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
Canonical name	MorphismOfSchemesInducesAMapOfPoints
Date of creation	2013-03-22 14:11:02
Last modified on	2013-03-22 14:11:02
Owner	archibal (4430)
Last modified by	archibal (4430)
Numerical id	4
Author	archibal (4430)
Entry type	Result
Classification	msc 14A15