Higher pushforwards along finite morphisms

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. 1 thought on “Higher pushforwards along finite morphisms”

1. Dan on said:

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!