Functor represented by a scheme ($h_X(R)=\mathrm{Hom}(\mathrm{Spec}R,X)$)

The functor of points describes a scheme through its morphisms from test schemes. On commutative rings, it is covariant because a ring homomorphism reverses the corresponding affine-scheme map. A functor on rings is represented by $X$ in the algebraic-geometric sense when it is naturally isomorphic to $h_X$. The representing scheme need not be affine; this is different from demanding a representable functor in the category of rings itself.

// Target: module-theory.bigb

Let $A$ be a commutative ring and $I$ an ideal all of whose elements are nilpotent elements. Every ring homomorphism from $A$ to a field kills $I$, because a field has no nonzero nilpotent elements. Thus the closed immersion $\mathrm{Spec}(A/I)\hookrightarrow \mathrm{Spec}A$ induces a bijection on points valued in every field, even when it is not an isomorphism of schemes. The dual numbers provide the smallest example. Testing the functor of points on all affine schemes does detect this information, by affine test schemes determine scheme morphisms.

Restricting the functor of points to affine schemes gives a fully faithful functor from schemes to covariant set-valued functors on commutative rings. Given a natural transformation, apply it to the inclusions of an affine open cover of $X$. Naturality on affine open subschemes of overlaps makes the resulting maps to $Y$ agree, so they glue. For an arbitrary map from an affine scheme to $X$, cover its inverse images of these charts by affine open subschemes and apply naturality again. This proves that the glued map induces the prescribed natural transformation, and uniqueness follows from the original cover. In particular, the restricted functor of points determines the scheme up to an isomorphism of schemes.

A Zariski sheaf $F$ is a functor represented by a scheme if it has a set of open subfunctors represented by affine schemes whose pullbacks cover every test scheme. The overlaps are open subschemes in both charts, and their identifications satisfy the compatibility on triple overlaps. Gluing of schemes along open subschemes produces a representing scheme; the Zariski sheaf condition identifies its functor of points with $F$. Conversely, every affine open cover of a scheme supplies such a presentation. A family of local charts covers in this sense without requiring every global element to belong to a single chart.

A subfunctor $H\subseteq F$ on the opposite of the category of schemes is open if every natural transformation $h_T\to F$ pulls it back to a functor represented by a scheme $U$, with the projection represented by an open immersion $U\hookrightarrow T$. Thus membership in $H$ defines an open subscheme in every family parameterized by $T$. This condition is preserved by further pullbacks.