By Jean-Pierre Serre

ISBN-10: 1461210356

ISBN-13: 9781461210351

ISBN-10: 1461269938

ISBN-13: 9781461269939

Precis of the most Results.- Algebraic Curves.- Maps From a Curve to a Commutative Group.- Singular Algebraic Curves.- Generalized Jacobians.- classification box Theory.- staff Extension and Cohomology.- Bibliography.- Supplementary Bibliography.- Index.

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0 Corollary 3. Suppose that f : G -+ G' is injective. In order that the algebraic group structure of G be induced by that of G' it is necessary and sufficient that f* : s( G') -+ s( G) be surjective. PROOF. After replacing G' by a subgroup, we can suppose that tive, and we are then reduced to the preceding corollary. f is bijec0 One could go further and define the adjoint linear representation of G, bi-invariant differential forms, the p-th power operation on the Lie algebra (in characteristic p), etc.

P, = vp,(g')f'(PI), by the very definition of f'. Condition iii) is thus fulfilled, and the same is evidently true of condition iv). 0 c) Local symbol of a norm. Let 11' : X ---+ X' be a ramified covering, let SI be a finite subset of XI, and let f' : XI - SI ---+ G be a map from XI - SI to a commutative group G. We put S = 1I'-1(SI). g the norm of 9 in the extension k(X)jk(XI) defined by 11'. We then have the following proposition, analogous to proposition 3, but where the roles of f and 9 have been permuted: Proposition 4.

One can even extend it to inseparable coverings using a suitable definition of the trace of a differential (the definition used above no longer applies). We will come back to this in chap. III, no. 3. 13. Proof of lemma 5 As in proposition 5, the question is local. We have a field of formal power series K and a finite separable extension L of K. If t (resp. u) denotes a uniformizing parameter of L (resp. K), we want to establlish the formula ReSt (f du) = Res .. (Tr(f) du) Moreover, we can restrict to the case where for all IE L.

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