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!