Post 10: Perverse Sheaves Part 5

10 Jan 2024

In this post, I’ll discuss sheaves.

Intuition

Sheaves are objects associated to a topological space $X$. First, pick some category $\mc C$. Then a sheaf of objects in $\mc C$, call it $F$, assigns an object $F(U)$ of $\mc C$ to each open subset $U$ of $X$. These objects are required to have some compatibilities that we’ll get to in a moment.

The usual intuition people give for sheaves is that they encode “local data”. For instance, take a point $p$ in $X$. Then consider the objects $F(U)$ for all open sets $U$ that contain $p$. There is a way to get a single object, denoted $F_p$, which can be thought of, and in fact formalized, as taking the limit of $F(U)$ as $U$ gets smaller and smaller. For example, if there is a smallest open set $U$ containing $p$, then $F_p=F(U)$. The object $F_p$ is called the stalk of $F$ at $p$.

Another aspect of sheaves that makes them “local” is the concept of gluing. I mentioned this in Part 1 as well. I like to think that there are two kinds of gluing for sheaves: internal and external. The internal gluing is part of the definition, while the external gluing is a consequence of the definition. In slightly more concrete terms, internal gluing says that you can take “compatible data” about a sheaf and produce a new piece of data that “agrees” with your starting data. External gluing says you can take “compatible” sheaves and produce a new sheaf that “agrees” with your starting sheaves.

We are first going to define presheaves, and we will define a lot of concepts for presheaves. Sheaves are then presheaves with extra conditions, and all the things we define for presheaves work exactly the same.

Presheaves

As I mentioned before, you can consider sheaves valued in any category $\mc C$. However, for our purposes, we are going to fix $\mc C$ to be the category of $R$-modules for a commutative ring $R$.

Next, we are going to form a category out of the open subsets of a topological space $X$. I’ll call it $Op(X)$. The objects of $Op(X)$ are, of course, the open subsets of $X$. Then, for any pair $U,V\in Op(X)$, there is either no morphism $U\to V$, or exactly one morphism $U\to V$, depending on whether $V$ isn’t or is a subset of $U$, respectively.

A presheaf (of $R$-modules) is nothing but a functor $F:Op(X)\to R\text{-mod}$. If $V\subset U$, the morphism $F(U)\to F(V)$ is called restriction from $U$ to $V$ and sometimes denoted $res_{U, V}$.

A morphism of presheaves is just a morphism of functors, i.e. a natural transformation. Unfortunately, I haven’t defined this outside of Part 2.5, so let me do that now:

Let $F,G:\mc C\to \mc D$ be two functors. A natural transformation $\eta:F\to G$ is a collection of morphisms $\eta(A):F(A)\to G(A)$ for each $A\in\mc C$, with the requirement that for any $f:A\to B$ in $\mc C$, we have $G(f)\circ \eta(A)=\eta(B)\circ F(f)$. The morphisms $\eta(A)$ are called the components of $A$. Sometimes the components are denoted $\eta_A$ for less parentheses.

With a notion of morphisms of presheaves, we have a category $Pre(X,R)$ of presheaves of $R$-modules on $X$.

Since $F(U)$ is a module, it has an underlying set. I won’t bother to distinguish the module and the set. An element of $F(U)$ is called a section. If $V\subset U$, we often denote $res_{U,V}(s)$ by $s|_V$.

I’m going to set up some terminology here that will make some definitions much easier to parse. Suppose $U, V$ are open sets and $W$ is an open set contained in $U\cap V$. Let $s\in F(U)$ and $t\in F(V)$. We say that $s$ and $t$ agree on $W$ if $s|_W = t|_W$.

As I mentioned before, we can consider all of the modules $ F(U)$ where $U$ contains a point $p$, and form a new module, $ F_p$, called the stalk of $ F$ at $p$. The definition is a bit long-winded, so I won’t include it here. However, stalks are absolutely important.

Since $F_p$ is a module, it has a $0$ element, and the module itself can be the $0$ module, ${0}$. Correspondingly, we define:

The support will show up in the definition of a perverse sheaf.

Sheaves

A sheaf is a presheaf $F$ that satisfies the following two conditions:

  1. (Gluing) Let ${U_i}$ be an open covering of an open subset $U$ of $X$, and choose sections $s_i\in F(U_i)$. Suppose that for all $i,j$, the sections $s_i$ and $s_j$ agree on $U_i\cap U_j$. Then there exists a section $s\in F(U)$ such that $s|_{U_i}=s_i$ for all $i$.

  2. (Local identity) Let ${U_i}$ be an open covering of an open subset $U$ of $X$. If sections $s,t\in F(U)$ agree on $U_i$ for all $i$, then $s=t$.

A morphism of sheaves is the same as a morphism of presheaves. We then get a category $Sh(X,R)$ of sheaves of $R$-modules on $X$.

A consequence of the axioms of a sheaf is that $ F(\varnothing) = 0$. Do you see why? Hint in this footnote.1

To better get a feel for the definitions, convince yourself that the data of a sheaf on the one point space, denoted pt, is just a choice of $R$-module. Furthermore, a morphism of sheaves on pt is just a morphism between the corresponding modules.

Sheafify your presheaves

By definition, a sheaf is a presheaf, and a presheaf may not be a sheaf. However, given a presheaf $ F$, there is a “canonical” way to make a sheaf $ F^+$. I won’t go over the definition in this section, as it is both technical and not enlightening. One nice fact is that, given a point $p$, the stalks $ F_p$ and $ F^+_p$ are isomorphic. If $ F$ is a sheaf, then $ F^+$ is isomorphic to $F$.

Sheafification often shows up when we define certain operations on sheaves in the “obvious” way, but the result may not be a sheaf, so we have to sheafify it.

The return of categorical algebra

As I’ve hinted at before, perverse sheaves are certain elements in the derived category of sheaves. Before we can have a derived category, we need an abelian category. Indeed:

As a result, we can define the derived category of $Sh(X,R)$, which we denote by $D(X,R)$. Similarly, there are the bounded variants $D^+(X,R)$, $D^-(X,R)$, and $D^b(X,R)$ that will show up when we consider derived functors. I will generally denote elements of $D(X,R)$ by $\mc F$.

Formalities

Stalks

There are two ways to define stalks. The first way is very explicit:

  1. Let $\ol{ F}_p$ be the set of all pairs $(U,s)$, where $U$ is an open set containing $p$, and $s\in F(U)$.

  2. Two pairs $(U,s), (V,t)\in \ol{ F}_p$ are equivalent if there is a pair $(W,u)\in\ol{ F}_p$ such that $W\subset U\cap V$ and $s|_W = u = t|_V$. For example, if $V\subset U$, then $(U,s)$ is equivalent to $(V,s|_V)$.

  3. The stalk $ F_p$ is the set of equivalence classes of pairs in $\ol{ F}_p$. We denote such an element of $ F_p$ by $[(U,s)]$.

  4. The stalk $ F_p$ has the structure of an $R$-module as follows. The $0$ element is $[(X,0)]$. Given $[(U,s)],[(V,t)]\in F_p$, their sum is $[(U\cap V, s|_{U\cap V}+t|_{U\cap V})]$. Given $r\in R$ and $[(U,s)]\in F_p$, we define $r[(U,s)]=[(U,rs)]$.

If $p\in U$, the canonical map $F(U)\to F_p$ sends $s$ to $s_p:=[(U,s)]$.

The second way is less explicit, but cleaner. Namely, $F_p$ is the direct limit of the modules $F(U)$ taken over the directed set of opens $U$ containing $p$. I haven’t defined (co)limits, let alone direct limits, and I probably won’t. So, if you know you know.

Sheafification

Let $F$ be a presheaf. The sheafification $F^+$ is defined as follows. For an open $U$, $F^+(U)$ consists of functions $s:U\to \bigsqcup\limits_{p\in U} F_p$ such that, for all $p\in U$, there is an open subset $V$ with $p\in V\subset U$ and a section $t\in F(V)$ such that $s(y)=t_y$ for all $y\in V$.

In other words, the elements of $F^+(U)$ are functions that locally are given by sections. Of course, one must verify that $F^+$ is in fact a sheaf.

What’s next?

The end is in sight. We need to discuss some functors between categories of sheaves, and constructibility, and then we will be ready to define perverse sheaves. If you’ve made it this far, thanks for sticking through!

Footnotes:

  1. Consider the empty cover of the empty set. Recall properties of “vacuous truths”. ↩