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%	TITLE PAGE
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\title[Spectra of commutative rings]{\textsc{Schemes on the horizon} \\ Spectra of commutative rings} % The short title appears at the bottom of every slide, the full title is only on the title page

\author{Michael L. Baker} % Your name
\institute[Pure Mathematics] % Your institution as it will appear on the bottom of every slide, may be shorthand to save space
{
University of Waterloo \\ % Your institution for the title page
\medskip
\textit{mlbaker.org} % Your email address
}
\date{June 4, 2013} % Date, can be changed to a custom date

\begin{document}

\begin{frame}
\titlepage % Print the title page as the first slide
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\begin{frame}
\frametitle{Overview} % Table of contents slide, comment this block out to remove it
\tableofcontents % Throughout your presentation, if you choose to use \section{} and \subsection{} commands, these will automatically be printed on this slide as an overview of your presentation
\end{frame}

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%	PRESENTATION SLIDES
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\section{Motivation}

\begin{frame}%[fragile]
\frametitle{Motivation}
To any affine variety $X$ we associated the coordinate ring $\Gamma(X) = A(X)$. $A(X)$ is always a rather special kind of ring: a finitely generated, reduced $k$-algebra. We now seek to ``go the other way'': to any commutative ring $A$, we wish to associate some kind of geometric object. These objects, which will generalise varieties, are called ``affine schemes''. \\~\\ Recall that ideals of $A(X)$ correspond to ideals of $A$ above $I(X)$. Under the hypothesis of algebraic closure, the Nullstellensatz tells us that maximal ideals of $A(X)$ correspond \emph{exactly} to points of $X$. This suggests one can recover the variety as the set of maximal ideals of $A(X)$.
\end{frame}

\begin{frame}
\frametitle{Motivation}
It turns out to be more fruitful (for example, from a functorial standpoint) to consider the larger class of \emph{prime} ideals. \\~\\ To elaborate, suppose we had a ring homomorphism $f : A \to A'$. Then if we defined $\text{Spec } A$ to be the set of all \emph{maximal ideals} of $A$, we wouldn't be able to get a reasonable map $\text{Spec } f : \text{Spec } A' \to \text{Spec } A$: your first guess is to take a maximal ideal $\mathcal{M} \in \text{Spec } A'$ and ``pull it back'' along $f$, that is, to define $(\text{Spec } f)(\mathcal{M}) = f^{-1}(\mathcal{M})$. However we may not always have $f^{-1}(\mathcal{M}) \in \text{Spec } A$: it is in general \emph{false} that preimages of maximal ideals (under ring homomorphisms) are maximal! \\~\\Our choice to consider \emph{prime} ideals means we will have some ``extra elements'' which correspond not to points of $X$, but rather to subvarieties. These may seem unsettling, but they will turn out to be rather convenient.
\end{frame}

\section{Underlying set}

\begin{frame}
\frametitle{The set $\text{Spec } A$}

Let $A$ be a commutative ring.

\begin{block}{Definition}
Denote by $\text{Spec } A$ the set of all prime ideals $\mathcal{P} \subseteq A$.
\end{block}

We observe that each $f \in A$ gives rise to a ``function'' on $\text{Spec } A$: given $\mathcal{P} \in \text{Spec } A$, we write $f(\mathcal{P})$ for the image of $f$ under the natural map
\[ A \to A/\mathcal{P} \to K(A/\mathcal{P}) \]
noting that $A/\mathcal{P}$ is an integral domain, so it makes sense to talk about its field of fractions $K(A/\mathcal{P})$. 

\begin{block}{Example}
A good example to keep in mind is when $A=\Gamma(X)=A(X)$ is the coordinate ring of some affine variety $X$ over an algebraically closed field. Then $\text{Spec } A$ is basically just $X$ (plus a few extra points corresponding to subvarieties), and each $f \in \Gamma(X)$ is viewed as a ``function'' on $X$.
\end{block}
\end{frame}

\begin{frame}
\frametitle{Examples of $\text{Spec } A$}

\begin{block}{Example}
\begin{center}
\setlength{\fboxsep}{0px}
\setlength{\fboxrule}{0px}
\fbox{\includegraphics[width=300px]{specZ.jpg}}

$\text{Spec } \mathbb{Z}$
\end{center}
We have one point $(p)$ for each prime $p \in \mathbb{Z}$ in addition to an extra point $(0)$, usually called the \emph{generic point}.
\end{block}

It is common to scale the points $\mathcal{P}$ of $\text{Spec } A$ according to the size of the \emph{residue field}, $\kappa(\mathcal{P}) = K(A/\mathcal{P})$. You may also see a point $\mathcal{P}$ drawn as ``smudged'', as above, to suggest that (\emph{geometrically} speaking!) it contains other points in the diagram. \\~\\ We now give some examples of $\text{Spec } A$ for $A = \mathbb{Z}$, $\mathbb{C}[x]$, $\mathbb{Z}[x]$. Another interesting example is the ring of integers of an algebraic number field.
\end{frame}

\begin{frame}
\frametitle{Examples of $\text{Spec } A$}

\begin{block}{Example}
\begin{center}
\setlength{\fboxsep}{0px}
\setlength{\fboxrule}{0px}
\fbox{\includegraphics[width=210px]{Speccx.jpg}}

$\text{Spec } \mathbb{C}[x]$
\end{center}
$\mathbb{C}$ is algebraically closed. Also, $k[x]$ is a PID when $k$ is a field, so all nonzero primes are maximal: its Krull dimension is 1. Thus $\text{Spec } \mathbb{C}[x]$ is just a ``complex line'' of maximal ideals sitting above the prime $(0)$.
\end{block}
\end{frame}

\begin{frame}
\frametitle{Examples of $\text{Spec } A$}

\begin{block}{Example}
\begin{center}
\setlength{\fboxsep}{0px}
\setlength{\fboxrule}{0px}
\fbox{\includegraphics[width=200px]{mumford-21.jpg}}

Mumford's drawing of $\text{Spec } \mathbb{Z}[x]$, the \textbf{arithmetic surface}
\end{center}
The Krull dimension of $\mathbb{Z}[x]$ is 2.
\end{block}
\end{frame}

\begin{frame}
\frametitle{Examples of $\text{Spec } A$}

\begin{block}{Example (``polynomial remainder theorem'' from high school)}
Let $A = \mathbb{C}[x]$ and $p \in \mathbb{C}[x]$. If $\alpha \in \mathbb{C}$, then $(x-\alpha)$ is prime in $\mathbb{C}[x]$, and one can identify $\kappa((x-\alpha))$ with $\mathbb{C}$ naturally, so that the value of $p$ at the point $(x-\alpha) \in \text{Spec } \mathbb{C}[x]$ is the number $p(\alpha)$.
\end{block}

\begin{block}{Example}
More generally, if $A$ is the coordinate ring of an affine variety $X$ over an algebraically closed field $k$, and $\mathcal{P}$ is the maximal ideal corresponding to $x \in X$, then $\kappa(\mathcal{P}) = k$ and $f(\mathcal{P})$ is the value of $f$ at $x$ in the usual sense.
\end{block}

Note that $f \in A$ takes values in fields that \emph{vary} from point to point. Moreover, $f$ is not determined by its values: take the ``$\mathbb{C}$-dual numbers'' $A = \mathbb{C}[x]/(x^2)$ and take $f=x \in A$. Then $0 \neq f$ vanishes on all of $\text{Spec } A$.

\begin{block}{Definition}
Elements of $A$ are called \textbf{regular functions} on $\text{Spec } A$.
\end{block}
\end{frame}

\section{Topology}

\begin{frame}
\frametitle{Topology on $\text{Spec } A$}
An element $f \in A$ such that $f \in \mathcal{P}$ corresponds to a ``function'' that is zero on $\mathcal{P}$. We now use regular functions to define a topology on $\text{Spec } A$. Note the obvious analogue with the ``zero sets'' of the classical case. \\~\\ The idea is that we want to make each $f \in A$ behave like a continuous function (even though this doesn't make sense). The fields $K(A/\mathcal{P})$ certainly all contain a 0; if $f$ is to be ``continuous'' then $f^{-1}( \{ 0 \} )$ should definitely be a closed set.
\begin{block}{Definition}
For each $S \subset A$, we define
\begin{align*}
V(S) & = \{ x \in \text{Spec } A : f(x) = 0\text{ for all } f \in S \} \\
& = \{ \mathcal{P} \in \text{Spec } A : \mathcal{P} \supset S \}.
\end{align*}
\end{block}

Of course, $V(S) = V(\mathcal{I})$ if $\mathcal{I}$ is the ideal generated by $S$ in $A$.
\end{frame}

\begin{frame}
\frametitle{Topology on $\text{Spec } A$}
The following proposition from Hartshorne assures us that the sets of the form $V(\mathcal{I})$ do indeed constitute the closed sets of a topology on $\text{Spec } A$.

\begin{block}{Proposition}
\begin{enumerate}
\item[(a)] $V(\mathcal{IJ}) = V(\mathcal{I}) \cup V(\mathcal{J})$ (closure under finite unions).
\item[(b)] $V(\sum \mathcal{I}_\alpha) = \bigcap V(\mathcal{I}_\alpha)$ (closure under arbitrary intersections).
\item[(c)] $V(\mathcal{I}) \subseteq V(\mathcal{J})$ if and only if $\sqrt{\mathcal{I}} \supseteq \sqrt{\mathcal{J}}$.
\end{enumerate}
\end{block}

\begin{block}{Proof}
\begin{enumerate}
\item[(a)] If $\mathcal{P} \supseteq \mathcal{I}$ or $\mathcal{P} \supseteq \mathcal{J}$, then clearly $\mathcal{P} \supseteq \mathcal{IJ}$. Conversely if $\mathcal{P} \supseteq \mathcal{IJ}$ but $\mathcal{P} \not\supseteq \mathcal{J}$, then $\exists b \in \mathcal{J}$ with $b \notin \mathcal{P}$. Now for any $a \in \mathcal{I}$ we have that $ab \in \mathcal{P}$ and therefore $a \in \mathcal{P}$ by primality. We conclude $\mathcal{P} \supseteq \mathcal{I}$.
\item[(b)] $\mathcal{P} \supseteq \sum \mathcal{I}_\alpha$ if and only if $\mathcal{P} \supseteq \mathcal{I}_\alpha$ for all $\alpha$, simply because $\sum \mathcal{I}_\alpha$ is the smallest ideal containing all the ideals $\mathcal{I}_\alpha$.
\item[(c)] Note that $\sqrt{\mathcal{I}}$ is the intersection of all prime ideals containing $\mathcal{I}$.
\end{enumerate}
\end{block}
\end{frame}

\section{Structure sheaf}

\begin{frame}
\frametitle{Sheaf of rings on $\text{Spec } A$}
The final piece of data carried by $\text{Spec } A$ is its sheaf of rings (``structure sheaf''), which we now define. Recall $A_{\mathcal{P}}$ denotes the localisation of $A$ at $\mathcal{P}$, that is, $(A \setminus \mathcal{P})^{-1} A$. 
\begin{block}{Definition}For each open $U \subseteq \text{Spec } A$, we define $\mathcal{O}(U)$ to be the set of functions $s : U \to \bigsqcup_{\mathcal{P} \in U} A_{\mathcal{P}}$ such that $s(\mathcal{P}) \in A_{\mathcal{P}}$ for each $\mathcal{P}$, and $s$ is ``locally a quotient of elements of $A$'', that is, for each $\mathcal{P} \in U$, there is a neighbourhood $V$ of $\mathcal{P}$, with $V \subseteq U$, and $a,f \in A$ such that for each $\mathcal{Q} \in V$, $f \notin \mathcal{Q}$, and $s(\mathcal{Q}) = a/f$ in $A_{\mathcal{Q}}$. \end{block} This is similar to the definition of regular functions on a variety, except instead of considering functions to a field, we look at functions into the various local rings.
\end{frame}

\begin{frame}
\frametitle{Sheaf of rings on $\text{Spec } A$}
Note that $\mathcal{O}(U)$ is a commutative ring with 1. With the natural restriction maps $\mathcal{O}(U) \to \mathcal{O}(V)$, $\mathcal{O}$ becomes a sheaf on $\text{Spec } A$.

\begin{block}{Definition}
$\mathcal{O}$ is called the \textbf{structure sheaf} (or the \textbf{sheaf of regular functions}) on $X = \text{Spec } A$.
\end{block}
With this definition, the relationship between $\text{Spec } A$ and $A$ generalises that between an affine variety and its coordinate ring. In particular, we will see that the ring $\Gamma(\text{Spec } A, \mathcal{O})$ of global sections of $\mathcal{O}$ is precisely $A$.
\end{frame}

\begin{frame}
\frametitle{The spectrum of $A$}
\begin{block}{Definition}
The \textbf{spectrum} of $A$ is the pair $(\text{Spec } A, \mathcal{O})$.
\end{block}

The open sets in the topology on $X = \text{Spec } A$ are just the complements of sets $V(S)$. The ones corresponding to \emph{singletons} $S$ will play a special role: they form a \emph{base} for the topology, and they are again spectra of rings.

\begin{block}{Definition}
For any $f \in A$, let $X_f = D(f)$ be the complement of $V(f)$. This is called the \textbf{distinguished} (or \textbf{basic}) \textbf{open subset associated with $f$}.
\end{block}

The points of $X_f$ (prime ideals of $A$ not containing $f$) are in bijection with the prime ideals of $A_f$ via sending $\mathcal{P} \subset A$ to $\mathcal{P}A_f \subset A_f$. Hence we identify $X_f$ with the points of $\text{Spec } A_f$. \\~\\An ``affine scheme'' will be defined as something isomorphic to a spectrum.
\end{frame}

\begin{frame}
\frametitle{Sheaf of rings on $\text{Spec } A$}
The following proposition gives us some information about the structure sheaf $\mathcal{O}$ on $\text{Spec } A$.

\begin{block}{Proposition (2.2 in Hartshorne)}
Let $A$ be a ring, and let $(\text{Spec } A, \mathcal{O})$ be its spectrum.
\begin{enumerate}
\item[(a)] For any $\mathcal{P} \in \text{Spec } A$, the stalk $\mathcal{O}_{\mathcal{P}}$ of the sheaf $\mathcal{O}$ is isomorphic to the local ring $A_{\mathcal{P}}$.
\item[(b)] For any element $f \in A$, the ring $\mathcal{O}(D(f))$ is isomorphic to the localisation $A_f$.
\item[(c)] In particular, $\Gamma(\text{Spec } A, \mathcal{O}) \cong A$.
\end{enumerate}
\end{block}

\begin{block}{Proof}
Long. See Hartshorne, p.\ 71.
\end{block}
\end{frame}

\section{Functoriality of $\text{Spec}$}
\begin{frame}
\frametitle{Ringed spaces}
To each commutative ring $A$, we have now associated its spectrum $(\text{Spec } A, \mathcal{O})$. We want to get a functor out of this, so we need a suitable category of spaces with sheaves of rings on them. The appropriate notion is that of a \emph{locally ringed space}, which will be treated next time. We conclude with the following preliminary. \\~\\ Recall from last week the concept of a \emph{direct image sheaf} $f_* \mathcal{F}$.

\begin{block}{Definition}
A \textbf{ringed space} is a pair $(X, \mathcal{O}_X)$ consisting of a topological space $X$ and a sheaf of rings $\mathcal{O}_X$ on it. A \textbf{morphism} of ringed spaces 
\[ (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y) \] is a pair $(f, f^{\#})$ consisting of a continuous map $f : X \to Y$ and a morphism $f^{\#} : \mathcal{O}_Y \to f_* \mathcal{O}_X$ of sheaves of rings on $Y$.
\end{block}
\end{frame}
\end{document} 