# Grothendieck spectral sequence without spectral sequences

Last semester, I taught a course Sheaves in topology, about the cohomology of topological spaces via sheaves. An important and often-used tool is the Grothendieck spectral sequence: if

are left exact functors of abelian categories, with composition , where and have enough injectives, and takes injectives to -acyclic objects, then there is a spectral sequence

for any .

However, I do not want to assume spectral sequences, nor spend multiple lectures developing them from scratch. Here is a variant that is easy to prove and often sufficient:

Lemma. Let and as above, and let .

1. If for all and all , then

for all .

2. If for all and all , then

for all .

Proof. For both parts, let be an exact sequence with injective. The long exact sequences for and give

and isomorphisms and for . Write for the image of , so that we get short exact sequences

(1)

Since is -acyclic, we get

and isomorphisms for .

We prove (1) by induction on . For , there is nothing to prove. Since , , and are -acyclic, so are and by the short exact sequences of (1), and we get exact sequences

These glue to an exact sequence , which by the exact sequence for gives . This proves the result for . We also see that is acyclic for all : for , we proved this above, and for this follows from . Thus satisfies the same hypotheses as , and the isomorphisms

for prove the result by induction on .

Likewise, we prove (2) by induction on . Again, for , there is nothing to prove. The long exact sequence of gives for all since for all . We obtain an exact sequence

and isomorphisms for . Comparing with the long exact sequence for gives , showing the result for . Moreover, we have for all and all , so satisfies the same hypotheses as . Then the isomorphisms

show the result for by induction.

Corollary. If is exact and takes injectives to -acyclic objects, then

for all . Likewise, if is exact, then the acyclicity assumption on is automatic, and

for all .

Example. This is used many times in sheaf cohomology. Some examples:

• If is a closed subset, then for all abelian sheaves on and all .
• If is a locally closed subset, then for all abelian sheaves on .
• If is the derived functor of the inclusion from sheaves into presheaves, then .
• If is a sheaf on , then the sheafification of is for .
• The higher pushfoward along a map is the sheafification of .
• If is a map and for , then . This is used for the projection (with a constant sheaf) in the proof of homotopy invariance of .
• If is a sheaf and is an open cover such that for all and all finite subsets , then for all . (Here, we write for .)
• If is a topological space such that for implies for all , then for all . (It’s not hard to show that this hypothesis is satisfied if is paracompact Hausdorff.)

It’s a good exercise to go through all of these and see how they follow from the corollary.

# Internal Hom

This is an introductory post about some easy examples of internal Hom.

Definition. Let be a symmetric monoidal category, i.e. a category with a functor that is associative, unital, and commutative up to natural isomorphism. Then an internal Hom in is a functor

such that is a left adjoint to for any , i.e. there are functorial isomorphisms

Remark. In the easiest examples, we typically think of as ‘upgrading to an object of ‘:

Example. Let be a commutative ring, and let be the category of -modules, with the tensor product. Then with its natural -module structure is an internal Hom, by the usual tensor-Hom adjunction:

The same is true when is the category of -bimodules for a not necessarily commutative ring .

However, we cannot do this for left -modules over a noncommutative ring, because there is no natural -module structure on for left -modules and . In general, the tensor product takes an -bimodule and a -bimodule and produces an -bimodule . Taking gives a way to tensor a right -module with a left -module, but there is no standard way to tensor two left -modules, let alone equip it with the structure of a left -module.

Example. Let . Then is naturally a set, making it into an internal Hom for :

When is the categorical product , the internal (if it exists) is usually called an exponential object, in analogy with the case above.

Example. Another example of exponential objects is from topology. Let be the category of locally compact Hausdorff topological spaces. Then the compact-open topology makes into an internal Hom of topological spaces. (There are mild generalisations of this beyond the compact Hausdorff case, but for an arbitrary topological space the functor does not preserve colimits and hence cannot admit a right adjoint.)

Example. An example of a slightly different nature is chain complexes: let be a commutative ring, and let be the category of cochain complexes

of -modules (meaning each is an -module, and the are -linear maps satisfying ). Homomorphisms are commutative diagrams

and the tensor product is given by the direct sum totalisation of the double complex of componentwise tensor products.

There isn’t a natural way to ‘endow with the structure of a chain complex’, but there is an internal Hom given by

with differentials given by

Then we get for example

since a morphism is given by an element such that , i.e. , meaning that is a morphism of cochain complexes.

Example. The final example for today is presheaves and sheaves. If is a topological space, then the category of abelian sheaves on has an internal Hom given by

with the obvious transition maps for inclusions of open sets. This is usually called the sheaf Hom. A similar statement holds for presheaves.

# 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.