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.- team Extension and Cohomology.- Bibliography.- Supplementary Bibliography.- Index.

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**Extra info for Algebraic Groups and Class Fields**

**Example text**

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. Let b1 , ... , br be the coefficients of the these equations and let C = k[bt, . , br 1be the subalgebra of B generated by the bj . The Xi are integral over C and generate A; this implies that A is a C-module of finite type.

Interpretation in terms of idetes. The preceding can easily be translated to Chevalley's language of "ideles". , the multiplicative group of invertible elements in the ring of repartitions (chap. II, no. 5). Write F for the field k(X) and, for every P E X, denote by Up the subgroup of F* formed by the functions 9 such that vp(g) = 0; if n 2': 1, denote by U~n) the subgroup of Up formed by the functions such that vp(l- g) 2': n. With these notations, an idele a is nothing other than a family {ap} PEX of elements of F* such that ap E Up for almost all P.

The maximal ideal m' of A' generates a primary ideal m' A in A and = ep = eA( m' A) = the multiplicity of the ideal m' A in A, in the sense of Chevalley-Samuel [70]. 52 III. Maps From a Curve to a Commutative Group One can also show that ep is equal to the alternating sum ofthe dimensions ofthe k-vector spaces Torr (A, k); this is a particular case of the "Tor formula" . In the case of curves, A and A' are discrete valuation rings, and ep is equal to the ramification index of the corresponding valuations.

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