用户: Scavenger/Recording Some Facts/Basics for Perfectoid Spaces

I collect here some results used implicitly or explicitly in [1], whose proofs are not given and I find them hard to be found in the literature.

Notations and Conventions

For notions concerning perfectoid spaces, we follow [1]. In particular, a perfectoid space is an adic space locally isomorphic to $\mathrm{Spa}(A,A^{+})$ where $A$ is a perfectoid Tate ring.

For notions concerning adic spaces, we follow [3], which in turn follows the original works of Huber.

1Basics Facts

Notion 1.1. Let $X$ be an analytic adic space and let $x\in |X|$ be a point of the underlying topological space of $X$. Let $k$ be the residue field of the stalk at $x$. $k$ is equipped with a natural valuation $v$ by the definition of adic spaces. We let $k(x)$ denote the completion of $k$ with respect to this valuation, and we let $k(x{)}^{+}\subseteq k(x)$ be the valuation ring corresponding to $v$. Since $X$ is analytic, $v$ is microbial, thus $(k(x),k(x{)}^{+})$ is a complete affinoid ring.

Proposition 1.2. Let $X$ be a perfectoid space and let $X\in |X|$ be a point. Then $k(x)$ is a perfectoid field.

Proof. $k(x)$ is clearly a complete uniform Tate ring, and it remains to find a pseudo-uniformizer $\varpi \in k(x)$ such that ${\varpi }^p|p$ in $k(x{)}^{\circ }$ and the frobenius on $k(x{)}^{\circ }/{\varpi }^p$ is surjective. We may assume $X=\mathrm{Spa}(A,A^{+})$ where $A$ is a perfectoid Tate ring. We choose $\varpi \in A$ to be a pseudo-uniformizer such that ${\varpi }^p|p$ in $A^{+}$ and $\varpi$ admits a sequence of $p$-power roots. We use the same notation to denote the image of $\varpi$ in $k(x)$.

Let $O,O^{+}$ denote the stalks of ${\mathcal{O}}_X,{\mathcal{O}}_X^{+}$ at $x$, respectively. Let $m$ be the maximal ideal of $O$, then it is clear that $m\subseteq {\varpi }^n{O}^{+}$ for every positive integer $n$ since $\varpi$ is a unit of $O$. It follows that $k(x{)}^{+}$ is isomorphic to the $\varpi$-adic completion of $O^{+}$, and $k(x{)}^{+}/\varpi \cong {\mathrm{colim}}_{x\in U}{\mathcal{O}}_X^{+}(U)/\varpi$, where $U$ runs through all rational subsets containing $x$, ordered by reverse inclusion. It follows that the frobenius on $k(x{)}^{+}/\varpi$ is surjective. Using this it can easily be shown that $k(x)$ is perfectoid.

$\square$

Proposition 1.3. Let $(A,A^{+})$ be an affinoid ring where $A$ is a perfectoid Tate ring. Then for $a\in A$, $v(a)=0$ for all $v\in \mathrm{Spa}(A,A^{+})$ if and only if $a=0$.

Proof. Let $a$ be such an element, and let $\varpi$ be a pseudo-uniformizer. Then $v({\varpi }^{-n}a)=0\le 1$ for all positive integer $n$ and all $v\in \mathrm{Spa}(A,A^{+})$, meaning that $a\in {\varpi }^n{A}^{+}$ for all $n$, and $a$ has to be 0 since $A^{+}$ is $\varpi$-adically complete.

$\square$

2Fiber Products

Theorem 2.1. Let $X\to Y\leftarrow Z$ be perfectoid spaces, then the fiber product $X{\times }_{Y}Z$ exists in the category of adic spaces, and is a perfectoid space.

This theorem is an immediate consequence of the following:

Proposition 2.2. Let $(A,A^{+})\leftarrow (B,B^{+})\to (C,C^{+})$ be affinoid rings (or Huber pairs) such that $A,B$ and $C$ are perfectoid Tate rings. Consider the topology on $A{\otimes }_{B}C$ making it into a topological ring such that the image of $A^{+}{\otimes }_{B^{+}}{C}^{+}$ in $A{\otimes }_{B}C$ is open and it has the $\varpi$-adic topology (as an open subring with the subspace topology), where $\varpi \in B$ is an arbitrary pseudo-uniformizer (It is not hard to check that this topology exists and is unique). Let $D$ be the completion of $A{\otimes }_{B}C$ with respect to this topology, let $E\subseteq A{\otimes }_{B}C$ be the integral closure of the image of $A^{+}{\otimes }_{B^{+}}{C}^{+}$ in $A{\otimes }_{B}C$, and let $D^{+}\subseteq D$ be the completion of $E$. Then $D$ is a perfectoid Tate ring, and $(D,D^{+})$ is a pushout of $(A,A^{+})\leftarrow (B,B^{+})\to (C,C^{+})$ in the category of complete affinoid rings.

Proof. It is clear that $(D,D^{+})$ is a complete affinoid ring, and it is a pushout of $(A,A^{+})\leftarrow (B,B^{+})\to (C,C^{+})$ in the category of complete affinoid rings, so it remains to show $D$ is a perfectoid Tate ring.

We first prove the case where $pB=0$. In this case we only need to show that $D$ is perfect by virtue of [1] Proposition 3.5. This can be easily checked by hands using the fact that $A{\otimes }_{B}C$ and $A^{+}{\otimes }_{B^{+}}{C}^{+}$ (and hence the image of the latter in the former) are perfect.

We now turn to the general case. Let $(A^{♭},A^{♭+})$ be the tilt of $(A,A^{+})$, and similarly for $(B,B^{+})$ and $(C,C^{+})$, respectively. Let $\varpi \in B$ be a pseudo-uniformizer with $\varpi |p$ in $B^{+}$ which admits a sequence of $p$-power roots ${\varpi }^{1/p^n}$ giving rise to a pseudo-uniformizer ${\varpi }^{♭}$ of $B^{♭}$. Let $I\subseteq W(B^{♭+})$ be an ideal primitive of degree 1, such that $(B,B^{+})$ is the untilt of $(B^{♭},B^{♭+})$ with respect to $I$, in the sense of [1] Theorem 3.17. Then $(A,A^{+})$ is the untilt of $(A^{♭},A^{♭+})$ with respect to $I\cdot W(A^{♭+})$ and similarly for $C$. By convention if $(R,R^{+})$ is a perfectoid affinoid ring with tilt $(R^{♭},R^{♭+})$ we denote the natural quotient map $W(R^{{♭}^{+}})\to R^{+}$ by $\theta$.

Let $(D^{♭},D^{♭+})$ be the perfectoid ring obtained by applying the construction in the proposition to the diagram $(A^{♭},A^{♭+})\leftarrow (B^{♭},B^{♭+})\to (C^{♭},C^{♭+})$. Let $(D^{\prime },D^{\prime +})$ be the untilt of $(D^{♭},D^{♭+})$ with respect to $I\cdot W(D^{♭+})$, then $D^{\prime }$ is a perfectoid Tate ring, and we have a commutative diagram

We only need to prove $D$ and $D^{\prime }$ are isomorphic as topological rings. We do this by constructing two continuous homomorphisms being inverses to each others. There is a natural continuous homorphism $D\to D^{\prime }$. We now construct the other.

We first consturct a ring homomorphism $t:W(A^{♭+}{\otimes }_{B^{♭+}}{C}^{♭+})\to D$ by hand. For $a\in A^{♭+}$ and $c\in C^{♭+}$ we require $[a\otimes c]$ to be mapped to $\theta (a)\otimes \theta (c)\in A^{+}{\otimes }_{B^{+}}{C}^{+}\to D$. Every element of $W(A^{♭+}{\otimes }_{B^{♭+}}{C}^{♭+})$ can be written as an infinite sum of elements of the form $p^n[a\otimes c]$, such that for $N\in {\mathbb{Z}}_{+}$ there are only finitely many terms $p^n[a\otimes c]$ with $n\le N$. Such an infinite sum converges when being mapped into $D$ and we only need to check that the above construction is well defined. This can be done by hand by contemplating tensor product and Witt vector arithmetic.

Notice that the kernel of $A^{♭+}{\otimes }_{B^{♭+}}{C}^{♭+}\to A^{♭}{\otimes }_{B^{♭}}{C}^{♭}$ is precisely the ${\varpi }^{♭}$-torsion part of $A^{♭+}{\otimes }_{B^{♭+}}{C}^{♭+}$, so $t$ descends to a map $W(F^{♭})\to D$, where $F^{♭}$ is the image of $A^{♭+}{\otimes }_{B^{♭+}}{C}^{♭+}\to A^{♭}{\otimes }_{B^{♭}}{C}^{♭}$. Let $G^{♭}\subseteq D^{♭}$ be the ${\varpi }^{♭}$-adic completion of $F^{♭}$, then it is rather easy to check that $t$ natually extends to a map $W(G^{♭})\to D$.

Since $D^{♭+}$ is bounded in $D^{♭}$, there exists a positive integer $n$ such that ${\varpi }^{♭n}{D}^{♭+}\subseteq G^{♭}\subseteq D^{♭+}$. As a consequence, if we regard $W(D^{♭+})$ and $W(G^{♭})$ as subrings of $W(D^{♭})$, then we have $[{\varpi }^{♭}{]}^{n}W(D^{♭+})\subseteq W(G^{♭})\subseteq W(D^{♭+})$, thus $W(D^{♭+})[\frac{1}{[{\varpi }^{♭}]}]=W(G^{♭})[\frac{1}{[{\varpi }^{♭}]}]$. Since $t([{\varpi }^{♭}])$ is a unit in $D$, we see that $t$ extends to a map $W(D^{♭+})[\frac{1}{[{\varpi }^{♭}]}]\to D$. It is easy to check that $t$ maps $I$ to 0, so $t$ descends to a map $D^{\prime }=W(D^{♭+})[\frac{1}{[{\varpi }^{♭}]}]/I\to D$.

We check that the map constructed above is continuous. This is clear since the topology on $D^{\prime }$ is determined by the $[{\varpi }^{♭}]$-adic topology on the image of $W(G^{♭})$, and $t$ maps $W(G^{♭})$ into the completion of the image of $A^{+}{\otimes }_{B^{+}}{C}^{+}\to A{\otimes }_{B}C$. It is then easy to check that the two maps are mutual inverses using continuity.

$\square$

We also record an application of the above proposition.

Lemma 2.3. Let $A$ be a commutative ring, let $B$ and $C$ be $A$-algebras equipped with a valuation $v$ (respectively $w$) such that their restrictions to $A$ are equivalent valuations. Then there exists a valuation on $B{\otimes }_{A}C$ extending $v$ and $w$.

Proof. We may assume $A,B$ and $C$ are fields by a simple step of reduction.

In terms of valuation rings the question is equivalent to the following: given valuation rings $A,B,C$ and injective local ring homomorphisms $A\to B$ and $A\to C$ there exists a ring map $B{\otimes }_{A}C\to D$ where $D$ is a valuation ring such that $B\to D$ and $C\to D$ are injective local ring homomorphisms.

To prove this, it suffices to find a specialization $x^{\prime }⇝x$ of points of $\mathrm{Spec}(B{\otimes }_{A}C)$ such that $x^{\prime }$ maps to the generic points of $\mathrm{Spec}(B)$ and $\mathrm{Spec}(C)$ and such that $x$ maps to the closed points of $\mathrm{Spec}(B)$ and $\mathrm{Spec}(C)$. Namely, then we can apply [2] tag 01J8 to find $D$.

Denote ${\kappa }_A$ the residue field of $A$ and similarly for $B$ and $C$. Since ${\kappa }_B{\otimes }_{{\kappa }_A}{\kappa }_C$ is not the zero ring, there exists a point $x$ of $\mathrm{Spec}(B{\otimes }_{A}C)$ mapping to the closed points of $\mathrm{Spec}(B)$ and $\mathrm{Spec}(C)$. Pick any maximal point $x^{\prime }$ of $\mathrm{Spec}(B{\otimes }_{A}C)$ specializing to $x$, in other words, $x^{\prime }$ corresponds to a minimal prime ideal of $\mathrm{Spec}(B{\otimes }_{A}C)$. Since $A\to B$ and $A\to C$ are flat ring maps (as torsion free $A$-modules are flat), the ring maps $B\to B{\otimes }_{A}C$ and $C\to B{\otimes }_{A}C$ are flat as well. By going down for flat ring maps, we see that $x^{\prime }$ maps to the generic points of $\mathrm{Spec}(B)$ and $\mathrm{Spec}(C)$. This finishes the proof.

$\square$

Corollary 2.4. Let $(L,L^{+})\stackrel{f}{\leftarrow }(K,K^{+})\stackrel{g}{\to }(M,M^{+})$ be maps of affinoid rings such that $K,L,M$ are perfectoid fields, $K^{+},L^{+},M^{+}$ are valuation rings, and $f^{-1}(L^{+})=g^{-1}(M^{+})=K^{+}$. Then there exists a perfectoid field $N$ together with an open and bounded valuation subring $N^{+}\subseteq N$, and maps of affinoid rings $(L,L^{+})\stackrel{h}{\to }(N,N^{+})\stackrel{i}{\leftarrow }(M,M^{+})$ such that $h^{-1}(N^{+})=L^{+},i^{-1}(N^{+})=M^{+}$.

Proof. Form the pushout $(D,D^{+})$ of $(L,L^{+})\leftarrow (K,K^{+})\to (M,M^{+})$ in the category of complete affinoid rings as in Proposition 2.2. Choose a valuation $v$ on $L{\otimes }_{K}M$ whose restrictions to $L$ and $M$ are the given ones by Lemma 2.3. Let $\varpi$ be a pseudo-uniformizer of $K$. Let ${\Gamma }_v$ be the value group of $v$, and let $z\in {\Gamma }_v$ be $v(\varpi )$. Let $Q\subseteq {\Gamma }_v$ be the convex subgroup generated by $z$. It is clear that $v(L)$ and $v(M)$ are contained in $Q\cup \{0\}$ since $L$ and $M$ are nonarchimedean fields. Let $Q^{\prime }=\{0\}\cup \{x\in {\Gamma }_v:\text{there exists an }y\in Q\text{ such that }x\le y\}$. It is clear that $v^{-1}(Q^{\prime })$ is a subring of $L{\otimes }_{K}M$ containing $L$ and $M$, so it is equal to $L{\otimes }_{K}M$. Let $v^{\prime }$ be the valuation on $L{\otimes }_{K}M$ defined by $$v^{\prime }(x)=\{\begin{array}{rlrl}& 0& & \text{if }v(x)\notin Q\\ & v(x)& & \text{if }v(x)\in Q\end{array}$$(1)for $x\in L{\otimes }_{K}M$. It is clear that the restriction of $v^{\prime }$ to $L$ and $M$ coincides with the given ones, and $v^{\prime }$ is continuous with respect to the topology on $L{\otimes }_{K}M$ given in Proposition 2.2. $v^{\prime }$ thus extends to a valuation on $D$, which is in fact an element of $\mathrm{Spa}(D,D^{+})$. Considering the residue field of $\mathrm{Spa}(D,D^{+})$ at $v^{\prime }$ we obtain the perfectoid field we want.

$\square$

Corollary 2.5. Let $X\to Y\leftarrow Z$ be a diagram of perfectoid spaces, then the natural map $|X{\times }_{Y}Z|\to |X|{\times }_{|Y|}|Z|$ is surjective, where $|\bullet |$ means underlying topological space.

References

[1]	Peter Scholze, “Etale Cohomology of Diamonds”, available on the author’s homepage.
[2]	The Stacks project authors, “The Stacks project”, https://stacks.math.columbia.edu
[3]	Torsten Wedhorn, “Adic Spaces”, arXiv:1910.05934v1