Past exam of the introduction to schemes course of the University of Oxford / 2023 / 1 / a

Write $h_T(S)=\mathrm{Hom}(S,T)$ for the functor of points of a scheme $T$. A subfunctor $H\subseteq F$ is an open subfunctor if every natural transformation $h_T\to F$ has a pullback represented by an open subscheme $U\subseteq T$:The displayed natural isomorphism must identify the projection to $h_T$ with the natural transformation induced by $U\hookrightarrow T$. Equivalently, for each $\xi \in F(T)$, there is an open subscheme $U\subseteq T$ such that a morphism of schemes $a:S\to T$ factors through $U$ exactly when $F(a)(\xi )\in H(S)$. The pullback is taken in the functor category, so this tests arbitrary families, rather than just underlying points.

A Zariski cover of a scheme $T$ is a family of open immersions $(j_i:U_i\hookrightarrow T{)}_{i\in I}$ whose images cover its underlying topological space:There is no finiteness requirement. Intersections used in the sheaf on a site condition are the pullbacks $U_i{\times }_T{U}_j$, identified with the corresponding open subschemes $U_i\cap U_j$.

A presheaf $F:{\mathsf{Sch}}^{\mathrm{op}}\to \mathsf{Set}$ is a Zariski sheaf when every Zariski cover $(U_i\to T)$ gives an equalizer diagramThus compatible local elements glue to a unique element of $F(T)$. Compatibility means that the two restrictions agree on every intersection. This includes the empty Zariski cover of the empty scheme, so $F(\varnothing )$ is a singleton.

The open affine representability criterion says that $F$ is a functor represented by a scheme exactly when $F$ is a Zariski sheaf and admits a family of open subfunctors $F_i\subseteq F$, each represented by an affine scheme, which covers $F$ in the following sense: for every scheme $T$ and every $\xi \in F(T)$, the open subschemes representing $h_T{\times }_F{F}_i$ cover $T$. In particular, the covering condition does not demand that every $\xi$ lie in one $F_i(T)$.

In symbols, for affine schemes $X_i$,The family is indexed by a set. For necessity, take an affine open cover of a representing scheme; morphisms of schemes glue over a Zariski cover. For sufficiency, $F_i{\times }_F{F}_j$ is represented by an open subscheme of each $X_i,X_j$. The two descriptions identify these overlaps and satisfy the compatibility on triple overlaps because all maps come from the inclusions into $F$. Apply gluing of schemes along open subschemes to obtain $X$. Local identification gives $h_X\to F$, and the Zariski sheaf condition makes this a natural isomorphism: its elements can be constructed and compared on the pulled-back open cover.