Download E-books Commutative Group Schemes (Lecture Notes in Mathematics) PDF

By F. Oort

We limit ourselves to 2 features of the sphere of staff schemes, within which the consequences are relatively entire: commutative algebraic staff schemes over an algebraically closed box (of attribute assorted from zero), and a duality concept challenge­ ing abelian schemes over a in the community noetherian prescheme. The prelim­ inaries for those issues are introduced jointly in bankruptcy I. SERRE defined houses of the class of commutative quasi-algebraic teams via introducing pro-algebraic teams. In char8teristic 0 the location is obvious. In attribute various from 0 info on finite workforce schemee is required in an effort to deal with workforce schemes; this knowledge are available in paintings of GABRIEL. within the moment bankruptcy those rules of SERRE and GABRIEL are prepare. additionally extension teams of simple team schemes are decided. a tenet in a paper by way of MANIN gave crystallization to a fee11ng of symmetry relating subgroups of abelian kinds. within the 3rd bankruptcy we end up that the twin of an abelian scheme and the linear twin of a finite subgroup scheme are similar in a truly typical method. Afterwards we turned conscious certain case of this theorem used to be already recognized through CARTIER and BARSOTTI. functions of this duality theorem are: the classical duality theorem ("duality hy­ pothesis", proved by means of CARTIER and by means of NISHI); calculation of Ext(~a,A), the place A is an abelian sort (result conjectured via SERRE); an explanation of the symmetry (due to MANIN) in regards to the isogeny kind of a proper staff connected to an abelian kind.

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