## Locally Ringed Spaces

Sorry about February, got rather caught up in schoolwork. My posts should resume, though still not with the frequency they used to. So, anyway, we now have sheaves and their morphisms, and even some morphisms that we get when we have continuous maps of our spaces. Today, we’re going to put it all together and look at spaces with a special sheaf of rings.

So now we will take $X$ to be a topological space and $\mathscr{O}_X$ to be a sheaf of rings. So now we define the pair $(X,\mathscr{O}_X)$ to be a ringed space. We call it a locally ringed space if each stalk is a local ring. A morphism of locally ringed spaces is then a pair $(f,f^\sharp)$ such that $f:X\to Y$ is continuous and $f^\sharp:\mathscr{O}_Y\to f_*\mathscr{O}_X$ is a morphism of sheaves such that the induced morphism at each stalk is a local homomorphism (that is, the inverse image of the maximal ideal in the target is the maximal ideal in the domain). And, naturally, we define an isomorphism to be a morphism with a two-sided inverse.

Generally, we make a slight error in notation and call $X$ a locally ringed space. We also should notice that locally ringed spaces form a category, and includes all of the usual geometric and topological categories. Let’s look at some examples:

If $V$ is a variety, then $(V,\mathscr{O}_V)$ is a locally ringed space.

If $M$ is a manifold (of whatever sort), then $(M,\mathscr{O}_M)$ is a locally ringed space, where $\mathscr{O}_M$ consists of functions to $M$ which are of whatever class the manifold is.

Next time we’re going to talk about the locally ringed spaces that we’re mostly going to be concerned with (at least for the near future), in the meantime, I’ll just link back to an old post where I talked about locally ringed spaces (and also about abstract varieties, which I’ll write more about next time).