# 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. # Flat and projective

See the previous post for the notion of -finitely presented modules.

Lemma. Let be a -finitely presented flat module. Then is projective.

Proof. For every prime , the module is finitely presented and flat, hence free (use Nakayama). In particular, it is projective over , hence for all . By our previous lemma, we conclude that for any -module , as is -finitely presented. Since is arbitrary, this forces for any -module . Hence is projective. Remark. Using the equational criterion for flatness, one can in fact prove that any finitely presented flat module is projective. However, I thought the above proof was nice enough to make up for this slight loss of generality.

Remark. The Stacks project gives an example of a finitely generated (but not finitely presented) flat module that is not projective.

# Ext and localisation

This post and the next are related, but I found this result interesting enough for a post of its own.

Lemma. Let be a finitely presented -module, and let be a multiplicative subset. Then Proof. The result is true when is finite free, since whereas Now consider a finite presentation of . Since is left exact and localisation is exact, we get a commutative diagram with exact rows (where the in the bottom row is over ). The right two vertical maps are isomorphisms, hence so is the one on the left. Definition. Let be an -module. Then is -finitely presented if there exists finite free modules and an exact sequence For example, is finitely generated if and only if it is -finitely presented, and finitely presented if and only if it is -finitely presented. Over a Noetherian ring, any finitely generated module is -finitely presented for any .

[I do not know if this is standard terminology, but it should be.]

Corollary. Let , let M be a -finitely presented module, and let be a multiplicative subset. Then Proof. Given an exact sequence with finite free, let be the kernel of . Then is -finitely presented, and we have a short exact sequence Now the result follows by induction, using the long exact sequence for . Remark. As Sebastian pointed out to me, we never used any specific properties of localisation, and the same result (with the same proof) works for any flat -algebra.