This file contains general information about the lecture notes aswell as the configuration for my Commutative Algebra notes.

** Structure

This file is an Emacs Org Mode file which is edited and then /tangled/ with the ~header.tex~ file, which I do not touch manually.

The benefit of this is that comments, which would have been written in plaintext inside ~header.tex~ can now be written using Org Mode Formatting, allowing for literate configuration.

For example, I can use headings, use formatted text, insert tables, source code blocks or even images, which would not be possible in LaTeX comments.

I structure my notes as follows:

Instead of a ~main.tex~ file, I have a ~main.org~ file which I then export to latex via ~M-x org-export-dispatch l o~.

Since I have just started using Emacs, my configuration does not quite allow me to take notes in real-time.

This means I still take my notes in Vim by writing ~.tex~ files and putting them in the ~./files~ directory, from where they will be sourced and can be compiled using my existing vim-tex setup.

** How to use

After changing the configuration, place the point at the top of this document (~g g~ in evil-normal mode) and do ~C-c C-c~ to refresh. Then execute ~M-x org-babel-tangle~, which will update ~header.tex~.

It should output ~Tangled n code blocks from README.org~

To use source code blocks without having them exported to ~header.tex~, use the ~:tangle no~ option.

** TODO

- I plan on trying out a language-agnostic solution where I can export this document to ~.html~ format and upload it on my webpage.

- As of now, I use different configuration files for every lecture. This means that there is alot of code redundancy as the configuration only differs by about 5% (if at all) between different lecture notes.

But since the README.org files are only a couple kilobytes in size each, I should be fine.

The lecture content is mostly guided by Antoine Chamber-Loir's Book (Mostly) Commutative Algebra.

We will not follow it one-to-one, but chapter numbers will be provided on the go.

There will be some extra topics covered that are not in the book.

Definitions and theorems enclosed with parenthesis are not part of the lecture and (mostly) come from the exercise sheets.

\section{Overview}

\subsection{Introduction/Motivation}

\subsubsection*{What is it?}

Commutative Algebra is the study of commutative Rings and related objects such as ideals, modules etc.

We will use the convention that rings, (unless specified otherwise) are all commutative with unit.

We will also require that ring morphisms preserve the unit.

In particular, $2\Z\subseteq\Z$ is not a ring and $\Z\to\Z, x \mapsto0$ is not a morphism of rings.

\subsubsection*{Why study this?}

One reason to study commutative algebra is that it is the ``local'' side of algebraic geometry, which is the study of geometric objects such as solution sets of systems of polynomial equations.

One reason to study those is that they are a natural generalisation of linear algebra.

However there are other good motivations, such as the tensor product which proves itself as a universally useful construction.

\subsubsection*{What are some results?}

\begin{thm}[Hilbert/Noether]

Let $n \geq1$ be an integer.

Let $(P_i)_{i \in I}$ be a family of polynomials $P_i \in\C[X_{1}, \ldots, X_{n}]$ with $I$ arbitrary.

There exist finitely many polynomials $Q_{1}, \ldots, Q_{l}\in\C[X_{1}, \ldots, X_{n}$ such that both families shares the same solution set i.e.

An example of such a set would be the set of odd numbers or the numbers with remainder $12$ modulo $17$.

The fibonnaci sequence is such an example as it satisfies $u_{n+1}- u_{n+1}- u_{n}=0$.

\subsection{Some quick guidelines}

There are many definitions that at by themselves aren't hard to understand, but often seem unmotivated at first sight.

However that is not the case. Many of the concepts we introduce are motivated by the following:

\begin{center}

We want to study more complicated objects using properties of simple ones we understand well.

In the case of Commutative Algebra, we understand linear algebra quite well. For example we know what a dimension of a kernel of a linear map is.

The generalisation is the Krull dimension.

The second object we understand well are PID's such as $K[X]$ or $\Z$.

\end{center}

\subsection{Reminders and Notation}

\begin{itemize}

\item The natural numbers should index the possible dimensions of vector spaces. Since $\{0\}$ is a vector space. $\N$ should contain $0$. $\N=\{0,1,2,\ldots\}$

\item For a ring $R$, we denote the group of units by $R^{\times}$.

\item An element $x \in R$ is \textbf{nilpotent}, if $\exists n \in\N: x^{n}=0$

\item A ring called \textbf{reduced}, if it has no non-zero nilpotent elements.

\item An $R$-module $M$ is an abelian group $(M,+)$ with ring-multiplication

\begin{align*}

R \times M \to M, (x,m) \mapsto x \cdot m

\end{align*}

such that $1_R \cdot m = m, 0_R \cdot m =0_M, x \cdot0_M =0_M$.

With distributivity and associativity.

\item An ideal $I \subseteq R$ is an $R$-submodule of $R$.

From this point of view, it is more flexible to study modules instead of ideals.

\item Let $I \subseteq R$ be an ideal. The following are equivalent

\begin{itemize}

\item$I$ is \textbf{prime}.

\item$I \neq R$ and $R/I$ is an inegral domain

\item$I \neq R$ and $rs \in I \implies$$r \in I$ or $s \in I$.

\end{itemize}

\item The following are equivalent:

\begin{itemize}

\item$I \subseteq R$ is \textbf{maximal}

\item$R/I$ is a field.

\item$I \neq R$ and there is no ideal other than $R$ and $I$ itself that contains $I$.

\end{itemize}

\end{itemize}

\subsubsection*{Facts}

\begin{itemize}

\item If $f: R \to S$ is a ring morphism, $P \subseteq S$ is a prime ideal, then $f^{-1}(P)$ is also prime.

To prove it, all we need is to see that $f$ induces an injective map $\faktor{R}{f^{-1}(P)}\stackrel{f}{\hookrightarrow}\faktor{S}{P}$.

This fact is false for maximal ideals. Consider the inclusion map $f: \Z\hookrightarrow\Q$.

As $\Q$ is a field $\{0\}$ is a maximal ideal, but $f^{-1}(\{0\})=\{0\}$ is not.

\item Krull: Let $I \neq R$ be an ideal of $R$. Then there exists a maximal ideal containing $I$. (See [ACL p. 57]

\end{itemize}

\begin{dfn}[Algebra]

Let $R$ be a ring.

An \textbf{algebra over $R$} (or $R$-algebra) is a ring-morphism $\phi: R \to S$.

Often, we just say that $S$ is an $R$-algebra and call $\phi$ the \emph{structure morphism} of the algebra.

If $R \stackrel{\psi}{\to}T$ is another $R$-algebra, a morphism of $R$-algebras is a ring homomorphism $f: S \to T$ such that the following diagram commmutes

\begin{center}

\begin{tikzcd}[column sep=0.8em]

S \arrow[]{rr}{f}&& T\\

& R \arrow[]{ul}{\phi}\arrow[swap]{ur}{\psi}

\end{tikzcd}

\end{center}

\end{dfn}

In practice: We can multiply elements of $S$ by those of $R$ by writing

\begin{align*}

r \cdot s := \phi(r) s

\end{align*}

using this notion, then $f: S \to T$ is an $R$-algebra morphism if and only if

What this means in practice is that if some theorem holds for algebras, then it also holds for rings.

\item If $R \subseteq R$ is a subring, then $S$ is an $R$-algebra, where the structure morphism is the inclusion mapping.

In particular, $\C$ is an $\R$-algebra and a $\Q$-algebra. More generally, if $L/K$ is a field extension, then $L$ is a $K$-algebra.

\item Warning! On a given ring $S$, there may be more than on $R$-algebra structure on $S$ (structure morphisms $R \to S$)

For example $\C$ can be seen as $\C$-algebra with the following strucutre morphisms:

\begin{align*}

z \cdot w = zw (\phi(z) = z), \quad z \cdot w = \overline{z} (\phi(z) =\overline{z})

\end{align*}

more generally, every automorphism defines a structure morphism.

\end{enumerate}

\end{xmp}

\section{The language of Categories and Functors [ACL A.3]}

When looking at common features ``mathematical strucutres'' such as Groups, Fields, Top. Spaces, Hilbert spaces, etc., one finds that it is very hard to define them using the common method of specifying properties of subsets of sets, or sets of subsets of sets etc.

The Eilenberg-MacLane notion of a category is as follows:

\begin{dfn}[Category]

A category $\textsf{C}$ consists of the following data:

\begin{itemize}

\item A ``collection'' of \textbf{objects}$X$ of $\textsf{C}$.

\item For any objects $X,Y$ of $\textsf{C}$, we define a \emph{set}$\Hom_{\textsf{C}}(X,Y)$ of \textbf{morphisms} from $X$ to $Y$ in $\textsf{C}$.

Instead of writing $f \in\Hom(X,Y)$, we will usually write $X \stackrel{f}{\to}Y$ or $f: X \to Y$.

\item For every object $X$ of $\textsf{C}$, there exists an \textbf{identity-morphism}$\id_X: X \to X$.

\item Morphisms can be composed. For objects $X,Y,Z$, there exists a map

We are intentionally being obtuse when talking about ``collection'' of objects.

The reason is that we want to talk about the ``collection'' of \emph{all} sets.

Using the normal set theory we know from the first semester, we quickly run into problems as there is no such thing as a set of all sets.

There are a number of solutions to this though.

\begin{itemize}

\item There are set theories that make use of \emph{classes} to enlarge the notion of set.

\item Restrict what sets we are allowed to talk about. For example using \emph{Grothendieck universes}.

\item Ignore the problem.

\end{itemize}

We will chose the third option.

If you still feel unfomfortable, one can say that there is some logical formuale $\Phi_{\textsf{C}}(X)$ in one variable, which evaluates to \texttt{true}, if $X$ is an object in the category $\textsf{C}$.

\end{rem}

\begin{xmp}[]

When we say that something is a category, we usually specify what the objects are and what the morphisms (and composition) look like.

Feel free to skip those you are not familiar with.

\begin{itemize}

\item The category of sets $\Set$, where the objects are sets and morphisms are functions.

\item The category $\Grp$, where the objects are groups and morphisms are group homomorphisms.

\item$\Top$ the category of topological spaces, where the objects are topological spaces (with their topology) and where the morphisms are \emph{continuous} functions.

\item$\Top_{\ast}$, with Topological spaces with a basepoint $(X,\tau,x_0)$ as objects and where the morphisms are continous, base-point preserving functions.

\item$\Ring$ with Rings as objects and ring homomorphisms as morphisms.

\item Given a fixed ring $R$, the category $R-\textsf{Mod}$ of $R$-modules and linear maps.

\item\textsf{Graph} has graphs as objects and graph homomorphisms as morphisms.

\item For a fixed field $K$, the category $\textsf{Vec}_K$ has $K$-vector spaces as objects with linear maps as morphisms.

\item One rather weird category is the category of fields with field homomomorphisms.

\end{itemize}

The examples above are categories were objects were \emph{sets with structure} and the morphisms were \emph{structure preserving} maps.

But the objects need not be sets, and morphisms need not be functions!

\begin{itemize}

\item The trivial category, which consists of a single object $\ast$ and its identity morphism $\ast\stackrel{\id}{\to}\ast$.

\item A group (or monoid) $(G,\cdot,e)$ can be viewed as a category with a single object $\ast$, where the morphisms $g: \ast\to\ast$ are the elements $g \in G$ and where composition of morphisms is defined as the multiplication in $G$: $g \circ h := g \cdot h$.

\item A poset $(P,\leq)$ is a category where the objects are the elements of $P$ and there exist unique morphisms $x \to y$ if and only if $x \leq y$.

\item\textsf{Hask}, the idealized cateogory of Haskell types and functions.

\item Given a category $\textsf{C}$, we can talk about its \textbf{arrow category}, where the objects are morphisms $(X \stackrel{f}{\to} Y)$ in $\textsf{C}$ and a morphism between objects $(X \stackrel{f}{\to} Y), (X' \stackrel{g}{\to}Y')$ are pairs of $\textsf{C}$-morphisms $\alpha: X \to X'$ and $\beta: Y \to Y'$ such that the following diagram commutes:

\begin{center}

\begin{tikzcd}[ ] %\arrow[bend right,swap]{dr}{F}

X \arrow[]{r}{f}\arrow[]{d}{\alpha}& Y \arrow[]{d}{\beta}\\

X' \arrow[]{r}{g}& Y'

\end{tikzcd}

\end{center}

i.e. such that $g \circ\alpha=\beta\circ f$.

\end{itemize}

\end{xmp}

In general, always specify what category you are working in.

The real numbers $\R$ are a Set, Topological space, Group, Field, Poset etc.

\begin{rem}[]

We say that a triangle

\begin{tikzcd}[column sep=0.8em]

&\cdot\arrow[]{dr}{h}\\

\cdot\arrow[]{ur}{g}

\arrow[]{rr}{f}

&&\cdot

\end{tikzcd}

\textbf{commutes}, if $g \circ h = f$.

A diagram is commutative, if every triangle of that diagram commutes.

This includes non-obvious triangles, such as those formed by morphisms obtained through composition or the identity morphisms.