# Representability of zariski functors on schemes over a base by gluing

Suppose $F$ is a Zariski functor $(Sch/S)^{opp}\rightarrow Set$ where $S$ is a scheme. Suppose that I can show that for every affine $j:U\subseteq S$, I can show that the functor $F_U:(Sch/U)^{opp}\rightarrow Set$ defined by

$F_U(T)=F(j^{-1}(U))= F(U\times_ST)$

is representable, where $T\rightarrow S$ is a $S$ scheme.

The meaning of this is that oftentimes, $F$ is defined explicitly, and I can show it for the special case when $S$ is affine. Thinking about it a bit, the above seems to be right expression of `showing representability in the case $S$ is affine' that is independent of an explicit definition for $F$.

How do then do I use the formalism of open subfunctors to show that $F$ is representable?

Specifically, I seem to be having trouble making each $F_U$ a subfunctor of $F$ for the map $U\times_ST\rightarrow T$ only induces a map $F(T)\rightarrow F(U\times_ST)$ which is the wrong direction.

• – Krish Aug 29 '17 at 8:07
• Thanks, there seems to be a subtle difference between my construction and the poster's formulation there. I'd appreciate now if anyone could let me know if my construction is the wrong, or they are equivalent. – Victor Zhang Aug 29 '17 at 8:42