8
votes
1answer
345 views

Which local ringed spaces are schemes?

(This was originally asked on math.stackexchange, but didn't get any responses. I figured it might be worthwhile to move it here and try again.) This paper gives a proof that the underlying ...
3
votes
3answers
450 views

Are schemes pushouts of neighbourhoods and formal neighbourhoods?

Hello, I have two questions, the first less important. Let $X$ be a scheme, $x \in X$ a schematic point. What is an elegant way of defining/characterizing the map $\operatorname{Spec}(O_{X,x}) ...
1
vote
2answers
838 views

Closed subschemes and pulling back the structure sheaf via the inclusion map

I would just like a clarification related to closed subschemes. If $(X,{\cal O}_X)$ is a locally ringed space and $A\subset X$ is any subset with the subspace topology then $i^{-1}{\cal O}_X$ will be ...
4
votes
2answers
403 views

Given a morphism from X to Y, when is the morphism from O_Y to the pushforward of O_X injective

I would like to know under what condition the morphism $\mathcal{O}_Y\longrightarrow f_\ast \mathcal{O}_X$ induced by a morphism $f:X\longrightarrow Y$ of schemes is injective. Let me give an example ...