Perfectoid Spaces: Theory and Applications

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1 Perfectoid Spaces: Theory and Applications Caleb McWhorter April 18, Historic Background In 2007 when Scholze attended Bonn as an undergraduate, Michael Rapoport gave him the following problem to consider: Let X be a smooth projective scheme over Q p. Fix i 0 and let l p be prime. Consider the Gal(Q p /Q p ) representation V = H i ét (X Q p, Q l ). It is known that there is a weight decomposition: if Φ Gal(Q p /Q p ) is a geometric Frobenius, then V = 2i j=0 V j, where Φ acts through Weil numbers of weight j on V j. If X has semistable reduction, this is due to Rapoport Zink while the general result (via reduction to the semistable case) is due to de Jong. There is also a monodromy operator N : V V(±1), the Tate twist, coming from the action of the inertia subgroup. In particular, N : V j V j 2 as Weil numbers on the Tate twist are of weight 2. The conjecture (known as the Weight Monodromy Conjecture) is for j = 0,..., i, N j : V i+ j V i j is an isomorphism. The Weight Monodromy Conjecture is critical to understanding local factors of Hasse Weil zeta functions at places of bad reduction. The conjecture is known in dimension 1 (by reducing to abelian varieties/curves and using Néron models/semistable models) and dimension 2 due to Rapoport Zink Jong. It is also known in equal characteristic p, i.e. over F p ((t)), proved in Deligne s Weil paper and uses the fact that L functions over function fields have nice properties. Rapoport suggested to try to reduce to the case of equal characteristic after some base change to a very ramified extension K/Q p. The idea is that if X/O K is integral (or semistable) model of X Qp K, then X Spec OK Spec O K /p lives over O K /p F q [t]/t e, where e is the ramification index of K/Q p. So in some sense, if e is sufficiently large, then O K /p is almost F p t. But this cannot be the whole story as even if e is large, one still does not know how to deform X Spec OK Spec O K /p from O K /p = F p [t]/t e to F p t (though there is no obstruction in dimension 1 modulo some technicalities). One would need to know how to relate V = H i ét (X Q p, Q) acting on Gal(Q p /Q p ) to H i ét (X F p ((t)), Q l) acting on Gal ( F p ((t)) sep /F p ((t)) ), where X /F p ((t)) is the generic fiber of the deformation. In the semistable case, one could relate them using log geometry. Scholze then read the following theorem of Fontaine Wintenberger which suggested (after a great deal of thinking and after considering some work of Fontaine and Gabber Ramero) what to do in the relative case. Theorem 1.1 (Fontaine Wintenberger). The absolute Galois groups of Q p (p 1/p ) and F p ((t)) are canonically isomorphic. 1 1 To be explicit, Q p (p 1/p ) = Q p ({ pn p: n 1}) and Fp ((t)) = Frac(F p t ). 1

2 So somehow after adjoining lots of pth power roots to a mixed characteristic object one obtains an object in equal characteristic. The rough idea of the proof is to begin with the completion of Q p (ζ p ) and n 1 F p ((t 1/pn )) under the p adic and t adic norm, respectively. Using Krasner s Lemma and the fact that absolute Galois groups are not changed by purely inseparable extensions, one can show that the absolute Galois groups of Q p (p 1/p ) and Q p (ζ p ) are isomorphic. This follows mutatis mutandis for F p ((t)) and n 1 F p ((t 1/pn )). The proof then constructs a functor ( ) from finite extensions of k to finite extensions of k and showing this is an equivalence of categories. Since this functor will be discussed later, we say a bit more about its construction here. Let K be a finite extension of k. Since k is Henselian, the valuation (equivalently, the norm) on k extends uniquely to a valuation on K. Thus, we can write K = O K = {x K : x 1} = {x K : v(x) 0}. This is a local ring with p m. Furthermore, char K /p = p so that Frob : K K is a surjective ring homomorphism. Then define K := O K = lim K /p = Frob (x n) O K /p: x p n+1 = x n so that K is a topological ring using the inverse limit topology (simply give each K /p the discrete topology). Consider the map ( ) : K K via (x n ) n lim x n pn, where x n K is any lift of x n K /p. To see that this is well defined, suppose x y mod p n. But then x = y + p n t so that it must be that x p y p mod p n+1. Therefore by simply using the Binomial Theorem ( ) ( ) ( ( ) ) p p 1 p x p = y p + y p 1 (p n t) + + y(p n t) p 1 + (p n t) p = y p + p n+1 y p 1 t p 1 Then if x n, x n are two lifts of (x n ) K, then x n pn x n pn mod p n+1. But then x n pn is a Cauchy sequence (hence convergent) limit independent of the choice of lifts. Furthermore, ( ) induces an isomorphism of topological monoids K lim K. Hence, K is a domain and we are able to write K = Frac(K ). Frob The exciting theorem of Scholze s initial paper 2 Introduction There are many geometric spaces that one learns about in their mathematical career. 2 First, one typically encounters topological spaces. These are the most basic geometric spaces. The typical examples one sees early on are R n and C n. While basic topology provide general language and framework for the various properties of spaces and can short cut results which could be shown more directly, they lack the power of alternative theories. Next, one typically sees manifolds which equip more general spaces with a geometric structure by requiring they locally look like some model space R n in the real manifold case and C n in the complex manifold space. This has the added benefit of possessing a more natural language to discuss the properties and restrictions of functions on the space. Indeed, the structures themselves can be phrased in terms of transition functions between charts on the manifold. However, it is tedious to talk about the spaces in terms of charts. Next, one learns the more general language of ringed spaces, a pair 2 This introduction closely follows the wonderful background motivation of [Wei17] for the Arizona Winter School, which the author was lucky to attend. For more details, see this paper. n 0 2

3 (X, O X ), where X is a topological space and O X is a sheaf of rings on X such that locally on X the pair (X, O X ) is isomorphic to some desired geometric space (the model space) with its sheaf of functions. This has the natural advantage of making it simpler to define morphisms between two ringed spaces. This naturally leads to the notion of schemes. Typically, one first encounters affine schemes which are spectra of rings. At first, these spaces seem strange as they are typically not Hausdorff. But there are advantages to working with these spaces. For example, integral schemes have a generic point. This reduces proving many results to simply proving it to some generic fiber as many properties true on the generic fiber are generically true on the whole scheme. This is common in number theory: in Spec Z, if a property holds over the generic point Spec Q, then it holds at almost all special points of Spec F p. One should also note the role (or in many cases lack of) nilpotents in schemes, aka reducedness. For example: (i.) A scheme is reduced if and only if none of the stalks have nonzero nilpotents (ii.) If A is a reduced ring, then Spec A is reduced (iii.) Two functions on global sections on a reduced scheme which agree on points are equal. (iv.) A scheme is integral if and only if it is reduced and irreducible. Nilpotents in a ring place restrictions on the associated ringed space. The reader should note the parallels of nilpotents in the ring to the role of nilpotents to come in Huber rings. Next came formal schemes, where the model spaces are Spf A, the formal spectrum, where A is separated and complete for the I adic topology for some (finitely generated) ideal I. That is, as topological rings, we have A lim A/In, e.g. Z p or Z T. [The ideal I need be finitely generated, otherwise problems can arise. For this reason, formal schemes are typically only defined in the noetherian case.] These allow one to study what happens in infinitesimal neighborhoods of schemes. Moreover, formal schemes come with natural notion of completions. Finally, one encounters rigid analytic spaces. Consider the nonarchimedean fields Q p, C p, k((t)). Each is a complete metric field. One would then expect there to be a good theory of manifolds for these spaces. Which ringed spaces (X, O X ) should then be the model spaces? One could naïvely say that X should locally look open subsets of K n and then define O X to be its sheaf of continuous K valued functions. There is a clear disadvantage with this approach: since the space X is totally disconnected, it becomes too easy to glue functions together. If one defined X = P 1 were defined this way, then H 0 (X, O X ) K and Hét 0 (X, A) A, violating GAGA and the Comparison Isomorphisms, respectively. The model spaces for rigid analytic spaces are Spm A, the maximal spectrum, where A is an affinoid K algebra 3 The topology that Tate placed on these spaces was not the topology coming from K n. Indeed, the topology was not a topology at all but rather a Grothendieck topology: a collection of admissible opens and a notion of admissible coverings. A rigid analytic space over K is then a pair (X, O X ), where X is a set carrying a Grothendieck topology and O X is a sheaf of K algebra. This sheaf of K algebras is locally isomorphic to the model Spm A. This theory was successful in recreating desired theorems: a rigid analytic GAGA theorem, a comparison theorem, and various theorems involving curves, Shimura varieties, and moduli spaces. However, there are shortcomings to this approach mainly resulting from the approach being made specifically for 3 A Tate algebra, K T 1,..., T n is simply the K algebra of power series in K T 1,..., T n with coefficients tending to 0, i.e. the completion of K[T 1,..., T n ] with respect to the Gauss norm. An affinoid K algebra a quotient of a Tate algebra. 3

4 the K affinoid algebra which Tate studied. The issues in rigid analytic spaces begin with the topologies themselves. Take the examples given by Weinstein in [Wei17]: Example 2.1. Let X = Spm K T is the rigid analytic closed unit disk and Y be the disjoint union of the open unit disc U along with the circle S = Spm K T, T 1. There is an open immersion Y X which is a bijection on the level of points. However, the immersion is not an isomorphism since U is not a finite union of affinoid subdomains, meaning it cannot be an admissible open in X. Example 2.2. Let X = Spm K T and α K be transcendental over K. Define Y X to be the union of all affinoid subdomains U which do not contain α, i.e. α does not satisfy the collection of inequalities among power series defining U. The open immersion Y X is a bijection on points but fails to be an isomorphism. In each example, there is an open immersion which is a bijection on points but is not an isomorphism. The issue is that there are hidden points of X which Y is missing. It will turn out that as adic spaces, Y is simply the complement of X of a single point. In fact, adic spaces will encompass the categories of rigid analytic spaces as well as both ordinary and formal schemes. This is especially advantageous as one can therefore easily pass between these categories. But what kind of model space do we want for a theory of adic spaces? We certainly will want to allow Z p T, Q p T, or generally and ring with the discrete topology. In the case of Z p T, the topology is generated by a finitely generated ideal. In the case of Q p T, there is an open subring, namely Z p T, whose topology is generated by p. These basic examples serve as a model for the types of (topological) rings we want to serve as our model spaces. 3 Huber and Tate Rings Recall that a topological ring R is a ring which is also a topological space such that both addition and multiplication are continuous maps R R R, where R R is given the product topology. That is, R is a additive topological group and a multiplicative topological semigroup. Unless otherwise stated, let A denote a topological commutative ring. Definition 3.1 (Valuation Ring). Let K be a field and Γ a ordered abelian group. A valuation on K is a map : K Γ {0} satisfying (i) x = 0 x = 0. (ii) xy = x y (so induces a group homomorphism K Γ). (iii) x + y max{ x, y }. The total ordering on Γ is extended to Γ {0} by declaring 0 γ for γ Γ and γ 0 = 0 = 0 γ for all γ Γ {0}. The group Γ is called the value group of. Valuation rings are an example of local non noetherian rings. R := {x K : x 1} is a local ring with maximal ideal m := {x K : x < 1}. The ring R is the valuation ring corresponding to the valuation satisfying the property that for all x K, x or x 1 is in R. R is noetherian if and only if Γ is trivial (in which case K = K) or Γ is order isomorphic to Z. 4

5 Remark 3.1. Valuations are written multiplicatively instead of additively because we want to think of valuations as higher rank analogues of non Archimedean norms (these correspond to valuations rings that have Krull dimension 1). The term higher rank norm is less common in literature. Definition 3.2 (Adic Ring). A is called an adic ring if there exists an ideal I of A such that {I n : n 0} forms a basis of open neighborhoods of 0. The ideal I is called the ideal of definition. Remark 3.2. In EGA, adic topological rings are complete by definition a restriction we do not require here. Example 3.1. (i) Any commutative ring A can be made into an adic ring. Simply give A the discrete topology and set I = (0). (ii) One can extend the previous example, given any commutative ring A with an ideal I, there is a unique topology on A making it into a topological ring such that {I n : n > 0} forms a neighborhood basis of 0. (iii) The completion of a local ring (R, m) with respect to the m adic topology is an adic space. (iv) As an example which one will understand later, if K is a perfectoid field with valuation ring K, then K is an adic ring. [Later, we shall see that this means K is a Huber ring.] The subring K is an open subring of K (despite being closed in K) under the natural metric topology on K (this follows by the nonarchimedean property). The induced topology on K is then adic with ideal of definition (a), where a K. Hence, ideals of definition need not be unique. Remark 3.3. The set of open prime ideals of an adic ring A, denoted Spf(A) is the local object for the study of formal schemes. There is a fully faithful functor from the category of schemes to the category of formal schemes carrying Spec(A) to Spf(A) by considering A as an adic ring with the discrete topology. Definition 3.3 (Huber Ring). A topological ring A is called a Huber ring 4 if there exists a open subring A 0 of A such that the induced topology on A is adic for some finitely generated ideal I of A 0. The ring A 0 is called the ring of definition of A and I is an ideal of definition of A. Remark 3.4. Called f adic rings by Huber. Scholze redefined. f stands for finite. Remark 3.5. I is an ideal of definition: we mean I is an ideal of some open subring of A whose induced topology is the I adic topology. Example 3.2. (i) Any commutative ring A with the discrete topology is a Huber ring with ideal of definition I = (0). (ii) Any adic ring with a finitely generated ideal of definition is automatically a Huber ring. Furthermore, any adic ring which is Huber possesses a finitely generated ideal of definition, though this take far more work. 4 Huber in [Hub93] originally called these f adic rings. However, Scholze called these Huber Rings in [? ] and the terminology has caught on. 5

6 (iii) If K is a perfectoid field, then K is a Huber ring with ring of definition K and ideal of definition taken to be the ideal generated by any choice of pseudouniformizer. Explicitly, equipping K with the topology induced by the nonarchimedean norm, K is an open subring of K whose topology is π adic, where 0 π K is arbitrary. (iv) If A is any nonarchimedean normed K algebra, where K is perfectoid, then A is Huber with ring of definition A 0 = {x A: x 1} and ideal of definition pa 0 for any p K. In particular, the Tate algebra is Huber: K x 1,..., x n = f K x 1,, x n : f = a n x n with a n 0 Definition 3.4 (Tate Ring). A Huber ring is called a Tate ring if it has a topologically nilpotent unit. Proposition 3.1. Let A be a Tate ring with ring of definition A 0 and topological nilpotent ϖ. Then (i) ϖ n A 0 for n 0. (ii) If ϖ A 0, then the topology on A 0 is the ϖ n A 0 adic topology. (iii) A = A 0 [ ϖ 1 ] for ϖ n A 0. If in addition the topology on A is Hausdorff, then n>0 ϖ n A 0 = {0} so that we can define a norm : A R 0 via a = inf{2 n : a ϖ n A 0, n Z}. Then if A is in addition complete in the ϖ n A 0 adic topology, then A is a Banach ring (a complete normed topological ring). Definition 3.5 ((Power) Bounded). A subset S A is bounded if for every open neighborhood U of 0, there is an open neighborhood V of 0 such that {vs: v V, s S } U. An element a A is power bounded if the set {a n : n > 0} is a bounded set. The set of power bounded elements is denoted A. The set of topologically nilpotent elements of A is denoted A. In particular, if K is a rank 1 valued field, then K is the valuation ring of K and K is the maximal ideal. Though we shall not prove it here, the following facts are worth noting: 1. If A is adic then A is bounded. 2. If A is a Huber ring then any ring of definition is bounded. 3. Any topologically nilpotent element of A power bounded. Furthermore, A is a ring (its is the filtered direct limit of bounded open subrings of A) that is integrally closed in A. Example 3.3 (Boundedness). (i) Every finite subset of A is bounded. (ii) If A = C, then A = {z C: z 1} and A = {z C: z < 1}. (iii) If K is a valued field, we can give it the topology endowed by. The R is a boiunded subring of K. n 6

7 (iv) Every subset of a (power) bounded subset is (power) bounded. (v) A bounded topologically nilpotent subset of A is power bounded. Furthermore, every finite union of (power) bounded subsets is (power) bounded. Proposition 3.2. Let A denote a Huber ring and A 0 a subring of A. Then the following are equivalent: (i) A 0 is a ring of definition of A. (ii) A 0 is an open and bounded subring of A. Then A is the union of all rings of definition of A. Remark 3.6. Note by the preceding proposition, we can take A to be the ideal of definition whenever A is bounded. However, this is not generally the case. For example, take A = Q p [t]/(t 2 ), the ring of dual numbers over Q p. In this case, the ring of definition is Z p Z p t and A = Z p Q p t. But then A is not bounded despite A being Huber. Note also that the proposition implies that a topological ring A is Huber if and only if any subring of A is Huber. 4 Continuous Valuations and Spa Definition 4.1 (Continuous Valuation). A continuous valuation is a valuation : A Γ {0} satisfying the additional condition that for all γ Γ, {a A: a < γ} is open in A. Definition 4.2 (Equivalent). We say that two valuations, : A Γ {0} are equivalent if for all a, b A, a b if and only if a b. Note kernel of is a prime ideal of A depending only on its equivalence class. Definition 4.3 (Cont). Let Cont(A) denote the equivalence class of continuous valuations of A. For an element x Cont(A), denote f f (x) denote a continuous valuation representing x. Give Cont(A) the topology generated by the subsets of the form {x: f (x) g(x) 0} with f, g A. For x Cont(A), the rank of x is the rank of the ordered abelian group generated by the image of a continuous valuation representing x. Note that sets of the form { g(x) 0} are open as are sets of the form { f (x) 1}. This blends features of the Zariski topology on schemes and topology on rigid spaces. Furthermore, Cont(A) is quasicompact. Definition 4.4 (Rational Subsets). Let s 1,, s n A and let T 1,, T n A be finite subsets such that T i A A is open for all i. Define the subset ({ }) ( Ti T1 U = U,..., T ) n = {x X : t i (x) s i (x) 0 for all t i T i } s i s 1 s n Open as intersection of finite collection of sort of opens defining the topology on X. Theorem 4.1. Let U Spa(A, A + ) be a rational subset. Then there is a complete Huber pair (A, A + ) (O X (U), O + X (U)) such that the map Spa(O X(U), O + X (U)) Spa(A, A+ ) factors over U and is universal for such maps. Moreover, this map is a homeomorphism onto U. In particular, U is quasicompact. 7

8 Definition 4.5. Define a preheaf O X of topological rings on Spa(A, A + ) as follows: if U X is rational, O X (U) is as in Theorem 4.1. For a general open subset W X, define O X (W) := Proposition 4.1. For all U X = Spa(A, A + ), lim O X (U) U W rational O + X (U) = { f O X(U): f (x) 1 for all x U}. In particular, O + X is a sheaf if O X is. If (A, A + ) is complete, then O X (X) = A and O + X (X) = A+. Definition 4.6 (Adic Space). An adic space consists of the following data: a topological space X, a sheaf of topological rings O X, and the continuous valuations on O X,x for each x X. We require that X be covered by open subsets of the form Spa(A, A + ), where each (A, A + ) is a sheafy Huber pair. Theorem 4.2. A Huber pair (A, A + ) is sheaf when A is discrete. Thus, there is a functor from schemes to adic spaces sending Spec A to Spa(A, A). 2. A is a finitely generated algebra over a noetherian ring of definition. Thus, there is a functor from noetherian formal schemes to adic spaces sending Spf A to Spa(A, A). 3. A is Tate and strongly noetherian. That is, A X 1,..., X n = i=(i 1,...,i n ) 0 a i T i : a i A, a i 0 is noetherian for all n 0. Thus, there is a functor from rigid spaces over a nonarchimedean field K to adic spaces over Spa K sending Spm A to Spa(A, A ) for an affinoid K algebra A. 5 Perfectoid Fields Definition 5.1. Let K be a nonarchimedean field of residue characteristic p. Then K is a perfectoid field if 1. K is nondiscrete. 2. Frob : K /p K /p is surjective. Example 5.1. The completions of Q p (µ p ) and Q p (p 1/p ) are perfectoid. Furthermore, the completion of any strictly arithmetically profinite extension is perfectoid. Example 5.2. If k is a nonarchimedean field with characteristic p, then k is perfectoid if and only if it is perfect. As an example, k((t 1/p )), where k/f p is a perfect field the completion of the perfectoid field k((t)). In fact, if K is a perfectoid field with characteristic p with residue field k, then K contains k((t 1/p )), where t K is any element with 0 < t < 1. 8

9 Definition 5.2 (Perfectoid Ring). If A is a Huber ring, then A is a perfectoid ring if 1. A is Tate 2. A is uniform 3. A contains a pseudouniformizer ϖ such that ϖ p p in A and such that Frob : A/ϖ A/ϖ p is an isomorphism. Theorem 5.1. Let (A, A + ) be a Huber pair with A perfectoid. Then (A, A + ) is sheafy so that X = Spa(A, A + ) is an adic space. Furthermore, O X (U) is a perfectoid ring for every rational subset U X. Definition 5.3 (Perfectoid Space). A perfectoid space is an adic space which is covered by affinoids of the form Spa(A, A + ), where A is perfectoid. References [Ber90] V.G. Berkovich. Spectral theory and analytic geometry over non-archimedean fields. Mathematical Surveys and Monographs, 33, American Mathematical Society, Providence, RI, [Fal88] G. Faltings. p-adic hodge theory. J. Amer. Math. Soc., 1, no. 1: , [FW79] J.M. Fontaine and J.P. Wintenberger. Extensions algébrique et corps des normes des extensions APF des corps locaux. C.R. Acad. Sci. Paris Sér, 288: no. 8: , [GR03] O. Gabber and L. Ramero. Almost ring theory. volume 1800 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, [Hub93] R. Huber. Continuous valuations. Math. Z, 212, no. 3: , [Hub94] R. Huber. A generalization of formal schemes and rigid analytic varieties. Math. Z, 217, no. 4: , [Jon96] A.J. de Jong. Smoothness, semi-stability and alterations. Publications mathématiques de Institut des Hautes Études Scientifiques, 83:51 93, [KLa] K.S. Kedlaya and R. Liu. Relative p-adic hodge theory, I: Foundations. [KLb] K.S. Kedlaya and R. Liu. Relative p-adic hodge theory, II: (ϕ, Γ)-modules. [RZ82] M. Rapoport and T. Zink. Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik. Invent. Math, 68, no. 1:21 101, [Sch12] P. Scholze. Perfectoid spaces. Publications mathématiques de Institut des Hautes Études Scientifiques, 116: , [Sch13] P. Scholze. Perfectoid spaces: A survey. Current Developments in Mathematics 2012, International Press,

10 [Sch14] P. Scholze. Perfectoid spaces and their applications. In ICM lecture, Seoul, [Sch17] P. Scholze. Historic Remarks about [the] genesis of [the] paper Perfectoid Spaces, March [Tat71] J. Tate. Rigid-analytic spaces. Invent. Math., 12: , [Wei17] J. Weinstein. Arizona Winter School 2017: Adic Spaces. March 7,