By James R. Milgram
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Additional resources for Algebraic and Geometric Topology
Representation of functors. 1) Setup. Given a scheme S and a contravariant functor F from schemes over S to sets. All schemes and morphisms will be taken over S. Given a scheme X over S, we denote by hX the contravariant functor from schemes over S to sets which sends a scheme T to the set of S–homomorphisms hX (T ) = HomS (T, X) from T to X, and to a morphism h: U → T associates the map hX (h): hX (T ) → hX (U ) given by hX (h)(g) = gh, for all morphisms g: T → X. 2) Note. There is a natural bijection between elements in F (X) and morphisms of functors H: hX → F .
1) is an isomorphism for m ≥ m0 . 3) n choose a surjection OX → K(m0 ). We have that the the map FU → h∗XT G facT n tors via FU if and only if the composite map OX → KU (m0 ) → FU (m0 ) → U ∗ hXT G(m0 ) is zero.
2)(2) and (3) that we can choose an m0 such that K(m) and G(m) are generated by global sections, and such that H i (XT , G(m)) = 0 for i > 0 and for m ≥ m0 . 1) is an isomorphism for m ≥ m0 . 3) n choose a surjection OX → K(m0 ). We have that the the map FU → h∗XT G facT n tors via FU if and only if the composite map OX → KU (m0 ) → FU (m0 ) → U ∗ hXT G(m0 ) is zero.
Algebraic and Geometric Topology by James R. Milgram