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\textbf{Recall:} $p \in X$ is \gls{sing} if
$\dim_{\KK} \tspace_p X > \dimX X = \inf\{\dim
\tspace_p X\}$.

The above lemma tells us that the set of \gls{sing}
points of $X$ is a \emph{proper} \gls{zzclosed}
subset.

\begin{example*}
  $y_2 - x^3 = 0$, $J = (2y, -3x^2)$
  \begin{center}
    \includegraphics[width=0.3\linewidth]{images/b8e6e7f54efd461c.png}
  \end{center}
  vanishes when $x = y = 0$.
\end{example*}

\begin{example*}
  $x^2 + y^2 - z^2 = 0$, $J = (2x, 2y, -2z)$,
  vanishing at the origin.
  \begin{center}
    \includegraphics[width=0.3\linewidth]{images/bbcce1225e884745.png}
  \end{center}
\end{example*}

\subsubsection*{Intrinsic characterisation of the
tangent space}

\glssymboldefn{varphip}{$\varphi_p$}{$\varphi_p$}
Let $X$ be an \gls{affvty}. For $p \in X$, define
$\varphi_p : \cring A(X) \to \KK$ to be the
\Kalgebra{\KK} homomorphism given by $\varphi_p(f)
= f(p)$.

\begin{flashcard}[derivation-centred-at-defn]
\begin{definition*}[Derivation centred at $p$]
  \glsnoundefn{dcat}{derivation}{derivations}
  \glssymboldefn{Der}{Der$(A(x), p)$}{Der$(A(x),
  p)$}
  \glsnoundefn{Lrule}{Leibniz rule}{N/A}
  \cloze{A \emph{derivation centred at $p$} is a map
  $D : \cring A(X) \to \KK$ such that
  \begin{enumerate}[(1)]
    \item $D(f + g) = D(f) + D(g)$
    \item $D(fg) = \varphip_p(f) D(g) + D(f)
      \varphip_p(g)$ (the $\RHS$ can also be
      written as $f(p)D(g) + g(p)D(f)$). (Leibniz
      rule).
    \item $D(a) = 0$ for $a \in \KK$.
  \end{enumerate}
  Denote $\Der(\cring A(X), p)$ to be the set of
  derivations centred at $p$.}
\end{definition*}
\end{flashcard}

\begin{note*}
  $\Der(\cring A(X), p)$ is a
  $\KK$-vector space (check $D_1 + D_2$, $aD$ are
  derivations if $D_1, D_2, D$ are derivations).
\end{note*}

\begin{flashcard}[TpX-Der-iso-vspace-thm]
\begin{theorem*}
  $\tspace_p X \cong \Der(\cring A(X), p)$ as
  $\KK$-vector spaces for $p \in X$.
\end{theorem*}

\begin{proof}
  \cloze{Given $(v_1, \ldots, v_n) \in \tspace_p X$, so
  if $\I(X) = \langle f_1, \ldots, f_r \rangle$,
  $\sum_i v_i \pfrac{f_j}{x_i} (p) = 0$ for all
  $j$. Define
  \begin{align*}
    \KK[x_1, \ldots, x_n]
    &\to \KK
    f
    &\mapsto \sum_i v_i \pfrac{f}{x_i}(p)
  \end{align*}
  This vanishes on elements of $\I(X)$, which are
  of the form $f = \sum_{j = 1}^r g_j f_j$ for
  $g_j \in \KK[x_1, \ldots, x_n]$. Then
  \begin{align*}
    f
    &\mapsto \sum_{i = 1}^n v_i \left( \sum_{j =
    1}^r \left( \pfrac{f_j}{x_i} \cdot g_j +
    \pfrac{g_j}{x_i} f_j \right) (p) \right)
    &&\text{($f_j(p) = 0$ for all $j$, since $p \in
    X$)} \\
    &= \sum_{i, j} \left( v_i \pfrac{f_j}{x_i}
    g_j(p) \right) \\
    &= \sum_j g_j(p) \left( \sum_i v_i
    \pfrac{f_j}{x_i} (p) j\right) \\
    &= 0
  \end{align*}
  Thus we get a well-defined $\KK$-linear map
  \[ D_r : \frac{\KK[x_1, \ldots, x_n]}{\I(X)} =
  \cring A(X) \to \KK .\]
  Check easily that this is a \gls{dcat}. Given $D
  \in \Der(\cring A(X), p)$, define $v_i =
  D(x_i)$. By repeated use of the \gls{Lrule},
  \[ D(f) = \sum_{i = 1}^n v_i \pfrac{f}{x_i} (p)
  .\]
  Example:
  \begin{align*}
    D(x_1 x_2)
    &= D(x_1) \cdot x_2(p) + x_1(p) D(x_2) \\
    &= v_1 x_2(p) + v_2 x_1(p) \\
    &= v_1 \pfrac{(x_1 x_2)}{x_1}(p) + v_2
    \pfrac{(x_1 x_2)}{x_2} (p)
  \end{align*}
  Thus $D(f_j) = \sum_i v_i \pfrac{f_j}{x_i}(p)$,
  but $f_j \in \I(X)$, so $D(f_j) = 0$. Thus
  $\sum_i v_i \pfrac{f_f}{x_i}(p) = 0$ for all
  $j$, so $(v_1, \ldots, v_n) \in \tspace_p X$.}
\end{proof}
\end{flashcard}

\begin{remark*}
  \glsref[sing]{Singular} points and
  \glspl{tspace} are intrinsic to \glspl{affvty}.
\end{remark*}

\begin{flashcard}[local-ring-defn]
\begin{definition*}[Local ring]
  \glsnoundefn{locring}{local ring}{local rings}
  \glssymboldefn{OXp}{$O_{X, p}$}{$O_{X, p}$}
  \cloze{Let $X$ be a \gls{vty}, $p \in X$. We define the
  \emph{local ring} to $X$ at $p$ to be
  \[ \mathcal{O}_{X, p} = \{(U, f) \st \text{$U$
  is an open neighbourhoof of $p$, $f : U \to \KK$
  a \gls{gregf} function}\} / \sim \]
  where $(U, f) \sim (V, g)$ if $f|_{U \cap V} = g_{U \cap
  V}$. This is a subring of $\gK(X)$, the field of
  fractions.}
\end{definition*}
\end{flashcard}

\begin{example*}
  \phantom{}
  \begin{enumerate}[(1)]
    \item $X \subseteq \AA^n$ is an \gls{affvty},
      \[ \OXp_{X, p} = \left\{ \frac{f}{g} \in
      \gK(X) ~\Bigg|~ g(p) \neq 0, f, g \in \cring A(X)
      \right\} .\]
    \item $X \subseteq \P^n$ a \gls{projvty}. Then
      \[ \OXp_{X,p} = \left\{ \frac{f}{g} ~\Bigg|~
      \substack{f, g
      \in \KK[x_1, \ldots, x_n] / \I(X), g(p) \neq
      0, \\
      f, g \text{ \gls{homog} of the same degree}
      }\right\} \]
      which is a subring of
      \[ \gK(X) = \left\{ \frac{f}{g} \st
      \substack{f, g \in
       \KK[x_0, \ldots, x_n] / \I(X), g \neq 0 \\
       \text{$f, g$ \gls{homog} of the same degree}
      }\right\} \]
  \end{enumerate}
\end{example*}

\begin{remark*}
  The definition of $\OXp_{X, p}$ makes it
  intrinsic, i.e. not dependent on the embedding.
\end{remark*}

\begin{remark*}
  $\OXp_{X, p}$ is a ring ($(U, f) + (V, g) = (U
  \cap V, f|_{U \cap V} + g|_{U \cap V})$ etc). We
  define
  \[ m_p = \{(U, f) \in \OXp_{X, p} \st f(p) = 0\}
  .\]
  This is an ideal, and every element of $\OXp_{X,
  p} \setminus m_p$ is invertible. Thus $m_p$ is
  the \emph{unique} maximal ideal of $\OXp_{X,
  p}$.
\end{remark*}

\begin{flashcard}[local-ring-defn]
\begin{definition*}[Local ring]
  \glsnoundefn{locring}{local ring}{local rings}
  \cloze{A ring $A$ with a unique maximal ideal is called
  a \emph{local ring}.}
\end{definition*}
\end{flashcard}