Algebraic Geometry Notes

Introduction #

Algebraic geometry is a subject that always felt frightening to me as a mathematician. I studied algebra, specifically representation theory, and algebraic geometry always felt more like “real maths”

Background #

Category theory #

\cite{Vakil} Ch 1

Sheaves #

\cite{Vakil} Ch 2

Complex analysis #

\cite{Kirwan} Appendix B

Riemann surfaces #

\cite{Kirwan} Ch 5

Commutative algebra #

Localisation #

\cite{Eisenbud} Ch 2

Varieties #

\cite{Hartshorne} Section I
\cite{Fulton} Ch. 1,2

Schemes #

We follow \cite{Hartshorne} Section II, starting in II.2, but use \cite{EisenbudHarris} and \cite{Vakil} (Ch 3-17) as additional references to support the terse exposition of Hartshorne. Much of the material given in exercises in Hartshorne is expanded upon in the other sources.

Affine schemes #

We will define schemes as topological spaces $X$ together with a sheaf $\mathcal{O}$, locally isomorphic to an affine scheme. So first we need the definition of an affine scheme associate to a ring $R$.

\begin{definition} The {\em spectrum} of the commutative ring $R$, denoted $\mathrm{Spec}\;R$, is the set of all prime ideals of $R$. \end{definition}

\begin{example} {\bf Spectra of rings}. \begin{itemize} \item[(a)] (\cite{EisenbudHarris} Exercise I-1(a)). $\mathrm{Spec}\;\mathbb{Z}$. $\mathbb{Z}$ has a prime ideal $[\mathfrak{p}] = (p)$ for every prime $p\in \mathbb{Z}$, as well $(0)$. \item[(b)] (\cite{EisenbudHarris} Exercise I-1(b)). $\mathrm{Spec}\;\mathbb{Z}/(3)$. $\mathbb{Z}/(3)$ is an integral domain, so its spectrum is a single point $(0)$. \item[( c)] (\cite{EisenbudHarris} Exercise I-1( c)). $\mathrm{Spec}\;\mathbb{Z}/(6)$. There are three ideals in $\mathbb{Z}/(6)$: $(0), (2), (3)$. To see which are prime we can easily form the quotient in each case; $(0)$ is {\em not} prime since $\mathbb{Z}/(6)$ itself is not an integral domain, but since $(\mathbb{Z}/(6))/(2)\cong \mathbb{Z}/(2)$ and $(\mathbb{Z}/(6))/(3)\cong \mathbb{Z}/(3)$ are both integral domains (indeed, fields), the ideals $(2)$ and $(3)$ are prime. \end{itemize} \end{example}