By Jean-Pierre Serre

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

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13). 8. Proof in characteristic p > 0: case a) We are going to need the following two elementary lemmas: Lemma 6. Every regular map from the projective line minus one point to the multiplicative group G m is constant. PROOF. We can suppose that the point in question is the point at infinity. , a constant. 0 Lemma 7. Let V be a non-singular variety and let Q be the sheaf of differential forms of degree 1 (cf chap. II, no. 7). Suppose that, for every P E V, the Op-module Q p is generated by elements of HO(V, Q).

H == 1 mod ml on SI - pl. S = = L L p .... P' (110 1I',h)p (110 1I',g)p. o 33 §1. Local symbols 3. Example of a local symbol: additive group case From now on, we limit ourselves to the case where the commutative group G is a connected algebraic group, the map I : X - S -+ G being a regular map. We can then consider I as a rational map from X to G, regular away from S. Unless otherwise stated, we suppose that S is the smallest subset of X having this property, in other words it is the set of points where I is not regular.

Elements The ring A is a k-algebra which is generated by a finite number of Xl, ... ,x n . These elements are integral over A", as the equation 49 §3. Auxiliary results of integral dependence II (x - x") = 0 "Eg shows. The following lemma then shows that A g is a k-algebra of finite type: Lemma 10. Let A be an algebra of finite type over a commutative Noetherian ring k and let B be a subalgebra of A such that every element of A is integral over B. Then B is a k-algebra of finite type. PROOF. Let Xi, 1 :::; i :::; n, be generators of the algebra A; each of these elements satisfies an equation of integral dependence over B, say fi (Xi) = O.