Representable Functors

The most important sheaves are the representable functors, i.e. functors isomorphic to one of the form... 

Relative dimension, 2-5,2-12 Representable functor, 1-1 Ruled surface, 11-9... 

Such F are called representable, and one says that A represents F. We can now officially define an affine group scheme over k as a representable functor from k-algebras to groups. 

Thinking of the usual axioms for groups, we can see that A is the most important part of a Hopf algebra structure on an algebra A. For suppose we have a representable functor G and a map A A A <8> A giving a composition law on the G(li). If they happen to be groups, the unit and inverses are... 

Theorem. Let k be an infinite field. The closed subsets of k , with polynomial maps, are precisely equivalent to certain representable functors. The equivalence preserves products, and rakes closed subsets to closed subfunctors represented by quotient rings). Closure in a larger C corresponds to base extension. 

We have not yet settled the question which representable functors actually do arise from closed sets. The only answer to this in general is the following. 

Let G be an affine group scheme over a ring k, and let AT be a representable functor on which G acts (3.1). Following the usual definition for groups, we say the action is simply transitive, or X is formally principal homogeneous, if for each pair of points in there is a unique element in G(R) taking the second to the first. In other words, the map G x X - X x X sending (g, x) to (gx, x) should be bijective. 

Representable functor 5 Representation, see Linear representation Representation of a Lie algebra 96 Ring of functions on S 30 Root system 98... 

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