Perfectoid spaces and their applications

Transcription

1 Perfectoid spaces and their applications February 17-21, 2014 Contents 1 Almost ring theory Almost mathematics Almost commutative algebra Unramified and etale morphisms Almost étale extensions Adic spaces Affinoid algebras Valuations Adic spectra Glueing of affinoid adic spaces Almost ring theory 2: perfectoid rings Tilting for fields Tilting more general algebras Comparing K-Perf and K a -Perf Some deformation theory Comparing K a -Perf and (K a /π)-perf Tilting étale covers Adic spaces 2: perfectoid rings Tilting perfectoid algebras Almost purity Perfectoid spaces Introduction to p-adic comparison theorems Rings of Witt vectors Tilting revisited Tilting and analytic extensions Period rings Notes taken by Daniel Miller 1

2 6 Adic spaces Étale topology for perfectoid spaces Open problems Generic fibers of formal schemes The weight-monodromy conjecture Tilting Comparing perfectoid spaces with noetherian spaces Proving the weight-monodromy conjecture Lubin-Tate spaces The Lubin-Tate deformation problem Formal vector spaces Connection to p-adic Hodge theory The Fargues-Fontaine curve Affine adic curves Gluing Examples The pro-étale site Basic definitions Perfectoid setting Contractible objects Relative padic Hodge theory Goals of relative p-adic Hodge theory Local systems via perfectoid spaces Robba rings and Q p -local systems p-adic Hodge theory for rigid spaces Comparison theorem Proof of auxiliary result Lubin-Tate spaces Some explicit formulas Period maps Some conjectures Shimura varieties and perfectoid spaces 1: completed cohomology Some definitions The setting Main example Future directions 1: formal Q p -vector spaces of slope > Crystals and Dieudonné modules The universal cover

3 16 padic Hodge theory for rigid spaces Hodge-Tate spectral sequence Relative period rings Shimura varieties and perfectoid spaces Moduli spaces of abelian varieties Sketch of proof of main theorem Shimura varieties and perfectoid spaces A theorem on Hecke actions

4 1 Almost ring theory 1 Recall that there is a tilting correspondence relating objects in characteristic zero to objects in characteristic p. This correspondence works not at an integral level, but at an almost integral level, i.e. using the language of almost mathematics. A good general reference for almost mathematics is the book [GR03] of Gabber and Ramero. 1.1 Almost mathematics Throughout, a non-archimedean field K is a field equipped with a rank-one valuation : K R >0. If K is any non-archimedean field, K denotes the valuation ring {x K : x 1} of K. Definition A perfectoid field K is a complete non-archimedean field such that: 1. The residue characteristic of K is p > The associated rank-one valuation is non-discrete. 3. The Frobenius map Φ : K /p K /p is surjective. Example The p-adic completion of Q p (p 1/p ) is perfectoid. Example The p-adic completion C p of Q p is perfectoid. Example If k is a non-archimedean field of positive characteristic, then the perfectification of k is perfectoid. Here the perfectification of k is the completion of the colimit lim k, where Φ is Frobenius. The perfectification of F p((t)) is the t-adic completion of Φ F p ((t))(t 1/p ). Example The field Q p is not perfectoid because its valuation is discrete. Let K be a perfectoid field, and denote by m K the unique maximal ideal. Since the valuation on K is non-discrete, m is not finitely generated. One has m m m = m 2, the first isomorphism coming from the flatness of m. Let Σ K -Mod be the full subcategory consisting of m-torsion modules. Since m 2 = m, Σ is a (thick) abelian Serre subcategory. Definition Let K be a perfectoid field. 1. A K -module M is almost zero if M Σ, i.e. mm = The category of K a -modules is the Serre quotient K -Mod/Σ. For example, the residue field K /m is almost zero. On the other hand, K /p (or K /t for any t m) is not almost zero. Write ( ) a for the localization functor K -Mod K a -Mod. A crucial fact is that ( ) a has both left and right adjoints, denoted N N! and N N, respectively. This allows us to easily compute hom-sets in K a -Mod. Indeed, Bhargav Bhatt (Feb. 17) hom K (M!, N) = hom K a(m a, N a ) = hom K (M, N ) 4

5 so the problem of computing hom K a(m a, N a ) is reduced to that of computing M! and N. Fortunately, the functors ( )! and ( ) can be explicitly described. One has (T a )! = m T (T a ) = hom K (m, T ). For a K a -module M, one calls M the module of almost elements of M. This notation is motivated by topology. If j : U X is the inclusion of an open subset into a topological space, then the restriction functor j : Sh(X) Sh(U) has left and right adjoints j! and j. This suggests that K a -Mod is the category of sheaves on some subscheme of Spec(K ) that lies in between the special point and generic fiber, even though no such subscheme exists. It is an easy exercise (and purely formal) to prove that N N! is an exact functor. Moreover, if N is a K a -module, then (N! ) a = N = (N ) a, from which we see that N N! and N N are fully faithful. (This too is an exercise in pure category theory.) While ( )! and ( ) are sections of ( ) a, one does not generally have (M a )! = M or (M a ) = M, for M a K -module. For example, if M = K, then (M a )! = m M. Similarly, if M = m, then (M a ) = K M. The subcategory Σ K -Mod is an ideal in the sense that it is closed under taking tensor products with arbitrary K -modules. Thus K a -Mod inherits a tensor product from K -Mod, so we can talk about algebras in K a -Mod. For example, any K -algebra A induces a K a -algebra A a. Using the functor ( ), one can show that every K a -algebra is of this form. Moreover, with the obvious definition of a module over a K a -algebra A a, one sees that all A a -modules are of the form M a, for M an A-module. The abelian tensor category (K a -Mod, ) has an internal hom-functor. For K a - modules M, N, the hom-set hom K a(m, N) is naturally a K -modules, so we can put alhom(m, N) = hom K a(m, N) a. 1.2 Almost commutative algebra As before, let K be a perfectoid field. Definition Let A be a K a -algebra, M an A-module. 1. M is flat if M A is an exact functor. 2. M is almost projective if the functor alhom(m, ) is exact. 3. Assume A = R a and M = N a. Then M is almost finitely generated if for all ɛ m, there exists a finitely generated R-module N ɛ with a map f ɛ : N ɛ M such that ker(f ɛ ) and coker(f ɛ ) are killed by ɛ. 4. If the number of generaters of the N ɛ can be taken to be bounded, we say that M is uniformly finitely generated There is an obvious notion of an almost finitely presented A-module along the same lines. Let A = R a be an almost K a -algebra, and suppose M = N a. Then M is almost flat if and only if Tor R i (N, ) takes values in almost zero modules for all i > 0. Similarly, M is almost projective if and only if Ext i R(M, ) takes values in almost zero modules for all i > 0. 5

6 It is not the case that an almost A-module is almost projective if and only if it is projective in the categorical sense. In fact, it is a good exercise to show that if M is a K a -module that is projective, then M = 0. Any finitely generated ideal I K is an almost finitely generated K a -module. In fact, such I are uniformly almost finitely generated. Fix r R >0 such that r / K. Then the ideal I r = {f K : f < r} is not finitely generated, but is uniformly almost finitely generated. 1.3 Unramified and etale morphisms Suppose A B is a finite étale map of commutative rings. Then the diagonal Spec(B) Spec(B A B) is clopen. It follows that there is a unique idempotent e B A B, called the diagonal idempotent such that 1. e 2 = e 2. µ(e) = 1, where µ : B A B B is the multiplication map 3. ker(µ) e = 0. For example, if A B is Galois with group G, then B A B g G B via (b 1 b 2 ) (b 1 g(b 2 )) g G. In this setting, e is the element (1, 0,..., 0). If one writes e = N i=1 x i y i with x i, y i B, then tr(e) = tr(x i y i ) = 1. Moreover, consider the maps α : B A N and β : A N B defined by α(b) = (tr(bx i )) i β(a i ) i = i a i y i. One has β α = 1 B, so B is projective. 1.4 Almost étale extensions As before, let K be a perfectoid field. Definition Let f : A B be a morphism of K a -algebras. 1. f is unramified if there exists e (B A B) such that e 2 = e, µ(e) = 1, and ker(µ) e = f is étale if it is flat and unramified. 3. f is finite étale if it is étale and B is an almost finitely presented projective A-module. If A is a K a -algebra, write A fét for the category of finite étale A-algebras. There is a good deformation theory for finite étale extensions. For example, if I A is nilpotent, then the natural functor A fét (A/I) fét is an equivalence of categories. Suppose K is a perfectoid field of characteristic p > 0. Choose an element 0 t m. Let A be a flat K -algebra which is integrally closed inside A[1/t]. Let B be a finite étale A[1/t]-algebra. Let B be the integral closure of A in B. Proposition If A is perfect, then A a B a is finite étale as a map of K a -allgebras. 6

7 Proof. We have a diagonal idempotent e B A B. There exists some N > 0 such that t N e B A B. Everything in sight is perfect, so we can apply the inverse of Frobenius to conclude that (t N ) 1/pn e B A B for all n > 0. Thus e (B A B) is almost integral, so B is almost étale. To see that B is almost finitely presented, fix ɛ m. Write ɛ e = N i=1 x i y i. The composite B A N B of b (tr(bx i )) i and (a i ) a i y i is multiplication by ɛ, so B is uniformly almost finitely presented projective. 7

8 2 Adic spaces 1 The main goal is to introduce the category of adic spaces over a non-archimedean base field, which generalizes rigid geometry. We want this framework to be able to handle more general affinoid rings (no finiteness conditions). The structure sheaf should be defined on all opens, not just with respect to a Grothendieck topology. 2.1 Affinoid algebras Fix a non-archimedean base field k, i.e. k is a complete topological field, with the topology defined by a nontrivial rank-one valuation : k R 0. Main examples include Q p, F q ((t)), and any perfectoid field. Definition An f-adic ring is a topological ring R such that R contains an open subring R 0 (called the ring of definition) such that the topology on R 0 is adic with respect to a finitely generated ideal of definition. Definition An f-adic ring R is called a Tate ring if there exists a topologically nilpotent unit in R 0. If R is a topological k-algebra, then R is a Tate ring if the sets {ar 0 : a k } form a basis for the topology on R. Example Consider the ring k T 1,..., T n = k [T 1,..., T n ] [ 1 π ], where ( ) denotes π-adic completion for some π k with π < 1. An element a R is called power-bounded if the set {a n : n 0} is a bounded subset of R. In this context, a set S R is bounded if for any open neighborhood U of 0, there exists a k such that au S. Write R for the subring of R consisting of power-bounded elements. Definition An affinoid algebra is a pair (A, A + ) with A an f-adic ring and A + A an open integrally closed subring. We say that an affinoid algebra (A, A + ) is of finite type over k if A is a quotient of k T 1,..., T n for some n, and A + = A. From now on, let A denote either an f-adic ring or a Tate ring. 2.2 Valuations Definition A valuation on a ring A is a map v : A Γ {0}, where Γ is a totally ordered abelian group (written multiplicatively), such that 1. v(0) = 0 and v(1) = 1 2. v(ab) = v(a)v(b) 3. v(a + b) max{v(a), v(b)} Eugene Hellman (Feb. 16) 8

9 Here as is typical, we put γ > 0 for all γ Γ, and define γ 0 = 0 γ = 0. If v is a valuation on A, let supp(v) = {x A : v(x) = 0}; this is an ideal in A. Let Γ v be the subgroup of Γ generated by {v(x) : x A, v(x) 0}. We say that two valuations v, v are equivalent if for all a, b A, one has v(a) v(b) v (a) v (b). Two valuations v, v are equivalent in this sense if and only if supp(v) = supp(v ) and the associated valuation rings in Frac(A/ supp(v)) are the same. Suppose v is a valuation on a topological ring A. We call v continuous if for all γ Γ v, there exists an open neighborhood U of zero in A such that v(x) < γ for all x U. Definition Let A be an f-adic ring. Define Spv(A) to be the set of equivalence classes of valuations on A. Definition If A is an f-adic ring, Cont(A) is the set of equivalences of continuous valuations on A. Definition Let (A, A + ) be an affinoid ring. Define Spa(A, A + ) to be the set of continuous valuations v on A such that for all a A +, v(a) 1. We equip all the spaces with a topology generated by the sets for f, g A. U f,g = {v : v(f) v(g) 0} Theorem The spaces Spv(A), Cont(A) and Spa(A, A + ) are spectral spaces. 2.3 Adic spectra Let (A, A + ) be an affinoid ring. For x Spa(A, A + ) corresponding to a valuation v, and f A, put f(x) = v(f). If A is a Tate k-algebra, then the rational subsets ( ) f1,..., f n U = {x Spa(A, A + ) : f i (x) g(x) for all i} g for f 1,..., f n A with (f 1,..., f n ) = A, form a basis for the topology on Spa(A, A + ). Example Let A be a Tate k-algebra of finite type. If m A is a maximal ideal, then we get a valuation on A by composing A A/m with the unique extension of k to A/m. This gives a canonical inclusion Sp(A) = {m Spec(A) : m maximal} Spa(A, A ). If A is a finite-type Tate k-algebra, then we have a corresponding statement for the corresponding rigid analytic space associated to A. The topology on Spa(A, A ) induces the standard Grothendieck topology on the standard rigid-analytic space. One has bijections {q-c opens in Spa(A, A )} {q-c opens in Sp(A)} {coverings by q-c opens} {admissible covers by q-c admissible opens} 9

10 We d like to construct a structure ( sheaf on ) X = Spa(A, A + ). For simplicity s sake, we assume (A, A + ) is Tate. Let U = U f1,...,f n g X be a standard rational subset; we need to define O X (U). Let A f1 g,..., fn f1 g be the completion of A[ g,..., fn g ] with respect to the topology generated by opens of the form a A 0 [ f1 g,..., fn g ], for a k. (Here A 0 A is any ring of definition.) Let A f1 g,..., fn g + be the completion of the integral closure of A + [ f1 g,..., fn g ] with respect to the same topology. Proposition The canonical map Spa(A fi fi g, A g + ) (Spa A, A + ) induced by ϕ : (A, A + ) (A fi fi g, A g + ) factors over U. Moreover, ϕ is universal for maps (A, A + ) (B, B + ) with B complete such that Spa(B, B + ) Spa(A, A + ) factors over U. For U as above, we define O X (U) = A f1 g,..., fn g and O+ f1 X (U) = A g,..., fn g +. For general U, we put O X (U) = lim W U U rational O X (W ). It is not known whether O X as defined is a sheaf in general. This is easily circumvented by the following definition. Definition Let (A, A + ) be as above. Then Spa(A, A + ) is called an affinoid adic space with sheaf O X if the presheaf O X is a sheaf. Theorem If A is strongly noetherian (i.e. A T 1,..., T n is noetherian for all n 0) then O X is a sheaf on X = Spa(A, A + ). For example, Tate algebras of finite type over any non-archimedean field k are strongly noetherian. Example (The closed unit ball). Let k be a non-archimedean field which is complete and algebraically closed, for example C p. Let X = Spa(k T, k T ). The space X can have up to five different types of points, depending on whether k is spherically complete. 1. These are the classical points coming fromo rigid geometry. Every element x k induces a valuation f f(x). The next two types of points are constructed in the same way. Let x k, r R >0. The valuation v x,r is defined by either of the following formulas: ( ) v x,r (T x) i = sup a i r i i0 v x,r (f) = sup f(y) y D r(x) where D r (x) = {y k : x y r} is the closed ball of radius r about x. 2. If r k, we say that v x,r X is type If r / k, we say that v x,r X is type Suppose k is not spherically complete, i.e. there exists a sequence D = D 1 D 2 of open balls such that i D i =. The valuation v D (f) = inf i1 sup y D i f(y) 10

11 is said to be of type Points of type 1, 3, and 4 are closed and rank one. Points of type 2 are not closed (but have rank one) they have (rank two) specializations of the following type, illustrated by an example. The point v x,1 has specialization ξ x,1, which takes values in R >0 γ, where we put γ < R >0 (i.e. use the lexicographic order). Put ξ x,1 (f) = sup a i γ i R >0 γ Z. i1 2.4 Glueing of affinoid adic spaces One glues adic spaces in the category V consisting of triples (X, O X, {v x : x X}), where (X, O X ) is a locally ringed space, O X is a sheaf of complete topological rings on X, and for each x X, v x is a valuation on O X,x. Morphisms in V are required to respect all the data. If (A, A + ) is an affinoid ring such that Spa(A, A + ) is an adic space, then for x X = Spa(A, A + ), there is a canonical extension of the valuation v x to O X,x. It is necessary to keep track of the valuations {v x : x X} to be able to recover O + X. The corresponding category of triples (X, O X, O + X ) is equivalent to V. Definition An adic space is an object of V that is locally isomorphic to an affinoid adic space. 11

12 3 Almost ring theory 2: perfectoid rings Here we treat the affine case of the tilting correspondence for perfectoid spaces. Unless anything is said to the contrary, all theorems are due to Scholze. If R is a ring of characteristic p, let Φ : R R be the Frobenius x x p. 3.1 Tilting for fields Let K be a perfectoid field. Definition The tilt of K has valuation ring K = lim Φ K /p. So K = Frac(K ). Theorem (Fontaine, Wintenberger). If K is perfectoid, so is K, and tilting induces an equivalence of categories K fét K fét. One proves this by starting with K, passing to the integral level via K, and noting that K K /π K /p for some π with p π < 1. Our goal is to generalize this by defining categories of perfectoid algebras over K and K, and proving that these categories are equivalent via a generalized tilting functor. It is not so obvious, but every perfectoid field of characteristic p > 0 is the tilt of a perfectoid field of characteristic zero. 3.2 Tilting more general algebras Let K be a perfectoid field, and choose some nonzero π m with p π < 1. Definition A Banach K-algebra R is perfectoid if R R is open and bounded, and Φ : R /p R /p is surjective. Let K-Perf be the category of perfectoid K-algebras and continuous morphisms of K-algebras. 2. A perfectoid K a -algebra is a flat K a -algebra R that is π-adically complete and Frobenius induces an isomorphism R/π 1/p R/π. Let K a -Perf be the category of perfectoid algebras over K a. 3. A perfectoid (K a /π)-algebra R is a flat (K a /π)-algebra R such that Frobenius induces an isomorphism R/π 1/p R. Let (K a /π)-perf be the category of perfectoid (K a /π)-algebras. Example Let R = K T 1/p in the sense of Tate, i.e. R = K [T 1/pn ] n1 [ 1 p ]. Example Similarly, A = (K T 1/p ) a is a perfectoid K -algebra. Example If K has characteristic p > 0, then perfectoid K-algebras are simply Banach K-algebras A such that A is open and bounded. Bhargav Bhatt (Feb. 17) 12

13 Theorem (Tilting equivalence). There are natural equivalences of categories K-Perf (a) K a -Perf (b) (K a /π)-perf K -Perf (d) K a -Perf (c) (K a /π)-perf One proves equivalences (a) and (d) using almost mathematics, while (c) and (d) require some deformation theory in particular the cotangent complex. 3.3 Comparing K-Perf and K a -Perf Lemma Let M be a K a -module, N a K -module. Then 1. M is flat in over K a if and only if M is flat over K. 2. If N is flat over K, then N a is flat, and { [ ] } 1 N a = x N : ɛ x N for all ɛ m π 3. If M is flat over K a, then M is π-adically complete if and only if M is π-adically complete Theorem The functors K-Perf K a -Perf R (R ) a are an equivalence of categories. K a -Perf K-Perf A A[ 1 π ] The proof of this theorem relies in part on the following proposition. Proposition If R is perfectoid over K, then R a is perfectoid over K a. Proof. By assumption, Φ : R /π R /π is surjective. We show that its kernel is π 1/p R. Let x R such that x p π R. Then x p = π y for some y R. Thus (x/π 1/p ) p = y R, so x/π 1/p R, whence x π 1/p R. To see that R a is π-adically complete, refer to Lemma Proposition Let A be a perfectoid K a -algebra. Define R = A [ 1 π ], with the Banach structure making A open and bounded. Then A = R and R is perfectoid. Proof. First, note that Φ : A /π 1/p A /π is injective by Lemma Alternatively, we know that it is an almost isomorphism. So for x A with x p π A, we have ɛ x π 1/p A for all ɛ m. It follows that x ((π 1/p A ) a ) = π 1/p A, whence ker(φ) = 0. Next, we assume x R has x p A and show that x A. Write y = π k/p x A for some k 1. Raise both sides to the p power, to obtain y p = π k x p πa. By the injectivity we have already proved, we get y π 1/p A. Thus π (k 1)/p x A, so we can apply induction. 13

14 Finally, we show that Φ : A /π 1/p A /π is surjective. Because the map is already almost surjective, it is enough to show that the composite A /π 1/p A /π A /m is surjective. Choose x A. By almost surjectivity, we can write π 1/p x y p mod πa for some y. Let z = y/π 1/p2 R. Then z p = y p /π 1/p = x mod π 1 1/p A, so z p A. We already have shown that this implies z A, so z p = q mod π 1/p A implies that x has a p-th root modulo m. 3.4 Some deformation theory We ll start with a review of the cotangent complex as it appears in standard algebraic geometry. If A B is a ring homomorphism, then the cotangent complex L B/A is an object in D 0 (B-Mod). If A B is smooth, then L B/A = Ω 1 B/A [0]. Most importantly, L B/A controls the deformation theory of A B regardless of whether B is smooth over A. Example Suppose A = F p and B is any perfect F p -algebra. It turns out that L B/A = 0. This is easy to prove. By assumption, the Frobenius map Φ : B B is an isomorphism, so the induced morphism dφ : L B/A L B/A is an isomorphism. But dφ = 0 because d(y p ) = py p 1 dy = 0. Corollary There exists a unique flat (Z/p n )-algebra W n (B) lifting B from F p to Z/p n. One typically calls W n (B) the ring of p-typical Witt vectors in B of length n. 3.5 Comparing K a -Perf and (K a /π)-perf If A B is a map of K a -algebras, then [GR03] define an object of D(B-Mod) that they call L B/A. This L B/A is actually constructed as an honest complex in D(B -Mod). The complex L B/A satisfies all the properties one would hope for. Lemma Let A be a perfectoid (K a /π)-algebra. Then L A/(K a /π) = 0. Proof. Essentially, this comes down to the fact that the relative Frobenius is an isomorphism. Consider the following commutative diagram: A A/π 1/p A K a /π K a /π 1/p K a /π The argument from Example completes the proof. Lemma implies that (K a /π)-perf (K a /π n )-Perf K -Perf. Explicitly, start with A (K a /t)-perf. Then A = lim Φ A is a perfectoid K a -algebra. 14

15 3.6 Tilting étale covers Let A be a perfectoid algebra over K. categories: A fét (a) A a fét We have the following chain of equivalences of (b) (A a /π) fét A fét (d) A a fét (c) (A a /π) fét Essentially, we have seen that (b) and (d) follow from deformation theory, while (c) follows from the last example. 15

16 4 Adic spaces 2: perfectoid rings 4.1 Tilting perfectoid algebras Fix a perfectoid base field K. This field contains the subring K = O K of integral elements, which contains a unique maximal ideal m. Recall that the tilt of K is K OK = lim O K /p m Φ Choose a nonzero ϖ m, ϖ m such that O K /ϖ O K /ϖ. Definition An affinoid K-algebra (R, R + ) is perfectoid if R is a perfectoid K- algebra. Note that m R R + R, and R + R is an almost isomorphism. Also, the rings R +, R carry the ϖ-adic topology, and are complete with respect to this topology. (This is true because R R is bounded.) Proposition There is a tilting equivalence {(R, R + ) perfectoid K-algebras} {(S, S + ) perfectoid K -algebras} where a perfectoid K-algebra R is mapped to R = lim R x x p as a multiplicative monoid. One has R R + = lim x x p R+. There are natural isomorphisms R + /ϖ R + /ϖ. We have a diagram K O K O K /ϖ K O K O K /ϖ From Bhargav s talk, we know that R = lim R /ϖ, and there is an obvious multiplicative Φ projection lim Φ R lim R /ϖ. Φ It turns out that this is an isomorphism, so we can use it to transfer the additive structure of lim R /ϖ to lim Φ x x p R. For an arbitrary sequence ( x 0, x 1,... ) lim R /ϖ, choose lifts x i of x i to R, and put x i = lim ( x i+n) pn R n It is not hard to show that this gives a well-defined continuous multiplicative homomorphism Peter Scholze (Feb. 18) R = lim x x p R R f f. 16

17 Proposition There exists a continuous map Spa(R, R + ) Spa(R, R + ), x x defined by f(x ) = f (x) (f R ). Proof. First, we need to check that x is actually a valuation. All the properties but the triangle inequality are clear. But f f is not additive, so it is not a priori clear that x satisfies the triangle inequality. Let f, g R. Rescaling by an element of K, we may assume that f, g R +, but are not both in ϖ R +. Note that f mod ϖ = f mod ϖ, as elements of R + /ϖ = R + /ϖ. It follows that f + g (f + g) mod ϖ, i.e. ( ) is additive modulo ϖ. We can now compute for n 0. desired. (f + g)(x ) 1/pn = (f 1/pn + g 1/pn ) (x) max( π, ((f ) 1/pn + (g ) 1/pn )(x) ) max( π, f (x) 1/pn, g (x) 1/pn ) = max( f (x), g (x) ) 1/pn, Raising to the p n -th power, we get (f + g)(x ) = max( f(x ), g(x ) ), as Theorem Let (R, R + ) be a perfectoid affinoid K-algebra with tilt (R, R + ). Then 1. The map ( ) : X = Spa(R, R + ) Spa(R, R + ) = X is a homeomorphism, preserving rational subsets. 2. The structure presheaves O X, O + X, O X, O+ X are sheaves. 3. For all U X rational, (O X (U), O + X (U)) is a perfectoid affinoid K-algebra with tilt (O X (U), O + (U)). X 4. For all x X, the completed residue field k(x) is perfectoid with tilt k(x ). We have already seen that if R is strongly noetherian, then the structure presheaves on Spa(R, R + ) are sheaves. In this theorem, there are no finiteness conditions it is the perfectoidness of R that is used in showing that O X etc. are sheaves. The proof is based on an approximation lemma: given f R, we want there to exist g R such that f g is small. In some sense, this happens, but see Caraiani s talk for more details. First we show that O X is a sheaf if K has characteristic p. Definition We say that (R, R + ) is p-finite if there is a reduced Tate K-algebra (S, S + ), topologically of finite type, such that R + is the ϖ-adic completion of lim S +, and Φ R = R + [ 1 ϖ ]. Proposition In this situation, the map X = Spa(R, R + ) Y = Spa(S, S + ) is a homeomorphism preserving rational subsets. Moreover, for any U X rational, (O X (U), O + X (U)) is p-finite, and is the perfect completion of O Y (U), O + Y (U)). Corollary Let X be as in the proposition. Let X = U i be a finite cover by rational U i X. Then 0 R + O + X (U i) O + X (U ij) is almost exact (i.e. its cohomology is killed by m). 17

18 Proof. We already know that O Y is a sheaf, so 0 S OY (U i ) OY (U ij ) is exact. We can apply Banach s open mapping theorem to show that in the complex 0 S + O + Y (U i) O + Y (U ij) all the cohomology groups are killed by a (single) power of ϖ. Take the direct limit over Frobenius, all cohomology groups are almost zero. Complete everything in sight, and we get the result. As in Bhargav s talks, the main idea is that properties true for the generic fiber are almost true (i.e. true in the almost context). Proposition Suppose K is a perfectoid field of characteristic p. Any perfectoid affinoid K-algebra (R, R + ) is a completed filtered direct limit of p-finite ones. This is a generalization of the fact that any Z-algebra is a direct filtered direct limit of Z-algebras of finite type, and the proof runs similarly. Note that a filtered direct limit of almost exact sequences is almost exact. This gives us the following corollary. Corollary For any X = Spa(R, R + ), where (R, R + ) is a perfectoid affinoid K- algebra and K has characteristic p, the structure presheaves (O X, O + X ) are sheaves. 4.2 Almost purity Fix a perfectoid K-algebra R. In Bhargav s talk we had the following diagram: R fét (R a /ϖ) fét Rfét a Rfét Rfét a (R a /ϖ ) fét We get a functor R fét R, inverse to the tilting functor. Theorem There is a natural equivalene R fét R fét. Equivalently, for all finite étale R-algebras S, S is perfectoid, and S a is finite étale over R a. This is the almost purity theorem of Faltings, motivated by the classical Zariski-Nagata purity theorem. Take any such S. Choose R + = R, and let X = Spa(R, R + ), and X = Spa(R, R + ). If U X is rational, we have S(U) = S R O X (U)., a finite étale O X (U)-algebra. Lemma Fix a point x X. Then 2-lim U x O X (U) fét k(x) fét. 18

19 Proof. We compute 2-lim (lim O X (U)) fét. U x But lim O + X (U) is henselian along ϖ because each O+ X (U) is complete. General theory tells us that (O X,x ) fét (Ô+ X,x [ 1 ϖ ]) fét But in the eact sequence 0 I O + X,x k(x) + 0 the ideal I is a K-vector space. Indeed, if f I, then f ϖ, so f lies in an open neighborhood of x. Thus f ϖ lim (O+ X,x k(x)+ ) = I. Take ϖ-adic completion, and I vanishes, so Ô+ X,x = k(x) + ). It follows that (Ô+ X,x [ 1 ϖ ]) fét = k(x) + fét. Corollary lim O X (U) fét k(x) fét k(x ) fét 2-lim O X (U) fét U x U x It follows that locally, S(U) is in the image of O X (U) fét O X (U) fét. A basic gluing argument yields the result. 4.3 Perfectoid spaces Definition A perfectoid space over K is an adic space over K that is locally isomorphic to Spa(R, R + ), where (R, R + ) is a perfectoid affinoid K-algebra. Corollary There is an equivalence of categories {perfectoid spaces over K} {perfectoid spaces over K } given by X X. Moreover, X X, O X tilts to O X when evaluated on affinoid perfectoid opens U X. 19

20 5 Introduction to p-adic comparison theorems Unfortunately, titles of talks were not given by the speakers. So this talk will contain nothing about p-adic comparison theorems. A better title would be constructing universal rings. 5.1 Rings of Witt vectors Fix a prime number p > 0. Cuntz and Denninger discovered a very beautiful way of constructing the ring of Witt vectors W (R), for W a perfect F p -algebra. It involves some naive multiplicative deformation theory. For more details, see their preprint [CD13]. Let Λ be a commutative ring, R a Λ-algebra n > 0 an integer. Recall that a Λ- infinitesimal thickening of order n of R is a pair (A, I), where A is a Λ-algebra, I A is an ideal, together with an isomorphism A/I R of Λ-algebras, such that I n+1 = 0. It is natural to ask whether the category of such thickenings has an initial object. The answer is no in general, so we consider the category of triples (A, I, σ), where (A, I) is a Λ-thickening of R and σ : R A is a multiplicative section of A R. The category of such triples has an initial object which can be constructed explicitly. Start with the monoid ring Λ[R] consisting of formal sums x R λ x[x], with all but finitely may λ x = 0. There is a ring homomorphism (called the augmentation map) ε : Λ[R ] R, sending λ x [x] to λ x x. Put U n,λ (R) = Λ[R ]/(ker ε) n+1. Clearly (U n,λ (R), ker ε) is a Λ-infinitesimal thickening of R of order n. There is an obvious multiplicative section σ : R U n,λ (R) given by r [r]. It is easy to check that (U n,λ (R), ker ε, σ) is the universal Λ-infinitesimal thickening of order n with section of R. If R is a perfect F p -algebra, then U n,z (R) = U n,zp (R) is W n (R), the ring of length-n Witt vectors of R, so W (R) = lim U n,z (R). If R is a perfect F p -algebra, then given any n Λ-infinitesimal thickening (A, I) of R, we can form a multiplicative section σ : R A by x ( x p m ) pm, for x p m a lifting of x to A, and m Tilting revisited Let s start by defining the class of rings we ll be working with. Definition A Banach ring A is a topological ring containing a pseudo-uniformizer (topologically nilpotent invertible element) π, and an open subring A 0 π, such that A = A 0 [ 1 π ], and such that A 0 lim A 0 /π n is an isomorphism of topological rings. Note that a Banach ring is simply a complete Tate algebra (in the sense of Huber). Recall that if A is a Banach ring, then a set S A is called bounded if for all neighborhoods U of 0, there exists n 1 so that π n S U. An element a A is power-bounded if the set {a n : n 0} is bounded, and we write A A for the subring of power-bounded elements. Definition A spectral ring is a Banach ring A such that A is bounded. If A is a Banach ring, then A is spectral if and only if there exists a power-multiplicative norm inducing the topology. Recall that a power-multiplicative norm on A is a map : A R 0 such that ab a b, 1 = 1, and a n = a n for n 0. If A is complete with respect to, one has A = {a A : a 1}. If A is a Banach ring, we can construct a Jean-Marc Fontaine (Feb. 18) 20

21 power-multiplicative norm directly. Choose π A as in Definition Also choose ρ R with 0 < ρ < 1. For a A, put { } r as v π (a) = sup : r Z, s > 0 and s π r A. One has v π (a) R 0, so it makes sense to define a = ρ vπ(a). Definition A perfectoid ring is a spectral ring A, such that there exists a powerbounded unit π A such that p π p A and the Frobenius map A /π p A is surjective. So a perfectoid field is a perfectoid ring which is a field, and which admits a multiplicative norm. If A is a spectral ring of positive characteristic, then A is perfectoid if and only if it is perfect. Moreover, if π A is any pseudo-uniformizer, then A is naturally an algebra over the perfectoid field F p ((π 1/p )) = Fp ((π))(π 1/p ). If the characteristic of A is not p, then A is a Banach p-adic algebra, so Q p A, but there is no canonical perfectoid subfield of A. Let A be a perfectoid ring of characteristic zero. One can define a new ring A, called the tilt of A, in two different but equivalent ways. First, choose a pseudo-uniformizer π A with p πa, and put A = lim A /π Φ where Φ : x x p is Frobenius. Define A = A [ 1 π ]. Alternatively, we could have defined A as a set to be A = lim { (x (n) ) n0 : (x (n+1) ) p = x (n)}. A = x x p This has an obvious multiplication operation, but the addition is not obvious. We define it directly by (x + y) (n) = lim m (x(n+m) + y (n+m) ) pm. Define ( ) : A A given by (x (n) ) n0 x (0). If A is a perfectoid ring of characteristic zero, put R = A ; this is a perfectoid ring of characteristic p. There is a natural surjection θ A : W (R ) A, defined by ) ( θ A p i [a i ] i=0 = i=0 p i a (0) i = p i a i where each a i = (a (n) i ) n0. The kernel of θ A is a primitive ideal of degree one. In other words, it can be generated by an element of the form [π]+pη, where π is a pseudo-uniformizer of R, and η W (R ). The operation A (A, ker θ A ) is a functor from the category of perfectoid rings of characteristic zero to the category of perfectoid pairs. Here a perfectoid pair is a pair (R, I), where R is a perfectoid ring of characteristic p > 0, and I W (R ) is a primitive ideal of degree one. Theorem This functor is an equivalence of categories. Proof. We construct an inverse. Given a perfectoid pair (R, I), define A = (R, I) by A = W (R )/I and A = A [ 1 p ]. Checking that ( ) is an inverse to A (A, ker θ A ) is a straightforward exercise. Alternatively, see Kedlaya s note [Ked13]. i=0 21

22 5.3 Tilting and analytic extensions We know that for any perfectoid field K of characteristic zero, the tilt K is a perfectoid field of characteristic p. On the other hand, suppose we start out with a perfectoid field F of characteristic p. Then to any primitive ideal I W (F ), the above construction yields a perfectoid field (F, I) of characteristic zero, whose tilt is F. Thus every perfectoid field of characteristic p is the tilt of a (explicitly constructed) perfectoid field of characteristic zero. Suppose K 0 is a finite extension of Q p. Let L be an analytically profinite extension of K 0. In other words, L could be an algebraic extension of K 0 which is infinitely ramified and such that the Galois group of the Galois closure of L/K 0 is a p-adic Lie group. Clearly any analytically profinite extension of K 0 gives a p-adic representation of G K0. Let K be the completion of L. To K we associate a norm field, which is isomorphic to k L ((t)), where k L is the residue field of L. The Fontaine-Wintenberger theorem is that L and k L have the same étale sites. Moreover, K is perfectoid and K is the perfectoid completion of k L ((t)). If we took K = C p, then we get an algebraically closed perfectoid field F of characteristic p. It is unknown if all the untiltings of this field are isomorphic. Let K be a fixed perfectoid field of characteristic zero. Then K is a perfectoid field of characteirstic p. We know that the kernel of θ K : W (K ) K is principal; let ξ be any generator. If A is any perfectoid K-algebra, then (A, (ξ)) is a perfectoid pair. 5.4 Period rings Let k be a perfect field of characteristic p, and fix an ultrametric field E with residue field k. For any perfect k-algebra R of characteristic p, we will construct a bounded period ring B b E (R). Start by defining W E (R) to be the completed tensor product E W (k) W (R) = lim (E /π n ) W (k) W (R), where π E is any pseudo-uniformizer. The surjection W E (R) R has a multiplicative section [ ] : R W E (R), given by x 1 [x]. If E has characteristic p, this section is additive and we can view R as a subring of W E (R) = E k R. Suppose R is a perfectoid k-algebra. (Since k is not perfectoid, we simply assume k R.) Let ϖ be a pseudo-uniformizer of R, and define [ 1 B b E(R) = W E (R ) π, 1 [ϖ] If E is discretely valued, we can choose π to be a uniformizer of E, and { } B b E(R) = [a i ]π i : a i R bounded i In general, given (a n ) n0, with {a n } R bounded, and (ν n ) n0, with ν n E converging to zero, the infinite sum n=0 [a n]ν n B b E (R), but this decimal expansion is not unique. Choose an absolute value on E, and a power-multiplicative norm on R. Up to equivalence, these are unique. Proposition For all ρ R, 0 < ρ < 1, there exists a unique power multiplicative norm ρ on B b E (R) such that [a] ρ = a for all a R, and such that π = ρ. ]. 22

23 Choose a non-empty closed interval I = [ρ 1, ρ 2 ], with 0 < ρ 1 ρ 2 < 1. Define α I = sup{ α ρ1, α ρ2 } = sup α ρ. ρ 1ρρ 2 Let B E/I (R) be the completion of B b E (R) with respect to the norm I. This is a spectral E-algebra. If E is a perfectoid field, then B E/I (R) is a perfectoid E-algebra. The tilt of B E/I (R) is B E /I(R). Now we can define some period rings. For any perfectoid Q p -algebra R, we can define B + cris (A) and B+ dr (A). The morally correct definition of B+ dr is B + dr (A) = lim B n (A). n0 Here the inverse limit is taken in the category of Banach Q p -algebras, and B n (A) is the universal divided-power thickening of A of order n + 1. For a more explicit definition, start with the homomorphism θ : W (A )[ 1 p ] A, whose kernel is a principal ideal (ξ). It turns out that B n (A) = W (A )[ 1 p ]/ξn, so [ ] 1 B + dr (A) = lim W (A ) /(ker θ) n. p This is a discrete valuation ring with uniformizer ξ, so we can define B dr (A) to be the field of fractions of B + dr (A), i.e. B dr(a) = B + dr (A)[ 1 ξ ]. To define A cris (A), start by defining A ϕ cris (A) to be the sub-w (A )-algebra of W (A ) generated by {ξ n /n : n 1}. Let A cris (A) be the p-adic completion of A ϕ cris (A), and define B + cris (A) = A cris(a)[ 1 p ]. 23

24 6 Adic spaces Étale topology for perfectoid spaces Note that perfectoid rings (i.e. spectral rings, in Fontaine s vocabulary) are always reduced. If x R is nilpotent, then any ϖ n x R, so K x R, but this is impossible if R is bounded. So we can t use an infinitesimal lifting criterion to define étaleness. However, we have the following proposition. Proposition For adic spaces of finite type over K, open embeddings and finite étale covers generate the étale site. Note that this is not true for schemes. One defines finite étale covers using Bhargav s definition for global sections. Definition A map f : Y X of perfectoid spaces is etale if it is locally of the form Y Z W X, where Y Z is an open embedding, Z W is finite étale, and W X is open. One can check that this notion of étaleness satisfies the expected properties (e.g. closure under composition). Thus we can define the étale site Xét. Moreover, if a map becomes étale after a change of the ground field, it was étale to begin with. Proposition If X is a perfectoid space with tile X, then 1. Xét X ét 2. Let X be affinoid perfectoid; X = Spa(R, R + ). Then H i (Xét, O + X ) = { R + if i = 0 almost zero if i > 0 Part 1 follows from the fact that X X, combined with the almost purity theorem for finite étale covers. If X = Spa(R, R ) is affinoid of finite type over K, then H i (Xét, O X ) is R for i = 0, and is 0 for i > 0. On the other hand, H i (Xét, O + X ) can contain lots of torsion, e.g. it could contain K/O K. This torsion gets killed on perfectoid covers. In Niziol s talks, we will see the following theorem. Let K be algebraically closed of characteristic 0, and let X be a proper smooth variety over K. We can look at RΓ(Xét, O + X ). One has RΓ(Xét, O + X )[ 1 p ] = RΓ(X ét, O X ) (usual coherent cohomology). On the other hand, RΓ(Xét, O + X ) L OK O K /p n is almost isomorphic to RΓ(Xét, Z/p n ) L Z/p n O K /p n. This gives us a connection between cohomology O + X on a perfectoid space, and more familiar étale cohomology. It is natural to ask whether there is a good category containing both rigid-analytic varieties and perfectoid spaces. In some sense, the answer is yes because the category of adic spaces accomplishes this. On the other, hand, this is not a satisfactory answer because the category of adic spaces is not as nice as one might hope. The problem is that it is very hard to determine whether O X (or quasi-coherent modules) are sheaves. Peter Scholze (Feb. 18) 24

25 Example (Rost, Buzzard). Consider R = Q p pt, pt 1 ; the algebra of functions on the strip p 1 T p. Let M be the Banach R-module with Banach Q p -basis {p n T n, p n T n : n 0}. In other words, M is the p-adic completion of the submodule of Q p T ±1 generated by these elements. Let R 1 = Q p pt, T 1, R 2 = Q p T, pt 1. Proposition The element 1 M dies in M R R i for i = 1, 2. In fact, M R R i = 0 for each i. Proof. We can assume i = 1. Write p n = (p n T n ) T n M R 1. One has p n T n 1 and T n 1. By the definition of the norm on M R R i, one gets p n 1 for all n, i.e. 1 p n for all n, whence 1 = 0, so 1 dies. Theorem R M (M a square-zero ideal) violates the sheaf property. Proof. In fact, there is an element which is globally non-zero, but is locally zero. Note that R M is not spectral in the sense of Fontaine, because it has nilpotents. Proposition If R is spectral, then for any cover X = Spa(R, R + ) = U i with the U i rational, one has R O X (U i ). In fact, R x X k(x). Proof. This follows from a theorem of Berkovich to the effect that the pullback of the supremum norm on k(x) makes R the 1 subring. 6.2 Open problems A natural question is: are there counterexamples to the sheaf property for spectral rings? Also, is there spectral R with U Spa(R, R + ) such that O X (U) is not spectral? Finally, could these two phenomena occur simultaneously? The following question is due to Rapoport: is perectoid a local property? Let K be a perfectoid field. Let X be a perfectoid space over K. Assume X = Spa(R, R + ) is affinoid. Is R perfectoid? We know that there is a covering of X by rational subsets which are perfectoid, but it is not clear that this implies the perfectoidness of X. So we have to distinguish between affinoid and perfectoid affinoid subsets of a perfectoid space. One can consider inverse limits. Let (X i ) i0 be a tower of reduced adic spaces, all of finite type over a field K. Moreover, we assume that the transition maps are finite. Let X be a perfectoid space, and {f i : X X i } i0 a compatible system of maps. Definition Say that X is a naive inverse limit if all X i = Spa(R i, R + i ) and X = Spa(R, R + ), everything is affinoid perfectoid, and R + is the π-adic completion of lim R + i. 2. Say X lim X i if this is satisfied locally. The category of affinoid rings does not admit filtered direct limits. If {(R i, R + i )} i is a direct system, what topology should we put on lim R + i? With the direct limit topology, lim R+ i is not affinoid, and it is not clear what other topology to impose. On the other hand, if the R i are spectral, we can let R + be the ϖ-adic completion of lim R + i and R = R + [ 1 ϖ ]. Proposition If X lim X i, then X = lim X i in the category of locally spectral adic spaces. 25

26 Another question: does it make sense to develop the theory of spectral adic spaces? Spectral adic spaces are reduced, so they don t see tangent spaces, at least not via K[ε]/ε 2 - valued points. For example, the isomorphism of Lubin-Tate and Drinfel d towers, we have lim M i M N lim N i But is Ω 1 M 0 M Ω 1 N 0 N? This is not known. 6.3 Generic fibers of formal schemes This section is preparation for Weinstein s talk. Let R be complete for the I-adic topology for some finitely generated ideal I R. One can consider Spa(R, R), which comes equipped with a structure-presheaf. Assume R is a Z p -algebra, and p I. Then Spa(R, R) lies over Spa(Z p, Z p ) = {s, η}, where s is the special point and η is the generic point. One has η = Spa(Q p, Z p ). Definition The generic fiber of Spf R is Spa(R, R) η = Spa(R, R) Spa(Zp,Z p)spa(q p, Z p ). The generic fiber is an open subset of Spa(R, R). In any reasonable setup, Spf R Spa(R, R) is a fully faithful functor {formal schemes} {adic spaces}. Example Suppose R = Z p T. Then the closed unit disk. Spa(R, R) η = Spa(R[p 1 ], R) = Spa(Q p T, Z p T ) Example If R = Z p T with I = (p, T ), then Spa(R, R) η is the open unit disk, hence is not affinoid. Example Let R = O K T 1/p, the (p, T )-adic completion of OK [T 1/p ], where K is some perfectoid field. Then Spa(R, R) η is the perfectoid open unit ball. Proposition Fix f 1,..., f k I such that I = (p, f 1,..., f k ). Then where we assume f i p 1/n. Spa(R, R) η = n1 Spa(R f n i p,..., f n k p [1 p ]) Proposition Assume R is a flat commutative O K -algebra, where K is a perfectoid field. Moreover, assume Φ : R/p 1/p R/p is an isomorphism. Then Spa(R, R) η is a perfectoid space over K. 26

27 7 The weight-monodromy conjecture The goal is to understand Scholze s proof of the weight-monodromy conjecture for hypersurfaces in projective space. We will do this by comparing étale topoi in characteristic zero and characteristic p. 7.1 Tilting Recall the tilting equivalence. Let K be a perfectoid field (i.e. a complete non-archimedean field with non-discrete valuation such that Frobenius K /π K /π is surjective, where π K has p π < 1). The tilt of K is K, which as a set is K = lim K. x x p There is a section map ( ) : K K that is multiplicative and continuous, but not additive or surjective. We say that Spa(K, K ) tilts to Spa(K, K ). A baby case of the almost purity theorem tells us that there is an equivalence of categories K fét K fét. This leads to an isomorphism of Galois groups Gal( K/K) Gal( K /K ). This can be generalized to perfectoid spaces over K. Recall that a perfectoid space over K is an adic space X over K that is locally of the form Spa(R, R + ), where R is a perfectoid K-algebra and R + R is an open integrally closed subring. The space X tilts (locally) to Spa(R, R + ), yielding a functor X X from perfectoid spaces over K to perfectoid spaces over K. Recall that we have a map R = lim x x p R R given by (x i) i1 x 0. This map is continuous, multiplicative, but not surjective. However, it induces an isomorphism R + /π R + /π. This lets define a homeomorphism X X, x x, where for f R, we have f(x ) = f (x). This homeomorphism preserves rational subsets. To prove this, one uses an approximation lemma: given f R +, there exists g R + such that f + g except when both are very small (i.e. except when f < ε, in which case we only require g < ε). The structure sheaves (O X, O + X ) of X tilt to sheaves (O X, O+ X ) on X. Recall that an étale morphism of perfectoid spaces is a composite of a finite étale morphism with an open immersion. Using the almost purity theorem, we get an isomorphism Xét Xét. Recall that a key point here is that locally, Ô+ X,x [ 1 π ] k(x), and both are perfectoid fields. This reduces the proof of the almost purity theorem to the case for perfectoid fields. This isomorphism on stalks actually holds more generally for analytic adic spaces. The sheaf property is crucial here. 7.2 Comparing perfectoid spaces with noetherian spaces To apply any of this to the weight-monodromy conjecture, we need to compare étale sites and underlying spaces of perfectoid spaces with the étale sites and underlying spaces of locally noetherian adic spaces. For example, let (P n K )ad be the adic projective space glued out of Spa(K T 1,..., T n, K T 1,..., T n ) in the usual way. Also define (P n K )perf by glueing copies of Spa(K T 1/p 1,..., Tn 1/p, K T 1/p 1,..., Tn 1/p ) Ana Caraiani (Feb. 18) 27