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.