@@ -15,12 +15,12 @@ This gives rise to the definition of a functor.
\item$F (\id_X)=\id_{F X}$
\item$F (g \circ f)= Fg \circ Ff$
\end{enumerate}
the second condition is equivalent to saying that for any morphism $f: X \to Y$ in $\textsf{C}$, the following ``diagram commutes'' \footnote{This is technically not a diagram in the sense that we are drawing two categories ($\textsf{C}$ and $\textsf{D}$) next to eachother, but that is ignored here.}.
the second condition is equivalent to saying that for any pairs of composable morphisms: $(X \stackrel{f}{\to} Y), (Y \stackrel{g}{\to}Z)$ in $\textsf{C}$, the following diagram commutes
\begin{center}
\begin{tikzcd}[] %\arrow[bend right,swap]{dr}{F}
X \arrow[]{rr}{F}\arrow[]{d}{f}&& F X \arrow[]{d}{Ff}\\
Y \arrow[]{rr}{F}\arrow[]{d}{g}&& F Y \arrow[]{d}{Fg}\\
Z \arrow[]{rr}{F}&&F Z
F X \arrow[]{r}{Ff}\arrow[bend right]{rr}{F(g \circ f)}\arrow[bend left]{rr}{Fg \circ Ff}&
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
Insert definition of functors and natural transformations.
\end{center}
We will omit parenthesis for functor application unless necessary so instead of $F(X)$ or $F(f)$, we write $FX$ or $Ff$.
\begin{exr}[]
Let $\Vec_\C$ be the category of vector spaces over $\C$ and $\Set$ the category of sets.
Let $F$ be the forgetful functor $\Vec_\C\to\Set$.
\begin{enumerate}
\item Show that one can define a functor $G: \Set\to\Vec_\C$ by $G(X)=\C^{(X)}$ (the vector space with basis $X$, or the vector space of functions from $X$ to $\C$ which are zero for all but finitely many $x \in X$),
with $G(X \stackrel{f}{\to}Y)$ the linear map $\C^{(X)}\to\C^{(Y)}$ such that the basis vector $x \in X$ of $\C^{(X)}$ is mapped to the basis vector $f(x)\in Y$.
\item Show that for any set $X$ and vector space $V$, there is a bijective map of sets
\begin{align*}
\Hom_{\Vec_\C}(GX,V) \to\Hom_{\Set}(X,FV)
\end{align*}
\end{enumerate}
\end{exr}
\begin{proof}[Solution]
To avoid a clash of notation, we will sometimes enclose elements of $\C^{(X)}$ with square brackets to differentiate them from elements of $X$, so $x \in X$ and $[x]\in\C^{(X)}$.
\begin{enumerate}
\item We first check that it is well-defined and then show the functor laws.
For any set $X$, the image $G(X)=\C^{(X)}$ is -- by construction -- indeed a $\C$-vector space.
Linearity follows by definition as we defined $G(f)$ on a basis, so a general element $\bm{x}\in\C^{(X)}$ gets mapped as follows
\begin{align*}
\bm{x} = \sum_{x \in X}' \lambda_x x \stackrel{Gf}{\mapsto}\sum_{x \in X}' \lambda_x f(x) \quad\text{for}\quad\lambda_x \in X
\end{align*}
where the sum goes over finitely many $x \in X$.
By linearity, it is sufficient to check the functor laws on basis elements.
