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M 2 --? M® f -l( 13) - 0 is exact as desired. As noted earlier, (iii) f°(N ' ) = M 0 this and part (a) above that f°(N 9 m) = M that is locally free of rank m over p E M . Since of f ; and it follows from m m . NCO mlf(U)) = M 0 m , so that is locally free of rank m over M . (iv) over of Finally, suppose that A is a coherent analytic sheaf N , and consider a point f1 p E M .
With these observations in mind, Theorem 6 can be extended as follows. jective line Let J be any coherent analytic sheaf over the proI P, represented by an exact sequence of analytic sheaves of the form where a and -1 are locally free. I , considered as a sheaf of -modules, it follows readily from the condition that g and are locally free that the following is also an exact sequence of analytic sheaves. 'm- 0 in (7) is represented by an ml Xm non- singular matrix ® of rank m1 over the field P(p , m ) ; and after choosing coordinates such that this matrix has the form ® = (®1,0) , where ®1 c GL(m1, ®d "'(p ) , it follows from (7) that m1 J m Thus, upon tensoring with ryyi, every coherent analytic sheaf over IP reduces to a free sheaf of -modules.