This is a continuation of my previous post on local schemes. Here is a ridiculous lemma.
Lemma. Let be a local scheme, and let be any abelian sheaf on . Then for all .
Proof. It suffices to show that the global sections functor is exact. Let be a surjection of abelian sheaves on , and let be a global section. Then can be lifted to a section of in an open neighbourhood of . But the only open neighbourhood of is . Thus, can be lifted to a section of .
What’s going on is that the functors and are naturally isomorphic, due to the absence of open neighbourhoods of .
Remark. It seems believable that there are suitable site-theoretic versions of this lemma as well. For example, a strictly Henselian local ring has no higher cohomology in the étale topology. The argument is essentially the same: every open neighbourhood of the closed point has a section; see e.g. the proof of Tag 03QO.