For \(X \in \Sch\slice S\) and \(T \in \Sch\slice S\), a \(T\)-valued point of \(X\) is a morphism \(T \to X\) over \(S\). The assignment \(T \mapsto X(T) \da \Hom_S(T, X)\) is the functor of points of \(X\), and \(X \mapsto X(-)\) is fully faithful into \(\Sets^{\opcat{\Sch\slice S}}\) by Yoneda.
Morphisms \(\Spec L \to X\), for \(L\) a field, correspond to pairs: a point \(x \in X\) and a field embedding \(\kappa(x) \injects L\).
Morphisms \(\Spec k[\eps]/\eps^2 \to X\) for \(X\) over \(k\) correspond to pairs: a \(k\)-rational point \(x\), and a tangent vector in \(\dualof{(\mfm_x/\mfm_x^2)}\).
The functor of points is what makes the fibre product’s universal property usable, and it is how you recover the naive picture: \(X(k)\) for \(k\) algebraically closed is the classical variety, and \(X(\ZZ)\) or \(X(\QQ)\) is what a number theorist means by the solutions.
These two computations explain the standard test schemes: a field detects a point together with a residue-field embedding, while the dual numbers detect a tangent vector. Replacing \(k[\eps]/\eps^2\) by \(k[t]/t^n\) detects an \(n\)-th order arc, which is where jet schemes come from.
The usable slogan is that a scheme is determined by what maps into it, so constructing a scheme can be replaced by writing down a functor and proving it representable — this is how Hilbert and Picard schemes are defined.
Dependencies
D-SCHFPRThe fibre product of schemesuses
Backlinks
FE-SCHREPFUNCTORSThe functors of points of $\AA^n$, $\AA^n \setminus \{0\}$ and $\PP^n$usesD-OPENSUBFUNCTORZariski sheaves, open subfunctors, and representability by gluinguses