By Shafarevich I.R. (ed.)

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By a chart map from B—>F, we mean a pair (b,j) where j e A(F) and b e Mor^(B,F(j)). The set of chart maps from B to F is denoted Ch(B,F). od and c = p k od }. Then R is an equivalence relation on Ch(B,F). The set of R-equivalence classes is denoted by Cc(B,F). For (b,j),(c,k) e Ch(B,F), write (b,j) - (c,k) if ((b,j),(c,k)) e R. For (b,j) e Ch(B,F), the class of (bJ) is called its chart 39 Passage from Local to Global class and is denoted by lb,j] or lb]. In w h a t follows, fix B e C and F e Can(C).

The lemma follows directly. d) on M o r C a n ( B 0 , C 0 ) x M o r C a n ( A 0 , B 0 ) - - * M o r C a n ( A 0 , C 0 ) , > c0ob0 th e identity assignment A 0 •—> £ A e Mor£an(A0,A0). c) on AP(1,1), P l =p2=lA. For b : A—>B a C - m o r p h i s m , let bP denote t h e Can-morphism {(l,[b,l])} . The following r e m a r ks are e l e m e n t a r y : (A) For A , Be C, t h e function b—>bP is a bijection M o r c ( B , A ) — > M o r C a n ( B p , A P ) . (B) For A G C and B 0 ^ Can, t h e function b0»—>b0(l) determines a bijection Mor^ a n (AP,B 0 )—>Cc(A,B 0 ).

Suppose A e D , ot12:G(l,2)—>A and ot 23:G(2,3)—>A are morphisms such that (G(2);p12,p23) is a fibered product for (G(l,2),cx12)>

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Algebraic geometry I-V by Shafarevich I.R. (ed.)


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