Mathematics – Algebraic Geometry
Scientific paper
2009-04-14
Mathematics
Algebraic Geometry
36 pages. Added new results on affine functors
Scientific paper
A functor of sets X over the category of K-commutative algebras is said to be an affine functor if its functor of functions, A_X, is reflexive and X = Spec A_X. We prove that affine functors are equal to a direct limit of affine schemes and that affine schemes, formal schemes, the completion of affine schemes along a closed subscheme, etc., are affine functors. Let G be an affine functor of monoids. We prove that A^*_G is the enveloping functor of algebras of G and that the category of G-modules is equivalent to the category of A^*_G-modules. Moreover, we prove that the category of affine functors of monoids is anti-equivalent to the category of functors of commutative proquasi-coherent bialgebras. Applications of these results include Cartier duality, neutral Tannakian duality for affine group schemes and the equivalence between formal groups and Lie algebras in characteristic zero.
Navarro Jesus
Sancho Carlos
Sancho Pedro
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