Is Hartshorne's Remark III.2.9.1 actually a valid argument?

Question

In Hartshorne's book Algebraic Geometry, following Proposition III.2.9 which states that on a Noetherian topological space $X$ that cohomology of abelian sheaves commutes with direct limits he follows with Remark III.2.9.1 which says

"As a special case we see that cohomology commutes with infinite direct sums."

Now I don't doubt that infinite directs sums commute with cohomology, by all means they may. I don't see how this is a special case of the above proposition however, a direct sum is not a direct limit unless you are taking the direct sum of a single object, it's defined as a colimit over a discrete category which could not be directed if there's more than a single object. Is there something that I'm missing, or is this a completely bogus proof?

Answer 1

The key is to get more inventive with your diagram. Instead of taking the colimit all at once, build it up in pieces: if $I$ is an infinite set, then we can consider the diagram made up of finite subsets $J\subset I$ where the arrows are inclusions. Then $\bigoplus_{i\in I} A_i$ is the colimit of $\bigoplus_{j\in J} A_j$ as $J$ varies over that diagram, and as cohomology commutes with finite direct sums and direct limits, we have a proof.

Answer 2

An infinite direct sum is naturally the direct limit of its finite subsums (that is, $\bigoplus_{i\in I}A_i$ is the direct limit of $\bigoplus_{i\in F}A_i$ where $F$ ranges over the directed set of finite subsets of $I$). Since cohomology preserves finite direct sums and direct limits, it follows that it preserves infinite direct sums.