My main motivation is in trying to intuitively understand schemes. As I currently understand them, schemes are tools which allow us to recover points (in the traditional sense) as morphisms or encode infinitesimal data via nilpotents, among other things, using only some initial data (a base ring and an algebra over it). One sees both in the tangent bundle of an affine (appropriately qualified) scheme Spec $A[\epsilon]$, where $\epsilon^2=0$ and $A$ is a $k$-algebra, which has the same number of points as Spec $A$ topologically, but whose morphisms Spec $A[\epsilon] \to $ Spec $A$ correspond to the tangent vector fields, yielding far more "geometric points" than topological points. (Do we refer to the scheme Spec $A[\epsilon]$ as the tangent bundle or Hom$_k(\text{Spec} A[\epsilon], \text{Spec} A)$ which are its “points”?) As one sees in the above example, the underlying topological spaces (one generic point) don’t seem to do much since it’s the maps on the structure sheaves that do the heavy lifting and determine tangent vectors. So of what value is the underlying top space? Wouldn’t it be easier to just work with the sheaves of local data and consider morphisms on those?
More generally, is there any value that the topological spaces of (locally) ringed spaces add that’s not seen through just the structure sheaf?
[Edit]: If I recall correctly, Hom$_k(\text{Spec} A[\epsilon], \text{Spec} A)$ are the tangent vector fields because they correspond to the derivations on $A$, but aren't the actual "points"/"tangent vectors" given by the morphisms Hom$_k(\text{Spec} k, \text{Spec} A[\epsilon])$? I think I'm confusing something here.
Best Answer
We need topological spaces to define sheaves. That is enough of a reason to be honest, but we still use many topological properties like connectedness, irreducibility etc. Imagine you want to define algebraic curves or surfaces. That in particular means that you want to have the notion of dimension which is given by the krull dimension of the underlying topological space. This on the other hand uses the notion of noetherian topological spaces to make sure that we have a nice notion of dimension. Now we can define an algebraic scheme to be a separated $k$-scheme of finite type. An algebraic curve now is an algebraic scheme of pure dimension $1$ (all irreducible components are of dimension $1$) , algebraic surfaces as algebraic schemes of pure dimension $2$, etc.
I also feel like it is just very natural in the sense that one often can compare to the situations in differential geometry with their manifolds.