New exercises for commutative algebra course

exercises.tex

@@ -2509,6 +2509,46 @@ \section{Filtered derived category}
 \section{Regular functions}
 \label{section-regular-functions}
 
+\begin{exercise}
+\label{exercise-extra-function}
+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 $U$.
+\end{exercise}
+
+\begin{exercise}
+\label{exercise-no-extra-function}
+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.
+\end{exercise}
+
+\begin{exercise}
+\label{exercise-cone}
+Let $X \subset \mathbf{C}^n$ be an affine variety. Let us say $C$ is a
+{\it 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 $x_1, \ldots, x_n$.
+\end{exercise}
+
+\begin{exercise}
+\label{exercise-extra-function-cone}
+Give an example of am affine variety $X \subset \mathbf{C}^n$
+which is a cone (see Exercise \ref{exercise-cone})
+and a regular function $f$ on $U = X \setminus \{(0, \ldots, 0)\}$
+which is not the restruction of a polynomial
+function on $\mathbf{C}^n$.
+\end{exercise}
 
 \begin{exercise}
 \label{exercise-regular-functions}

tags/tags

@@ -17843,3 +17843,7 @@
 0E9A,moduli-curves-lemma-stable-separated
 0E9B,moduli-curves-lemma-stable-quasi-compact
 0E9C,moduli-curves-theorem-stable-smooth-proper
+0E9D,exercises-exercise-extra-function
+0E9E,exercises-exercise-no-extra-function
+0E9F,exercises-exercise-cone
+0E9G,exercises-exercise-extra-function-cone