Tilting is the key construction for perfectoid rings which allows to naturally associate a perfectoid (perfect) ring in characteristic to a perfectoid ring of any characteristic (in particular, ). Thus, some of the geometrical problems in characteristic might be translated to characteristic , solved there and the result can be translated again to characteristic . This was done with the weight-monodromy conjecture by Peter Scholze. The following blog post (as was the previous) is based mostly on notes of Jared Weinstein from Scholze’s lectures at Berkeley.
Let be a perfectoid Tate ring. The tilt of is
with the inverse limit topology. A priori this is only a topological multiplicative monoid, but we can endow it with a ring structure, by defining the addition
We have to check that it is well-defined.
Lemma 33 The limit exists and defines a ring structure, which makes into a topological -algebra that is a perfect uniform Tate ring. The subset of power-bounded elements is given by the topological ring isomorphism
where is a pseudo-uniformizer which divides in . Moreover there exists a pseudo-uniformizer with in which admits a sequence of -th power roots , giving rise to an element
which is a pseudo-uniformizer of . We have .
Proof: By definition is perfect. Let be a pseudo-uniformizer of . Let us check that the maps
are isomorphisms. The key is that any sequence lifts uniquely to a sequence . This is standard: , where is any lift of . Hence is a well-defined ring.
Now assume in . We construct . In fact the preimage of under is an element with the right properties. It is congruent to modulo and therefore it is also a pseudo-uniformizer. Then the projection of to the -th coordinate is the desired pseudo-uniformizer of .
As in the last line of the proof above, we have a continuous multiplicative (but not additive) map
which projects onto the -th coordinate. We denote it by . This projection defines a ring isomorphism . Let us note the following fact which follows from this isomorphism, the nilpotence conditions and a fact that an open subring is integrally closed in (resp. ) if it its image in the quotient (resp. ) is integrally closed.
Lemma 34 The set of rings of integral elements is in bijection with the set of rings of integral elements by . Also .
We say that a Huber pair is perfectoid if is perfectoid. We have the following theorem due to Scholze (see also Kedlaya-Liu’s work)
Proposition 35 Let be a perfectoid Huber pair. Then for all rational subsets , the is again perfectoid. In particular is stably uniform, hence sheafy (by Buzzard-Verberkmoes, as explained earlier).
The following two fundamental theorems are due to Scholze. We leave the without proof. A reader should consult Scholze’s “`Perfectoid spaces”‘.
Theorem 36 Let be a perfectoid Huber pair with tilt . There is a homeomorphism
which sends to such that that . This homeomorphism preserves rational subsets.
Theorem 37 Let be a perfectoid ring with tilt . Then there is an equivalence of categories between perfectoid algebras and perfectoid -algebras via .