\documentclass{amsart}
\newenvironment{emquote}{\begin{quote}\em}{\end{quote}}
\newcommand{\superset}{\supset}
\newcommand{\bbP}{{\mathbb P}}
\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\Proj}{Proj}
\DeclareMathOperator{\Ann}{Ann}
\DeclareMathOperator{\Fin}{{\bf Fin}}
\usepackage{hyperref}
\begin{document}
\title[Assignments]{The end of section assignments}
\author{Kapil Hari Paranjape}
\maketitle
\section{1, 2, 3, 4 \dots }
\dots bend down and touch the floor. I have collected a bunch of what I
think are instructive exercises.
\subsection{Finite Rings}
Give a structure theorem for finite rings of order $p^3$.
Is it possible to ``easily'' generalise this for $p^n$ for all $n$?
\subsection{Hensel's lemma}
Use Hensel's lemma to show that any matrix $A$ over a field of
characteristic 0 can be written as the sum $A=S+N$ with $S$ and $N$
commuting with each other, $N$ nilpotent and the minimal polynomial of
$S$ is separable (``has distinct roots''). Can you generalise this to
characteristic $p$?
\subsection{Affine schemes}
Show that there cannot be an isomorphism between
$\Spec(k[X,Y]/(X^2-Y^3))$ and $\Spec(k[T])$ by considering these as
functors on the category of finite dimensional $k$ algebras.
\subsection{Finite morphisms}
Find a finite morphism $\Spec(k[X,Y]/(XY(X+Y)-1))\to \Spec(k[T])$.
Your proof should work over any field $k$!
\subsection{Closed morphisms}
Show that the morphism $\Spec(k[X,Y])\to\Spec(k[X])$ is not closed.
\subsection{Hilbert basis theorem}
Consider the ideals defining circles with centres along points of the
$y$-axis. What is a finite basis for the ideal defining the intersection
of the corresponding closed subschemes?
\subsection{Primary decomposition}
Let $\Spec(R/I)$ be a closed subscheme of a scheme $\Spec(R)$. If $I$ is
contained in a component of $\Spec(R)$ then show that every element of
$I$ is a zero=divisor. What about the converse?
\subsection{Segre embedding}
Consider the natural transformations of functors
$\bbP^n\times\bbP^m\to\bbP^{nm+n+nm}$
which, for each finite ring $A$, has the form
\[
( (a_0,\dots, a_n); (b_0,\dots,b_m)) \mapsto
(a_ib_j)_{i=0,j=0}^{i=n,j=m}
\]
Show that this is a bijection onto the projective variety in
$\bbP^{nm+n+m}$ defined by the equations $Z_{ij}Z_{kl}=Z_{il}Z_{kj}$ as
$i$, $k$ (respectively $j$, $l$) vary from 0 to $n$ (respectively from 0
to m).
\subsection{Empty projective schemes}
Let $R$ be a finitel generated graded ring which is a finite over the
subring $R_0$ of elements of degree 0. Show that $\Proj(R)(A)$ is empty
for every finite ring $A$. What about the converse?
\subsection{Dimension theory}
Let $R$ be a graded $k$-algebra with $R_0=k$ such that the graded
dimension of $R$ is $r$. Let $k[Z_0,\dots,Z_r]\to R$ be a graded
homomorphism such that it induces a morphism $\Proj(R)\to\bbP^r$. Show
that the homomorpihsm is finite.
\end{document}