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\centerline{\Kopfont Commutative Algebra}

\smallskip
\centerline{Excercises 1}

\bigskip\noindent
{\bf The Spectrum of a ring}\enspace [Reference: EGA I, Chapter 0 \S 2]

\smallskip\item{\bf -2.} Compute $\Spec \mZ$ as a set and describe its topology.

\smallskip\noindent
Let $A$ be any ring. Let $X$ be any topological space.

\smallskip\item{\bf -1.} For $f\in A$ we define $D(f):= \Spec A\setminus V(f)$. Prove that the open subsets $D(f)$ form a basis of the topology of $\Spec A$.

\smallskip\item{\bf 0.} Prove that the map $I\mapsto V(I)$
defines a natural bijection
$$\{\hbox{$I\subset A$ with $I=\sqrt{I}$}\}\longrightarrow \{\hbox{$T\subset \Spec A$
closed}\}$$

\smallskip\noindent
A topological space $X$ is called {\it quasi-compact}
if for any open covering $X=\bigcup_{i\in I} U_i$ there is a finite
subset $\{i_1,\ldots,i_n\}\subset I$ such that $X=U_{i_1}\cup\ldots
U_{i_n}$. 


\smallskip\item{\bf 1.} Prove that $\Spec A$ is quasi-compact for any ring $A$.

\smallskip\noindent
A topological space $X$ is said to verify the separation axiom $T_0$
if for any pair of points $x,y\in X$, $x\not=y$ there is an open
subset of $X$ containing one but not the other.
We say that $X$ is {\it Hausdorff} if for any pair $x,y\in X$, $x\not=y$
there are disjoint open subsets $U, V$ such that $x\in U$
and $y\in V$.

\smallskip\item{\bf 2.} Show that $\Spec A$ is {\bf not} Hausdorff in general.
Prove that $\Spec A$ is $T_0$. Give an example of a topological
space $X$ that is not $T_0$.

\smallskip\noindent
Remark: usually the word compact is reserved for quasi-compact and
Hausdorff spaces. A topological space $X$ is called {\it irreducible} if $X$ is not empty
and if $X=Z_1\cup Z_2$ with $Z_1, Z_2\subset X$ closed, then either
$Z_1=X$ or $Z_2=X$. A subset $T\subset X$ of a topological space
is called irreducible if it is an irreducible
topological space with the topology induced from $X$. 
This definitions implies $T$ is irreducible if and only 
if the closure $\bar T$ of
$T$ in $X$ is irreducible.

\smallskip\item{\bf 3.} Prove that $\Spec A$ is irreducible if and only if
$Nil(A)$ is a prime ideal and that in this case it is the unique
minimal prime ideal of $A$.

\smallskip\item{\bf 4.} Prove that a closed subset $T\subset \Spec A$
is irreducible if and only if it is of the form $T=V(\gp)$ for 
some prime ideal $\gp\subset A$.

\smallskip\noindent
A point $x$ of an irreducible topological space $X$ is called
a {\it generic point} of $X$ if $X$ is equal to the closure of 
the subset $\{x\}$. 

\smallskip\item{\bf 5.} Show that in a $T_0$ space $X$ every irreducible closed
subset has at most one generic point. 

\smallskip\item{\bf 6.} Prove that in $\Spec A$ every
irreducible closed subset {\it does} have a generic point.
In fact show that the map $\gp \mapsto \overline{\{\gp\}}$ is
a bijection of $\Spec A$ with the set of irreducible closed
subsets of $X$.

\smallskip\item{\bf 7.} Give an example to show that an irreducible
subset of $\Spec \mZ$ does not neccesarily have a generic point.

\smallskip\noindent
A topological space $X$ is called {\it Noetherian} if any
decreasing sequence $Z_1\supset Z_2 \supset Z_3\supset \ldots$
of closed subsets of $X$ stabilizes.
(It is called {\it Artinian} if any increasing sequence of closed
subsets stabilizes.)

\smallskip\item{\bf 8.} Show that if the ring $A$ is Noetherian then 
the topological space $\Spec A$ is Noetherian. Give an 
example to show that the converse is false. (The same for
Artinian if you like.)

\smallskip\noindent
A maximal irreducible subset $T\subset X$ is called an
{\it irreducible component} of the space $X$. Such an irreducible
component of $X$ is automatically a closed subset of $X$.

\smallskip\item{\bf 9.} Prove that any irreducible subset
of $X$ is contained in an irreducible component of $X$.

\smallskip\item{\bf 10.} Prove that a Noetherian topological space $X$
has only finitely many irreducible components, say $X_1,\ldots, X_n$,
and that $X=X_1\cup X_2\cup\ldots\cup X_n$. (Note that
any $X$ is always the union of its irreducible components, but that
if $X=\mR$ with its usual topology for instance then the irreducible
components of $X$ are the one point subsets. This is not
terribly interesting.)

\smallskip\item{\bf 11.} Show that irreducible components of $\Spec A$
correspond to minimal primes of $A$.

\smallskip\noindent
A point $x\in X$ is called closed if $\overline{\{x\}}=\{ x\}$.
Let $x,y$ be points of $X$. We say that $x$ is a {\it specialization}
of $y$, or that $y$ is a {\it generalization} of $x$ if
$x\in \overline{\{y\}}$.

\smallskip\item{\bf 12.} Show that closed points of $\Spec A$ 
correspond to maximal ideals of $A$.

\smallskip\item{\bf 13.} Show that $\gp$ is a generalization of $\gq$ 
in $\Spec A$ if and only if $\gp\subset \gq$. Characterize closed points,
maximal ideals, generic points and minimal prime ideals in terms of
generalization and specialization. (Here we use the terminology that a point
of a possibly reducible topological space $X$ is called a generic point
if it is a generic points of one of the irreducible components of $X$.)

\smallskip\item{\bf 14.} Let $I$ and $J$ be ideals of $A$.
What is the condition for $V(I)$ and $V(J)$ to be disjoint?

\smallskip\noindent
A topological space $X$ is called {\it connected} if it is not the union
of two nonempty disjoint open subsets. A {\it connected component}
of $X$ is a (nonempty) maximal connected subset. Any point of $X$ is contained
in a connected component of $X$ and any connected component of $X$ is
closed in $X$. (But in general a connected component need not be open in $X$.)

\smallskip\item{\bf 15.} Show that $\Spec A$ is disconnected
iff $A\cong B\times C$ for certain nonzero rings $B,C$.

\smallskip\item{\bf 16.} Let $T$ be a connected component
of $\Spec A$. Prove that $T$ is stable under generalization.
Prove that $T$ is an open subset of $\Spec A$ if $A$ is Noetherian.
(Remark: This is wrong when $A$ is an infinite product of copies of
${\bbb F}_2$ for example. The spectrum of this ring consists of infinitely
many closed points.)

\smallskip\item{\bf 17.} Compute $\Spec k[x]$, i.e., describe
the prime ideals in this ring, describe the possible specializations, and
describe the topology. (Work this out when $k$ is algebraically closed but
also when $k$ is not.)

\smallskip\item{\bf 18.} Compute $\Spec k[x,y]$, where $k$ is algebraically
closed. [Hint: use the morphism $\varphi: \Spec k[x,y]\to\Spec k[x]$;
if $\varphi(\gp)=(0)$ then localize with respect to $S=\{f\in k[x]
\mid f\not=0\}$ and use result of lecture on localization and $\Spec$.]
(Why do you think algebraic geometers call this affine 2-space?)

\smallskip\item{\bf 19.} Compute $\Spec \mZ[y]$. 
[Hint: as above.] (Affine 1-space over $\mZ$.)

\bye
