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Section 7.6 Rigid GAGA (Aash)

Definition 7.6.1.

Let \((Z, \sheaf O_Z)\) be a \(k\)-scheme of locally finite type. A rigid analytification is a rigid space \((Z^{rig}, \sheaf O_{Z^{rig}})\) together with a morphism of locally ringed \(G\)-spaces

\begin{equation*} (i, i^*) \colon (Z^{rig}, \sheaf O_{Z^{rig}}) \to (Z, \sheaf O_Z) \end{equation*}

satisfying: Given \((Y, \sheaf O_Y)\) a rigid \(k\)-space and a morphism

\begin{equation*} (Y, \sheaf O_Y) \to (Z, \sheaf O_Z) \end{equation*}

this factors through \((i, i^*)\) via a unique morphism \((Y, \sheaf O_Y)\to(Z^{rig}, \sheaf O_{Z^{rig}})\text{.}\)

Example 7.6.2. Affine space.

Recall we had maps

\begin{equation*} \overbrace{k \langle \zeta _{1, j+1}, \ldots, \zeta _{n, j+1}\rangle}^{T_{n,j+1}} \to k \langle \zeta _{1, j}, \ldots, \zeta _{n, j}\rangle \end{equation*}
\begin{equation*} \zeta _{i, j+1} \mapsto c\zeta _{i,j} \end{equation*}

for some \(|c| \lt 1\text{.}\) Glue along these maps

\begin{equation*} B_j \subseteq B_{j+1} \end{equation*}

as larger balls. This is an admissible covering so we have

\begin{equation*} \aff^{k, rig} = \bigcup _{j=0}^\infty B_j\text{.} \end{equation*}

Consider \(k \lb \zeta _1, \ldots, \zeta _n\rb \) mapping to each of \(T_{j}\) compatibly. This induces an inclusion of max specs

\begin{equation*} \Sp(T_{n,0}) \subseteq \Sp(T_{n,1}) \subseteq \cdots \Max k\lb \zeta \rb \end{equation*}

claim that for

\begin{equation*} \ideal m \subseteq k\langle \zeta \rangle \end{equation*}

a maximal ideal, then

\begin{equation*} \ideal m' \ideal m \cap k\lb \zeta \rb \end{equation*}

s.t.

\begin{equation*} \ideal m = \ideal m'k \langle \zeta \rangle\text{.} \end{equation*}

Additionally claim given \(\ideal m' \subset k \lb \zeta \rb \text{,}\) there exists \(i_0 \in \NN\) s.t. \(\forall i \ge i_0, \ideal m'k\langle x^i \zeta \rangle\) is maximal in \(k\langle c^i \zeta \rangle = T_{n,i}\text{.}\) So all \(T_{n,i} \xrightarrow\phi k \lb \zeta \rb /\ideal m'\) and the maximal spectra of \(k\lb \zeta \rb \) equals \(\bigcup B_i\text{.}\)

More generally given an affine scheme \(Z = \Spec k\lb \zeta \rb /a\) and glue

\begin{equation*} T_{n,0}/(a) \leftarrow T_{n,1}/(a) \leftarrow \cdots \end{equation*}

and \(k\lb \zeta \rb / a\) maps to each. Giving \(\Spm (T_{n,0}/a) \hookrightarrow \Spm (T_{n,1}/a) \hookrightarrow \)

\begin{equation*} \Spm(k\lb \zeta \rb/a) = \bigcup_{j=0}^\infty \Spm(T_{n,j}/a) \end{equation*}

In order to check the properties of this construction, we note that

\begin{equation*} Z^{rig} \to Z \end{equation*}

via

\begin{equation*} k\lb \zeta \rb /a \to T_{n,i}/a \end{equation*}

locally, giving

\begin{equation*} \sheaf O_Z(Z) \to \sheaf O_{Z^{rig}}(Z^{rig})\text{.} \end{equation*}

So we have

\begin{equation*} (i, i^*) \colon (Z^{rig}, \sheaf O_{Z^{rig}}) \to (Z, \sheaf O_Z) \end{equation*}

need to check universal property: WLOG let \((Y, \sheaf O_Y)\) be an affinoid space

\begin{equation*} (Y, \sheaf O_Y) \to (Z, \sheaf O_Z) \end{equation*}

gives

\begin{equation*} k\lb \zeta \rb /a \xrightarrow \sigma B \end{equation*}

wts

\begin{equation*} k\lb \zeta \rb /a \to T_{n,i}/a \to B \end{equation*}

choose \(i\) big enough s.t.

\begin{equation*} | \sigma (\bar \zeta _j)| \le \frac{1}{|c|^i} \end{equation*}

\(\sigma \) will extend uniquely though

\begin{equation*} T_{n,i}/a\text{.} \end{equation*}

We get morphisms and hence a functor for rigidification by universality. Call this the GAGA functor.

It respects fibre products.

\(\sheaf O_{Z^{rig},z}\) is the completion at \(z\in Z^{rig}\) is the same as the completion of \(\sheaf O_{Z,z}\) at \(z\text{.}\)

GAGA is faithful but not full.

Definition 7.6.4.

We have a sheaf \(\sheaf F\) associated to \(A\) modules \(M\)

\begin{equation*} \sheaf F = M\otimes _A \sheaf O_X \end{equation*}

this functor is fully faithful commutes with kernels, cokernels, images and tensor products.

A coherent module has finite type, in that there exists a covering with

\begin{equation*} \sheaf O_X^{s_i}|_{X_i} \to \sheaf F|_{X_i} \to 0 \end{equation*}

and also the kernel here is finite type.

Cohomology, we have a section functor

\begin{equation*} \Gamma (X, -) \colon \sheaf F \to \sheaf F(X) \end{equation*}

and

\begin{equation*} \phi \colon X\to Y \end{equation*}
\begin{equation*} \phi _*\colon \sheaf F \to \phi _*\sheaf F \end{equation*}

is left exact, need an injective resolution.

An object \(\sheaf F\) is injective if given

\begin{equation*} 0 \to \sheaf E ' \to \sheaf E \to \sheaf E'' \to 0 \end{equation*}
\begin{equation*} 0 \to \Hom(\sheaf E ', \sheaf F) \to \Hom(\sheaf E, \sheaf F) \to \Hom(\sheaf E'', \sheaf F) \to 0 \end{equation*}

Cech cohomology \(\varinjlim_U H(U, \sheaf F)\) the limit over admissible coverings, ordered by refinement.

\begin{equation*} C^q(U, \sheaf F) = \prod_{i_0, \ldots, i_q \in I} \sheaf F(\bigcap _k U_{i_k}) \end{equation*}

have a coboundary map which makes this a complex.

Definition 7.6.8.

\(\phi \colon X\to Y\) is called a closed immersion if there exists an admissible affinoid covering \((V_j)_j\) s.t. for all \(j \in J\)

\begin{equation*} \phi _j \colon \phi \inv (V_j) \to V_j \end{equation*}

is a morphism of affinoid spaces with corresponding algebra map

\begin{equation*} B_j \twoheadrightarrow A_j\text{.} \end{equation*}
Definition 7.6.9.

\(\phi \colon X\to Y\) is called a separated if

\begin{equation*} \Delta \colon X \to X\times_X X \end{equation*}

is a closed immersion.

In rigid geometry we do not have that for \(\phi \colon X \to Y\) with \(\Delta \colon X\to X\times_Y X\) locally closed then sep iff closed immer.

Definition 7.6.11. Properness.

A map \(f: X \rightarrow Y\) of rigid spaces is proper if it is separated and quasi-compact and there exists an admissible affinoid open covering \(\left\{U_{i}\right\}\) of \(Y\) and a pair of finite (necessarily admissible) affinoid open coverings \(\left\{V_{i j}\right\}_{j \in J_{i}}\) and \(\left\{V_{i j}^{\prime}\right\}_{j \in J_{i}}\) (same index set \(J_{i}\) of \(j^{\prime}\) s! of \(f^{-1}\left(U_{i}\right)\) such that two conditions hold: \(V_{i j} \subseteq V_{i j}^{\prime}\) for all \(j,\) and for all \(j \in J_{i}\) there is an \(n \geq 1\) and a closed immersion \(V_{i j}^{\prime} \hookrightarrow U_{i} \times \mathbf{B}^{n}\) over \(U_{i}\) such that \(V_{i j} \subseteq U_{i} \times\left\{\left|t_{1}\right|, \ldots,\left|t_{n_{i}}\right| \leq r\right\}\) for some \(0\lt r\lt 1\) with \(r \in \sqrt{| k}^{\times} | .\) (Equivalently, by the Maximum Modulus Principle, we can replace \(* \leq r "\) with \({*}\lt 1\) ".)