Exercise 111.31.1. Consider the affine curve $X$ given by the equation $t^2 = s^5 + 8$ in $\mathbf{C}^2$ with coordinates $s, t$. Let $x \in X$ be the point with coordinates $(1, 3)$. Let $U = X \setminus \{ x\} $. Prove that there is a regular function on $U$ which is not the restriction of a regular function on $\mathbf{C}^2$, i.e., is not the restriction of a polynomial in $s$ and $t$ to $U$.

## 111.31 Regular functions

Exercise 111.31.2. Let $n \geq 2$. Let $E \subset \mathbf{C}^ n$ be a finite subset. Show that any regular function on $\mathbf{C}^ n \setminus E$ is a polynomial.

Exercise 111.31.3. Let $X \subset \mathbf{C}^ n$ be an affine variety. Let us say $X$ is a *cone* if $x = (a_1, \ldots , a_ n) \in X$ and $\lambda \in \mathbf{C}$ implies $(\lambda a_1, \ldots , \lambda a_ n) \in X$. Of course, if $\mathfrak p \subset \mathbf{C}[x_1, \ldots , x_ n]$ is a prime ideal generated by homogeneous polynomials in $x_1, \ldots , x_ n$, then the affine variety $X = V(\mathfrak p) \subset \mathbf{C}^ n$ is a cone. Show that conversely the prime ideal $\mathfrak p \subset \mathbf{C}[x_1, \ldots , x_ n]$ corresponding to a cone can be generated by homogeneous polynomials in $x_1, \ldots , x_ n$.

Exercise 111.31.4. Give an example of an affine variety $X \subset \mathbf{C}^ n$ which is a cone (see Exercise 111.31.3) and a regular function $f$ on $U = X \setminus \{ (0, \ldots , 0)\} $ which is not the restriction of a polynomial function on $\mathbf{C}^ n$.

Exercise 111.31.5. In this exercise we try to see what happens with regular functions over non-algebraically closed fields. Let $k$ be a field. Let $Z \subset k^ n$ be a Zariski locally closed subset, i.e., there exist ideals $I \subset J \subset k[x_1, \ldots , x_ n]$ such that

A function $\varphi : Z \to k$ is said to be *regular* if for every $z \in Z$ there exists a Zariski open neighbourhood $z \in U \subset Z$ and polynomials $f, g \in k[x_1, \ldots , x_ n]$ such that $g(u) \not= 0$ for all $u \in U$ and such that $\varphi (u) = f(u)/g(u)$ for all $u \in U$.

If $k = \bar k$ and $Z = k^ n$ show that regular functions are given by polynomials. (Only do this if you haven't seen this argument before.)

If $k$ is finite show that (a) every function $\varphi $ is regular, (b) the ring of regular functions is finite dimensional over $k$. (If you like you can take $Z = k^ n$ and even $n = 1$.)

If $k = \mathbf{R}$ give an example of a regular function on $Z = \mathbf{R}$ which is not given by a polynomial.

If $k = \mathbf{Q}_ p$ give an example of a regular function on $Z = \mathbf{Q}_ p$ which is not given by a polynomial.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)