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Representable functor

The most important sheaves are the representable functors, i.e. functors isomorphic to one of the form... [Pg.9]

Relative dimension, 2-5,2-12 Representable functor, 1-1 Ruled surface, 11-9... [Pg.209]

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. [Pg.15]

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... [Pg.20]

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. [Pg.41]

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. [Pg.42]

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. [Pg.150]

Representable functor 5 Representation, see Linear representation Representation of a Lie algebra 96 Ring of functions on S 30 Root system 98... [Pg.88]


See other pages where Representable functor is mentioned: [Pg.203]    [Pg.10]    [Pg.193]    [Pg.14]    [Pg.15]    [Pg.21]    [Pg.22]    [Pg.54]    [Pg.54]    [Pg.125]    [Pg.8]    [Pg.12]    [Pg.28]    [Pg.28]    [Pg.91]    [Pg.95]    [Pg.98]    [Pg.149]    [Pg.6]   
See also in sourсe #XX -- [ Pg.60 , Pg.63 , Pg.83 , Pg.101 , Pg.102 , Pg.109 ]

See also in sourсe #XX -- [ Pg.60 , Pg.63 , Pg.83 , Pg.101 , Pg.102 , Pg.109 ]

See also in sourсe #XX -- [ Pg.5 ]




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Functor

Representable Functors

Representable Functors

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