Post 12: Perverse Sheaves Part 6

16 Jan 2024

---

In this post, I’ll discuss some necessary functors between (derived) categories of sheaves, and some important concepts, such as the dualizing complex.

As this post is rather long, I’ll save some technical details for the next post. In other words, this will be more on the intuition side of things.

---

To define perverse sheaves, we need two functors. One is the derived sheaf Hom. The other is “the right adjoint to derived proper push-forward”. As you might expect, the latter is much more complicated than the second. In fact, just constructing the second functor is so complicated I’m not going to attempt it, even in the formalities section. Unfortunately, this was a major oversight when I decided to take on this project. Oh well; I’m already this far in!

Sheaf Hom

First, given a sheaf $F$ on a topological space $X$ and an open subset $U$ of $X$, there is a sheaf $F|_U$ on $U$ defined by $F|_U(V)=F(V)$ for any open subset $V$ of $U$. Then, given two sheaves $F,G$ on $X$, we define the sheaf Hom by $\H om(F,G)(U)=\Hom_{Sh(U)}(F|_U,G|_U)$. One has to check that this is actually a sheaf, which I will not do.

Since $Sh(U)$ is abelian, the outputs of $\H om(F,G)$ are at least abelian groups. One can show that there is a “canonical” way to make $\H om(F,G)(U)$ an $R$-module, so $\H om(F,G)$ is a sheaf of $R$-modules.

Hopefully you see why this is called the sheaf Hom. It is essentially a way of turning the set $\Hom_{Sh(X)}(F,G)$ into a sheaf. It is also sometimes called the internal Hom for sheaves.

If you fix a sheaf $F$, then $\H om(F,-)$ is a functor $Sh(X)\to Sh(X)$. It is left exact. This functor has a right adapted class given by “injective” sheaves, which I will not discuss. Therefore, you can define the right derived functor $R\H om(F,-):D^+(X)\to D^+(X)$. In fact, for any $\mc F\in D^-(X)$, there is a functor $R\H om(\mc F, -):D^+(X)\to D^+(X)$.

Upper shriek

As I mentioned before, the other sheaf functor we need is much more complicated to define, so I will only talk about it vaguely. If you have a continuous map $f:X\to Y$, there are three functors $^\circ f_*$, $^\circ f^*$, $^\circ f_!$ one can define. The functors $^\circ f_*$ and $^\circ f_!$ go from $Sh(X)$ to $Sh(Y)$, and the functor $^\circ f^*$ goes from $Sh(Y)$ to $Sh(X)$.

It turns out that $^\circ f^*$ is exact, while $^\circ f_*$ and $^\circ f_!$ are both left exact. It follows that $^\circ f^*$ induces a functor $f^*:D(Y)\to D(X)$. As with sheaf Hom, the “injective” sheaves are a right adapted class for both $^\circ f_*$ and $^\circ f_!$, so they have right derived functors $D^+(X)\to D^+(Y)$ that we denote by $f_*$ and $f_!$.

As you might see, there is something asymmetrical about what I’ve just said. You might think that there should be a functor $^\circ f^!$ or $f^!$ that goes in the same direction as $f^*$, i.e. from (derived) sheaves on $Y$ to (derived) sheaves on $X$. It turns out that, with “mild” assumptions on the spaces $X,Y$, the ring $R$, and even the function $f$, there is a functor $f^!:D^+(Y)\to D^+(X)$ that is related to $f_!$. However, part of what makes this functor so complicated is that there is not a “non-derived” counterpart $^\circ f^!$.

However, once the functor has been shown to exist, it satisfies a very special property that actually determines it. Namely, it is a right adjoint to $f_!$: for all $\mc F\in D^-(X)$ and $\mc G\in D^+(Y)$, there is a “natural”¹ isomorphism $\Hom_{D(Y)}(f_!(\mc F),\mc G)\cong \Hom_{D(X)}(\mc F,f^!(\mc G))$.

Note: The astute reader might object to the usage of $f_!(\mc F)$ for $\mc F\in D^-(X)$. As I wrote above, $f_!$ is a right derived functor, so it is defined on $D^+(X)$. This is where some of the assumptions on $X,Y,R,$ and $f$ come in. In particular, in some cases, $f_!$ can be extended to take values in $D^-(X)$.

Interlude: the moral dilemma of black-boxing

This discussion raises a very good question: if the construction of this functor is complicated, how can we expect to work with it?

To be completely honest, I don’t know the details of the construction of $f^!$. However, I can still work with it, because it satisfies a handful of nice properties that relate it to the other sheaf functors. In fact, Pramod Achar’s book “Perverse Sheaves …” has a “quick reference” at the end. It is a few pages that is jam packed with a bunch of useful facts and formulas that get used over and over when using perverse sheaves.

This procedure of using facts without knowing the technical details is known as “black-boxing”. It can be very dissatisfying to some people, especially if what drew you into math was the fact that you could prove everything. I for one was very conflicted by it. Of course, no one is stopping you from going through the technical details. However, if you want to pursue math research, you will inevitably reach a point where some things are not worth proving. We must learn to “stand on the shoulders of giants” if we want to become giants ourselves.

Back to the math

With our derived functors in hand, we can introduce the main player in our story.

Given a topological space $X$, there is a unique continuous map $a_X:X\to pt$, where $pt$ is a one-point topological space. Recall also that the data of a sheaf of $R$-modules on $pt$ is just the data of an $R$-module. Let $\ul R$ be the sheaf corresponding to the $R$-module $R$. Let $\ul{\mc R}^\bullet$ be the chain complex of sheaves on $pt$ with $\ul{\mc R}^0 = \ul R$ and $\ul{\mc R}^i=0$ for $i\neq 0$.

We define the dualizing complex on $X$ to be $\omega_X=a_X^! (\ul{\mc R}^\bullet)$.

Before we can define the next functor, either recall or learn the following definition:

• Given a category $\mc C$, the opposite category $\mc C^{op}$ has the same objects as $\mc C$, but a morphism $f:X\to Y$ in $\mc C^{op}$ is by definition a morphism $f:Y\to X$ in $\mc C$.

In other words, the opposite category is the result of swapping the direction of the morphisms.

Here are some important facts for an abelian category $\mc A$:

1. $\mc A^{op}$ is abelian.

2. $D(\mc A^{op}) = (D(\mc A))^{op}$.

3. $D^+(\mc A^{op}) = (D^-(\mc A))^{op}$.

Note that I will write $D(\mc A)^{op}$ and $D^-(\mc A)^{op}$ instead of $(D(\mc A))^{op}$ and $(D^-(\mc A))^{op}$.

Just as $\H om(F,-)$ is a functor $Sh(X)\to Sh(X)$, it turns out that $\H om(-,G)$ is a functor² $Sh(X)^{op}\to Sh(X)$. This is left exact, and there is a right derived functor $R\H om(-,G):D^-(X)^{op}\to D^+(X)$. Just as before, if $\mc G\in D^+(X)$, we also have a functor $R\H om(-,\mc G):D^-(X)^{op}\to D^+(X)$.

The two notions of $R\H om$ agree with each other up to isomorphism; applying $R\H om(\mc F,-)$ to $\mc G$ gives you an object isomorphic to $R\H om(-, \mc G)$ applied to $\mc F$, so we denote either of these isomorphic objects by $R\H om(\mc F,\mc G)$.

Finally, we can define the Verdier duality functor by $\bb D = R\H om(-,\omega_X):D^-(X)^{op}\to D^+(X)$.

At this point, I should mention that we generally restrict our attention to $D^b(X)$, since objects in $D^b(X)$ are in both $D^+(X)$ and $D^-(X)$, and we don’t have to worry about which functors take inputs from $D^+(X)$ and which take inputs in $D^-(X)$. In fact, more than just being lazy, perverse sheaves will be defined as elements of $D^b(X)$. So, below, imagine that all sheaf functors have been restricted to $D^b(X)\to D^b(X)$ or $D^b(X)^{op}\to D^b(X)$.

To give an indication of why we use the word “duality”, here are some properties that $\bb D$ has in some cases³ :

1. For any $\mc F\in D^b(X)$, $\mc F$ is isomorphic to $\bb D(\bb D(\mc F))$.

2. For any $\mc F,\mc G\in D^b(X)$, $R\H om(\mc F,\mc G)$ is isomorphic to $R\H om(\bb D(\mc G), \bb D(\mc F))$.

3. Let $f:X\to Y$ be a continuous map, and by abuse of notation, let $\bb D$ denote the Verdier duality functor for both $X$ and $Y$. The functors $f_*\circ \bb D$ and $\bb D\circ f_!$ are isomorphic⁴. Similarly, $f_!\circ \bb D\cong\bb D\circ f_*$, $f^!\circ\bb D\cong \bb D\circ f^*$, and $f^*\circ \bb D\cong \bb D\circ f^!$.

These properties aren’t really relevant to defining perverse sheaves, so don’t worry if they seem like to much to grasp. However, the definition of perverse sheaves does directly involve $\bb D$, so we really do need it.

---

Sneak peek

I will state here the definition of perverse sheaves, in the case where the coefficient ring $R$ is a field. We are missing the definitions of some key words, which I will put in bold, but it will show you how close we are to being done.

• A perverse sheaf is a complex $\mc F\in D^b(X)$ satisfying the following conditions:

1. For all integers $i$, the sheaf $H^i(\mc F)$ is constructible; roughly, this means it is “well-behaved” on some subspaces of $X$.

2. For all integers $i$, we have $\text{dim}\left (\text{supp}\left ( H^i(\mc F) \right )\right )\leq -i$; supp is the support (see Part 5 for the definition), and dim means topological dimension.

3. For all integers $i$, we have $\text{dim}\left (\text{supp}\left ( H^i(\bb D(\mc F)) \right )\right )\leq -i$.

As I mentioned above, this definition assumes $R$ is a field. Conditions 1 and 2 are the same when $R$ is not a field, but condition 3 must be changed. In particular, instead of taking the dimension of the support of $H^i(\bb D(\mc F))$, one must take the modified dimension of support, denoted mdsupp. The definition of mdsupp is, at least in my opinion, rather unintuitive, and involves a detour in commutative algebra that I would rather not have us take. For that reason, I’m going to stick to assuming $R$ is a field.

---

Footnotes: