By Prof. Dr. Michel Lazard (auth.)
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Additional info for Commutative Formal Groups
C H A P T E R III THE G E N E R A L E Q U I V A L ~ C E i. 12). 11) . ca! groups, ~K(G_m) , w i t h the same u n d e r l y i n g uniform space, n a m e l y the set of m o r p h i s m s ~(DK,DK) with its (simple or order) -58topology (once the topological matter which one - there space and a uniform space; remember K . 2), G any curve as a formal series the ~dentity m o r p h i s m ~(~a ) and ~(_G_m) respectiy 9- any occurence some group ~(G) of ¥~ (resp. shall mean that (resp. 4 the formula ¥ = En~$ means group law an ~ K .
Deg. (n+l) Any morphism the same n-bud n-bud. can f~ w e take representative of degree There x+y bilinear and as . e. of and bv morphisms~ rings ~ 2' over has Jn f = J n f' ' defines is a r e p r e s e n t a t i v e a formal module with vanishing o f the structure, we homogeneous . e. 6 : V In the c a s e w h e r e components by f' m f(x,f(y,z)) theorem. a qroup of morphisms, concerning changes of homomorphisms. Every n-bud, for any l a w as a r e p r e s e n t a t i v e . 7 The tension using ~ theorem theorem n-buds.
Zn~ P : K Let become G=m over respecti- n ~! (-i) n - i ~ K group be a s u r j e c t i v e rinq O homQ~orphism, and K. ~ e D is a q r o u p such there that We formal ~,G o, the the lift groups: any formal for a s u i t a b l e Let K the by taking for by fl have K we the for some such that the g r o u p law KO , but it h o l d s for to K° lift theorem, we o b t a i n coefficients , where modulo series f = fo in ~*fo ~(x) embedding set ~ x K± defined + ~(y) formal in ~ K o. e. ~ = Id). by which the sum on the make K° D (I) up fl by writing In t h i s case, we c a n c o n s i d e r law o o(I) ~ n(I) KI Ki , the c o e f f i c i e n t s words, K.
Commutative Formal Groups by Prof. Dr. Michel Lazard (auth.)