# morphism of schemes induces a map of points

Let $f:X\to Y$ 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 $\varphi :T\to X$. So examine the following diagram:

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

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

Title | morphism of schemes induces a map of points |
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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 |