fractions of A. The rank of a linear map is the rank of its image. The length of a module

M is indicated by »(M ).

The notations of homological algebra concerning Hom, —, and their derived functors

seem to be completely standardized; for them we refer to [Rt]. Let A be a ring, M and

N A-modules, and f : M ’ N a homomorphism. We put

M — = HomA (M, A)

and

f — = HomA (f, A) : N — ’ M — .

M — and f — are called the duals of M and f .

For the symmetric and exterior powers of M (cf. [Bo.1] for multilinear algebra) we

use the symbols

i

M and Sj (M )

i i

resp. Sometimes we shall have to refer to bases of F — , F and F — , given a basis

e1 , . . . , en of the free module F . The basis of F — dual to e1 , . . . , en is denoted by e— , . . . , e— .

1 n

For I = (i1 , . . . , ik ) the notation eI is used as an abbreviation of ei1 § · · · § eik , whereas

e— expands into e—1 § · · · § e—k . (The notation eI will be naturally extended to arbitrary

i i

I

families of elements of a module.)

We need some combinatorial notations. A subset I ‚ Z also represents the sequence

of its elements in ascending order. For subsets I1 , . . . , In ‚ Z we let

σ(I1 , . . . , In )

denote the signum of the permutation I1 . . . In (given by iuxtaposition) of I1 ∪. . .∪In rela-

tive to its natural order, provided the Ii are pairwise disjoint; otherwise σ(I1 , . . . , In ) = 0.

A useful formula:

σ(I1 , . . . , In ) = σ(I1 , . . . , In’1 )σ(I1 ∪ . . . ∪ In’1 , In ).

For elements i1 , . . . , in ∈ Z we de¬ne

σ(i1 , . . . , in ) = σ({i1 }, . . . , {in }).

The cardinality of a set I is denoted |I|. For a set I we let

S(m, I) = {J : J ‚ I, |J| = m}.

Last, not least, by

1, . . . , i, . . . , n

we indicate that i is to be omitted from the sequence 1, . . . , n.

3

B. Minors and Determinantal Ideals

B. Minors and Determinantal Ideals

Let U = (uij ) be an m — n matrix over a ring A. For indices a1 , . . . , at , b1 , . . . , bt

such that 1 ¤ ai ¤ m, 1 ¤ bi ¤ n, i = 1, . . . , t, we put

«

···

u a 1 b1 u a 1 bt

¬ . ·.

[a1 , . . . , at |b1 , . . . , bt ] = det . . .

. .

···

u a t b1 u a t bt

We do not require that a1 , . . . , at and b1 , . . . , bt are given in ascending order. The

symbol [a1 , . . . , at |b1 , . . . , bt ] has a twofold meaning: [a1 , . . . , at |b1 , . . . , bt ] ∈ A as just

de¬ned, and

[a1 , . . . , at |b1 , . . . , bt ] ∈ Nt — Nt

as an ordered pair of t-tuples of non-negative integers. Clearly [a1 , . . . , at |b1 , . . . , bt ] = 0

if t > min(m, n). For systematic reasons it is convenient to let

[…|…] = 1.

If a1 ¤ · · · ¤ at and b1 ¤ · · · ¤ bt we say that [a1 , . . . , at |b1 , . . . , bt ] is a t-minor of U . Of

course, as an element of A every [a1 , . . . , at |b1 , . . . , bt ] is a t-minor up to sign. We call t

the size of [a1 , . . . , at |b1 , . . . , bt ].

Very often we shall have to deal with the case t = min(m, n). Our standard assump-

tion will be m ¤ n then, and we use the simpli¬ed notation

[a1 , . . . , am ] = [1, . . . , m|a1 , . . . , am ].

The m-minors are called the maximal minors, those of size m ’ 1 the submaximal

minors. (In section 9 the notion “maximal minor” will be used in a slightly more general

sense.)

The ideal generated by the t-minors of U is denoted

It (U ).

The reader may check that It (U ) is invariant under invertible linear transformations:

It (U ) = It (V U W )

for invertible matrices V, W of formats m — m and n — n resp.

Sometimes we will need the matrix of cofactors of an m — m matrix:

Cof U = cij ,

cij = (’1)i+j [1, . . . , j, . . . , m|1, . . . , i, . . . , m].

4 1. Preliminaries

C. Determinantal Rings and Varieties

Let B be a commutative ring, and consider an m — n matrix

«

X11 · · · X1n

¬. .·

X= . .

. .

Xm1 · · · Xmn

whose entries are independent indeterminates over B. The principal objects of our study

are the residue class rings

Rt (X) = B[X]/It (X),

B[X] of course denoting the polynomial ring B[Xij : i = 1, . . . , m, j = 1, . . . , n]. The ideal

It (X) is generated by the t-minors of X, cf. B. Whenever we shall discuss properties of

Rt (X) which are usually de¬ned for noetherian rings only (for example the dimension or

the Cohen-Macaulay property), it will be assumed that B is noetherian.

Over an algebraically closed ¬eld B = K of coe¬cients one can immediately associate

a geometric object with the ring Rt (X). Having chosen bases in an m-dimensional vector

space V and an n-dimensional vector space W one identi¬es HomK (V, W ) with the mn-

dimensional a¬ne space of m — n matrices, of which K[X] is the coordinate ring. Under

this identi¬cation the subvariety de¬ned by It (X) corresponds to

Lt’1 (V, W ) = { f ∈ HomK (V, W ) : rk f ¤ t ’ 1 }.

We want to associate the letter r with “rank”, and so we replace t by r + 1. Furthermore

we put L(V, W ) = HomK (V, W ).

It is not surprising that the geometry of Lr (V, W ) re¬‚ects certain properties of the

linear maps f ∈ Lr (V, W ). Let us consider the following two elementary statements

which will lead us quickly to some nontrivial information on Lr (V, W ): (a) The map f

can be factored through K r . (b) Let U ‚ V be a vector subspace of dimension r and

U a supplement of V , i.e. V = U • U; if f |U is injective, then there exist unique linear

maps g : U ’ U , h : U ’ W such that f (u • u) = h(u) + h(g(u)) for all u ∈ U , u ∈ U

(in fact, h = f |U ).

Statement (a) shows that the morphism

L(V, K r ) — L(K r , W ) ’’ Lr (V, W ),

given by the composition of maps, is surjective. Being an epimorphic image of an irre-

ducible variety, Lr (V, W ) is irreducible itself. An application of (b): It is easy to see that

the subset

M = { f ∈ Lr (V, W ) : f |U injective }

is a nonempty open subvariety of Lr (V, W ): One chooses a basis of V containing a

basis of U ; then M is the union of subsets of Lr (V, W ) each of which is de¬ned by the

non-vanishing of a determinantal function. By property (b) we have an isomorphism

L(U , U ) — L(U, W ) \ Lr’1 (U, W ) ’’ M.

5

C. Determinantal Rings and Varieties

Since the variety on the left is an open subvariety of L(U , U ) — L(U, W ), we conclude at

once that

dim Lr (V, W ) = dim M = dim L(U , U ) — L(U, W ) = (m ’ r)r + rn

= mr + nr ’ r2 .

Furthermore M is non-singular. Varying U one observes that all the points f ∈ Lr (V, W )\

Lr’1 (V, W ) are non-singular:

(1.1) Proposition. (a) Lr (V, W ) is an irreducible subvariety of L(V, W ).

(b) It has dimension mr + nr ’ r 2 .

(c) It is non-singular outside Lr’1 (V, W ).

The only completely satisfactory information on Rr+1 (X) we can draw from (1.1),

is its dimension:

dim Rr+1 (X) = mr + nr ’ r 2

Part (a) only shows that the radical of Ir+1 (X) is prime, and unfortunately there seems

to be no easy way to prove that Ir+1 (X) is a radical ideal itself (over every reduced

ring B of coe¬cients). Once this is known one can of course directly reverse (c): The

generators of the ideal of Lr (V, W ) have all their partial derivatives in Ir (X), and the

Jacobi criterion (or the de¬nition of non-singularity, depending on ones point of view)

implies in conjunction with (c) that Lr’1 (V, W ) is the singular locus of Lr (V, W ).

Proposition (1.1) and its proof have been included not only in order to enrich these

introductory considerations by some substantial results. We shall encounter algebraic

versions of the ideas underlying its proof several times again.

It would be very di¬cult (for us, at least) to investigate the rings Rt (X) without

viewing them as the most prominent members of a larger class of rings of type B[X]/I

which we call determinantal rings. Their de¬ning ideals I can be described as follows:

Given integers

1 ¤ u1 < · · · < up ¤ m, 0 ¤ r1 < · · · < rp < m,

and

1 ¤ v1 < · · · < vq ¤ n, 0 ¤ s1 < · · · < sq < n,

the ideal I is generated by the

(ri + 1)-minors of the ¬rst ui rows

and the

(sj + 1)-minors of the ¬rst vj columns,

i = 1, . . . , p, j = 1, . . . , q. Later on we shall introduce a systematic notion for determi-

nantal rings which is hard to motivate at this stage.

In order to relate the general class of determinantal rings just introduced to the ge-

ometric description of Rr+1 (X) given above, one chooses bases d1 , . . . , dm and e1 , . . . , en

of V and W resp., K being an algebraically closed ¬eld, V and W vector spaces of

dimensions m and n. Let

k k

—

Ke—

Vk = Kdi and Wk = i

i=1 i=1

(e— , . . . , e— is the basis dual to e1 , . . . , en , cf. A above).

1 n

6 1. Preliminaries

Then the ideal I above de¬nes the determinantal variety

rk f |Vui ¤ ri , rk f — |Wvj ¤ sj ,

—

{ f ∈ HomK (V, W ) : i = 1, . . . , p, j = 1, . . . , q }.

The reader may try to ¬nd and to prove the analogue of (1.1) for the variety just de¬ned.

It will of course be included in the main results of the Sections 5 and 6.

D. Schubert Varieties and Schubert Cycles

In the sections 4“9 we shall treat a second class of rings simultaneously with the

determinantal rings: the homogeneous coordinate rings of the Schubert varieties (gener-

alized to an arbitrary ring of coe¬cients) which we call Schubert cycles for short. There

are two reasons for our treatment of Schubert cycles: (i) They are important objects of

algebraic geometry. (ii) Their combinatorial structure is simpler than that of determi-

nantal rings, and most often it is easier to prove a result ¬rst for Schubert cycles and to

descend to determinantal rings afterwards. Algebraically one can consider every determi-

nantal ring as a dehomogenization of a Schubert cycle (cf. 16.D and (5.5)). In geometric

terms one passes from a (projective) Schubert variety to an (a¬ne) determinantal variety

by removing a hyperplane “at in¬nity”.

The ¬rst step in the construction of the Schubert varieties is the description of the

Grassmann varieties in which they are embedded as subvarieties. While a projective

space gives a geometric structure to the set of one-dimensional subspaces of a vector

space, a Grassmann variety does this for the set of m-dimensional subspaces, m ¬xed.

Let K be an algebraically closed ¬eld, V an n-dimensional vector space over K, and

e1 , . . . , en a basis of V . In a ¬rst attempt to assign “coordinates” to a vector subspace

W , dim W = m, one chooses a basis w1 , . . . , wm of W and represents w1 , . . . , wm as

linear combinations of e1 , . . . , en :

n

wi = xij ej , i = 1, . . . , m.

j=1

Unfortunately the assignment W ’ (xij ) is not well-de¬ned, since (xij ) depends on the

basis w1 , . . . , wm of W . Exactly the matrices

T · (xij ), T ∈ GL(m, K),

represent W . However, the Pl¨cker coordinates

u

p = ([a1 , . . . , am ] : 1 ¤ a1 < · · · < am ¤ n)

formed by the m-minors of (xij ) remains almost invariant if (xij ) is replaced by T · (xij );

it is just replaced by a scalar multiple: The point of projective space with homogeneous

coordinates p depends only on W ! Thus one has found a well-de¬ned map

n

P : { W ‚ V : dim W = m } ’’ PN (K), ’ 1.

N=

m

It is called the Pl¨cker map.

u

7

D. Schubert Varieties and Schubert Cycles

This construction can of course be given in more abstract terms. With each subspace

W , dim W = m, one associates the embedding

iW : W ’’ V.