The four functors of Grothendieck in examples
Tuesday, May 01st, 2012 | Author: Konrad Voelkel
This post will discuss the definition of the four functors "pushforward" , "pullback"
, "pushforward with compact support"
and "exceptional pullback"
of sheaves of abelian groups, associated to a continuous morphism
of topological spaces
and
. Then we will look at maps
which are open immersions or closed immersions, and calculate in the example of
and its closed complement
exactly what happens. This is intended to give some intuition what the general four functor calculus is about.
The four functor formalism arises as part of the six functor formalism (add Hom and Tensor to make it six) in certain (co)homological set-ups. Where I encountered it first was in a paper I tried to read, about the stable motivic homotopy category, but most likely you'll see this stuff in papers dealing with perverse sheaves or motives and their realisations.
Disclaimer: We'll stay in the topological category for this post, i.e. the objects are topological spaces and the morphisms continuous maps. Sheaves are ordinary sheaves of abelian groups (no fancy Grothendieck topology necessary here), not -modules of some sort. However, the discussion doesn't change too much if you translate into the algebraic category, so this should be a good exercise for the bored reader.
Pushforward
Pushforward of sheaves is straightforward: given a space , a sheaf
on
and a continuous map
, the sheaf
on
should be a sheaf that does on open subsets
what
had done on the corresponding open subsets of
, i.e.
. Check that this definition gives again a sheaf. Observe that the constant map
yields
, so
is almost the global section functor and we should think of any
as some kind of generalized global section functor.
Pullback
I want to define the pullback functor as the left adjoint to
. Of course, I have to show existence.
If would be an open embedding, we would have
open in
for all open subsets
of
, and it would be natural to define
. To see that we indeed have a left adjoint by this definition is up to you, but it fails for a general
, since
needn't be open in general.
So, given a sheaf on
I define a new presheaf on
by
, where the limit ranges over all open subsets
such that
contains
. By this "trick" we circumvent the given problem (and introduce new behaviour) and it turns out that this is a correct definition, in the technical sense that we really have found a left adjoint to
.
Proof of the adjunction :
for an open subset of
, a homomorphism from
to
is just a homomorphism from
to
and a homomorphism from
to
is just a homomorphism from
to
. So you see, if we have homomorphisms
for all
, this gives in the limit homomorphisms
.
For the other direction, observe that if we have homomorphisms for all
, we certainly have this for all
, where the limit is just
, i.e. where we have just
.
Pushforward with compact support
We have already seen how pushforward generalizes global sections. As global sections give (as derived functor) cohomology of sheaves, there is a global section with compact support functor, which gives cohomology with compact support. For the locally constant sheaf this gives back "singular" cohomology with compact support, as it appears in Poincaré duality. I will explain this in some more detail now, although I won't explain how to move from global sections to cohomology.
Poincaré duality states, for a smooth compact complex n-dimensional manifold X
and if X is not compact, there is still Poincaré duality:
where

which is related to the functor of global sections with compact support,
just as ordinary cohomology is related to the ordinary global section functor.
The functor of global sections with compact support is defined as
By analogy, we define the pushforward with compact support as a subfunctor of
(which just means that
will be a subsheaf of
for every
, which in turn just means that
is a subset of
for every open set
).
This really gives a sheaf and for

the values are exactly

An example:
Let be an open embedding
, then
is just the "extension by zero", i.e. the stalks at all points of
are just the same as those of
, and all other/new stalks (over
) are plain
.
Another example:
Let be a proper map
, then
, as you can see from the definition.
A comprehensive example:
If can be factored into
with
an open embedding and
proper, we have
, which gives a very explicit description of
.
Exceptional inverse image
We define a functor called
, if it exists. We should say straightforward, that it doesn't exists, in general, on the level of sheaves and this is one of the things that makes working with complexes of sheaves necessary (in fact, the derived category).
However, for innocent maps , we can actually define a functor that is right adjoint to
and thus deserves to be called
.
For f an open embedding , we have just
, i.e. the functor
is the left adjoint to
and also the right adjoint to
.
The proof is similar to the proof of the adjointness of with
, so I leave it out.
Now I want to make clear why a right adjoint to doesn't exist (on the level of sheaves) in general, for categorical reasons.
Every left adjoint functor preserves colimits, since an adjunction like
means that one can compute


Now being right-exact is a special case of preserving colimits, since it means to preserve cokernels (which are special colimits). Clearly, is not right-exact, since it has cohomology: let
be a compact space and
the constant map to a point. Then for
to be right-exact, the cohomology on
must vanish.
The salvation consists of enlarging the category of sheaves to the category of chain complexes of sheaves, only to make it smaller again by introducing the appropriate definition of morphisms, which in the end gives what is called the exists.
Concrete examples for four functors
Let us look at the embedding and its closed complement
.
First we will look at a skyscraper sheaf on with stalk some abelian group
over
. We denote the skyscraper sheaf by
. By definition, we have
a skyscraper sheaf with stalk
over
. Now
, since
throws away all information from the stalk over
.
Okay, let's look at a local system on , i.e. a locally constant sheaf
.
This is the same data (an equivalent category) as the monodromy representation of the fundamental group, in this case .
We have as a sheaf with stalks just
where
, and
, since every section with compact support is away from an arbitrarily small ball around the origin.
The sheaf has the same stalks
where
but it has a new one at the origin, given by the usual stalk-limit-formula you would write down - and in general, this is non-zero.
Cleary vanishes, since
picks the stalk at the origin and throws away everything else. Of course,
contains exactly the "new" stalk which might be interesting.
Thinking about it, the sheaves and
are both zero, by the same argument we had for
. Here you can also use the adjunction for reasoning!
Last words
The nice thing about this setting is that it generalizes to give the following:
Take a closed subspace in
and
its open complement, then you have an open embedding
and a closed embedding
which behave very much like our
and
from the last examples. It presents the category of sheaves on
as an extension of the sheaves on
by the sheaves on
. The same happens for the derived category. The magic word for this situation is "Recollement".
2018-03-26 (26. March 2018)
In 4th line of your "Pullback" section where the notion is defined do you mean to write: "(f^*G)(U):= G(f(U)" where G is a sheaf of abelian groups over Y?
2018-03-26 (26. March 2018)
Corrected, thanks!