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This has H = ?, V = ?
Just parts d,e
Work on the level of presheaves. You clearly have an injection . And there are natural restriction maps . So there is at least a map
Now let be a rational function on an open set . Since we're dealing with , after some automorphism we can assume there are no poles at infinity, so write the point is that is can poles at only finitely many points so if is not one of these points, then restricts to an element of hence maps to zero in .
I.e. the map is nonzero at only finitely many places, so factors through the direct sum, so there is a factorization on the level of presheaves, so by universal property of sheafification you get and its exact on stalks which gives the result.
For part e, just take a finite number of prescribed principle parts, and sum them, then this will constitute a lift, showing the required map is surjective.