# example of functor of points of a scheme

Let $X$ be an affine scheme^{} of finite type over a field $k$. Then we must have

$$ |

with the structure morphism $X\to \mathrm{Spec}k$ induced from the natural embedding $k\to k[{X}_{1},\mathrm{\dots},{X}_{n}]$.

Let ${k}^{\prime}$ be some field extension of $k$. What are the ${k}^{\prime}$-points of $X$? Recall that a ${k}^{\prime}$-point of $X$ is by definition a morphism $\mathrm{Spec}{k}^{\prime}\to X$ (observe that since we have an embedding $k\to {k}^{\prime}$ we have a morphism $\mathrm{Spec}{k}^{\prime}\to \mathrm{Spec}k$, so $\mathrm{Spec}{k}^{\prime}$ is natuarlly a $k$-scheme). Since $X$ is affine, this must come from a ring homomorphism^{}

$$ |

which takes elements of $k$ to themselves inside ${k}^{\prime}$. Such a homomorphism^{} is completely specified by specifying the images of ${X}_{1},\mathrm{\dots},{X}_{n}$; for it to be a homomorphism, these images must satisfy ${f}_{1},\mathrm{\dots},{f}_{m}$. In other words, a ${k}^{\prime}$-point on $X$ is identified with an element of ${({k}^{\prime})}^{n}$ satisfying all the polynomials ${f}_{i}$.

If ${k}^{\prime}$ is an algebraically closed field, a point on $X$ corresponds uniquely to a point on an affine variety^{} defined by the same equations as $X$. If ${k}^{\prime}$ is just any extension^{} of $k$, then we have simply found which new points belong on $X$ when we extend the base field. T

For an example of why schemes contain much more information than the list of points over their base field, take $$. Then $X$ has *no* points over $\mathbb{R}$, its natural base field. Over $\u2102$, it has two points, corresponding to $i$ and $-i$.

This suggests that schemes may be the appropriate adaptation of varieties^{} to deal with non-algebraically closed fields.

Observe that we never used the fact that ${k}^{\prime}$ (or in fact $k$) was a field. One often chooses ${k}^{\prime}$ as something other than a field in order to solve a problem. For example, one can take $$. Then specifying a ${k}^{\prime}$-point on $X$ amounts to choosing an image ${\kappa}_{i}+{\lambda}_{i}\u03f5$ for each ${X}_{i}$. It is clear that the ${\kappa}_{i}$ must satisfy the ${f}_{j}$. But upon reflection, we see that the ${\lambda}_{i}$ must specify a tangent vector to $X$ at the point specified by the ${\kappa}_{i}$. So the $$-points tell us about the tangent bundle to $X$. Observe that we made no assumption^{} about the field $k$ — we can extract these “tangent vectors” in positive characteristic or over a non-complete field.

The ring $$ and rings like it (often any Artinian ring) can be used to define and study infinitesimal deformations of schemes, as a simple case of the study of families of schemes.

Title | example of functor of points of a scheme |
---|---|

Canonical name | ExampleOfFunctorOfPointsOfAScheme |

Date of creation | 2013-03-22 14:11:07 |

Last modified on | 2013-03-22 14:11:07 |

Owner | archibal (4430) |

Last modified by | archibal (4430) |

Numerical id | 4 |

Author | archibal (4430) |

Entry type | Example |

Classification | msc 14A15 |