What is Sp(X) (Underlying topological space without scheme structure)?

Question

Even though I have spent a lot of time trying to know the concept of the Space of X, which denotes $ Sp(X)$ - the topological space without scheme structure, I think that I precisely do not understand this meaning. Thus, I have a question about that. As I already have referred the title, $ Sp(X)$ is an underlying topological space without scheme structure. For example, if $K$ is a field, $X= Spec ~K $ is a clearly affine scheme. then What is $Sp(Spec~ K )$ ? Since one of the conditions of the affine scheme is X is homeomorphic to some ring $R$, I decide that $Sp(X)$ is merely the topological space of X, but it is apparently weird. (it is just underlying topological space.) Hence probably I do not pick up on the sentence "without the scheme structure." Or, if such an example is not better to understand $ Sp(X)$ , can you give another example?)

Answer 1

A scheme $X$ is a locally ringed space that is covered by affine schemes. As a locally ringed space, it is a topological space equipped with a sheaf a rings such that the stalks are local. Now you just forget about the affine coverings and the sheaf and consider the topological space again.

For $X = \text{Spec}(k)$, the underlying topological space is a singleton since fields only have one prime ideal.

If now $R$ is a discrete valuation ring, the underlying topological space of $X = \text{Spec}(R)$ is the Sierpinski space, where the generic point is given by the zero ideal and the closed point is given by the prime ideal generated by a uniformizer.

If $X =\text{Spec}(R)$ is an arbitrary affine scheme, then the underlying space is the set of prime ideals equipped with the Zariski topology. That's it.

In general, you just apply the forgetful functor from the category of schemes to the category of topological spaces and then you get the space without the scheme structure.

If you are interested in the question: How much does the underlying topological space tell you about a scheme? Then I would recommend that you read the preprint "What determines a variety" by Kollar, which you can find on the arxiv.

Answer 2

A scheme consists of two data: A topological space $X$ and a sheaf of rings $\mathcal{O}_X: \tau(X)^\text{op} \rightarrow \mathsf{CRing}$ subject to some conditions (stalks are local rings, the whole thing is glued together by affine schemes etc...), where $\tau(X)$ denotes the poset of the topology of $X$ ordered by inclusion. The underlying space of the scheme is just the first part of the data, ie. the topological space on which the sheaf of rings is defined.

In the case of an affine scheme $(\operatorname{Spec} A, \mathcal{O}_A)$ the underlying space of this scheme hence is the topological space $\operatorname{Spec} A$, ie. the set of prime ideals of the ring $A$ equipped with the Zariski-topology.