### F. Oort's Commutative group schemes PDF

By F. Oort

We limit ourselves to 2 elements of the sector of workforce schemes, within which the implications are really whole: commutative algebraic staff schemes over an algebraically closed box (of attribute diversified from zero), and a duality thought hindrance ing abelian schemes over a in the neighborhood noetherian prescheme. The prelim inaries for those concerns are introduced jointly in bankruptcy I. SERRE defined homes of the class of commutative quasi-algebraic teams through introducing pro-algebraic teams. In char8teristic 0 the placement is apparent. In attribute varied from 0 info on finite staff schemee is required in an effort to deal with staff schemes; this knowledge are available in paintings of GABRIEL. within the moment bankruptcy those principles of SERRE and GABRIEL are prepare. additionally extension teams of effortless workforce schemes are decided. a guideline in a paper through MANIN gave crystallization to a fee11ng of symmetry pertaining to subgroups of abelian forms. within the 3rd bankruptcy we turn out that the twin of an abelian scheme and the linear twin of a finite subgroup scheme are similar in a truly normal method. Afterwards we grew to become conscious precise case of this theorem used to be already recognized by way of CARTIER and BARSOTTI. purposes 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 type (result conjectured by way of SERRE); an evidence of the symmetry (due to MANIN) about the isogeny kind of a proper crew hooked up to an abelian sort.