aa r X i v : . [ m a t h . AG ] F e b Untilting Line Bundles on Perfectoid Spaces
Gabriel Dorfsman-Hopkins
Abstract
Let X be a perfectoid space with tilt X ♭ . We build a natural map θ : Pic X ♭ → lim Pic X where the (inverse) limit is taken over the p -power map, and show that θ is an isomorphism if R = Γ( X, O X ) is a perfectoid ring. As a consequence we obtain a characterization of when thePicard groups of X and X ♭ agree in terms of the p -divisibility of Pic X . The main technicalingredient is the vanishing of higher derived limits of the unit group R ∗ , whence the main resultfollows from the Grothendieck spectral sequence. In [Sch12] Scholze introduced a class of algebro-geometric objects called perfectoid spaces which arisenaturally in the context of p -adic geometry. To a perfectoid space X over the p -adic numbers, onecan functorially assign a perfectoid space X ♭ in characterstic p , called the tilt of X . Remarkably, X and X ♭ share many algebraic and topological properties, enumerated through various tiltingequivalences , including that they are homeomorphic and have canonically identified ´etale sites.Perhaps surprisingly given their similarities, it is not true in general that X and X ♭ have isomorphicPicard groups (see [Heu20, Section 9.5] or Example 4.6). The goal of this paper is to describe exactlyhow different these two Picard groups can be.Observe that since X ♭ is perfect, p acts invertibly on its Picard group. We will show that the onlyobstruction to the Picard groups of X and X ♭ being isomorphic is the failure of p to act invertiblyon Pic X . To see this we construct a map θ : Pic X ♭ −→ lim Pic X where the (inverse) limit is taken along the p -power map on Pic X . A rough outline of the construc-tion is the following: the tilting equivalence identifies the ´etale sites of X and X ♭ [Sch12, Theorem7.9], so that we can view O ∗ X and O ∗ X ♭ as objects in the same category. The construction of tiltingthen provides an identification: O ∗ X ♭ ∼ = lim O ∗ X , where the limit is taken along the map f f p . One obtains θ by taking derived global sections ofthis isomorphism and analyzing the Grothendieck spectral sequence. The obstructions to θ being anisomorphism are evidently controlled by derived limits of O ∗ X and R ∗ = H ( X, O ∗ X ). Therefore, themain technical component of this paper is the study of these derived limits. We do this in Section2, showing the following. Proposition 1.1. If R is a perfectoid ring then R i lim R ∗ = 0 for all i > . We prove the main theorem in Section 3, using the Grothendieck spectral sequece to construct θ ,and explicitly computing the obstructions to injectivity and surjectivity in terms of derived limitsof O ∗ X and R ∗ . This allows us to leverage Proposition 1.1 to prove the main result.1 heorem 1.2. Let X be a perfectoid space with tilt X ♭ and assume that R = Γ( X, O X ) is aperfectoid ring. Then there is a natural isomorphism θ : Pic X ♭ ∼ −→ lim Pic X . Composing the target of θ with projection onto the first coordinate one obtains a characterizationof when the Picard groups of X and X ♭ agree. Corollary 1.3.
Let X be a perfectoid space with tilt X ♭ and assume that R = Γ( X, O X ) is aperfectoid ring. There is a natural map Pic X ♭ → Pic X which is an isomorphism if and only if Pic X is uniquely p -divisible. In fact, the kernel and cokernel of this map can be explicitly computed in terms of the action of p on Pic X . The image consists of line bundles which are p n -powers for every n , and the kernel canbe identified with the p -adic Tate module of Pic X .We continue by enumerating consequences and examples, including examples where the Picardgroups of X and X ♭ agree, and examples where they don’t. For instance, a consequence of Bhattand Scholze’s theory of prismatic cohomology is that the Picard groups of a large class perfectoidrings are uniquely p -divisible [BS19, Corollary 9.5], so that we may extract the following result. Corollary 1.4.
Let R be a perfectoid ring over a perfectoid field, and let R ♭ be its tilt. Then Pic R ♭ ∼ = Pic R . In the final section we apply these results to describe an example of a perfectoid cover of an abelianvariety over C p with good reduction, A ∞ → A (as in [PS16] and [BGH + A → Pic A ∞ is not surjective. This stands in stark contrast to the cases of theperfectoid cover of projective space and perfectoid covers of toric varieties where the analogousmap is an isomorphism ([DH19, Theorem 3.4] and [DHRW20, Theorem 4.1] respectively). To dothis we use analog of Theorem 1.2 to compare the N´eron-Severi groups of A , A ∞ , and A ♭ ∞ . This isnot exclusively a mixed characteristic phenomenon, forthcoming work of Ben Heuer [Heu20, Section9.1] constructs such an example in characteristic p , using a comparison between the Picard groupof A ∞ and the Picard group of the perfection of the special fiber of A . In both cases the analysisboils down the phenomenon of Picard rank potentially jumping on a special fiber.In [DH19] we constructed θ from a geometric perspective using the notion of projectivoid geometry .This construction required the existence of an ample line bundle on X ♭ to guarantee that onecan associate line bundles on X ♭ to maps to a perfectoid analog of projective space using [DH19,Theorem 4.6] which in turn are in one-to-one correspondance for X and X ♭ by the tilting equivalenceof Scholze [Sch12, Proposition 6.17], thereby giving θ . With this we gave a geometric proof that θ was injective [DH19, Theorem 5.13], although in less generality than this paper. The author is indebted to Bhargav Bhatt for communicating many of the main ideas of this proofover several conversations. The author also thanks Ben Heuer, Martin Olsson, and Peter Wear for en-lightening conversations and advice. The author is partially supported by NSF grant DMS-1646385as part of the Research Training Group in arithmetic geometry at the University of California,Berkeley. 2
Derived Limits of Unit Groups of Perfectoid Rings
For what follows we will denote by lim the functor which takes an abelian group (or sheaf ofabelian groups) A to the inverse limit of the system ( · · · → A → A ) whose transition maps are allmultiplication by p (or p -power if the group is multiplicative). We will denote the derived functorsby R i lim and the derived inverse limit by R lim. The main goal of this section is to establish thevanishing of derived limits of the unit groups of perfectoid rings, in both the integral and Tate cases.We begin with the integral case. Definition 2.1 (Integral Perfectoid Rings).
A topological ring R is called an integral perfec-toid ring if there exists a non-zero-divisor ̟ ∈ A such that the following conditions hold.(1) R has the ̟ -adic topology.(2) ̟ divides p .(3) Frobenius induces an isomorphism R/̟R ∼ −→ R/̟ p R . We will first show that if R is integral perfectoid then R ∗ has vanishing derived limits. The followinglemma sets up the argument. Lemma 2.2.
Let ( A i ) be an inverse system of abelian groups and let A = lim i A i . Suppose thefollowing 2 conditions hold.(1) For all i , lim A i = 0 .(2) For all i , R lim A i = 0 .Then R lim A = 0 . Proof.
We derive the identification lim(lim i A i ) ∼ = lim i (lim A i ) and consider the composition offunctors spectral sequence which induces the following:0 R lim(lim i A i ) R (lim ◦ lim i )( A i ) lim( R lim i A i )0 R lim i (lim A i ) R (lim i ◦ lim)( A i ) lim i R lim A i . ∼ = The bottom left corner is 0 by condition (1), and the bottom right corner is 0 by condition (2).Therefore the top left corner is 0, but this is R lim A , so we are done. (cid:3) We would like to apply this in the following case.
Lemma 2.3.
Suppose A is an abelian group with a complete descending filtration whose subquo-tients are p -torsion. Then R lim A = 0 . Proof.
We denote the filtration by · · · ⊆ F ⊆ F ⊆ F = A , let Q i = F i /F i +1 be the subquotients,and consider the inverse system ( · · · → Q i → Q i ) whose transition maps are multiplication by p − hence 0 since Q i is p -torsion. Therefore lim Q i = 0 and the system has stable images (which areall 0), so that it satisfies the Mittag-Leffler condition and R lim Q i = 0 as well.3et A i = A/F i . We have the following diagram of exact sequences for all i .0 F i +1 F i Q i F i +1 A A i +1 . Therefore by the snake lemma we obtain exact sequences0 −→ Q i −→ A i +1 −→ A i −→ , for all i . Passing to the limit along multiplication by p one obtains the following long exact sequence:0 → lim Q i → lim A i +1 → lim A i → R lim Q i → R lim A i +1 → R lim A i . Arguing inductively we may assume that lim A i = 0 = R lim A i (since A = Q ), so that lim A i +1 =0 = R lim A i +1 . Lastly, since we are assuming that A is complete with respect to the topologyinduced by the F i , we see that the natural map A → lim i A i is an isomorphism, so that by Lemma2.2 we may conclude that R lim A = 0. (cid:3) With this in hand we can take care of the main result of this section in the integral perfectoid case.
Proposition 2.4.
Let R be an integral perfectoid ring. Then R lim R ∗ = 0 Proof.
Since ̟ lives in the Jacobson radical of R , the projection map R → R/̟ remains surjectiveon unit groups, so that we have an exact sequence1 −→ ̟ −→ R ∗ −→ ( R/̟ ) ∗ −→ . As R is perfectoid, Frobenius surjects on R/̟ , so that the p -power map surjects on ( R/̟ ) ∗ . Inparticular, the system ( · · · → ( R/̟ ) ∗ → ( R/̟ ) ∗ ) satisfies the Mittag-Leffler condition and so R lim( R/̟ ) ∗ = 0. Considering the long exact sequence derived from applying lim to the sequenceabove therefore provides a surjection R lim(1 + ̟ ) R lim R ∗ , so that it suffices to prove the source vanishes. But 1 + ̟ has a complete filtration by 1 + ̟ n for n ≥
0, and the subquotients are 1 + ̟ n /̟ n +1 . Since ̟ | p , we observe that 1 + ̟ n /̟ n +1 is p -torsion.Indeed, given 1 + f ̟ n ∈ ̟ n /̟ n +1 , we have(1 + f ̟ n ) p = 1 + ( pf ̟ n + · · · ) = 1 + ̟ n +1 ( · · · ) ≡ ̟ n +1 . Therefore by Lemma 2.3 we deduce that R lim(1 + ̟ ) = 0 completing the proof. (cid:3) For the remainder of this section we extend this result to the perfectoid Tate case. We first recallthe definition.
Definition 2.5 (Perfectoid Tate Rings).
A complete topological ring R is Tate if there existsan open subring of definition R ⊆ R whose topology is ̟ -adic for some unit ̟ ∈ R ∩ R ∗ . Theelement ̟ is called a pseudouniformizer for R .A Tate ring R is called a perfectoid Tate ring if the subring R ◦ of power bounded elements in R isan integral perfectoid ring. emark 2.6. To a perfectoid ring R one can associate a perfectoid ring R ♭ of characteristic p called the tilt of R . As a multiplicative monoid R ♭ = lim x x p R so that it comes equipped with amultiplicative map ♯ : R ♭ → R which is identified with projection onto the first coordinate. R ♭ and R share many algebraic and topological properties, explored through the various tilting equivalences of Scholze [Sch12].Fix a perfectoid Tate ring R . We will reduce the study of the derived limits of R ∗ to that of( R ◦ ) ∗ (whose derived limits we already know vanish by Proposition 2.4) using the following exactsequence: 0 −→ ( R ◦ ) ∗ −→ R ∗ −→ R ∗ / ( R ◦ ) ∗ −→ . The right hand side of this sequence can be thought of as a generalized value group for R . It clearlysuffices to show that this generalized value group is p -divisible, and indeed, the argument is similarto showing that the value group of a perfectoid field is p -divisible. The crucial observation in thiscase is that a perfectoid Tate ring can be equipped with a nonarchimedean seminorm under which R ◦ is the unit ball. To set this up, we fix a pseudouniformizer ̟ ∈ R together with a coherentsystem of p -power roots ̟ /p n of ̟ . In this way, the symbol ̟ d makes sense for any d ∈ Z [1 /p ].One can do this, for example, by taking the image ( ̟ ♭ ) ♯ of a pseudouniformizer ̟ ♭ ∈ R ♭ in the tiltof R , under the untilting map ♯ : R ♭ → R . Definition 2.7 (The Spectral Seminorm).
Let R be a perfectoid Tate ring, and R ◦ its subringof power bounded elements. Let ̟ ∈ R be a pseudouniformizer equipped with a complete set of p -power roots. We define the spectral seminorm || · || : R → R ≥ by assigning to f ∈ R the value || f || = inf n − d : d ∈ Z [1 /p ] and f ∈ ̟ d R ◦ o . Proposition 2.8.
Let R be a perfectoid Tate ring and let || · || be the spectral seminorm.(1) The spectral seminorm is a seminorm. That is,(a) || || = 0 and || || = 1 (b) || f g || ≤ || f || · || g || for all f, g ∈ R (c) || f + g || ≤ max {|| f || , || g ||} for all f, g ∈ R (2) For all d ∈ Z [1 /p ] and f ∈ R , we have || ̟ d f || = || ̟ d || · || f || = 2 − d · || f || .(3) R ◦ = { f ∈ R : || f || ≤ } (4) R ◦∗ = { f ∈ R ∗ : || f || = 1 } Proof.
We make free use of following characterization of the spectral seminorm: || f || ≤ − d if andonly if f ∈ ̟ d R ◦ .We start with (1). Since 0 ∈ ̟ d R ◦ for all d , we see that || || ≤ − d for all d ∈ Z [1 /p ], so that itmust be 0. Since ̟ − d is never power bounded for any d >
0, we have 1 ∈ ̟ d R ◦ if and only if d ≤ || || = 1.Now let || f || = α and || g || = β . Any n ∈ Z [1 /p ] with 2 − n ≥ αβ can be written as d + e for d, e ∈ Z [1 /p ] such that 2 − d ≥ α and 2 − e ≥ β . This implies that f g ∈ ( ̟ d R ◦ )( ̟ e R ◦ ) = ̟ n R ◦ sothat || f g || ≤ − n . Passing to the infimum over all such n shows || f g || ≤ αβ establishing submulti-plicativity. 5o prove the ultrametric inequality, we may assume without loss of generality that α ≤ β . Thenfix any d with 2 − d ≥ β ≥ α and observe that f, g ∈ ̟ d R ◦ , meaning f + g ∈ ̟ d R ◦ . Therefore || f + g || ≤ − d . Taking the infimum over all such d shows || f + g || ≤ β = max {|| f || , || g ||} as desired.Therefore the spectral seminorm is indeed a seminorm.Part (2) follows immediately from the observation that f ∈ ̟ e R ◦ if and only if ̟ d f ∈ ̟ d + e R ◦ .We deduce part (3) observing that if || f || = α , then f ∈ R ◦ = ̟ R ◦ if and only if 2 = 1 ≥ α .From this we derive part (4), noting that for any f ∈ R ∗ ,1 = || f · f − || ≤ || f || · || f − || . But if f ∈ R ◦∗ then so is f − , so that in particular both || f || , || f − || ≤
1. Since their product is atleast 1, both factors must equal 1. (cid:3)
The spectral seminorm on a perfectoid Tate ring has the following normalization property.
Lemma 2.9.
Let R be a perfectoid Tate ring equipped with the spectral seminorm. Let f ∈ R ∗ be a unit. Then there is some p -power unit λ ∈ ( R ∗ ) p such that || λf || = 1 . Proof.
Let R ♭ be the tilt of R and ♯ : R ♭ → R the untilting map. Any element in the image of ♯ has a complete system of p -power roots, so in particular is a p -power. We will exhibit a normalizerfor f in the image of ♯ .We will first take care of the case where || f − || ≤ f and f − satisfies this since the spectral norm is a priori only submultaplicative). We fix somepositive integer d large enough so that f − / ∈ ̟ d R ◦ and || ̟ d f || <
1. Then ♯ descends to anidentification: ♯ : R ♭ ◦ / ( ̟ ♭ ) d ∼ −→ R ◦ /̟ d . Choose some g so that g ♯ = f − . Since f − is a unit, so is its image modulo ̟ d , therefore g isa unit. Futhermore, since R ♭ ◦ is ̟ ♭ -complete, the projection R ♭ ◦ ։ R ♭ ◦ / ( ̟ ♭ ) d remains surjectiveon unit groups, so we may choose g to be a unit. Since ♯ is multiplicative, g ♯ is a unit as well. Bydesign we have g ♯ = f − + ̟ d h for some h ∈ R ◦ so that we compute: || g ♯ f || = || ( f − + ̟ d h ) f || = || ̟ d f h || = 1 , where the last equality follows from the ultrametric property because || ̟ d f || <
1. Letting λ = g ♯ gives the desired normalization.In general it may be that || f − || >
1. Nevertheless, we may choose e so that || ̟ e f − || ≤
1. By theprevious paragraph we can find some p -power unit λ normalizing ̟ − e f . But ̟ − e is also a p -powerunit, so that λ̟ − e is a p -power unit that normalizes f . (cid:3) This machinery allows us to assert the p -divisibility of the generalized value group of R . Lemma 2.10.
Let R be a perfectoid Tate ring with ring of power bounded elements R ◦∗ . Then R ∗ /R ◦∗ is p -divisible. Proof.
Fix f ∈ R ∗ and suppose first that || f || = 1. As R is perfectoid, we know that Frobeniusis surjective on R ◦ /̟R ◦ , so that there are g, h ∈ R ◦ such that f = g p + ̟h . We first observe that g p ∈ R ◦∗ . Indeed, g p = f − ̟h = f (1 − ̟hf − ) . f is a unit is suffices to show 1 − ̟hf − is too. As || ̟h || < || f − || = 1 (by Proposition2.8(4)), their product has norm strictly less than 1. Therefore the geometric series11 − ̟hf − = X n ( ̟hf ) n converges. Thus g p is a unit in R ◦ so that f /g p ∈ R ◦∗ .Now let f ∈ R ∗ be any unit. By Lemma 2.9 there is some p -power unit λ = η p such that || λf || = 1.By the previous paragraph there is some g ∈ R ◦ such that λf /g p ∈ R ◦∗ . But: λfg p = fλ − g p = f ( η − g ) p . In particular, f ≡ ( η − g ) p mod R ◦∗ , i.e., that f is a p th power modulo R ◦∗ . Since f was arbitrary,we see that R ∗ /R ◦∗ is p -divisible. (cid:3) Now we can prove the main result of this section in the perfectoid Tate case.
Proposition 2.11.
Let R be a perfectoid Tate ring. Then R lim R ∗ vanishes. Proof.
Consider the exact sequence1 −→ R ◦∗ −→ R ∗ −→ R ∗ /R ◦∗ −→ , and apply lim, to obtain the long exact sequence, · · · −→ R lim R ◦∗ −→ R lim R ∗ −→ R lim R ∗ /R ◦∗ −→ · · · The left term vanishes by Lemma 2.4. Meanwhile, by Proposition 2.10 we know R ∗ /R ◦∗ is p -divisible,so that the inverse system ( · · · → R ∗ /R ◦∗ → R ∗ /R ◦∗ ) has surjective transition maps, and thereforesatisfies the Mittag-Leffler condition. This implies that the right term vanishes, and so the middleone must as well by exactness and we win. (cid:3) Definition 3.1.
A perfectoid space is an adic space with an open cover by affinoid adic spaces
Spa(
R, R + ) where R is a perfectoid Tate ring, or Spa(
R, R ) where R is an integral perfectoid ring.(In the literature the latter is often called a perfectoid formal scheme, but we will not distinguishbetween the two cases in what follows). Remark 3.2.
The tilting process of Remark 2.6 glues, so that to a perfectoid space X one canassociate its tilt : a perfectoid space X ♭ of characteristic p which is locally covered by the adicspectra of the tilts of the perfectoid affinoid rings defining the charts of X . The various tiltingequivalences referred to in Remark 2.6 globalize as well, for example X and X ♭ are homeomorphicand have equivalent ´etale sites [Sch12].Let X be a perfectoid space with tilt X ♭ . We begin this section by constructing the desired mapbetween the Picard groups of X and X ♭ . By [KL16, Theorem 3.5.8], vector bundles on X in theanalytic, ´etale, pro-´etale, and v-topologies all agree (and similarly for X ♭ ), so there is no confusion7y what we mean by the Picard group. Denote the sheaves of units of X and X ♭ by O ∗ X and O ∗ X ♭ respectively. As X and X ♭ have canonically isomorphic analytic, ´etale, pro-´etale, and v-sites, wecan view both as sheaves on X . With this perspective in mind, we have an identification: O ∗ X ♭ ∼ = lim O ∗ X . The desired map comes from considering the derived global sections of this identification andanalyzing the Grothendieck spectral sequence. We have the following diagram of functors.
ShAb ( X ) ShAb ( X ) AbAb
Γlim Γ lim . (1) Lemma 3.3. Γ takes injective sheaves to lim -acyclic abelian groups, and lim takes injective sheavesto Γ -acyclic sheaves. Proof.
In fact, the global sections functor takes injectives to injectives because it has an exactleft adjoint (given by the constant sheaf associated to an abelian group).To show lim takes injectives to acyclics we fix an injective module I , and observe that for every U , I | U is injective also (since restriction to U is right adjoint to extension by 0 which is exact). Theglobal sections of an injective sheaf is an injective abelian group and is therefore divisible. Thusthe multiplication by p -map on I is surjective. Furthermore, by injectivity we have H i ( U, I ) = 0for all i >
0, so that for any i and U the inverse system along multiplication by p (cid:0) · · · → H i ( U, I ) → H i ( U, I ) (cid:1) has surjective transition maps and therefore satisfies the ML-condition. Therefore, applying [GD63,Proposition 13.3.1] we observe that for all i the natural map:H i ( X, lim I ) → lim H i ( X, I ) , is an isomorphism. In particular, lim I is Γ-acyclic as desired. (cid:3) This allows us to harness the composition of functors spectral sequence to construct the morphismat the center of the manuscript.
Proposition 3.4.
Let X be a perfectoid space with tilt X ♭ . There is a natural homomorphism θ : Pic( X ♭ ) → lim Pic( X ) . Proof.
Consider the compositions in Diagram (1). Applying Lemma 3.3 and [Sta20, Tag 015M]one obtains natural isomorphisms of derived functors: R Γ( X, R lim( · )) ∼ ←− R (Γ( X, · ) ◦ lim( · )) ∼ = R (lim ◦ Γ( X, · )) ∼ −→ R lim R Γ( X, · ) . (2)8urthemore, the cohomology of the middle complex computes the sheaf cohomology of lim of asheaf. Indeed, if F is an abelian sheaf on X and I ∗ → F is an injective resolution, then byLemma 3.3 we know lim I ∗ → lim F is an acyclic resolution so that:H i ( R (Γ( X, · ) ◦ lim F )) ∼ = H i ( R Γ( X, lim F )) ∼ = H i ( X, lim F ) . Therefore, identifying O ∗ X ♭ ∼ = lim O ∗ X as abelian sheaves on X , as well as making the appropri-ate identifiations using Equation (2), we obtain the following 2 composition of functors spectralsequences: E p,q : H p ( X, R q lim O ∗ X ) = ⇒ H p + q ( X, R lim O ∗ X ) ,E ′ p,q : R p lim H q ( X, O ∗ X ) = ⇒ H p + q ( X, R lim O ∗ X ) . Considering the low degree terms gives us the following diagram whose rows are exact, and θ appears as the diagonal arrow.0 H ( X, O ∗ X ♭ ) H ( X, R lim O ∗ X ) Γ( X, R lim O ∗ X ) H ( X, O ∗ X ♭ )0 R lim Γ( X, O ∗ X ) H ( X, R lim O ∗ X ) lim H ( X, O ∗ X ) R lim Γ( X, O ∗ X ) . (3)If the global sections of X form a perfectoid ring then Diagram (3) provides the following immediatecorollary. Corollary 3.5.
Let X be a perfectoid space with tilt X ♭ . If the global functions on X form aperfectoid ring then θ fits into the following exact sequence. X ♭ ) lim Pic( X ) Γ( X, R lim O ∗ X ) . θ In particular, θ is injective. Proof.
We know derived limits of abelian groups have cohomology concentrated in degrees 0 and 1(see for example [Sta20, Tag 07KW]), so that the bottom right term of Diagram 3 is 0. Futhermore,since the global sections of O X are perfectoid R lim Γ( X, O ∗ X ) = 0 (by Proposition 2.4 or 2.11 inthe integral or Tate cases respectively). Therefore the bottom row of diagram (3) degenerates toan isomorphism between H ( X, R lim O ∗ X ) and lim H ( X, O ∗ X ) and the desired sequence emerges asthe top row. (cid:3) The surjectivity of θ reduces therefore to the vanishing of R lim O ∗ X . So far our arguments wereindependent of whether we were working in the analytic, ´etale, pro-´etale, or v-topologies. Thereforeif we can show that R lim O ∗ X vanishes locally in any of these topologies we will be done. Lemma 3.6.
Let X be a perfectoid space. Then R lim O ∗ X is the pro-´etale sheafification of thefunctor U H ( U, R lim O ∗ X ) . Proof.
We consider the distinguished trianglelim O ∗ X R lim O ∗ X τ ≥ R lim O ∗ X lim O ∗ X [1] , U ,H ( U, lim O ∗ X ) H ( U, R lim O ∗ X ) Γ( U, R lim O ∗ X ) H ( U, lim O ∗ X ) . The left side is the Picard group of U ♭ and the right side is the cohomological Brauer group of U ♭ .Both are pro-´etale locally trivial, so that as we vary U and sheafify both vanish. As sheafificationis exact, we obtain an isomorphism between the sheafifications of the middle two terms completingthe proof. (cid:3) Lemma 3.6 implies that the vanishing of R lim O ∗ X follows if the functor H ( · , R lim O ∗ X ) is pro-´etalelocally trivial on X , so we prove that next. Lemma 3.7.
The functor U H ( U, R lim O ∗ X ) is pro-´etale locally trivial on X . Proof.
We revisit the second spectral sequence from Proposition 3.4, ( E ′ p,q ), and consider againthe low degree terms, obtaining the following exact sequence for any pro-´etale open U :0 R lim Γ( U, O ∗ X ) H ( U, R lim O ∗ X ) lim H ( U, O ∗ X ) 0 . Since X is perfectoid, it has a basis of perfectoid affinoid opens U . On these opens Γ( U, O X ) is aperfectoid ring, so that by Propositions 2.4 or 2.11 (in the integral or Tate settings respectively)we conclude that R lim Γ( U, O ∗ X ) = 0, so the left side of the sequence is pro-´etale locally trivial.To conclude we show the right side is as well. We must show that given an inverse system of linebundles on U , there is a pro-´etale cover of U which simultaneously trivializes them all the linebundles. Let ( L , L , L , · · · ) ∈ lim Pic( U ). Then there is some cover V → U which trivializes L .Then we can take an ´etale cover V → V which trivializes L , and a cover V → V trivializing L .Continuing in this fashion and letting V = lim V i , we produce V → U , a pro-´etale cover trivializingeach L i .As both sides of the exact sequence are pro-´etale locally trivial, the middle is as well. (cid:3) The main theorem now follows.
Theorem 3.8.
Suppose X is a perfectoid space with tilt X ♭ , and suppose that Γ( X, O X ) is aperfectoid ring. Then θ : Pic X ♭ → lim Pic X is an isomorphism. Proof.
Lemmas 3.6 and 3.7 together imply that R lim O ∗ X = 0, so that the exact sequence fromCorollary 3.5 degenerates to the desired isomorphism. (cid:3) Let X be any perfectoid space. One can compose θ : Pic X ♭ → lim Pic X together with the pro-jection π : lim Pic X → Pic X onto the first coordinate to obtain the untilting homomorphism θ : Pic X ♭ → Pic X . The kernel and image of π are easy to understand:ker π = lim n Pic X [ p n ] =: T p Pic X, im π = \ n (Pic X ) p n =: (Pic X ) p ∞
10f Γ( X, O X ) is a perfectoid ring, then by Theorem 3.8 we have the following exact sequence:0 −→ T p Pic X −→ Pic X ♭ θ −→ (Pic X ) p ∞ −→ . In particular, we have the following corollary to Theorem 3.8.
Corollary 4.1.
Let X be a perfectoid space and suppose Γ( X, O X ) is perfectoid. Consider theuntilting homomorphism θ : Pic X ♭ → Pic X .(1) θ is injective if and only if Pic X is p -torsion free.(2) θ is surjective if and only if Pic X is p -divisible.(3) θ is an isomorphism if and only if Pic X is uniquely p -divisible. As a consequence of the ´etale comparison theorem in prismatic cohomology, Bhatt and Scholze show[BS19, Corollary 9.5] that a large class of perfectoid affinoid adic spaces have uniquely p -divisiblePicard groups. In particular, they show that if R is integral perfectoid, then R and R [1 /p ] haveuniquely p -divisible Picard groups. This gives us the following immediate corollary. Corollary 4.2.
Let R be an integral perfectoid ring. Then Pic R ♭ ∼ = Pic R and Pic( R [1 /p ] ♭ ) ∼ = Pic( R [1 /p ]) . In particular, θ is an isomorphism for any perfectoid algebra over a field. One observes that the same is true in a number of global examples.
Example 4.3.
In [DH19, Theorem 3.4] we computed the Picard group of projectivoid space , show-ing that that Pic P n, perf ∼ = Z [1 /p ], which is uniquely p -divisible. Therefore the untilting map forprojectivoid space is an isomorphism. Of course, since the value of Pic P n, perf is independent ofperfectoid base field (and therefore tilting) this is unsurprising. Example 4.4.
Generalizing the previous example, if X perf is the perfectoid cover of a smoothproper toric variety X (as in [Sch12, Section 8]), then Pic( X perf ) ∼ = Pic( X )[1 /p ], (by [DHRW20,Theorem 4.1]). In particular, p acts invertibly on Pic( X perf ) and so θ is an isomorphism. Example 4.5.
Let A be an abelian variety (over a perfectoid field), and [ p ] : A → A the multipli-cation by p map. Then passing to the inverse limit there is a perfectoid space A perf ∼ lim A (by[PS16, Lemme A.16] in the case where A has good reduction over an algebraically closed field, andby [BGH +
18, Theorem 4.6] in general). Work of Heuer [Heu20, Theorem 4.5] shows that if A hasgood reduction, then p acts invertibly in the Picard group of A perf , so that by Corollary 4.1 wemay conclude that Pic (cid:0) ( A perf ) ♭ (cid:1) ∼ = Pic( A perf ).We also include an example where θ is neither injective nor surjective, coming from a constructionof [BGH +
18] and [Heu20] of a perfectoid space whose Picard group is not p -divisble nor p -torsionfree. 11 xample 4.6. Let K be a perfectoid field of characteristic 0, and consider the Tate uniformizationof an elliptic curve E = G m /q Z , considered as an adic space over K . Suppose q has a coherent systemof p -power roots − for example, if q is in the image of ♯ : ( K ♭ ) ∗ → K ∗ . Fix such a system so that thesymbol q /p n makes sense for all n >
0. The p -power map induces a sequence of isogenies: · · · → G m /q /p → G m /q /p → G m /q, and by [BGH + E ∞ ∼ lim ←− n G m /q /p n . By [Heu20, Proposition 9.5] the Picard group of E ∞ fits into the following exact sequence.0 −→ K ∗ /q Z [1 /p ] −→ Pic E ∞ −→ Z [1 /p ] −→ . Consider multiplication by p on this sequence:0 K ∗ /q Z [1 /p ] Pic E ∞ Z [1 /p ] 00 K ∗ /q Z [1 /p ] Pic E ∞ Z [1 /p ] 0 Φ Ψ
Corollary 4.1 says that the injectivity and surjectivity of the untilting map θ : Pic E ♭ ∞ → Pic E ∞ is controlled by the kernel and cokernel of Ψ, and since Z [1 /p ] is uniquely p -divisible the snakelemma tells us that these are in turn isomorphic to the kernel and cokernel of Φ. But these are easyto compute. Consider the square whose rows are the p -power map K ∗ K ∗ K ∗ /q Z [1 /p ] K ∗ /q Z [1 /p ]Φ The kernel of the composition consists of functions f ∈ K ∗ such that f p = q d for some d ∈ Z [1 /p ].Thus f /q d/p ∈ µ p is a p ’th root of unity. In particular, f is given by an element of q Z [1 /p ] and one of µ p , and furthermore q Z [1 /p ] ∩ µ p = 1 so that the kernel is isomorphic to µ p × q Z [1 /p ] . We get the ker Φmodding out by the second factor, so that ker Φ ∼ = µ p . Arguing for successively higher powers of p we see that the kernel of the projection lim Pic E ∞ → Pic E ∞ is isomorphic to lim n µ p n = Z p (1), sothat by Theorem 3.8, we have a left exact sequence:0 −→ Z p (1) −→ Pic E ♭ ∞ θ −→ Pic E ∞ , (4)whose right exactness is controlled by coker Φ. The cokernel of the p -power map from K ∗ to itselfis K ∗ / ( K ∗ ) p , and we chose q so that q Z [1 /p ] ⊆ ( K ∗ ) p so that:coker Φ = (cid:16) K ∗ /q Z [1 /p ] (cid:17) / (cid:16) ( K ∗ ) p /q Z [1 /p ] (cid:17) ∼ = K ∗ / ( K ∗ ) p . Therefore Sequence 4 is right exact if and only if every element of K ∗ is a p ’th power. In particular,we see that it is right exact if K is algebraically closed, but is not in general − for example if K = Q p (cid:0) p /p ∞ (cid:1) ˆ. 12n fact, we can give a concrete description of the kernel and cokernel of θ for elements that arisefrom divisors on the Tate curve. Observe that the choices of p -power roots of q ∈ K ∗ determines aunique element q ♭ ∈ ( K ♭ ) ∗ . Then we can compute the tilt of E ∞ as( E ∞ ) ♭ ∼ lim ←− n ( G m ) ♭ / ( q ♭ ) /p n . Identifying colim Pic E as a subset of Pic E ∞ (and similarly for Pic E ♭ ∞ ), we can identify points of K ∗ /q Z [1 /p ] (respectively ( K ♭ ) ∗ / ( q ♭ ) Z [1 /p ] ) with certain degree 0 line bundles on E ∞ (respectively( E ∞ ) ♭ ). On these line bundles, θ descends from the untilting map ♯ .( K ♭ ) ∗ K ∗ ( K ♭ ) ∗ / ( q ♭ ) Z [1 /p ] K ∗ /q Z [1 /p ] . ♯θ With this presentation, we see that divisors in the kernel are precisely those with coordinates thatmap to 1 under ♯ (i.e., elements of Z p (1)), elements of the cokernel come from points in K ∗ whichdon’t have infintely many p -power roots. Let L , L , L , · · · be a system of line bundles on a perfectoid space X with L ⊗ pi +1 ∼ = L i . As weargued in the proof of Lemma 3.7, there is an obvious way to construct a pro-´etale cover of X which simultaneoulsy trivializes all of the L i , by further refining ´etale covers trivializing each L i individually and letting i go to infinity. Nevertheless, it isn’t immediately clear that there shouldbe an ´etale cover which simultaneoulsy trivializes all the L i . A consequence of Theorem 3.8 is thatthere is, and in fact there is even an analytic cover that does so. Corollary 4.7.
Let X be a perfectoid space and L , L , · · · , an system of line bundles on X with L ⊗ pi +1 ∼ = L i . Then there is an analytic cover U → X which simultaneoulsy trivializes all the L i . Proof.
Although the global sections of O X aren’t a priori perfectoid, there is an analytic opencover V → X where Γ( V, O V ) is. Therefore by Theorem 3.8, the inverse system ( L i | V ) ∈ lim Pic V is the untilt of a unique L ∈ Pic V ♭ . There is an analytic cover U ♭ → V ♭ trivializing L , which isthe tilt of an analytic cover U → V . By functoriality the inverse system ( L i | U ) ∈ lim Pic U is theuntilt of L U ♭ , which is trivial. Again by Theorem 3.8 each L i | U is trivial, completing the proof. (cid:3) ℓ -adic Cohomological N´eron-Severi Groups We’d like to study how the untilting maps θ and θ act on N´eron-Severi groups of perfectoid spaces.Since we don’t have representability of the Picard functor, we need a notion of N´eron-Severi groupsto make sense of this, so we take a cohomological approach. We begin by defining the ℓ -adic cycleclass map.Let X be an adic space, and ℓ a prime invertible in O X . From the long exact sequence on ´etalecohomology associated to the Kummer sequence 1 → µ ℓ n → G m → G m → Xℓ n Pic
X ֒ → H ( X, µ ℓ n ) . (5)13assing to the inverse limit among all n , one obtains a map Pic X ⊗ Z ℓ (1) ֒ → H ( X, Z ℓ (1)) andprecomposing with the natural map from Pic X , one obtains thet ℓ -adic cycle class map : c ℓ : Pic X → H ( X, Z ℓ (1)) . Definition 4.8.
Let X be an adic space and ℓ a prime invertible in O X . Then the ℓ -adic cohomo-logical N´eron-Severi group is defined to be the image of the ℓ -adic cycle class map. NS ℓ ( X ) := im( c ℓ ) . In certain nice situations, including the case of an abelian variety over an algebraically closed field,this definition agrees with the usual notion of N´eron-Severi groups.
Lemma 4.9.
Let A be a proper nonsingular variety over an algebraically closed field of character-isitic not equal to ℓ , and suppose further that NS( A ) is torsion-free. Then NS ℓ ( A ) ∼ = NS( A ) . Proof.
This is well known but we include the proof for completeness. We first observe that becausePic ◦ ( A ) is divisible, we have Pic ◦ A ⊆ ℓ n Pic A so that:Pic Aℓ n Pic A ∼ = Pic A/ Pic ◦ Aℓ n Pic A/ Pic ◦ A ∼ = NS( A ) ℓ n NS( A ) . In particular, the source of the injection from Equation 5 can be identified with NS( A ) /ℓ n NS( A )and passing the the inverse limit along n one obtains the following composition whose image agreeswith the image of the ℓ -adic cycle class map c ℓ :NS( A ) → NS( A ) ⊗ Z ℓ (1) ֒ → H ( X, Z ℓ (1)) . We finish by observing that the first map is injective because NS( A ) is torsion-free. (cid:3) The ℓ -adic cohomological N´eron-Severi group plays well with the untilting homomorphism that isthe subject of this paper. Proposition 4.10.
Let X be a perfectoid space and ℓ prime to the residue characteristic of X .Then the untilting homomorphism θ descends to an injection: NS ℓ ( X ♭ ) ֒ → NS ℓ ( X ) . Proof.
Identify the ´etale sites of X and X ♭ . Since the projection map ♯ : G ♭m ∼ = lim G m → G m ismultiplicative, for all n it induces a map of Kummer sequences1 µ ♭ℓ n G ♭m G ♭m µ ℓ n G m G m . Since the p -power map on µ ℓ n is an isomorphism, the projection µ ♭ℓ n → µ ℓ n is too. Taking the longexact sequences on ´etale cohomology fits θ into the following diagram.Pic X ♭ Pic X ♭ H ( X ♭ , µ ℓ n )Pic X Pic X H ( X, µ ℓ n ) θ ℓ n θ ≀ ℓ n n we obtain the following commutative square.Pic X ♭ H ( X ♭ , Z ℓ (1))Pic X H ( X, Z ℓ (1)) . θ c ♭ℓ ≀ c ℓ In particular, the vertical map on the right restricts to an injection between the images of the cycleclass maps, completing the proof. (cid:3)
Proposition 4.10 has the following immediate corollary, arguing analogously to Corollary 4.1.
Corollary 4.11.
Let X be a perfectoid space and ℓ prime to the residue characteristic of X . If Γ( X, O X ) is a perfectoid ring, then NS ℓ ( X ) is p -torsion free. Both [DH19, Theorem 3.4] and [DHRW20, Theorem 4.1] study examples of varieties X , togetherwith a ‘Frobenius like’ map Φ : X → X so that there is a mixed characteristic perfection , aperfectoid space X perf ∼ lim X . The content of the theorems in each case is that this limit commuteswith taking Picard groups, i.e., that the natural map colim Pic X → Pic X perf is an isomorphism.Forthcoming work of Heuer [Heu20, Section 9.1] shows that this is not true in general, giving anexample where this fails in characteristic p . We conclude by giving an example of this failure incharacteristic 0. In both cases, the counterexample consists of an abelian variety A over a perfectoidfield and its perfectoid cover A perf ∼ lim [ p ] A → A (as in Example 4.5) such that the induced mapcolim Pic A → Pic A perf is not surjective.The idea is to start with an abelian variety whose N´eron Severi rank jumps modulo p , and thenuse Proposition 4.10 to inject the (now larger) N´eron-Severi group of ( A perf ) ♭ into the N´eron-Severi group of A perf , thus exhibiting line bundles which cannot come from A . For the perfectoidcovers we will need to use the ℓ -adic cohomological N´eron-Severi groups of the previous section,observing by Lemma 4.9 that for abelian varieties over algebraically closed fields the two notionsare interchangable. We first confirm that the N´eron-Severi rank does not decrease when passing tothe perfectoid cover. Lemma 4.12.
Suppose A is an abelian variety over a perfectoid field, and let ℓ be prime to theresidue characteristic. Let A perf → A be its perfectoid cover. The induced map colim NS( A ) → NS ℓ ( A perf ) is injective. Proof.
The Kummer sequence 0 −→ µ ℓ n −→ G m −→ G m −→ A Pic A H ( A, µ ℓ n )colim Pic A colim Pic A colim H ( A, µ ℓ n )Pic A perf Pic A perf H ( A perf , µ ℓ n ) . ℓ n η n L i ) ∈ colim Pic A maps to 0 in NS ℓ ( A perf ). This means that for n large enough, it mapsto 0 in H ( A perf , µ ℓ n ). By [Sch12, Corollary 7.8], η n is an isomorphism, so that ( L i ) maps to0 in colim H ( A, µ ℓ n ). By exactness, L i is an ℓ n -th power for i >>
0, so that the class of L i inNS( A ) ∼ = Z ρ ( A ) is a multiple of ℓ n for all n >>
0, so that it must be 0. Therefore ( L i ) ∈ colim Pic ◦ A and we win. (cid:3) We now have a model for our counterexample.
Proposition 4.13.
Let K be an algebraically closed perfectoid field with tilt K ♭ . Suppose that A is an abelian variety over K , and B an abelian variety over K ♭ , whose perfectoid covers satisfy ( A perf ) ♭ = B perf . If ρ ( B ) > ρ ( A ) , then the map colim Pic A → Pic A perf is not surjective. Proof.
We remind the reader that for any abelian variety X over an algebraically closed field, if[ n ] : X → X is multiplication by n , then the pullback map [ n ] ∗ : NS X → NS X is multiplication by n (see for example [Mum70, 2.8(iv)]). Therefore, as NS( X ) ∼ = Z ρ ( X ) , we know colim [ n ] ∗ NS( X ) ∼ = Z [1 /n ] ρ ( X ) .Therefore, the assumptions of the proposition, together with Lemma 4.12 and Proposition 4.10 giveus the following chain of inequalities:rk Z [1 /p ] colim NS( A ) < rk Z [1 /p ] colim NS( B ) ≤ rk Z [1 /p ] NS ℓ ( B perf ) ≤ rk Z [1 /p ] NS ℓ ( A perf ) . In particular, colim NS( A ) → NS ℓ ( A perf ) cannot surject, and therefore neither can the map inquestion. (cid:3) We conclude by giving examples of abelian varieties which satisfy the assumptions of Proposition4.13. Let A be the integral model over Z p of an abelian variety with good reduction over Q p . Let C p be the completion of the algebraic closure of Q p , and let C ♭p be its tilt, (which obviously is anextension of F p ). Then one can consider the base change of A to C p and C ♭p , which we denote by A C p and A C ♭p respectively. Lemma 4.14.
In the setup of the previous paragraph, we have (cid:16) A perf C p (cid:17) ♭ ∼ = A perf C ♭p . Proof.
Let ̟ ♭ be a pseudouniformizer for C ♭p and ̟ = ( ̟ ♭ ) ♯ . By the tilting equivalence [Sch12],both A perf C p and A perf C ♭p are determined (up to almost isomorphism) by their models over O C ♭p /̟ ♭ = O C p /̟ =: R , which extends F p . In each case, we observe that this model must be the schemelim ←− [ p ] (cid:0) A F p × Spec F p Spec R (cid:1) . (cid:3) To construct abelian varieties satisfying the assumptions of Proposition 4.13, we may thereforestart with abelian varieties with good reduction over Q p . Abelian varieties over Q p whose N´eron-Severi ranks increase upon reduction modulo p are abundant. Take, for example, A = E × E where E is a non-CM elliptic curve over Q p with supersingular reduction. In this case we have ρ ( A C ♭p ) = 6 > ρ ( A C p ), giving the desired example.16 eferences [BGH +
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