Let be the category of perfectoid spaces in characteristic , equipped with its pro-étale topology. Any gives rise to a sheaf on . By definition, a diamond is a sheaf on which admits a relatively representable surjection which pulls back to a pro-étale cover of under any map ; we say admits a pro-étale cover by the representable sheaf . We can also view as the quotient of by the pro-étale equivalence relation (note that is representable, and that the maps are pro-étale). Morphisms of diamonds are simply sheaf maps . Let denote the category of diamonds.
Since the category of perfectoid spaces (in whatever characteristic) admits fiber products, it’s somewhat formal to check that for any maps of diamonds , the fiber product exists as a diamond. More strangely, however, we also have “baseless” products in :
Theorem. For any two diamonds , the product is canonically a diamond, where the product is taken in the category of sheaves of sets on .
This is another sense in which diamonds are a “topological” sort of gadget: one finds in practice that most categories of ringed spaces (e.g. schemes, complex analytic spaces, rigid analytic spaces, etc.) have a final object, so “baseless” fiber products can’t exist. But diamonds, like topological spaces, don’t have any sort of structure sheaf, and they don’t have a final object.
Curiously, the above theorem is never stated explicitly in the Berkeley notes; it’s certainly implicit, however, since e.g. the diamond is mentioned repeatedly. Anyway, the key point in the proof of this theorem is the following claim:
Proposition. Let , be two perfectoid Tate-Huber pairs in characteristic . Then for a certain perfectoid space whose formation is bi-functorial in the data of and .
Proof. We begin by defining . (Hat tip to Jared, who told me the definition of , from which I reverse-engineered everything else below.) Set , and let be the integral closure of in . Choose pseudouniformizers , and set (here we abbreviate by , and likewise for ; we continue to use this abbreviation in what follows). Let be the -adic completion of . Then we define
To see that is perfectoid, consider the open subsets
of . It’s easy to see that and . Furthermore, is a rational subset of , so we may describe it explicitly as an affinoid adic space. Precisely, let and let be the integral closure of in ; give the -adic topology, and give the topology making an open subring. (Note that the -adic, -adic, and -adic topologies on all coincide.) Let and be the completions of and for these topologies. Then
One checks directly that is a complete perfect uniform Tate ring, which exactly characterizes the perfectoid rings in characteristic . Therefore each is perfectoid, so is perfectoid.
Next we verify that has the claimed property at the level of sheaves. Note that we have natural continuous ring maps inducing morphisms and , and the latter morphisms restricted to factor through natural morphisms and , respectively.
Choose any , and suppose we’re given a morphism , i.e. an element of . Composing this morphism with the morphisms , we get morphisms , , i.e. an element of . This association is clearly functorial in , and so defines a map of sheaves
To go the other way, assume that is affinoid perfectoid, say for some perfectoid Tate-Huber pair in characteristic . Suppose we’re given an element of . This is equivalent to the data of a pair of continuous ring maps , such that . Consider the evident ring map . One checks directly that , using the obvious inclusion
together with the fact that is integrally closed in . Choose a pseudouniformizer ; replacing by if necessary, we may assume that divides both and in . This immediately implies that the ring map induces compatible ring maps , so passing to the inverse limit we get a continuous ring map . Passing to Spa’s, we may consider the composite
where the lefthand arrow is the evident open immersion. Since carries to a unit in , factors through the open subset , so we get a map , i.e. an element of . Summarizing our efforts in this paragraph so far, we’ve described a natural map of sets
for any affinoid perfectoid . One checks directly that for any map of affinoid perfectoids, the associated diagram
commutes, so extends to a morphism of sheaves. Finally, one checks that and are naturally inverse by staring hard at the above constructions.
Though it’s not strictly necessary for the above proof, let me note that is a perfectoid Huber ring in Gabber-Ramero’s sense, and hence is sheafy.