Are you dreading the social responsibilities that come with this time of the year? You can spice up the conversations with your relatives and friends using the following fun facts in algebraic geometry. Whenever a conversational lull comes up, drop one of these (more to come in later posts) and things will turn fun fun fun again!

Cohomology vanishing for quasicoherent sheaves

The following fact for schemes is well-known: if you can cover a quasicompact separated scheme using $n$ affine opens, then $\mathrm{H}^i(X,\mathcal{F})=0$ for all $i\geq n$. This is tag 01XI. Its proof is easy, and can easily be outlined on a napkin if your loved ones know a bit of Cech cohomology.

The fun fact that is less well-known that something similar can be done for algebraic spaces! Now you don't use an affine open cover in the usual sense, but you have to take an affine scheme with an étale surjective map to your algebraic space (let's take it quasicompact separated once more). Now you want to look at the cardinality of the fibers of this morphism, because if $d$ is an upper bound for these cardinalities, you know that the cohomological dimension for quasicoherent sheaves is bounded by this integer. In this form it is tag 072B.

But now your annoying little cousin asks: can you always find such an étale atlas? The good news is that (unlike for algebraic stacks, but we want to keep dinner conversation light) the answer is yes (whilst even improving the upper bound on the cohomological dimension). This can be found in tag 072C, which is not only adjacent in tag numbering but also in the actual text. Or you refer him to the section on separatedness conditions and universal boundedness for algebraic spaces.

Unfortunately the proof now requires a little more background, the proof requires 4 times as many tags, making it less napkin worthy.