prelim.tex

\documentclass[ag.tex]{subfiles}

\begin{document}

\chapter{Preliminaries}

\section{Commutative algebra}

We begin with some algebraic prerequisites.  For more see \cite{atiyah2018introduction}.

\begin{definition}
A \textit{commutative ring} $(R, +, \cdot, 0,1)$ is an abelian group $(R,+,0)$ along with an associative binary operator $\cdot$ and a unit element $1$ such that for all $x,y,z \in R$

\begin{enumerate}
\item $x \cdot y = y \cdot x$,
\item $x \cdot 1 = 1$, and
\item $x\cdot(y+z) = x\cdot y+x\cdot z$

\end{enumerate}

\end{definition}

We call a commutative ring $(R, +, \cdot , 0,1)$ simply a \textit{ring} and write it as $R$ when there is no ambiguity. Also we denote $x \cdot y$ by $xy$.

\begin{definition}
An \textit{ideal} of a ring $R$ is a subset $I \subset R$ that is closed under addition ($+$) and "absorbs" multiplication from elements of $R$: for all $x \in R$ and $y \in I$,  $xy \in I$.
\end{definition}

\begin{proposition}\label{primes_map_to_primes}
The images and preimages of a prime ideal under ring homomorphisms are prime.
\end{proposition}

\section{Categories and functors}

\begin{definition}
A \textit{category} $\mathcal{C}$ consists of a collection of \textit{objects}, a set $\mathcal{C}(A,B)$ of \textit{morphisms} between any two objects, an \textit{identity} morphism $id_A \in \mathcal{C}(A,A)$ for each object $A$, and a composition law

\begin{equation}
\circ : \mathcal{C} ( B , C ) \times \mathcal{C} ( A , B ) \to \mathcal{C} ( A , C )
\end{equation}

for each triple of objects $A, B, C$.  Composition must be associative, and identity
morphisms must behave as their names indicate: $h \circ (g\circ f)=(h\circ g)\circ f, id\circ f=f$, and $f\circ id=f$
whenever the the composites are defined.
\end{definition}

\begin{definition}\label{terminal object}
A \textit{terminal object} of a category $\mathcal{C}$ is an object $T$ to which there is a unique morphism from each object of $\mathcal{C}$.
\end{definition}

\begin{definition}
A \textit{functor} $F : \mathcal{C} \to \mathcal{D}$ assigns an object $F(A)$ of $\mathcal{D}$ to each object $A$ of $\mathcal{C}$ and a morphism $F(f) : F(A) \to F(B)$ of $\mathcal{D}$ to each morphism $f:A\to B$ of $\mathcal{C}$ such that 

\begin{equation}
F(id_A) = id_{F(A)} \text{ and } F(g \circ f) = F(g) \circ F(f).
\end{equation}

\begin{definition}
A \textit{natural transformation} $\alpha : F \to G$ between functors $F, G: \mathcal{C} \to \mathcal{D}$ consists of a morphism $\alpha_A : F(A) \to G(A)$ for each object $A$ of $\mathcal{C}$ such that the following diagram commutes for each morphism $f : A \to B$ of $\mathcal{C}$:

\begin{center}
\begin{tikzcd}
F(A) \arrow[r, "F(f)"] \arrow[d, "\alpha_A"]
& F(B) \arrow[d, "\alpha_B"] \\
G(A) \arrow[r, "G(f)"]
& |[]| G(B)
\end{tikzcd}
\end{center}
\end{definition}

\end{definition}

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