## Sheaves on the projective line and partial fractions

This post constitutes a solution to Exercise II.1.21(e) from Hartshorne’s book with a bit of extra discussion. Section II.1 is very important technically (and not just to algebraic geometers) but I thought this was the only genuinely interesting exercise. Thanks to Mariano Suárez-Alvarez over at the Mathematics Stack Exchange for a helpful hint.

Here’s the setup: write for the projective line over an algebraically closed field and for the function field. We denote by the constant sheaf on with values in , which is the same as the constant presheaf because is irreducible. Then the structure sheaf is naturally a subsheaf of , and it is not hard to see that the quotient is identified with , where denotes the skyscraper sheaf at with stalk .

Now for the interesting part. I claim that the map on global sections induced by the projection is actually surjective.

It suffices to show that this map surjects onto each summand, so given and we need to find such that and for all . Intuitively, we need to have at most a pole at (but nowhere else), and we require that has no pole at . So is the “singular part of at .”

For this, remove some point besides where does not have a pole, and without loss of generality suppose . So we are left with and we can write

where is the coordinate on . Note that since vanishes at infinity. If necessary, reorder the ‘s so that but for , then write out the partial fraction decomposition

I claim that

is the desired element of . Indeed, since we have for all , and clearly vanishes at so also. Since for any , we have as desired.

One interesting consequence is that that . To prove this, consider the short exact sequence

and pass to the long exact sequence in sheaf cohomology

.

Since the first map is surjective, the second is zero, whence the third map is injective. But is flasque, so , which implies .

You forgot to carry the three in the denominator…