# I.4: Structure sheaf and general adic spaces

Let ${(A,A^+)}$ be an affinoid ring. Let us define a rational subset

$\displaystyle U = U\left(\frac{f_1,...,f_n}{g}\right) = \{x \in \mathrm{Spa}(A,A^+) | \ \ |f_i(x)| \leq |g(x)| \not = 0 \ \ , i=1,...,n \}$

where the ${f_i}$‘s define an open ideal in ${A}$.

Recall that ${A \langle X_1,...,X_n \rangle}$ denotes a ring of formal power series

$\displaystyle \Sigma _{(i_1,...,i_n) \in \mathbb{N} ^n} a_{i_1,...,i_n} X^{i_1} _1 \cdot ... \cdot X^{i_n} _n$

such that ${|a _{i_1,...,i_n}| \rightarrow 0}$ for ${|i_1+...+i_n| \rightarrow \infty}$.

We define

$\displaystyle A\langle \frac{f_1}{g},..., \frac{f_n}{g} \rangle = A \langle X_1,...,X_n \rangle / \langle gX_i - f_i \rangle$

This is the completion of ${A[\frac{f_1}{g},...,\frac{f_n}{g}]}$ with respect to the ideal of definition of ${A}$.

Let ${B}$ be the integral closure of ${A^+[\frac{f_1}{g},...,\frac{f_n}{g}]}$ in ${A[\frac{f_1}{g},...,\frac{f_n}{g}]}$. Denote by ${\hat{B}}$ its completon. Then one can check that ${(A[\frac{f_1}{g},...,\frac{f_n}{g}], B)}$ is an affinoid ring and also ${(A\langle \frac{f_1}{g},...,\frac{f_n}{g}\rangle, \hat{B})}$ is an affinoid ring.

We can endow the topological space ${X= \mathrm{Spa}(A,A^+)}$ with presheaves ${(\mathcal{O}_X, \mathcal{O}_X ^+)}$, by putting for any rational subset ${U = U(\frac{f_1,...,f_n}{g})}$ of ${X}$:

$\displaystyle \mathcal{O}_X(U) = A\langle \frac{f_1}{g},...,\frac{f_n}{g}\rangle$

$\displaystyle \mathcal{O} _X ^+(U) = \hat{B}$

Hence ${\mathcal{O}_X(U)}$ depends only on ${U}$ and if ${V}$ is a rational subset of ${X}$ with ${V \subset U}$ then we get a natural continuous ring homomorphism ${\mathcal{O}_X(U) \rightarrow \mathcal{O}_X(V)}$. Thus for any open subset ${W}$ of ${X}$ we put

$\displaystyle \mathcal{O}_X(W) = \varprojlim _{U \subset W} \mathcal{O}_X(U)$

where the limit is taken over the rational subsets ${U \subset W}$ and similarly for ${\mathcal{O}_X ^+}$. Those are presheaves of complete topological rings.

Let ${\rho : (A,A^+) \rightarrow (A\langle \frac{f_1}{g},...,\frac{f_n}{g} \rangle, \hat{B})}$ be the standard map. We have the following lemma

Lemma 14 The map ${\mathrm{Spa} \rho : \mathrm{Spa} (A\langle \frac{f_1}{g},...,\frac{f_n}{g} \rangle, \hat{B}) \rightarrow \mathrm{Spa}(A,A^+)}$ factors through ${U}$. Moreover for any continuous homomorphism ${\phi: (A,A^+) \rightarrow (C,C^+)}$ from ${(A,A^+)}$ to a complete affinoid ring ${(C,C^+)}$ such that ${\mathrm{Spa} (\phi)}$ factors through ${U}$ there exists a unique continuous homomorphism ${\psi : (A\langle \frac{f_1}{g},...,\frac{f_n}{g} \rangle, \hat{B}) \rightarrow (C,C^+)}$ such that ${\psi \circ \rho = \phi}$.

Proof: See Lemma 8.1 in [Wed]. $\Box$

Let ${x \in X}$. If ${x \in U \subset X}$ then ${x : A \rightarrow \Gamma \cup \{0\}}$ can be extended to ${x : \mathcal{O}_X(U) \rightarrow \Gamma \cup \{0\}}$, thus also to the stalk ${\mathcal{O}_{X,x} = \varinjlim _{x \in U \textrm{ open}} \mathcal{O}_X(U)}$. Observe that ${\mathcal{O}_{X,x}}$ is a local ring with the maximal ideal ${\mathfrak{m} _x = \{ f \in \mathcal{O}_{X,x} | \ |f(x)| = 0 \}}$. Write ${k(x) = \mathcal{O}_{X,x} / \mathfrak{m}_x}$ for the residue field at ${x}$. It is equipped with a valuation ${f \mapsto |f(x)|}$. We write ${k(x)^+}$ for its valuation ring. Huber proves that

$\displaystyle \mathcal{O}_X ^+(U) = \{ f\in \mathcal{O}_X(U) | \ \ |f(x)| \leq 1 \ \forall x \in U \}$

It turns out that ${\mathcal{O}_X, \mathcal{O}_X ^+}$ are not in general sheaves. We say that ${X}$ is sheafy, when they are. When ${A}$ is strongly noetherian, i.e. ${A\langle X_1,...,X_n \rangle}$ is noetherian for all ${n}$, then ${\mathrm{Spa}(A,A^+)}$ is sheafy. Also ${\mathrm{Spa}(A,A^+)}$ is sheafy when ${A}$ is a perfectoid algebra. We shall return to this later on.

4.1. General adic spaces

Let us define the category ${\mathcal{V}}$ of tuples ${(X, \mathcal{O}_X, (v_x)_{x\in X})}$ where

1. ${X}$ is a topological space
2. ${\mathcal{O} _X}$ is a sheaf of complete topological rings on ${X}$ such that the stalk ${\mathcal{O}_{X,x}}$ of ${\mathcal{O}_X}$ is a local ring.
3. ${v_x}$ is an equivalence class of valuations on the stalk ${\mathcal{O}_{X,x}}$.

The morphisms are pairs ${(f, f^{\flat})}$ such ${f: X \rightarrow Y}$ is a continuous map of topological spaces and ${f^{\flat} : \mathcal{O}_Y \rightarrow f_{*} \mathcal{O}_X}$ is a morphism of pre-sheaves of topological rings (i.e. for all ${V \subset U}$ open, ${\phi _V \mathcal{O}_Y(V) \rightarrow \mathcal{O}_X(f^{-1}(V))}$ is a continuous ring homomorphism) such that for all ${x \in X}$ the induced ring homomorphism ${f ^{\flat} _x: \mathcal{O}_{Y, f(x)} \rightarrow \mathcal{O}_{X,x}}$ is compatible with the valuation ${v_x}$ and ${v_{f(x)}}$, that is ${v_{f(x)} = v_x \circ f^{\flat} _x}$ (this implies that ${f_x ^{\flat}}$ is local and ${\Gamma _{v_{f(x)}} \subset \Gamma _{v_x}}$).

Definition 15 An affinoid adic space is an object of ${\mathcal{V}}$ which is isomorphic to ${\mathrm{Spa}(A,A^+)}$ for some affinoid pair ${(A,A^+)}$.

Observe that a difference with what we had before is that we demand that ${\mathcal{O}_X}$ be a sheaf.

Definition 16 An adic space is an object of ${\mathcal{V}}$ which is locally an affinoid adic space.

Morphisms between adic spaces are the morphisms in ${\mathcal{V}}$. For example the morphisms of adic spaces ${X \rightarrow \mathrm{Spa}(A,A^+)}$ correspond bijectively to the continuous ring homomorphisms ${(A,A^+) \rightarrow (\mathcal{O}_X(X), \mathcal{O}_X ^+(X))}$.

Reference:

[Wed] T. Wedhorn, “`Adic spaces”‘, unpublished notes

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