II.1.11

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

HAPPY
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 $$ \Gamma(X, \varinjlim F_i) = \varinjlim \Gamma(X, F_i)$$