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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