How to understand a point of a scheme?

Question

I am not quite understand what a point of a scheme really is. It's not like the definition of a point in a usual topology, such as Euclidean space. In Hartshorne's, a point seems to be a prime ideal of a given ring, but I am not quite sure. I hope someone can show me a clear picture. Thanks!

Answer 1

There are two notions of a point in algebraic geometry. The first one, and the one I think you are most interested in is the following definition.

The first definition

First, let's recall the definition of a scheme. A scheme is a locally ringed space which is locally affine (i.e. locally isomorphic to $\newcommand\Spec{\operatorname{Spec}}\Spec A$ for a ring $A$). And as a reminder, a locally ringed space is a topological space, $X$, with a sheaf of rings, $\mathcal{O}_X$ whose stalks are local rings.

I've bolded the words topological space, since that part is the relevant part for talking about points of schemes (at least in this first sense). This gives us the definition:

A point of a scheme $(X,\mathcal{O}_X)$ is a point of the underlying topological space $X$. Thus points of a scheme are in fact the points of a topological space.

Now you may be wondering, well how do prime ideals enter the picture here? And the answer is, if $A$ is a ring, then the underlying topological space of $\Spec A$ is the set of all prime ideals of $A$ with the Zariski topology on them. Thus points of $\Spec A$ are precisely the prime ideals of $A$. Since general schemes are locally affine, we can think of the points of a general scheme as being a prime ideal in some affine open set containing that point, however the precise prime ideal depends entirely on the choice of neighborhood of the point and the isomorphism with the spectrum of some ring.

The other definition

If $X$ and $Y$ are schemes, we call a morphism from $Y\to X$ a $Y$-valued point of $X$, or if $Y=\Spec A$ is affine, we also say it is an $A$-valued point of $X$. The set of $Y$ valued points of a scheme $X$ is then just $\operatorname{Hom}(Y,X)$, and the functor $Y\mapsto \operatorname{Hom}(Y,X)$ is called the functor of points of the scheme $X$, since to a given scheme $Y$ it assigns the set of $Y$-valued points.

I'm not quite sure of the precise history of the usage of the term point for a morphism of schemes, but in many ways it makes sense. For example, if $Y=\Spec k$ where $k$ is a field, then a morphism from $Y\to X$ is determined by a choice of (topological) point $x\in X$ and a choice of map from $\kappa(x)\to k$, where $\kappa(x)$ is the residue field of $X$ at $x$, $\mathcal{O}_{X,x}/\mathfrak{m}_x$.

Anyway, I think this second definition is not likely to be what you meant, so I won't say too much more on it.