# Appendix 

## Notation

- $\ff(R)$ denotes the fraction field (or field of quotients) of $R$.
- $R\localize{S}$ is the ring $R$ localized at the multiplicative set $S \subseteq R$, i.e. the subset of the fraction field $\ff(R)$ with denominators only in $S$.
  This differs from the usual notation $S\inv R$.
- $\ZZpadic$ is the $p\dash$adic integers, i.e. the ring $R=\ZZ$ completed at the ideal $\gens{p}$.
  This differs from the usual notation $\ZZ_p$.

- $R\fps{t}$ is the topological ring of formal power series in $t$, i.e. infinite sums $\sum_{i\geq 0} r_i t^i$ with the $t\dash$adic topology.
- $R\fls{t}$ is the topological ring of formal Laurent series, i.e. half-infinite sums $\sum_{i\geq -N} r_i t^i$.
  - Note that $R \fls{t} = R\fps{t}\localize{S}$ where $S = \ts{1, x, x^2,\cdots}$.
  If $R$ is a field, $R\fls{t} = \ff(R\fps{t})$.

- If $A$ is a ring, then $\OO_A \da \OO_{\spec A}$ by abuse of notation.
- $\AA^n \da \AA^{n}\slice R \da \Spec R[x_1,\cdots, x_n], \PP^n\da \PP^n\slice R \da \Proj R[x_0,\cdots, x_n]$, where the ring $R$ is sometimes dropped from the notation.
- If $\cat C$ is a category and $x,y\in \cat C$ are objects, $\cat C(x, y) \da \Hom_{\cat C}(x, y)$.
  So for example, $\Sch(X, Y) \da \Hom_{\Sch}(X, Y)$.
  Hom sets are sometimes written as $\Mor_{\cat C}(x, y)$.

## Facts


:::{.remark}
Some useful facts:

- The equalizer diagram for a sheaf $\mcf$:

\begin{tikzcd}
    \cofinal
    \ar[r]
    &
    \mcf(U)
    \stackar{3}[r]
    &
    \displaystyle\prod_{i\in I} \mcf(U_{i})
    \stackar{5}[r]
    &   
    \displaystyle\prod_{i < j \in I} \mcf(U_{ij})
    \cdots
\end{tikzcd}

- The inverse image / pushforward ("direct image") adjunction:

\[
\adjunction{f_*}{f\inv}{\Sh(X)}{\Sh(Y)} \implies \Sh(X)(f\inv \mcg, \mcf) \iso \Sh(Y)(\mcg, f_* \mcf)
.\]


:::