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II.1.10

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This problem has Hartshorne Height 1.

Sketch

  • Fix a sheaf G and a system of compatible morphisms F_i \to G.

For every U you have the desired factorization on the level of presheaves F_i(U) \to \varinjlim F_i(U) \to G(U)

  • Show this is compatible with the restriction morphisms, i.e. for V\subset U and any F_i \to F_j in the directed system (possibly the identity) then
F_i(U) \to F_j(V) \to \varinjlim F_i(V) \to G(V)

is the same as

F_i(U) \to \varinjlim F_i(U) \to G(U) \to G(V)

(use the universal property of direct limit applied to the map F_i(U) \to G(V)).

  • Use Proposition-Definition II.1.2

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