# morphism of schemes induces a map of points

Let $f\colon X\to Y$ be a morphism of schemes over $S$, and let $T$ be a particular scheme over $S$. Then $f$ induces a natural function from the $S$-points of $X$ to the $S$-points of $T$.

Recall that a $T$-point of $X$ is a morphism $\phi\colon T\to X$. So examine the following diagram:

 $\xymatrix{T\ar[drr]^{\phi}\ar[ddrrr]\ar@{-->}[drrrr]^{\psi}&&&&\\ &&X\ar[rr]_{f}\ar[dr]&&Y\ar[dl]\\ &&&S&}$

Since all the schemes in question are $S$-schemes, the solid arrows all commute. The dashed arrow $\psi$ we simply construct as $f\circ\phi$, making the whole diagram commute. The $\psi$ is a $T$-point of $Y$.

Title morphism of schemes induces a map of points MorphismOfSchemesInducesAMapOfPoints 2013-03-22 14:11:02 2013-03-22 14:11:02 archibal (4430) archibal (4430) 4 archibal (4430) Result msc 14A15