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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