By David Mumford
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Then there exists a complex K' of finitely generated A-modules such that K. 0 (0) only if 0 w p < n and KP is free if 1 < p < n and a homomorphism of complexes that CD P(C), all i. K ' --. C' such that 0 induces isomorphisms H`(K') Moreover if the CP are A -flat, then K° will be A flat too. PROOF. We define, by descending induction on m, diagrams: am+l am } Km+1 Km > Km+2 --) ... j,#m+1 `Ym j,cm+s W Cm-1 Cm am Cm+1 am+1 s Cm+2 Put K9 = 0 for p > n. Suppose we have defined (KP, p> m + 1 such that the following conditions hold: aP) for 0) a'0P =0P+1aP, (p > m + 1).
We have I O(zo + w) I = 9(zo + d + u) I = 9(zo + d) I e and since ReH(zo+d,u)+ JH(u,u) =ReH(zo+d,w) - ReH(zo+d,d)+ JH(w,w) + JH(d, d) - ReH(w, d) _ JyH(w, w)+Re H(zo, w) + c(d, zo). Of the terms on the right, for fixed zo, the first is a real negative definite quadratic form in w, the second linear in w and the third is bounded (since d stays in a compact set K), so that the expression tends to - co as w-w in W, and applying the maximum principle to 9(zo w) as a function of w, we conclude that 9(zo + w) .
ALGEBRAIC THEORY VIA VARIETIES COROLLARY 3. ' be as above (unlike Corollary 2, Y need not be reduced). Assume for some p that H'(X5, Ft,) _ (0), all y e Y. Then the natural map RP-' f* (JF) ®ppk(y) - HP-1(X1 F,,) is an isomorphism for all y e Y. PROOF. Again assume Y = Spec (A), K' as in the theorem. For ally a Y, we know that dP-1 KP-1 (D k(y) ) K' (9 k(y) d9 ) K'+1 ® k(y) is exact. Split the vector space K' ® k(y) into W1 $ W2, where W1 = Image of K2'-1 ® k(y), and Wti is mapped injectively to K'+ 1® k(y).
Abelian Varieties, Second Edition by David Mumford