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 ${t}$-structure. If ${\mathcal{A}}$ is an abelian category, and ${\mathbf{D}(\mathcal{A})}$ 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 ${\mathbf{D}_{\geq 0}(\mathcal{A}), \mathbf{D}_{\leq 0}(\mathcal{A}) \subset \mathbf{D}(\mathcal{A})}$. Namely, a complex ${K^\bullet}$ can be said to be in ${\mathbf{D}_{\geq 0}(\mathcal{A})}$ if it has no cohomology in negative degrees, and similarly in ${\mathbf{D}_{\leq 0}(\mathcal{A})}$ if it has no cohomology in positive degrees. This is equivalent to saying that ${K^\bullet}$ is isomorphic to a complex all of whose nonzero terms are in nonnegative (resp. nonpositive) degrees because of the truncation functors ${\tau_{\geq 0}, \tau_{\leq 0}}$.

A ${t}$-structure is an axiomatization of this observation for the derived category.

Let ${\mathcal{D}}$ be a triangulated category. A ${t}$-structure on ${\mathcal{D}}$ is the data of two full subcategories ${\mathcal{D}_{\geq 0}, \mathcal{D}_{\leq 0}}$. We can define ${\mathcal{D}_{\leq n}}$ as ${\mathcal{D}_{\leq 0}[-n]}$ and similarly ${\mathcal{D}_{\geq 0} = \mathcal{D}_{\geq 0}[-n]}$, in accordance with what happens for the derived category.

The following axioms are required:

1. ${\mathcal{D}_{\leq 0} }$ is closed under translation ${X \mapsto X[1]}$.
2. If ${X \in \mathcal{D}_{\leq 0}}$, ${Y \in \mathcal{D}_{\geq 1}}$, then ${\hom_{\mathcal{D}}(X, Y) = 0}$.
3. If ${X \in \mathcal{D}}$, there is a triangle

$\displaystyle A \rightarrow X \rightarrow B \rightarrow X[1]$

such that ${A \in \mathcal{D}_{\leq 0}}$ and ${B \in \mathcal{D}_{\geq 1}}$.

The axioms for a ${t}$-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 ${\mathbf{D}(\mathcal{A})}$ of the abelian category ${\mathcal{A}}$ has a ${t}$-structure as above. The first condition is obvious. The third follows, because if ${K^\bullet}$ is a complex, then there is a short exact sequence

$\displaystyle 0 \rightarrow \tau_{\leq 0} K^\bullet \rightarrow K^\bullet \rightarrow \tau_{\geq 1} K^\bullet \rightarrow 0,$

where ${\tau_{\leq 0}, \tau_{\geq 1}}$ are truncation functors whose values are in the category of complexes vanishing outside ${[-\infty, 0)}$ (resp. ${[1, \infty)}$). 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 ${[-\infty, 0)}$ can’t map in a nonzero way to a complex in ${[1, \infty)}$, in the derived category. A special case of this is when the two complexes are ${A, B[-i]}$ for ${i > 0}$ for $A, B$ in $\mathcal{A}$; then

$\displaystyle \hom_{\mathbf{D}(\mathcal{A})}(A, B[i]) = \mathrm{Ext}^i(A, B) = 0 \quad \text{if } i < 0.$

To see this, let’s assume for convenience that ${\mathcal{A}}$ has enough injectives (though this is not necessary). Consider two complexes ${K^\bullet, L^\bullet}$ concentrated in ${(-\infty, 0]}$ and ${[1, \infty)}$. We can replace ${L^\bullet}$ by a complex of injectives in the same degrees by a standard argument, so let’s just assume that ${L^\bullet}$ consists of injectives. Then maps in the derived category ${K^\bullet \rightarrow L^\bullet}$ are homotopy classes of chain-complex homomorphisms ${K^\bullet \rightarrow L^\bullet}$, but these are obviously trivial.

It follows:

Proposition 1 The derived category ${\mathbf{D}(\mathcal{A})}$ has a ${t}$-structure given by ${\mathbf{D}_{\geq 0}(\mathcal{A}), \mathbf{D}_{\leq 0}(\mathcal{A})}$.

One may “translate” a ${t}$-structure: that is, one just translates the categories ${\mathcal{D}_{ \geq 0}, \mathcal{D}_{\leq 0}}$ by some factor.

2. Truncation

In the derived category, one frequently uses the truncation functors ${\tau_{\leq i}, \tau_{\geq i}}$ indicated above. In fact, these make sense in any triangulated category with a ${t}$-structure.

Let ${X \in \mathcal{D}}$. We know there is a triangle

$\displaystyle \tau_{\leq 0} X \rightarrow X \rightarrow \tau_{\geq 1} X \rightarrow \tau_{\leq 0}(X)[1].$

It is not obvious a priori that this triangle is unique at all, or that the ${\tau}$‘s are functorial: all that we are given is that ${\tau_{\leq 0} X \in \mathcal{D}_{\leq 0}}$ and ${\tau_{\geq 1}X \in \mathcal{D}_{\geq 1}}$.

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 ${X}$. The point is:

Proposition 2 The inclusion ${\mathcal{D}_{\geq 0} \rightarrow \mathcal{D}}$ has a left adjoint ${\tau_{\geq 0}}$. Similarly, the inclusion ${\mathcal{D}_{\leq 0} \rightarrow \mathcal{D}}$ has a right adjoint ${\tau_{\leq 0}}$.

This is the reason for functoriality of the truncation functors. Namely, the ${\tau_{\leq 0} X, \tau_{\geq 0} X}$, etc. have universal properties. The essential point of the argument is the following. Let’s say we want to do it for ${\mathcal{D}_{\geq 0} \rightarrow \mathcal{D}}$, and we have an object ${X}$. Let ${\tau_{\geq 0} X}$ be constructed as above; we don’t know yet that it’s unique (up to unique isomorphism).

The claim is that homming out of ${\tau_{\geq 0} X}$ is the same as homming out of ${X}$, if we’re homming into something in ${\mathcal{D}_{\geq 0}}$. That is, let ${Y \in \mathcal{D}_{\geq 0}}$; we claim that

$\displaystyle \hom(X, Y) = \hom(\tau_{\geq 0} X, Y).$

To see this, we use the long exact sequence in ${\hom}$ for the triangle ${\tau_{\leq -1} X \rightarrow X \rightarrow \tau_{\geq 0} X \rightarrow \tau_{\leq -1}(X)[1]}$. Here ${\tau_{\leq -1} X \in \mathcal{D}_{\leq -1}}$, and that this triangle exists follows from translating the axioms.

We get an exact sequence of abelian groups:

$\displaystyle \hom( \tau_{\leq -1}(X)[1], Y) \rightarrow \hom(\tau_{\geq 0} X, Y) \rightarrow \hom(X, Y) \rightarrow \hom(\tau_{\leq -1} X, Y).$

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

From this it follows that one can define the ${\tau}$‘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 ${\mathcal{D}_{\leq 0}}$ if and only if it has no components in ${\mathcal{D}_{>0} = \mathcal{D}_{\geq 1}}$.

Proposition 3 An object ${X \in \mathcal{D}}$ belongs to ${\mathcal{D}_{\leq 0}}$ if and only if ${\tau_{\geq 1} X = 0}$.

Indeed, this follows from the triangle

$\displaystyle \tau_{\leq 0} X \rightarrow X \rightarrow \tau_{\geq 1} X \rightarrow \tau_{\leq 0}(X)[1]$

and the fact that ${\tau_{\leq 0} X = X}$ if and only if ${X \in \mathcal{D}_{\leq 0}}$. In a triangle ${X \rightarrow Y \rightarrow Z \rightarrow X[1]}$, the first map is an isomorphism iff ${Z = 0}$.

In fact, another way of stating this result is that ${\mathcal{D}_{\leq 0}}$ and ${\mathcal{D}_{\geq 1}}$ are the “orthogonal complements” of each other. Namely, an object ${X}$ is in ${\mathcal{D}_{\leq 0}}$ if and only if it cannot map via a nonzero map into ${\mathcal{D}_{\geq 1}}$.

As a result of this observation, and the exact sequence in ${\hom}$ from a triangle, we find:

Corollary 4 ${\mathcal{D}_{\leq 0}}$ (resp. ${\mathcal{D}_{\geq 0}}$) is closed under extensions.

In other words, if there is a triangle

$\displaystyle A \rightarrow X \rightarrow B \rightarrow A[1]$

with ${A, B \in \mathcal{D}_{\leq 0}}$, then ${X \in \mathcal{D}_{\leq 0}}$.

There is another result in BBD about compositions of truncation functors. Namely, let ${a < b}$. It’s easy to see that ${\tau_{\leq a} \tau_{\leq b} = \tau_{\leq a}}$ (by considering the adjoints, for instance). It’s less obvious that

$\displaystyle \tau_{\geq a} \tau_{\leq b} = \tau_{\leq b} \tau_{\geq a}.$

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 ${\mathcal{A}}$ be an abelian category. Then the subcategory ${\mathbf{D}_{\leq 0}(\mathcal{A}), \cap \mathbf{D}_{\geq 0}(\mathcal{A}) \subset \mathbf{D}(\mathcal{A})}$ 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 ${\mathcal{A}}$, and in fact the natural functor

$\displaystyle \mathcal{A} \rightarrow \mathbf{D}_{\leq 0}(\mathcal{A}) \cap \mathbf{D}_{\geq 0}(\mathcal{A})$

is an equivalence of categories.

In particular, the intersection ${\mathbf{D}_{\leq 0}(\mathcal{A}) \cap \mathbf{D}_{\geq 0}(\mathcal{A})}$ is an abelian category. It turns out, however, that this fact is true of any ${t}$-structure on a triangulated fact, which is a big result in BBD.

Theorem 5 Let ${\mathcal{D}}$ be a triangulated category with a ${t}$-structure. Then ${\mathcal{D}_{\geq 0} \cap \mathcal{D}_{\leq 0}}$ is an abelian category ${\mathcal{C}}$.

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

For instance, BBD defines a “perverse” ${t}$-structure on the derived category of sheaves on a stratified topological space, which is quite far from the usual ${t}$-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 ${f: X \rightarrow Y}$ be a morphism between two objects in the heart. We can imbed this in a triangle

$\displaystyle X \stackrel{f}{\rightarrow} Y \rightarrow Z \rightarrow X[1].$

Now the problem is that ${Z}$ 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, ${Z}$ might not live in the heart.

Fix ${W \in \mathcal{C}}$. On the other hand, from rotating the triangle, we have an exact sequence

$\displaystyle 0 = \hom(W, Y[-1]) \rightarrow \hom(W, Z[-1]) \rightarrow \hom(W, X) \rightarrow \hom(W, Y).$

In other words, ${Z[-1]}$ would be a kernel if not for the fact that it probably isn’t in ${\mathcal{C}}$. This suggests that we should take the closest thing to it: a suitable truncation.

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

$\displaystyle Y \rightarrow Z \rightarrow X[1] \rightarrow Y[1].$

by rotation, which “sandwiches” ${Z}$ between ${Y, X[1]}$. This implies that ${Z \in \mathcal{D}_{\geq -1} \cap \mathcal{D}_{\leq 0}}$ at least because the two outer terms are there. In particular, ${Z[-1] \in \mathcal{D}_{\geq 0} \cap \mathcal{D}_{\leq 1}}$.

So if we take the truncation ${\tau_{\leq 0} (Z[-1])}$, then mapping into ${\tau_{\leq 0} (Z[-1])}$ from any ${W \in \mathcal{C}}$ is the same as mapping into ${Z[-1]}$. Also, because ${Z[-1] \in \mathcal{D}_{\geq 0} \cap \mathcal{D}_{\leq 1}}$, one sees (and I’m skipping some details) that ${\tau_{\leq 0} (Z[-1]) \in \mathcal{C}}$ 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, ${X \stackrel{f}{\rightarrow} Y}$ be an epic morphism. We imbed it in an exact triangle ${X \rightarrow Y \rightarrow Z \rightarrow X[1]}$. If ${W \in \mathcal{C}}$, we have an exact sequence

$\displaystyle 0 = \hom(X[1], W) \rightarrow \hom(Z, W) \rightarrow \hom(Y, W) \rightarrow \hom(X, W).$

This shows that ${\hom(Z, W) = 0}$ for each ${ W \in \mathcal{C}}$. In particular, ${\tau_{\geq 0} Z = 0}$. Since we saw that ${Z \in \mathcal{D}_{\geq -1} \cap \mathcal{D}_{\leq 0}}$, it follows that ${Z = \tau_{\leq -1} Z \in \mathcal{D}_{\leq -1} \cap \mathcal{D}_{\geq -1}}$.

That is, ${Z}$ is a translate of something in ${\mathcal{C}}$. When we consider the rotated triangle

$\displaystyle Z[-1] \rightarrow X \rightarrow Y \rightarrow Z,$

we find that ${Z[-1] \in \mathcal{C}}$, and we will show ${Y \rightarrow Z}$ is the cokernel of ${Z[-1] \rightarrow X}$ (which as above is the kernel of ${f}$). To see this, let ${W \in \mathcal{C}}$; then we have an exact sequence

$\displaystyle 0 = \hom(Z, W) \rightarrow \hom(Y, W) \rightarrow \hom(X, W) \rightarrow \hom(Z[-1], W) ,$

which states that mapping out of ${Y}$ is the same as mapping out of ${X }$ in such a way that the restriction to ${Z[-1]}$ is zero. This is precisely the claim we wanted.

Next time, I’ll describe a means of “gluing” $t$-structures, which will enable us to construct the category of perverse sheaves.