591 名前：his book and through which he calculates the Grothedieck cohomology of sheaves, the main virtue of his sheaves being that they are flabby, hence acyclic. So your choice of the letter G for this Godementification is excellent!
And you are perfectly right that the stalk Gx is much, much larger than the corresponding Fx: congratulations for not sharing the common misconception that these stalks are equal.
3 In the third line you write that taking stalks commutes with direct sums, but the actual construction of G is a direct product, not a direct sum (unless X is finite).
Even if X is finite, it won't be true that F=G unless all the points x are closed (so that X is actually discrete, in which case there is no difference between continuous and discontinuous sections).
The reason is that if x is not closed, then i?(Fx) has non-zero stalks not just at x, but at all the points in the closure of x. []
