用户: Scavenger/Recording Some Facts/Fiber Product of Perfectoid Spaces
Our definitions and conventions will follow [1].
Here is our main theorem:
Theorem 0.1. Let be perfectoid spaces, then the fiber product exists in the category of adic spaces, and is a perfectoid space.
which is an immediate consequence of the following:
Proposition 0.2. Let be affinoid rings (or Huber pairs) such that and are perfectoid Tate rings. Consider the topology on making it into a topological ring such that the image of in is open and it has the -adic topology (as an open subring with the subspace topology), where is an arbitrary pseudo-uniformizer (It is not hard to check that this topology exists and is unique). Let be the completion of with respect to this topology, let be the integral closure of the image of in , and let be the completion of . Then is a perfectoid Tate ring, and is a pushout of in the category of complete affinoid rings.
Proof. It is clear that is a complete affinoid ring, and it is a pushout of in the category of complete affinoid rings, so it remains to show is a perfectoid Tate ring.
We first prove the case where . In this case we only need to show that is perfect by virtue of [1] Proposition 3.5. This can be easily checked by hands using the fact that and (and hence the image of the latter in the former) are perfect.
We now turn to the general case. Let be the tilt of , and similarly for and , respectively. Let be a pseudo-uniformizer with in which admits a sequence of -power roots giving rise to a pseudo-uniformizer of . Let be an ideal primitive of degree 1, such that is the untilt of with respect to , in the sense of [1] Theorem 3.17. Then is the untilt of with respect to and similarly for . By convention if is a perfectoid affinoid ring with tilt we denote the natural quotient map by .
Let be the perfectoid ring obtained by applying the construction in the proposition to the diagram . Let be the untilt of with respect to , then is a perfectoid Tate ring, and we have a commutative diagram
We only need to prove and are isomorphic as topological rings. We do this by constructing two continuous homomorphisms being inverses to each others. There is a natural continuous homorphism . We now construct the other.
We first consturct a ring homomorphism by hand. For and we require to be mapped to . Every element of can be written as an infinite sum of elements of the form , such that for there are only finitely many terms with . Such an infinite sum converges when being mapped into 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 is precisely the -torsion part of , so descends to a map , where is the image of . Let be the -adic completion of , then it is rather easy to check that natually extends to a map .
Since is bounded in , there exists a positive integer such that . As a consequence, if we regard and as subrings of , then we have , thus . Since is a unit in , we see that extends to a map . It is easy to check that maps to 0, so descends to a map .
We also record an application of the above proposition.
Lemma 0.3. Let be a commutative ring, let and be -algebras equipped with a valuation (respectively ) such that their restrictions to are equivalent valuations. Then there exists a valuation on extending and .
Proof. See the discussion around this problem on mathoverflow: https://mathoverflow.net/questions/103945/valuations-on-tensor-products
I record the answer of user “Johan” (who being Johan de Jong, as I guess without evidence) verbatim for the sake of completeness:
(We should first make a step of reduction to assume and are fields.)
In terms of valuation rings the question is equivalent to the following: given valuation rings and injective local ring homomorphisms and there exists a ring map where is a valuation ring such that and are injective local ring homomorphisms.
To prove this, it suffices to find a specialization of points of such that maps to the generic points of and and such that maps to the closed points of and . Namely, then we can apply [2] tag 01J8 to find .
Corollary 0.4. Let be maps of affinoid rings such that are perfectoid fields, are valuation rings, and . Then there exists a perfectoid field together with an open and bounded valuation subring , and maps of affinoid rings such that .
Corollary 0.5. Let be a diagram of perfectoid spaces, then the natural map is surjective, where 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 |