I’ve been reading the Beilinson-Bernstein-Deligne (BBD) paper “Faisceaux pervers” as part of my summer project. To help myself understand it, I thought I would try to do a few blog posts about the material. The paper is ultimately about perverse sheaves, which are suitable complexes of sheaves on a (stratified) topological space satisfying certain cohomological conditions. The surprise is that perverse sheaves actually form an *abelian* category. The reason behind this is a bit of homological algebra, which I would like to discuss today to start.

**1. t-structures**

The first important notion in BBD is that of a -structure. If is an abelian category, and its derived category, we know that there is a notion of an object in *positive degrees* and one in *negative degrees*. There are two subcategories . Namely, a complex can be said to be in if it has no cohomology in negative degrees, and similarly in if it has no cohomology in positive degrees. This is equivalent to saying that is isomorphic to a complex all of whose nonzero terms are in nonnegative (resp. nonpositive) degrees because of the truncation functors .

A -structure is an axiomatization of this observation for the derived category.

Let be a triangulated category. A **-structure** on is the data of two full subcategories . We can define as and similarly , in accordance with what happens for the derived category.

The following axioms are required:

- is closed under translation .
- If , , then .
- If , there is a triangle
such that and .

The axioms for a -structure are simple, but they turn out to be quite powerful, because the axioms for a triangulated category are themselves quite powerful.

Let us first check that the derived category of the abelian category has a -structure as above. The first condition is obvious. The third follows, because if is a complex, then there is a short exact sequence

where are truncation functors whose values are in the category of complexes vanishing outside (resp. ). This exact sequence leads to a triangle in the derived category.

Finally, we have the more mysterious second axiom. It says that a complex concentrated in can’t map in a nonzero way to a complex in , in the derived category. A special case of this is when the two complexes are for for in ; then

To see this, let’s assume for convenience that has enough injectives (though this is not necessary). Consider two complexes concentrated in and . We can replace by a complex of injectives in the same degrees by a standard argument, so let’s just assume that consists of injectives. Then maps in the derived category are homotopy classes of chain-complex homomorphisms , but these are obviously trivial.

It follows:

Proposition 1The derived category has a -structure given by .

One may “translate” a -structure: that is, one just translates the categories by some factor.

**2. Truncation**

In the derived category, one frequently uses the truncation functors indicated above. In fact, these make sense in any triangulated category with a -structure.

Let . We know there is a triangle

It is not obvious a priori that this triangle is unique at all, or that the ‘s are functorial: all that we are given is that and .

But BBD show in fact (and it’s not terribly hard, but a bit of playing around with the axioms) that these are unique and completely functorial in . The point is:

Proposition 2The inclusion has a left adjoint . Similarly, the inclusion has a right adjoint .

This is the reason for functoriality of the truncation functors. Namely, the , etc. have universal properties. The essential point of the argument is the following. Let’s say we want to do it for , and we have an object . Let be constructed as above; we don’t know yet that it’s unique (up to unique isomorphism).

The claim is that homming out of is the same as homming out of , if we’re homming into something in . That is, let ; we claim that

This is precisely what the claim about the left adjoint is.

To see this, we use the long exact sequence in for the triangle . Here , and that this triangle exists follows from translating the axioms.

We get an exact sequence of abelian groups:

But the two terms on the outside vanish by the axioms.

From this it follows that one can define the ‘s as *functors*, because of their universal property. What’s more, one can use them to characterize the categories. The idea is that something will live in if and only if it has no components in .

Proposition 3An object belongs to if and only if .

Indeed, this follows from the triangle

and the fact that if and only if . In a triangle , the first map is an isomorphism iff .

In fact, another way of stating this result is that and are the “orthogonal complements” of each other. Namely, an object is in if and only if it cannot map via a nonzero map into .

As a result of this observation, and the exact sequence in from a triangle, we find:

Corollary 4(resp. ) is closed under extensions.

In other words, if there is a triangle

with , then .

There is another result in BBD about compositions of truncation functors. Namely, let . It’s easy to see that (by considering the adjoints, for instance). It’s less obvious that

The intuition is that this functor is truncation in both directions, and it doesn’t matter which way one does the truncation. The actual proof, however, relies on the octahedral axiom for a triangulated category, and I shall omit it.

**3. The heart**

Let be an abelian category. Then the subcategory consists of complexes cohomologically concentrated in degree zero, i.e. complexes quasi-isomorphic to complexes consisting of one nonzero element in degree zero. That is, such complexes are just elements of , and in fact the natural functor

is an equivalence of categories.

In particular, the intersection is an *abelian* category. It turns out, however, that this fact is true of any -structure on a triangulated fact, which is a big result in BBD.

Theorem 5Let be a triangulated category with a -structure. Then is an abelian category .

This abelian category is called the **heart** of the -structure. One reason this fact is so important is that, while the only example of a -structure we have given so far is the natural one on a derived category, there are *other* less obvious -structures (even on plain old derived categories).

For instance, BBD defines a “perverse” -structure on the derived category of sheaves on a stratified topological space, which is quite far from the usual -structure. The heart of this category is *not* the usual category of sheaves, but instead the category of so-called *perverse sheaves.*

Let’s indicate how this is proved. For instance, we can describe the construction of kernels. Let be a morphism between two objects in the heart. We can imbed this in a triangle

Now the problem is that is not really a cokernel or a kernel. One is supposed to think of it as the replacement that a triangulated category can offer, but it’s not quite the same thing. Moreover, might not live in the heart.

Fix . On the other hand, from rotating the triangle, we have an exact sequence

In other words, would be a kernel if not for the fact that it probably isn’t in . This suggests that we should take the closest thing to it: a suitable truncation.

Note that is still close to living in the heart. We can find an exact triangle

by rotation, which “sandwiches” between . This implies that at least because the two outer terms are there. In particular, .

So if we take the truncation , then mapping into from any is the same as mapping into . Also, because , one sees (and I’m skipping some details) that too. So we have the kernel.

The process of finding a cokernel is dual. To show that one has an abelian category, one should check that each monic is a kernel and every epic a cokernel. So let, for definiteness, be an epic morphism. We imbed it in an exact triangle . If , we have an exact sequence

This shows that for each . In particular, . Since we saw that , it follows that .

That is, is a translate of something in . When we consider the rotated triangle

we find that , and we will show is the cokernel of (which as above is the kernel of ). To see this, let ; then we have an exact sequence

which states that mapping out of is the same as mapping out of in such a way that the restriction to is zero. This is precisely the claim we wanted.

Next time, I’ll describe a means of “gluing” -structures, which will enable us to construct the category of perverse sheaves.

July 19, 2011 at 10:06 am

I heard once a claim that the proof of the decomposition theorem in BBD(G) is the hardest proof ever. Even if there are more difficult proofs out there this seems like quite a challenge for a (part of) summer poject! wow!

July 19, 2011 at 6:10 pm

BBD is certainly a very difficult paper. I haven’t gotten very far at this point (partially because my main goal is the Weil conjectures), but there is a lot to learn for me at this point.

July 20, 2011 at 9:45 pm

About weil conjecture,you can see Reinhardt Kiehl and rainer Weissauer’s book “weil conjectures,perverse sheaves and l’adic fourier transform”is uesful to me.

July 20, 2011 at 10:01 pm

Correct,It is useful to you,don’t understand to me,not useful.