Note (6/21): There is an issue with the proof of Lemma 1.7 below, but I’m pretty confident Kiran and I will find a fix. In any case, you should never take blog math too seriously!
In this post I want to announce some new foundational developments in p-adic geometry. More precisely, I want to introduce and motivate the category of sousperfectoid spaces defined below. This is a certain intrinsically defined full subcategory of analytic adic spaces over containing all perfectoid spaces and all smooth rigid analytic spaces. These spaces turn out to be extremely well behaved with respect to étale morphisms and pro-finite étale morphisms, and certain other favorable properties of sousperfectoid spaces allow us to non-trivially transfer consequences of the almost purity theorem from perfectoid spaces to smooth rigid spaces, with some striking consequences.
Some version of this blog post will appear in the next revision of my period maps paper; a more systematic development of sousperfectoid spaces and their applications (of which many are expected) will appear in a joint article of myself and Kedlaya. Since these spaces may be of general interest to the community, however, I wanted to describe them in public sooner rather than later; hence this post.
Nb. in what follows, KL1 and KL2 refers to Kedlaya-Liu’s Relative p–adic Hodge theory I & II papers, SW refers to Scholze-Weinstein’s paper on p-divisible groups, and HK refers to my forthcoming paper with Kedlaya.
OK, let’s do some math! In what follows, all Tate rings will be complete Tate rings over in which is topologically nilpotent. If is a uniform Tate ring, we always regard it as being normed by its spectral norm.
The key ring-theoretic definition is as follows.
Definition 1.1. A Tate ring is sousperfectoid if there exists a perfectoid Tate ring and a morphism of Tate rings which admits a continuous (or equivalently, bounded) -linear splitting .
The motivation for the name should be clear (if you speak a little French, at least): lives under some perfectoid Tate ring in a meaningful way. Admittedly, I also like the auditory suggestion of SUPERfectoid. Note that any perfectoid Tate ring is sousperfectoid.
Proposition 1.2. Any sousperfectoid Tate ring is stably uniform. In particular, any Huber pair with sousperfectoid is sheafy.
Proof: Let be as in the definition, so is perfectoid, and in particular stably uniform. The result now follows from the following observation of Kedlaya-Liu (KL1, Remark 2.8.12): if is a stably uniform Tate ring and is a morphism of Tate rings which admits a continuous -linear splitting, then is stably uniform as well.
Proposition 1.3. If is sousperfectoid, then each of the following rings is sousperfectoid as well:
i. The coordinate ring for any rational subset.
ii. Any finite étale -algebra .
iii. and .
iv. for any profinite set .
With regards to ii., we remind the reader that if is any uniform Tate ring, then any finite étale -algebra inherits a canonical compatible structure of uniform Tate ring such that is continuous: this follows immediately upon combining (KL1, Lemma 2.8.14) and (KL1, Prop. 2.8.16(b)). We also point out that for a general stably uniform Tate ring , the question of whether the stable uniformity (or even just the sheafyness) of enjoys permanence properties analogous to ii. and iii. here is wide open.
Proof: Choose , and witnessing the sousperfectoid nature of as in Definition 1.1.
For i., set , so induces a map . One easily checks that extends to a bounded splitting of the map induced by . Since is affinoid perfectoid, is a perfectoid Tate ring, and so the result follows.
For ii., we simply observe that extends uniquely to a bounded -linear splitting of the obvious map
induced by , while on the other hand is finite étale over and thus perfectoid by the almost purity theorem.
For iii., note that extends uniquely to a bounded -linear splitting of the obvious map
induced by , and that is perfectoid, so is sousperfectoid. To descend to , observe that admits a canonical bounded -linear splitting, so we get a bounded splitting of the composite map
Finally, iv. is easy and left to the reader (hint: for fixed , is functorial in).
Now we globalize the notion of being sousperfectoid in the obvious way.
Definition 1.4. A sousperfectoid space is an adic space which admits an open cover by affinoid adic spaces with each a sousperfectoid Tate ring.
By Proposition 1.3.i, any open subspace of a sousperfectoid space is sousperfectoid. We also caution the reader that if is a sheafy Tate-Huber pair and is an affinoid adic space which is sousperfectoid in this sense, it’s not clear whether the (completion of the) underlying Tate ring is necessarily sousperfectoid (though we don’t know any counterexample).
So far, this is mathematics which has been in my head since December 2015. The dam broke last Friday, though, when I realized how to prove Lemma 1.5 and Lemma 1.7 below.
Recall that for any adic space , Kedlaya-Liu have defined an étale site (KL1, Def. 8.2.19) and a pro-étale site (KL1, Def. 9.1.4). In general, the objects of are only preadic spaces étale over , and the objects of are certain (equivalence classes of) projective systems of preadic spaces étale over . (Since preadic spaces will only play an intermediate role in the following discussion, I won’t review them here; cf. (KL1, §8.2).) For sousperfectoid spaces, however, the étale and pro-étale sites turn out to be extremely well-behaved, as we now demonstrate.
Lemma 1.5. Let be a sousperfectoid space, and let be a preadic space equipped with an étale morphism . Then is an honest adic space, and moreover is sousperfectoid.
Proof: The claim is local on , so we may assume that factors as where the ‘s are morphisms of preadic spaces (a priori) such that is an open immersion, is finite étale, and is an open immersion. Now is open in the sousperfectoid space , and hence is honest and sousperfectoid. Arguing locally on , Proposition 1.3.ii and (KL1, Lemma 8.2.17(a)) together imply that is honest and sousperfectoid. But then is open in , so we’re done.
Proposition 1.6. Let be a nonarchimedean field which is sousperfectoid. Then any smooth rigid analytic space over is sousperfectoid.
There is also an obvious relative version of this result, whose statement and proof we leave to the interested reader.
Proof: Since is sousperfectoid by assumption, repeated use of Proposition 1.3.iii shows that any unit polydisk is sousperfectoid.
Now let be any smooth rigid space over , so we can choose a covering of by smooth affinoids together with étale maps . Since is étale with sousperfectoid target, the previous proposition implies that each is sousperfectoid as well, so is sousperfectoid as desired.
In (HK) we’ll show that every nonarchimedean field (with topologically nilpotent) is sousperfectoid; the idea is that admits a continuous splitting, which is not entirely obvious.
Next we bootstrap from étale to pro-étale morphisms. The key lemma is as follows.
Lemma 1.7. Fix a sousperfectoid Tate ring , and let be a directed system of Tate rings finite étale over . Let denote the completion of for the topology making open and bounded. Then is sousperfectoid.
Proof: Choose a perfectoid Tate ring , a continous ring map , and a bounded -linear splitting . Set , so is perfectoid by almost purity. Furthermore, extends canonically as before to a bounded -linear splitting of the evident map , compatibly with varying . Passing to the direct limit, we get the same properties for the evident maps
where we give the obvious topology; in particular, is bounded since it carries the open bounded subring into the bounded subset . Since everything is continous we may extend and uniquely to the completions of these rings, getting analogous maps
But is perfectoid, so these maps verify the sousperfectoid nature of
The next lemma is an easy weakening of (SW, Lemma 2.4.5), and accounts for all uniqueness statements we’ll make concerning inverse limits of adic spaces.
Lemma 1.8. Let be a filtered inverse system of adic spaces with qcqs transition maps, and let be an adic space with a compatible family of morphisms such that
in the sense of (SW, Def. 2.4.1). If is stably uniform, then
for any stably uniform adic space . In particular, a stably uniform adic space with is unique up to unique isomorphism.
Theorem 1.9. Let be a sousperfectoid space. Let be any object in , so we may choose a pro-étale presentation of as a cofiltered inverse limit of (pre)adic spaces étale over ; note that by Lemma 1.5 each is a sousperfectoid adic space étale over . Then there is a sousperfectoid space over with compatible maps such that
The space is unique up to unique isomorphism, independently of the choice of pro-étale presentation of . The functor defines a fully faithful embedding of into the slice category of adic spaces over .
Proof: Taking into account Lemma 1.5 and Lemma 1.7, this immediately follows from Lemma 1.8 by an easy gluing argument; we leave the details to the interested reader.
On account of this theorem, when is sousperfectoid there is no harm in conflating a formal pro-system with the associated space ; we will often write instead of .
Combining this theorem with Proposition 1.6 shows that the pro-étale site of any smooth rigid space over (for any nonarchimedean field ) is extremely well-behaved. In particular, we get the following theorem as a special case of Theorem 1.9.
Theorem 1.10. Fix a nonarchimedean field in which is topologically nilpotent, and let be a filtered inverse system of smooth rigid analytic spaces over with finite étale transition maps. Then there exists an adic space together with compatible morphisms to the tower of ‘s such that
The space is sousperfectoid. In particular, has an open covering by affinoid adic spaces associated with stably uniform Tate rings, and is unique up to unique isomorphism; furthermore, has a well-behaved étale site and pro-étale site, and it represents the functor on stably uniform adic spaces.
This result has many applications. For example, it instantly gives an infinite-level adic Shimura variety with
for any Shimura datum and any (reasonable) tame level . It also shows that the infinite-level Rapoport-Zink space constructed in (SW, Theorem D) is an honest adic space.
Next we record a necessary and sufficient condition for a Tate ring to be sousperfectoid which doesn’t mention perfectoid rings.
Definition 1.11. A morphism of Tate rings is faithfully profinite étale if is isomorphic to the completion of some filtered direct limit of faithfully finite étale -algebras for the spectral seminorm (or equivalently, for the topology making an open and bounded subring of ).
Note that if is faithfully profinite étale, then is uniform if and only if is injective.
Proposition 1.12. Let be a uniform Tate ring. Then is sousperfectoid if and only if every faithfully profinite étale morphism admits a continuous -linear splitting.
Proof: “If” is easy, since one can find such an with perfectoid. “Only if” will be proved in (HK).
The following result makes it easy to find examples of uniform Tate rings which are not sousperfectoid.
Proposition 1.13. If is any sousperfectoid Tate ring, then . In particular, is seminormal.
Proof: The former will appear in (HK). For the latter, use (KL2, Corollary 2.7.5).
I want to give a few more constructions and examples involving sousperfectoid spaces. Note that if is sousperfectoid, then by Proposition 1.3.iii the relative polydisk
is well-defined and sousperfectoid for any . This suggests the following definition.
Definition 1.14. A morphism of sousperfectoid spaces is smooth (resp. pro-smooth) if we can find open coverings and such that for all and , the restriction factors as a composition
for some where is the obvious projection and is étale (resp. realizes as for some ).
A morphism of sousperfectoid spaces is (pro-)smooth proper if it is (pro-)smooth, quasicompact and universally specializing.
When is a sousperfectoid rigid space, this specializes to the usual notions of smooth and smooth proper morphisms in rigid geometry. It’s easy to check that smooth and pro-smooth morphisms are well-behaved with respect to fiber products and are stable under base change along an arbitrary morphism of sousperfectoid spaces.
Example 1.15. Let be a smooth rigid space over a nonarchimedean field of characteristic zero, and let be a family of abeloid spaces, i.e. a family of smooth proper (connected) rigid analytic groups over . Then the multiplication-by-p maps on are finite étale, so we get the universal cover
as a sousperfectoid space over , and the natural morphism is pro-smooth proper. When is a geometric point and has good reduction, it’s not hard to check that is a perfectoid space. In particular, typically has perfectoid geometric fibers. However, the total space of need not be perfectoid.
Note that we can also form the “physical” Tate module
as a sousperfectoid space pro-finite étale over .
Example 1.16 (The dual of Example 1.15.) We can define the notion of a family of abeloid varieties over any sousperfectoid space : it is a smooth and proper morphism of sousperfectoid spaces with connected geometric fibers, together with morphisms
making into a commutative group object over . Using this notion, one can show that infinite-level adic Shimura varieties associated with PEL Shimura data actually represent an obvious functor on the category of sousperfectoid spaces.
More generally, we can make sense of families of smooth (or smooth proper) rigid spaces over an arbitrary sousperfectoid base.
Feel free to start improving your life with sousperfectoid spaces today!