\documentclass[12pt]{article}
\usepackage{amsfonts, amsthm, amsmath, stmaryrd, fullpage}

\include{preamble}

\begin{document}

\begin{center}
\textbf{
18.727, Topics in Algebraic Geometry (rigid analytic geometry)\\
Kiran S. Kedlaya, fall 2004 \\
More on coherent sheaves
}
\end{center}

\textbf{References:} [FvdP, Chapter 4]. Kiehl's original papers (in German)
are: Der Endlichkeitssatz f\"ur eigentliche Abbildungen in der
nichtarchimedische Funktionentheorie, \textit{Inv. Math.} \textbf{2}
(1967), 191--214; and
Theorem A und B in der nichtarchimedische Funktionentheorie,
\textit{Inv. Math.} \textbf{2} (1967), 256--273.

\subsection*{Addendum: comment about generic fibres}

I forgot to mention in the previous handout that the generic fibre
construction loses information in a very precise sense. If $P$
is a formal scheme of finite type over $\gotho_K$, and $P'$ is a blowup
of $\gotho_K$ within the special fibre, then the induced map from
the generic fibre of $P'$ to that of $P$ is an isomorphism.

Example: if $P = \Spf \gotho_K \langle x,y \rangle$, and $P'$
is the blowup along the ideal $(x,y,\pi)$ for some
$\pi \in \gothm_K$, then in both cases the generic fibre consists
of the closed points of the affine plane over $K$ represented by
geometric points $(a,b)$ with $|a| \leq 1$ and $|b| \leq 1$.

In fact, Raynaud proved an equivalence of categories between
a certain category of rigid spaces, and a certain category of
formal schemes localized at the blowups in the special fibre. (I
don't know the precise formulation offhand; will go look it up later.)
This cuts both ways: sometimes it's easier to work with rigid
spaces than formal schemes because you would just as soon ignore
the blowups in the special fibre. But sometimes you can't make
a construction in the rigid setting without keeping track
of some choice of an associated formal scheme (as in Laurent
Fargues's seminar talk a couple of weeks ago).

\subsection*{Coherent sheaves on affinoid spaces}

Last time I defined a coherent sheaf on a rigid space to be a sheaf
which, locally on some admissible affinoid covering, is isomorphic
to the sheaf generated by a finitely generated module over the
corresponding affinoid algebra. The Gerritzen-Grauert-Tate theorem
implies that you actually get a sheaf this way, but it doesn't
imply that this sheaf has ``enough'' sections. This is fixed
by the following theorem of Kiehl, whose proof I'll sketch here;
see [FvdP, Section 4.5] for details.
\begin{theorem}[Kiehl]
Let $\calF$ be a coherent sheaf on an affinoid space $X = \Max(A)$,
and put $M = \calF(X)$. Then $M$ is finitely generated,
and $\calF$ is isomorphic to the coherent sheaf associated to $M$.
\end{theorem}
\begin{proof}[Sketch of proof]
As in the Gerritzen-Grauert-Tate argument, one reduces to the case
of a Laurent covering given by a single $f$. (See [FvdP, Section 4.5]
for more on this reduction.) That puts us in the following situation.
We are given $f \in A$, finitely generated modules $M_+$ and $M_-$ over
$A\langle f \rangle $ and $A \langle f^{-1} \rangle$, respectively,
and an isomorphism between $M_+ \otimes A \langle f,f^{-1} \rangle$
and $M_- \otimes A \langle f,f^{-1} \rangle$.
For $M = \calF(X)$, we must show that the maps $M \otimes A \langle f \rangle
\to M_+$ and $M \otimes A \langle f^{-1} \rangle \to M_-$ are surjective.

One can easily reduce this to just checking for $M_-$ (as in
[FvdP, Lemma 4.5.5], but they seem to get it backwards; see below).
For this part, one needs a form of Cartan's lemma:
there exists $c>0$ such that any invertible $n \times n$ matrix
$U$ over $A \langle f,f^{-1} \rangle$ 
with $\|U - I_n\| < c$ (where the matrix norm is the maximum
over entries) factors as $U_+ U_-$, with $U_+$ invertible over
$A \langle f \rangle$ and $B$ invertible over $A \langle f^{-1} \rangle$.
See [FvdP, Lemma 4.5.3] for this calculation. (The basic idea is to
split $U$ \emph{additively} as $I + V_+ + V_-$, where $V_+$ has only
nonnegative powers of $V$, $V_-$ has only nonnegative powers of $V_-$,
and both $V_+$ and $V_-$ have small norm. Then you replace $U$
by $(1-V_+)U(1-V_-)$ and repeat; if $c$ is small enough, this process
converges to the desired factorization.)

To check surjectivity of
$M \otimes A \langle f \rangle \to M_+$ now, you choose a set of $n$
generators of $M_+$ and a set of $n$ generators of $M_-$ (for some $n$), and
write down
change-of-basis matrices $U$ and $V$ between the two sets of generators
over $A \langle f,f^{-1} \rangle$. 
Since the image of $A$ in $A \langle f \rangle$ is dense, we can approximate
$U$ closely by a matrix over $A$, so that $\|(U' - U)V\| < c$ with
$c$ as above. We can then factor $I_n - (U'-U)V = U_+ U_-$ as above,
and changing basis from $M_+$ via $U_+$ gives us a set of generators
which are defined on both subspaces, yielding the surjectivity.

Given the surjectivity, we can choose a finitely generated submodule
$M_1$ of $M$ which generates both $M_+$ and $M_-$. Let
$\calM_1$ be the associated sheaf: then $\calM_1 \to \calF$ is surjective.
Let $\calG$ be the kernel of that map; then $\calG$ is also coherent
and given by finitely generated modules on the two pieces of the cover,
so we can find a surjection $\calM_2 \to \calG$, where $\calM_2$ is the
coherent sheaf associated to the finitely generated module $M_2$. 
Hence $\calF$ is
the cokernel of the map $\calM_2 \to \calM_1$; by acyclicity, 
its global sections are precisely $M_2/M_1$, and the associated sheaf
is precisely $\calF$ because they match up on the two opens.
\end{proof}

This means that we can glue coherent sheaves on any admissible cover
of $X$ (I may have said this earlier without justification).

\subsection*{Example: the sheaf of differentials}

An important example of a coherent sheaf is the sheaf of continuous
differentials.
\begin{theorem}
Let $A$ be an affinoid algebra. There exists a finitely generated
$A$-module $\Omega_{A/K}$ equipped with a $K$-linear derivation
$d: A \to \Omega_{A/K}$ with the following universal property:
given any finitely generated $A$-module $M$ equipped with 
a $K$-linear derivation $D: A \to M$, there exists a unique
$A$-module homomorphism $l: \Omega_{A/K} \to M$ with
$D = l \circ d$.
\end{theorem}
\begin{proof}
In case $A = T_n = K \langle x_1, \dots, x_n \rangle$, this is easy
to check: take $\Omega_{A/K} = A dx_1 + \cdots + A dx_n$ together
with the formal total differential. That is,
\[
df = \sum_{i=1}^n \frac{\partial f}{\partial x_i} dx_i,
\]
where the partial derivation is done formally on the series.

For general $A$, pick a presentation $A \cong 
T_n / (f_1, \dots, f_m)$, and put
\[
\Omega_{A/K} = \Omega_{T_n/K} / (T_n df_1 + \cdots + T_n df_m),
\]
with the induced derivation from $T_n$.
\end{proof}
The module $\Omega_{A/K}$ is called the \emph{module of finite
differentials}, or the \emph{universal finite differential module}.
It is obviously unique up to unique isomorphism, so one gets from it
a coherent sheaf of differentials $\Omega_X$ on any rigid space $X$.

Note that the module of finite differentials is \emph{not} always the
same as the module of K\"ahler differentials, because the latter is
sometimes badly behaved. In fact, it can be shown (though I don't remember
a reference offhand) that the module of finite differentials is the
maximal separated quotient of the completion of the
ordinary module of K\"ahler differentials; it is thus sometimes called
the \emph{module of continuous differentials}, because the latter
description makes it universal for continuous derivations into any
Banach $A$-algebra.

Here are some fun facts about modules of differentials.
(See [FvdP, Theorem 3.6.3], although they mostly just refer to
Springer LNM 38 ``Differentialrechnung in der analytische 
Geometrie''):
\begin{prop}
Let $A$ be an affinoid algebra which is an integral domain, and let
$F$ be its fraction field.
\begin{enumerate}
\item[(a)] The dimension of $\Omega_{A/K} \otimes_A F$ over $F$ equals
the Krull dimension $d$ of $A$.
\item[(b)] Let $\gothm$ be a maximal ideal of $A$. Then the following
are equivalent.
\begin{enumerate}
\item[(i)] $A$ is smooth over $K$ at $\gothm$. (That means that locally
near $\gothm$, $A$ can be written as the vanishing locus in some affine
space $\Max T^n$ of some number $m$ of functions whose $m \times n$
matrix of partial derivatives has maximal rank. If $K$ is perfect,
this is equivalent to $A$ being regular at $\gothm$, i.e., the localization
$A_{\gothm}$ being a regular local ring.)
\item[(ii)] The localization $A_{\gothm} \otimes_A \Omega^{A/K}$
is free over $A_{\gothm}$ (of rank $d$).
\item[(iii)] The dimension of $\Omega_{A/K}/\gothm \Omega_{A/K}$
over $A/\gothm$ is $d$.
\end{enumerate}
\end{enumerate}
\end{prop}

\subsection*{Closed analytic subspaces}

Let $X$ be a rigid space over $k$, and let $\calI$ be a coherent sheaf
of ideals of $\calO$. Then we get a subspace of $X$ associated to
$\calI$ whose points are the support of the coherent sheaf
$\calO/\calI$ (with topology induced from $X$), and whose
structure sheaf is the restriction of $\calO/\calI$. On an affinoid
space, this of course corresponds to viewing the vanishing locus
of some ideal of functions as the Max of the quotient affinoid algebra.
Any such subspace of $X$ is called a \emph{closed analytic subspace}.

\subsection*{Next time: GAGA}

Next time, I'll talk about separatedness and properness for rigid spaces
and sketch the proof of Kiehl's rigid GAGA theorem.

\subsection*{Exercises}

\begin{enumerate}
\item
Assume that $K$ has characteristic zero.
Let $X$ be a \emph{smooth} rigid space, let $\calF$ be a coherent sheaf on $X$,
and suppose that there exists a $K$-linear connection
$\nabla: \calF \to \calF \otimes \Omega_{X/K}$. That is,
given a section $s$ of $\calF$ and a function $f$, we have
$\nabla(sf) = f \nabla(s) + s \otimes df$. Prove that 
$\calF$ must be locally free. (Hint: replace the local ring at a maximal
ideal by its completion and check there that the torsion submodule
must vanish.)
\end{enumerate}

\end{document}
