By F. Oort

We limit ourselves to 2 facets of the sphere of staff schemes, within which the implications are particularly entire: commutative algebraic team schemes over an algebraically closed box (of attribute diversified from zero), and a duality thought trouble ing abelian schemes over a in the community noetherian prescheme. The prelim inaries for those concerns are introduced jointly in bankruptcy I. SERRE defined houses of the class of commutative quasi-algebraic teams by means of introducing pro-algebraic teams. In char8teristic 0 the placement is apparent. In attribute varied from 0 info on finite workforce schemee is required with a purpose to deal with team 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 common workforce schemes are decided. a guideline in a paper via MANIN gave crystallization to a fee11ng of symmetry bearing on 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 comparable in a truly usual means. Afterwards we grew to become conscious specific case of this theorem used to be already identified by means of CARTIER and BARSOTTI. functions of this duality theorem are: the classical duality theorem ("duality hy pothesis", proved by means of CARTIER and through NISHI); calculation of Ext(~a,A), the place A is an abelian kind (result conjectured via SERRE); an evidence of the symmetry situation (due to MANIN) in regards to the isogeny form of a proper staff hooked up to an abelian kind.

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Spec(R) is surjective and the R-algebras C ⌦ B and C n are isomorphic. 2. Let R be a ring, and let A be a degree-n extension of R that is ´etale as an R-algebra. Let G be a subgroup of Sn , and let (B, ') be a G-closure of A over R. Then B is an ´etale degree-|G| extension of R. Proof. 1 to obtain an R-algebra C such that Spec(C) ! Spec(R) is surjective and C ⌦ A ⇠ = Cn as R-algebras. 3, we find that C ⌦B naturally has the structure of being a G-closure of C ⌦ A over C. 3, we must have C ⌦ B ⇠ = C |G| .

Then isomorphism classes of G-closures of A over R correspond to R-algebra homomorphisms R[x1 , . . , xn ]G ! R sending ek (x) 7! sk . Proof. 2, isomorphism classes of G-closures of A over R correspond to R-algebra homomorphisms R[x1 , . . , xn ]G ! R such that ek (x) 7! sk and R ! R[x1 , . . ,xn ]G is injective. 3, since |G| is a non-zerodivisor the latter condition is guaranteed to hold. 4 An-closures In this section we give generators and relations for R[x1 , . . , xn ]An as an R[x1 , . .

R. Then the R-algebra homomorphism (A⌦n )G ! 1, the corresponding G-closure of A is B. 2, we need criteria for proving that R ! R[x1 , . . ,xn ]G is injective. 3. Let R be a ring, let n be a natural number, and let G be a finite group acting on an R-algebra A. Let AG ! R be an R-algebra homomorphism, and N construct the R-algebra homomorphism R ! A AG R. If |G| is a non-zerodivisor in R, then this homomorphism is injective. Proof. Consider the AG -module homomorphism A ! AG given by p 7! p. The composite AG !

### Commutative group schemes by F. Oort

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