COHEN-LENSTRA FOR QUADRATIC FUNCTION FIELDS

5

arrangement is M(A) = V- UxEAX. The £-adic Poincare polynomial of M(A) is

Pe(M(A), t)

=

Li hi(M(A),

(}!e)ti.

Let C(A) be the set of nonempty intersections of elements of A, ordered by

reverse inclusion; if X, YEA, then X::::; Y if and only if X;:;? Y. The lattice C(A)

has a unique minimal element, V. The rank r A(X) of X E C(A) is the codimension

of X in V. We will say that a hyperplane H

C

Vis generic with respect to A if for

each X E C(A) we have dim(X

n

H)

=dim( X)- 1 if r(X)

n,

and X

n

H

=

0

if r(X) =

n.

Being generic with respect to

A

is an open condition on the space of

hyperplanes in

V.

If H

c

Vis a hyperplane and X

E

A, let XH = H

n

X; it is a hyperplane of

H. Let AH = {XH: X E

A};

it is a hyperplane arrangement inside H.

LEMMA

3.2.

Let A be an arrangement of hyperplanes inside an n-dimensional

vector space V over an algebraically closed field k, and let H

c

V be a hyperplane

which is generic with respect to

A.

Then

Pe(M(A), t)

=

Pe(M(AH ), t)

+

hn(M(A), cfl!e)tn.

PROOF. The Betti numbers of M(A) are independent of k and£, in the follow-

ing sense. Suppose that

k'

is an algebraically closed field,

V' / k'

is a vector space

of dimension

n,

£' is a rational prime invertible in

k',

and

A'

any arrangement of

hyperplanes in V'. If C(A')

~

C(A), then Pe(M(A), t) = Pe,(A', t) [6, Section 5].

Therefore, we may replace

A

by a combinatorially equivalent arrangement over C.

For a complex hyperplane arrangement the £-adic and topological Betti numbers

agree [18, Thm. 1.1], so that we may compute the Poincare polynomial using the

method of [21, Chapter 2].

Let

p,:

C(A) x C(A)---+ Z be the Mobius function of the lattice of subspaces of

A, and let p,(X) = p,(V, X). The Poincare polynomial of M(A) is then [21, Def.

2.48 and Thm. 5.93]

(3.1)

Pe(M(A),t)

=

L

p,(X)(-ttA(X)

XE.C(A)

(3.2)

='f)

-1)i (

L

p,(X))

ti.

i=O

XE.C(A):r A(X)=i

Note in particular that the

ith

Betti number depends only on those elements of

C(A) which have codimension at most i in

V.

Now let H

c

V be a hyperplane generic with respect to

A.

Recall that if

r A(X)

n,

then r AH (XH) = r A(X). The description

(3.2)

shows that the Poincare

polynomial of M(AH) is

Pe(M(AH),t)

=

~(-1)i

(

L

.p,(XH)) ti

t=O

XHE.C(AH):rAH(XH)=t

=

~(-1)i

(

L

p,(X))

ti

i=O

XE.C(A):r A(X)=i

n-1

=

L(

-1)ihi(M(A), cfl!e)ti. 0

i=O