LEFSCHETZ: THEORY OF ALGEBRAIC MANIFOLDS. 5 



whatever the manifold M. Proceed by induction. The formula 

 is easily verified when 



ikf = Ms = Ai A2 . . . . As , s>r . 

 Grant that it holds for s>s^, I say that it is also true for s = s^. 

 For /.M,^D = M,S; 



.'. \M,J1+Dy \ = lM,Ji+C) \. (6) 



But M, D'^ = M,.Z)'^-\ D. 



Hence from the assumptions made follows : 



1 



j M^^D'^'Ul+D) ;■ = ■; M.^D'^-^^+C) ' \ ;k>l 



This remains true when M^ is replaced by M^^ C^ . 



.-. \M,^D''\ = \M,^i(l+C)' -l)''\,k>l 

 From (6) follows 



(7) 



) 



k=0 



This together with (7) gives finally 



\M,^l+c)\ 



/ 1 

 1 



\_ 1 



]M, iiii + c)' -1 + iV +yM+D)-y.{i^-cy)\ 



1 



.-. j M,Sl + D) -M^Jl+O' \ =0, 

 as was to be proved. 



5. It remains to be shown that if >.D = Ci — C2 , then 



1 _ J 



jMa+D) S = jM(i+Ci)^(i+C2) M- 



If we set /.D = D' = Ci - C2, we have by (5) 



I 

 \M{1+D) \ = \Mil+D'y \ . 



(8) 



(9) 



