By A. Campillo
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Extra resources for Algebroid Curves in Positive Characteristic
GIG' admits a local differentiable (in tact, real analytic) section. G can be regarded in a natural way as a differentiable principal bundle over GIG' with structure group and fibre G' and projection u. 3. Let G' be a closed complex LIE subgroup 01 the complex LIE group G. ,. GIG' admits a local holomorphic section. G can be regarded in a natural way as a complex analytic principal bundle over GIG' with structure group and fibre G' and projection u. 3) can be proved by means of canonical coordinates in an open neighbourhood of e EG.
6)~ kernel (h~). 2 is the cohomology group HO(X. e). This proves the statement for q = O. Let ~p be the kernel of the homomorphism hP : 6 p -+ 6 p+l' The sequence (24) gives an exact sequence of sheaves over X (26) for each p ~ O. Since the cohomology groups H9 (X, 6 p ) vanish for q;;;;; 1 the exact cohomology sequence of (26) gives natural isomorphisms H9- 1 (X. ~Hl) ~ HIl (X. ~p) for q ;;;;; 2 . Since ~o = (27) e repeated application of (27) gives Hl(X. ~9-1) ~ H9(X. e) for q;;;;; 1. (28) The exact cohomology sequence of (26).
There is also a vector space Hom (A, B) over K, whose elements are the homomorphisms (linear maps) from A to B. For each finite dimensional vector space A over K the dual vector space A * of linear forms is defined. A* = Hom (A, K) by definition and dim(A*) = dim (A). The vector space AI' A of p-vectors is also defined. Vectors aI' as, ... , a p E A define a vector a l " as" ... " a p E AI' A, which depends linearly on each factor. A permutation of the factors av as, ... , ap multiplies a l " as " ...
Algebroid Curves in Positive Characteristic by A. Campillo