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
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