Just wondering about conventions dealing with the zero ring and the zero scheme.
- Does the category of schemes have an inital object?
- Is the zero ring considered local?
- For the purposes of scheme theory, is a map of sheaves that induces on stalks a map of the form $\mathcal{O}_{X,P}\to 0$ considered a "local" homomorphism on stalks?
https://en.wikipedia.org/wiki/Zero_ring
Wikipedia says that the zero ring is not local.
I am wondering how to square this with certain conventions in scheme theory. If $0$ is the zero ring, then conventionally (say in Hartshorne Chapter II, exercise 2.6) the category of schemes has $Spec(0)$ for an initial object; whose underlying space is $\emptyset$ and whose associated sheaf is the constant sheaf at zero. The direct image of this sheaf along the canonical map $\emptyset \to X$ would be, again, the constant zero sheaf, so the canonical natural transformation $\mathcal{O_X}\to 0$ would send every stalk to zero. It seems like this shouldn't count as a map of locally ringed spaces.
Best Answer
The empty scheme is initial in the category of schemes, and the zero ring is not a local ring, since it does not have a unique maximal ideal (it does not have any maximal ideal!). There is no special convention needed here--this all just follows from the general definitions.
In particular, there is no issue with what the unique map out of the empty scheme does on stalks. If $X$ and $Y$ are locally ringed spaces, then a morphism $X\to Y$ is a continuous map $f:X\to Y$ together with a morphism of sheaves of rings $\mathcal{O}_Y\to f_*\mathcal{O}_X$ such that for each $x\in X$ the induced map on stalks $\mathcal{O}_{Y,f(x)}\to \mathcal{O}_{X,x}$ is a local homomorphism. When $X$ is empty, there are no points $x\in X$ at which to check this condition, and so it holds vacuously.