m m m

W ’’

iW : V

m m

maps W onto a one-dimensional subspace of V which in turn corresponds to a point

m

in P( V ) ∼ PN (K).

=

It is easy to see that the Pl¨cker map is injective. Let p = P(W ) = P(W ). For

u

reasons of symmetry we may assume that the ¬rst coordinate of p is nonzero. Then we

can ¬nd bases w1 , . . . , wm and w1 , . . . , w m of W and W resp. such that

n n

wi = e i + xij ej , w i = ei + xij ej , i = 1, . . . , m.

j=m+1 j=m+1

Looking at the m-minors [1, . . . , i, . . . , m, k] of the m—n matrices of coe¬cients appearing

in the preceding equations one sees immediately that wi = w i for i = 1, . . . , m, hence

W = W.

It takes considerably more e¬ort to describe the image of P. The map P is induced

by a morphism P of a¬ne spaces; P assigns to each m—n matrix the tuple of its m-minors.

Let X be an m — n matrix of indeterminates, and let Y[a1 ,...,am ] , 1 ¤ a1 · · · < am ¤ n,

denote the coordinate functions of AN +1 (K). Then the homomorphism of coordinate

rings associated with P is given as

• : K[Y[a1 ,...,am ] : 1 ¤ a1 < · · · < am ¤ n] ’’ K[X],

Y[a1 ,...,am ] ’’ [a1 , . . . , am ],

[a1 , . . . , am ] specifying an m-minor of X now. We denote the image of • by

G(X);

it is the K-subalgebra of K[X] generated by the m-minors of X. By construction it is

clear that the a¬ne variety de¬ned by the ideal Ker • is the Zariski closure of Im P,

whereas the corresponding projective variety is the closure of Im P. Much more is true:

(1.2) Theorem. (a) P maps the set of m-dimensional subspaces of V bijectively

onto the projective variety with homogeneous coordinate ring G(X).

(b) P maps the mn-dimensional a¬ne space of m — n matrices over K surjectively onto

the a¬ne variety with coordinate ring G(X).

Part (a) obviously follows from (b). In order to prove (b) one ¬rst has to describe

the variety belonging to G(X) as a subvariety of AN +1 (K). This problem will be solved

8 1. Preliminaries

in (4.7). Secondly one has to show the surjectivity of P, a question which will naturally

come across us in Section 7, cf. (7.14).

The projective variety appearing in (1.2),(a) is usually denoted by Gm (V ) and called

the Grassmann variety of m-dimensional subspaces of V . (A di¬erent choice of a basis for

V only yields a di¬erent embedding into PN (K); all these embeddings are projectively

equivalent.)

The argument which showed the injectivity of P helps us to determine the dimension

of Gm (V ): the open a¬ne subvariety of Gm (V ) complementary to the hyperplane given

by the vanishing of Y[1,...,m] , is isomorphic to the a¬ne space of dimension m(dim V ’m),

hence

dim Gm (V ) = m(dim V ’ m).

(Note that we are using (1.2) here !) Varying the hyperplane one furthermore sees that

Gm (V ) is non-singular. The non-singularity of Gm (V ) can also be deduced from another

basic fact. The group GL(V ) of automorphisms of V acts transitively on Gm (V ), since

two m-dimensional subspaces of V di¬er by an automorphism of V only. On the other

m m

hand this action is induced by the natural action of GL(V ) on P( V ) (via V ); so

GL(V ) operates transitively as a group of automorphisms on the Grassmann variety

Gm (V ).

(1.3) Theorem. Gm (V ) is a non-singular variety of dimension m(dim V ’ m).

To de¬ne the Schubert subvarieties one considers the ¬‚ag of subspaces associated

with the given basis e1 , . . . , en of V taken in reverse order:

n

0 = V0 ‚ . . . ‚ Vn = V.

Vj = Kei ,

i=n’j+1

Let 1 ¤ a1 < · · · < am ¤ n be a sequence of integers. Then the Schubert subvariety

„¦(a1 , . . . , am ) of Gm (V ) is de¬ned by

„¦(a1 , . . . , am ) = { W ∈ Gm (V ) : dim W © Vai ≥ i for i = 1, . . . , m }.

The varieties thus de¬ned of course depend on the ¬‚ag of subspaces chosen. But the

automorphism group of V acts transitively on the set of ¬‚ags, and its action induced on

Gm (V ) makes corresponding Schubert subvarieties di¬er by an automorphism of Gm (V )

only. Hence „¦(a1 , . . . , am ) is essentially determined by (a1 , . . . , am ). It is indeed justi¬ed

to call „¦(a1 , . . . , am ) a variety:

(1.4) Theorem. „¦(a1 , . . . , am ) is the closed subvariety of Gm (V ) de¬ned by the

vanishing of all the coordinate functions

bi < n ’ am’i+1 + 1 i, 1 ¤ i ¤ m.

Y[b1 ,...,bm ] , for some

Proof: The proof is simpler if we dualize our notations ¬rst. Let ci = n ’ ai

and Wj = j Kek . Then V = Vn’j • Wj and there is a projection πj : V ’ Wj ,

k=1

Ker πj = Vn’j . By de¬nition

„¦(a1 , . . . , am ) = { W ∈ Gm (V ) : dim πci (W ) ¤ m ’ i for i = 1, . . . , m }.

9

E. Comments and References

After the choice of a basis w1 , . . . , wm , the subspace W is represented by the matrix

(xuv ), wu = n xuv ev . One obviously has

v=1

dim πci (W ) ¤ m ’ i ⇐’ Im’i+1 (xuv : 1 ¤ v ¤ ci ) = 0,

and in case this condition holds, every m-minor which has at least m’i+1 of its columns

among the ¬rst ci columns of (xuv ), vanishes. Thus all the coordinate functions named in

the theorem vanish on „¦(a1 , . . . , am ). Conversely, if Im’i+1 (xuv : 1 ¤ v ¤ ci ) = 0, then

there is an m-minor of (xuv ) di¬erent from zero and having at least m ’ i + 1 of its

columns among the ¬rst ci ones of (xuv ). ”

For arbitrary rings B of coe¬cients the Schubert cycle associated with „¦(a 1 , . . . , am )

is the residue class ring of G(X) with respect to the ideal generated by all the minors

[b1 , . . . , bm ] such that bi < n ’ am’i+1 + 1 for some i.

E. Comments and References

The references given below have been included to manifest the geometric signi¬cance

of determinantal and Schubert varieties. We have restricted ourselves to books (with one

exception) since any selection of research articles would inevitably turn out super¬cial

and random. (After all, the AMS classi¬cation scheme contains the keys “Determinantal

varieties” and “Schubert varieties”.)

The classical source for “the geometry of determinantal loci” is Room™s book [Rm]. It

gives plenty of information on the early history of our subject. The decisive treatment of

Schubert varieties has been given by Hodge and Pedoe in their monograph [HP]. Among

the recent books on algebraic geometry those of Arabello, Cornalba, Gri¬ths, and Harris

[ACGH], Fulton [Fu], and Gri¬ths and Harris [GH] contain sections on determinantal

and/or Schubert varieties. Kleiman and Laksov™s article [KmL] may serve as a pleasant

introduction.

2. Ideals of Maximal Minors

Though many of the results of this section are covered by the subsequent investiga-

tions, cf. Sections 4, 5, and 6, it seems worth to look for those properties of determinantal

rings which have been well known for a long time just as the methods they are proved

by. In particular one has a rather direct approach to the results concerning the residue

class ring B[X]/I where I is the ideal generated by the maximal minors of X.

The second part of the section deals with free resolutions of It (X) in two compar-

atively simple cases. The ¬rst one will be that of maximal minors and after it we shall

treat the case in which m = n, t = n ’ 1, digressing slightly from the title of this section.

A. Classical Results on Height and Grade

Let A be an arbitrary commutative ring and U = (uij ) an m — n matrix, m ¤ n,

of elements in A. As in Section 1 we denote by It (U ) the ideal in A generated by the

t-minors of U . There are two observations, simple but often used:

(i) It (U ) is invariant under elementary row or column transformations.

(ii) If the element umn is a unit in A, then It (U ) = It’1 (U ) where U = (uij ) is an

(m ’ 1) — (n ’ 1) matrix, uij = uij ’ umj uin u’1 .

mn

Our investigations concerning properties of It (X) begin with a height formula. There

is an upper bound which only depends on t and the size of the matrix.

(2.1) Theorem. Let A be a noetherian ring and U = (uij ) an m — n matrix of

elements in A. If It (U ) = A then

ht It (U ) ¤ (m ’ t + 1)(n ’ t + 1).

Proof: By induction on t. If t = 1, the inequality is Krull™s principal ideal theorem.

Let t > 1 and take a minimal prime ideal P of It (U ). We must show that ht P ¤

(m ’ t + 1)(n ’ t + 1). Localizing at P we may assume that A is a local ring with

maximal ideal P , It (U ) being P -primary.

If an element of U is a unit in A, the theorem follows from the inductive hypothesis

and the observation (ii) made above. We may therefore suppose that uij ∈ P for all i, j.

Let T be an indeterminate over A. We consider the m — n matrix

«

u11 + T u12 · · · u1n

u22 · · · u2n ·

¬ u21

U =¬ .·

. .

.

. .

. . .

um2 · · · umn

um1

Then It (U ) ‚ P A[T ] and It (U ) + T A[T ] = It (U )A[T ] + T A[T ]. From the lemma below

it follows that P = P A[T ] is a minimal prime ideal of It (U ). Because of ht P = ht P we

may replace P by P . After localizing the ring A[T ] at P , the element u11 + T becomes

a unit. As noticed above, the inequality then follows from the inductive hypothesis. ”

11

A. Classical Results on Height and Grade

(2.2) Remark. Theorem (2.5) will show that the bound in (2.1) cannot be im-

proved in general. However, under special circumstances one has much better estimates:

If It+1 (U ) = 0 and It (U ) = A, then

ht It (U ) ¤ m + n ’ 2t + 1.

The condition It+1 (U ) = 0 holds if U is a matrix of rank t. More generally, if U is a p — q

submatrix of U and u ≥ t, then

ht It (U )/Iu (U ) ¤ (m ’ t + 1)(n ’ t + 1) ’ (p ’ u + 1)(q ’ u + 1).

In a ring satisfying the saturated chain condition ([Ka], p. 99) the last inequality is

equivalent to

ht It (U ) ’ ht Iu (U ) ¤ (m ’ t + 1)(n ’ t + 1) ’ (p ’ u + 1)(q ’ u + 1).

Thus all the ideals Iu (U ) have their maximal height along with It (U ), in particular no

u-minor, u ≥ t, can be zero. We refer the reader to [Br.5] for these results. ”

(2.3) Lemma. Let A be a local ring with maximal ideal P and let I be a P -primary

ideal. In the polynomial ring A[T ], let I ‚ P A[T ] be an ideal which has I as residue

modulo T A[T ]. Then P A[T ] is a minimal prime ideal of I .

Proof: The hypothesis on I yields an isomorphism A[T ]/(I + T A[T ]) ∼ A/I. =

Therefore P A[T ] + T A[T ] is a minimal prime ideal of I + T A[T ]. Now let Q ⊆ P A[T ]

be a minimal prime ideal of I . In the ring A[T ]/Q the ideal (Q + T A[T ])/Q is a

principal ideal with (P A[T ] + T A[T ])/Q as one of its minimal prime ideals. From

ht(P A[T ] + T A[T ])/Q ¤ 1 and the chain of prime ideals Q ‚ P A[T ] ‚ P A[T ] + T A[T ]

we get Q = P A[T ]. ”

If U = X the inequality in (2.1) actually becomes an equality. This will be proved

by a localization argument frequently used in the sequel.

(2.4) Proposition. Let X = (Xij ) and Y = (Yij ) be matrices of indeterminates

over the ring B of sizes m — n and (m ’ 1) — (n ’ 1), resp. Then the substitution

’1

Xij ’’ Yij + Xmj Xin Xmn , 1 ¤ i ¤ m ’ 1, 1 ¤ j ¤ n ’ 1,

Xmj ’’ Xmj , Xin ’’ Xin

induces an isomorphism

B[X][Xmn ] ∼ B[Y ][Xm1 , . . . , Xmn , X1n , . . . , Xm’1,n ][Xmn ]

’1 ’1

=

which maps the extension of It (X), t ≥ 1, to the extension of It’1 (Y ). In particular this

isomorphism induces an isomorphism

Rt (X)[x’1 ] ∼ Rt’1 (Y )[Xm1 , . . . , Xmn , X1n , . . . , Xm’1,n ][Xmn ]

’1

mn =

where xmn denotes the residue class of Xmn in Rt (X).

Proof: The substitution given in the proposition of course induces a homomor-

phism

’1 ’1

• : B[X][Xmn ] ’’ B[Y ][Xm1 , . . . , Xmn , X1n , . . . , Xm’1,n ][Xmn ].

12 2. Ideals of Maximal Minors

’1 ’1

Analogously we get a homomorphism ψ : B[Y ][Xm1 , . . . , Xm’1,n ][Xmn ] ’’ B[X][Xmn ]

by substituting

’1

Yij ’’ Xij ’ Xmj Xin Xmn , Xmj ’’ Xmj , Xin ’’ Xin

Evidently • and ψ are inverse to each other. From the remark (ii) made above, it

’1 ’1

follows that It (X)B[X][Xmn ] = It’1 (X) where X = (Xij ’ Xmj Xin Xmn ). Clearly •

maps It’1 (X) to the ideal generated by It’1 (Y ). ”

(2.5) Theorem. Let X = (Xij ) be an m — n matrix of indeterminates over the

noetherian ring B. Then

grade It (X) = (m ’ t + 1)(n ’ t + 1)

if 1 ¤ t ¤ min(m, n) + 1.

Proof: In view of (2.1) we must only prove that (m ’ t + 1)(n ’ t + 1) is a lower

bound for grade It (X). The cases t = 1 and t = min(m, n) + 1 are trivial. Let 1 <

t ¤ min(m, n) and P be a prime ideal in B[X] containing It (X). We will show that

depth B[X]P ≥ (m ’ t + 1)(n ’ t + 1).

Certainly depth B[X]P ≥ mn > (m ’ t + 1)(n ’ t + 1) if P contains all the inde-

terminates Xij . Otherwise we may assume that Xmn ∈ P . Consider the isomorphism

/

’1 ∼ ’1

B[X][Xmn ] = B[Y ][Xm1 , . . . , Xm’1,n ][Xmn ] from (2.4). Using well-known grade formu-

las and the inductive hypothesis, we get

’1

depth B[X]P ≥ grade It [X]B[X][[Xmn ]

’1

= grade It [Y ]B[Y ][Xm1 , . . . , Xm’1,n ][Xmn ]

≥ grade It [Y ]

= (m ’ t + 1)(n ’ t + 1). ”

Though the following result is not covered by the title of this subsection, it is included

here since its proof is another e¬ective application of (2.4).