conditions hold:
(H0 ) A = i≥0 Ai is a graded B-algebra such that A0 = B, Π consists of homogeneous
elements of positive degree and generates A as a B-algebra.
(H1 ) The products ξ1 . . . ξm , m ∈ N, ξi ∈ Π, such that ξ1 ¤ · · · ¤ ξm are linearly inde-
pendent. They are called standard monomials.
(H2 ) (Straightening law) For all incomparable ξ, … ∈ Π the product ξ… has a representa-
tion
aµ ∈ B, aµ = 0, µ standard monomial,
ξ… = aµ µ,
satisfying the following condition: every µ contains a factor ζ ∈ Π such that ζ ¤ ξ, ζ ¤ …
(it is of course allowed that ξ… = 0, the sum aµ µ being empty).
The rather long notation “algebra with straightening law” 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 ∈ 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 Π outside
A and an injection Π ’ A. We have preferred to avoid this notational complication and
warn the reader that Π (or a subset of it) may be treated as a subset of di¬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 Π 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 de¬ned on
(a subset of) the set of monomials by reference to the factors of the monomials, then
such a de¬nition properly applies to the formal monomials.
Before we discuss an example, one simple observation: If A is a graded ASL over B
on Π and C a B-algebra, then A — C is a graded ASL over C on Π 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 2—2-matrix, δ its determinant.
We order the set Π of minors of X according to the diagram
sX22
 d
 d
  ds X
X12 s
d   21


sX
11

sδ.

The conditions (H0 ) and (H2 ) are obviously satis¬ed: Only X12 and X21 are incompa-
rable, and the straightening law consists of the single relation X12 X21 = X11 X22 ’ δ.
Replacing every occurence of the product X12 X21 in a monomial by X11 X22 ’ δ 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 δ k X22
i j’k l
if j ≥ k,
j
i k l
←’
X11 X12 X21 X22
X11 X21 δ j X22
i k’j l
if k > j.

Therefore the standard monomials must be linearly independent, and B[X] is an ASL
on Π. It is much more di¬cult to establish the analogous result for bigger matrices.
(4.1) Proposition. Let A be a graded ASL over B on Π. Then:
(a) The standard monomials generate A as a B-module, thus forming a B-basis of A.
(b) Furthermore every monomial µ = ξ1 . . . ξm , ξi ∈ Π, has a standard representation in
which every standard monomial contains a factor ξ ¤ ξ1 , . . . , ξm .
Proof: For ξ ∈ Π let u(ξ) = |{δ ∈ Π : ξ ¤ δ}| and w(ξ) = 3u(ξ) ; for µ = ξ1 . . . ξm ,
m
ξi ∈ Π, we put w(µ) = i=1 w(ξi ). (This is an example of a de¬nition properly applying
to the formal monomials.) Obviously w(ξ…) < w(µ) for all the monomials µ appearing
on the right side of the standard representation ξ… = aµ µ.
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 ξ1 , . . . , ξm of µ are comparable, µ
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 di¬erent from 0, have a greater value than µ 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(µ) ¤ d · 3|Π| for monomials µ 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. ”
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 “straightening” are performed. As a consequence the
straightening relations generate the de¬ning ideal of A:
(4.2) Proposition. Let A be a graded ASL over B on Π, and Tξ , ξ ∈ Π a family
of indeterminates over B. For each monomial µ = ξ1 . . . ξm , ξi ∈ Π, let Tµ = Tξ1 . . . Tξm .
Then the kernel of the epimorphism
• : B[Tξ : ξ ∈ Π] ’’ A, Tξ ’’ ξ,
is generated by the elements Tξ T… ’ aµ Tµ representing the straightening relations.
Proof: Let f ∈ Ker •, f = bµ Tµ , bµ ∈ B. If all the monomials µ are standard
monomials, bµ = 0 for all µ. 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 •, whose
successive terms di¬er by a multiple of such an element and for which fn , representing a
linear combination of standard monomials, is zero. ”

B. G(X) as an ASL
Let B be a commutative ring and X an m — n-matrix of indeterminates over B,
m ¤ n. As a B-algebra G(X) is generated by the set
“(X)
of maximal minors of X, cf. 1.D. “(X) is ordered partially in the following way:
[i1 , . . . , im ] ¤ [j1 , . . . , jm ] ⇐’ i 1 ¤ j 1 , . . . , im ¤ j m .
Only in the cases n = m and n = m+1 the set “(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]
 d
 d
  ds [2 3 5]
s [3 4] [1 4 5] s
d  d
d d
d  ds[2 3 4]
s [2 4] [1 3 5] s
 d  d  
 d  d 
[1 4]   ds [2 3] [1 2 5]   ds [1 3 4]
 
s s
d   d  
d  d 
d [1 3] d [1 2 4]
s s

s [1 2] s [1 2 3]
and
41
B. G(X) as an ASL

“(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 δ1 ¤ δ2 if and only if δ1 δ2 = δ 1 .
(4.3) Theorem. G(X) is a graded ASL on “(X).
Condition (H0 ) is obviously satis¬ed. The linear independence of the standard mono-
mials will be proved in Subsection C below. In the ¬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 Pl¨cker relations.
u
(4.4) Lemma. (Pl¨cker relations) For every m — n-matrix, m ¤ n, with elements
u
in a commutative ring and all indices a1 , . . . , ak , bl , . . . , bm , c1 , . . . , cs ∈ {1, . . . , n} such
that s = m ’ k + l ’ 1 > m, t = m ’ k > 0 one has

σ(i1 , . . . , is )[a1 , . . . , ak , ci1 , . . . , cit ][cit+1 , . . . , cis , bl , . . . , bm ] = 0.
i1 <···<it
it+1 <···<is
{1,...,s}={i1 ,...,is }



Proof: It su¬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 ± : C s ’ Z[X] be given by

±(y1 , . . . , ys ) = σ(π) det(Xa1 , . . . , Xak , yπ(1) , . . . , yπ(t) )
π∈Sym(1,...,s)

· det(yπ(t+1) , . . . , yπ(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 ± is a multilinear form on C s . When two of the vectors
yi coincide, every term in the expansion of ±, which does not vanish anyway, is cancelled
by a term of the opposite sign: ± is alternating. Since s > rk C, ± = 0.
We ¬x a subset {i1 , . . . , it }, i1 < · · · < it , of {1, . . . , s}. Then, for all π such that
π({1, . . . , t}) = {i1 , . . . , it } the summand corresponding to π in the expansion of ± equals

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

<<

. 9
( 47 .)



>>