Ok, I’m writing this now specifically so that I don’t have to do it in my Monday post, because I need it to even define what I’m talking about. Today’s topic of interest are locally ringed spaces, and they are rather important in general. In fact, any geometric or topological object can be interpreted in this language, so let’s get started.

Let $X$ be a topological space. In a previous post, I defined a category called $\mathrm{Cat}_X$ that turned the topological space into a category. In my next post, I defined what it meant for something to be a sheaf. Really, all it means is that $\mathscr{F}$ is a contravariant functor on $\mathrm{Cat}_X$ such that the elements of objects assigned to small open sets can be glued uniquely (when compatible) to elements of the objects on larger open sets.

Perhaps an example would be nice at this point. Let $X$ be a topological space. Then there is a sheaf of rings $\mathscr{O}_X$ with $\mathscr{O}_X(U)=\{f:U\to \mathbb{R}|f\mathrm{ continuous}\}$. All the sheaf is doing is keeping track of all the local and global information in a nice, convenient package. Another example is if $M$ is a differentiable manifold, say, $C^k$. Then there is a sheaf $\mathscr{O}_M$ which takes open sets to the ring of $C^k$ functions $f:U\to \mathbb{R}$.

This is rather suggestive. Perhaps there is some way of characterizing manifolds as being topological spaces with certain sheaves of rings? Well, as it turns out there is, we will need more definitions though, first. The biggest, most important, is that of a locally ringed space. A locally ringed space is a pair $(X,\mathscr{O}_X)$ such that $X$ is a topological space, $\mathscr{O}_X$ is a sheaf of rings, and the stalk of $\mathscr{O}_X$ at any point is a local ring. I guess this requires the definition of stalk. A stalk is $\varinjlim_{U\supset p}\mathscr{O}_X(U)$, so it’s the limit over all open sets containing a point $p$ of the values of the sheaf. Another way to think about it is as the ring of functions defined on SOME open set containing $p$.

So now we have a locally ringed space $X$ (it is often convenient to leave the sheaf out of the notation when it’s clear from context what it should be) and take another one, $Y$. A morphism of locally ringed spaces is a pair of functions $(f,f^*):(X,\mathscr{O}_X)\to (Y,\mathscr{O}_Y)$ with $f:X\to Y$ continuous and $f^*:\mathscr{O}_Y\to f_*\mathscr{O}_X$. Again, stuck with something I didn’t define in time. If $f:X\to Y$ is a continuous map and $\mathscr{F}$ is a sheaf on $X$, then $f_*\mathscr{F}$ is the sheaf on $Y$ defined by $f_*\mathscr{F}(U)=\mathscr{F}(f^{-1}U))$. Oh, and for technical reasons, we’ll also require that these “pullback” maps on sheaves induce local homomorphisms on the stalks, that is, maximal ideals are mapped into maximal ideals.

Now that we have morphisms, the notion of an isomorphism shouldn’t be too bad. It’s a morphism with a right and left inverse.

A standard notion from topology is a property holding locally. What this means is that for each point $p$ there exists an open set $U$ such that the property holds on $U$. So we say that a locally ringed space $X$ is locally isomorphic to a locally ringed space $Y$ if for each point $p\in X$, there exists an open set $U\subset X$ such that the locally ringed space $(U,\mathscr{O}_X|_U)$ is isomorphic to $Y$. This definition is wonderful for giving compact versions of the definitions of various geometric objects. For instance, the following:

Let $M$ be a locally ringed space. Then $M$ is a manifold if and only if $M$ is locally isomorphic to $(\mathbb{R}^n,\mathscr{O}_{\mathbb{R}^n})$ with its sheaf of differentiable functions and $M$ is Hausdorff and second countable.

So what advantage does this give? Well, locally ringed spaces give us (automatically!) a notion of tangent space. Let $x\in X$ be a point in a locally ringed space, and let $\mathscr{O}_{X,x}$ be the stalk at $x$ and $m_x$ the maximal ideal of $\mathscr{O}_{X,x}$. Then the tangent space to $X$ at $x$ is the $k_x=\mathscr{O}_{X,x}/m_x$-vector space that is dual to $m_x/m_x^2$.

Also, an affine variety can be seen to be a locally ringed space. Let $V$ be an affine variety with its Zariski topology (we can be working over any algebraically closed field $k$ here). The Zariski topology has a basis of open sets of the form $U_f=\{f(x)\neq 0\}$ for each $f$ a regular function on $V$. So if $I\subset k[x_1,\ldots,x_n]$ is the ideal of $V$, then $f\in k[x_1,\ldots,x_n]/I=k[V]$ defines a distinguished open set. Also of note is the fact that the Zariski topology on an affine variety is quasi-compact, that is, every open cover has a finite subcover (the quasi is to distinguish from the Hausdorff case, which is just called compact).

So any open set is a union of distinguished open sets. In fact, we can define a sheaf just on a basis of a topology, because the sheaf axioms force the rest of the structure upon us. So we define $\mathscr{O}_V(U_f)$ to be the localization of $k[V]$ at $f$. So then an affine variety is a locally ringed space $(V,\mathscr{O}_V)$ with $\mathscr{O}_V$ the sheaf of regular functions.

We now define a prevariety to be a locally ringed space $(X,\mathscr{O}_X)$ such that it is locally isomorphic to affine varieties. This is a bit more complex than before, as we require that open sets be isomorphic to SOME affine variety rather than all of them to the same one. To get a variety, we will use the diagonal map: $\Delta: V\to V\times V$. The map on topological spaces is given by $v\mapsto (v,v)$, and the map on sheaves $\mathscr{O}_{V\times V}\to \Delta_* \mathscr{O}_V$ by $(\alpha,\beta)\in\mathscr{O}_{V\times V}(\Delta^{-1}(U))$ maps to $\alpha\beta\in \mathscr{O}_V(U)$. We say that a prevariety is separated if the image of $\Delta$ is closed.

So just a bit of quick geometry of varieties to set up for my post on Monday. Let $V$ be a variety and let $p\in V$ a point. Then $\dim_p V=\dim_{k_p} (m_p/m_p^2)^*$ is the dimension of $V$ at $p$, which we take to just be the dimension of the tangent space at that point. We say that a point is smooth if $\dim_p V=\dim \mathscr{O}_{X,p}$ where the right hand side is the Krull dimension. We call a variety $X$ complete if, for any variety $Y$, $\pi_Y:X\times Y\to Y$ is a closed map (that is, the image of a closed set is closed).

Armed with these notions, on Monday, we’ll investigate elliptic curves a bit more thoroughly.

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