This post is about one of my favourite answers I have given on MathOverflow, although it seems to have gone by mostly unnoticed. In the post, Qixiao asks (essentially) the following:
Question. If is a finite morphism of schemes, is the pushforward
exact?
Note that this is true on the subcategory of quasicoherent sheaves because affine morphisms have no quasicoherent higher pushforwards. Also, in the étale topology the pushforward along a finite morphism is exact on the category of all abelian sheaves; see e.g. Tag 03QP.
However, we show that the answer to the question above is negative.
Example. Let be the spectrum of a DVR
, let
be a finite extension of domains such that
has exactly two primes
above
, and let
. For example,
and
, or
and
if you prefer a more geometric example.
By my previous post, the global sections functor is exact. If the same were true for
, then the global sections functor on
would be exact as well. Thus, it suffices to prove that this is not the case, i.e. to produce a surjection
of sheaves on
such that the map on global sections is not surjective.
The topological space of consists of closed points
and a generic point
. Let
and
; then
is open and
is closed. Hence, for any sheaf
on
, we have a short exact sequence (see e.g. Tag 02UT)
where and
are the inclusions. Let
be the constant sheaf
; then the same goes for
and
. Then the map
is given by the diagonal map , since
is connected by
has two connected components. This is visibly not surjective.
This is a nice example! Here is something maybe related: the pushforward along a finite morphism in the flat (fppf, big or small) topology need not be exact. An example is given by taking
to be a scheme of characteristic
and
the natural projection. Using the Kummer sequence, one can show that
.
Note that
is even a universal homeomorphism. I guess this really makes you appreciate the etale topology!