## Sheaves

Second in our survey of some technical tools are sheaves. Sheaves have a reputation for being terrifying and technical and some people I know have trouble seeing their purpose at all. In my opinion, they’re very nice, convenient ways to organize information. Sheaves do require a bit of topology to use, and I recommend John Armstrong’s exposition as a basic reference. I talked a bit about the topology relevant in algebraic geometry (which is rather different than that in most point set topology settings, note that we make use of the concept of irreducibility quite often) but the basics and the notion of connectedness are both still rather important.

So now for sheaves. Let $X$ be a topological space. We define a presheaf of abelian groups to be an assignment $\mathscr{F}$ which assigns to each open set an abelian group (called the group of sections) and for each inclusion $V\subseteq U$ we get a homomorphism of abelian groups $\rho_{UV}:\mathscr{F}(U)\to\mathscr{F}(V)$ such that $\mathscr{F}(\emptyset)=0$, $\rho_{UU}$ is the identity homomorphism, and if $W\subset V\subset U$ then $\rho_{UW}=\rho_{VW}\circ \rho_{UV}$.

If we replace “abelian group” in the definition we can get other sorts of presheaves. The most common choices we’ll be making there are sets, rings and modules, though I’ve heard of people using presheaves of other things as well. Also, a common convention that we’ll be using is that we’ll denote $\rho_{UV}(s)$ by $s|_V$ which should be read as “$s$ restricted to $V$“, because we want to try to think of these things as behaving like functions. Another useful notation is $\Gamma(U,\mathscr{F})=\mathscr{F}(U)$. Particularly, it is useful when we want to take sections over the same open set of several sheaves or presheaves.

So if we have defined presheaves, what’s a sheaf? A sheaf is simply a presheaf such that, for each open set $U$ and every open cover $\{U_i\}$ of $U$, we have two properties (which can be considered a uniqueness property and an existence property)

1. If $s\in \mathscr{F}(U)$ satisfies $s|_{U_i}=0$ for all $i$, then $s=0$. (Uniqueness)
2. If we have $s_i\in \mathscr{F}(U_i)$ such that for each $i$ and $j$, $s_i |_{U_i\cap U_j}=s_j|_{U_i\cap U_j}$, then there exists $s\in \mathscr{F}(U)$ such that $s|_{U_i}=s_i$ for all $i$.

Now that’s a lot to take in, but it can be remembered fairly simply: a presheaf is a sheaf if compatible sections glue together uniquely. This is all set up to make sections act as much like functions as possible. With that in mind, we should be able to take functions to be a sheaf…and we can! Probably the single most important sheaf on any variety $V$ takes $U\subset V$ open and assigns to it the collection of regular functions on $U$. This is the ring we earlier referred to as $\mathscr{O}_V(U)$. Now the reason for that notation should be getting a bit clearer, and we will call this sheaf $\mathscr{O}_V$, the structure sheaf of the variety. We’ll encounter quite a few other sheaves as we go along.

Other examples of sheaves include the continuous functions to $\mathbb{R}$ on any topological space, differentiable functions on a manifold and holomorphic functions on a complex manifold. Another class of examples can be constructed as follows: let $A$ be an abelian group. Define $\underline{A}(U)$ to be the direct product of copies of $A$, one for each connected component of $U$. This is called the constant sheaf, and consists of the locally constant functions $U\to A$.

So you might remember that for the local ring of a point $x\in X$ we used the notation $\mathscr{O}_{X,x}$. This suggests a connection to the structure sheaf and we’ll look at it now, first in the context of a general presheaf. First we take the collection of pairs $(U,s)$ with $x\in U$ and $s\in \mathscr{F}(U)$. We say that $(U,s)$ and $(V,t)$ are equivalent if there exists $W$ containing $x$ contained in the intersection such that $s|_W=t|_W$. This is called the stalk at $x$, denoted by $\mathscr{F}_x$, and will have whatever structure we put on the original presheaf. So for a sheaf of rings, like $\mathscr{O}_X$, we will get a ring (by performing the ring operation on the first open set on which the two sections are both defined). Sometimes the elements of the stalk are referred to as germs.

So the point is, sheaves collect local data all into one place, and let us look at what happens to that data as we focus on a single point. Though they are a bit technical, we all have intuition about them (whether or not we realize it) by just thinking of them as the functions on a space, and the stalks are just the functions that are defined in SOME neighborhood of the point.