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This problem has Hartshorne Height 2 and volume 2; I.1.7 can be used to solve it.


Using I.1.7 you have that X is quasi compact. So any cover can be taken to be finite. No elements of the direct limit sheaf are equivalence classes of elements: \langle a, F_i(U) \rangle \sim \langle b, F_j(U) \rangle if there is k bigger than i,j such that the images of a,b agree in F_k(U). In words, they agree if they agree far enough down the system.

  • Show that finite covers allow the sheaf axioms to be checked by checking them on a sheaf F_i of the system that is sufficiently far down in the system.
  • If \varinjlim F_i is already a sheaf then  \Gamma(X, \varinjlim F_i) = \varinjlim \Gamma(X, F_i)

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