Cohomology of (\varphi,Γ)-modules over pseudorigid spaces
CCOHOMOLOGY OF ( ϕ, Γ) -MODULES OVER PSEUDORIGIDSPACES REBECCA BELLOVIN
Abstract.
We study the cohomology of families of ( ϕ, Γ)-modules with coeffi-cients in pseudoaffinoid algebras. We prove that they have finite cohomology,and we deduce an Euler characteristic formula and Tate local duality. Weclassify rank-1 ( ϕ, Γ)-modules and deduce that triangulations of pseudorigidfamilies of ( ϕ, Γ)-modules can be interpolated, extending a result of [KPX14].We then apply this to study extended eigenvarieties at the boundary of weightspace, proving in particular that the eigencurve is proper at the boundaryand that Galois representations attached to certain characteristic p points aretrianguline. Introduction
In our earlier paper [Bel20], we began studying families of Galois representationsvarying over pseudorigid spaces, that is, families of Galois representations wherethe coefficients have a non-archimedean topology but which (in contrast to the rigidanalytic spaces of Tate) are not required to contain a field. Such coefficients arisenaturally in the study of eigenvarieties at the boundary of weight space.The theory of ( ϕ, Γ)-modules is a crucial tool in the study of p -adic Galois represen-tations. At the expense of making the coefficients more complicated, it lets us turnthe data of a Galois representation into the data of a Frobenius operator ϕ and a1-dimensional p -adic Lie group Γ. Moreover, Galois representations which are irre-ducible often become reducible on the level of their associated ( ϕ, Γ)-modules. Such( ϕ, Γ)-modules have played an important role in the p -adic Langlands program.In our previous paper [Bel20], we constructed ( ϕ, Γ)-modules associated to Galoisrepresentations varying over pseudorigid spaces. In the present paper, we turnto the study of the cohomology of ( ϕ, Γ)-modules over pseudorigid spaces havefinite cohomology, whether or not they come from Galois representations. Given a( ϕ, Γ)-module D , the Fontaine–Herr–Liu complex C • ϕ, Γ ( D ) is an explicit three-termcomplex which, when D arises from a Galois representation, computes the Galoiscohomology. We begin by proving that such families of ( ϕ, Γ)-modules have finitecohomology, extending the main result of [KPX14]:
Theorem 1.0.1.
Suppose D is a projective ( ϕ, Γ) -module over a pseudoaffinoidalgebra R . Then C • ϕ, Γ ( D ) ∈ D [0 , ( R ) . As a corollary, we deduce the Euler characteristic formula:
Corollary 1.0.2. If D is a projective ( ϕ, Γ K ) -module with coefficients in a pseu-doaffinoid algebra R , then χ ( D ) = − [ K : Q p ] . a r X i v : . [ m a t h . N T ] F e b R. BELLOVIN
This extends the result [Liu07, Theorem 4.3]. However, the method of proof isdifferent: Liu proved finiteness of cohomology and the Euler characteristic for-mula at the same time, making a close study of t -torsion ( ϕ, Γ)-modules to shiftweights. There is no element t in our setting, because p is not necessarily invert-ible. However, because we proved finiteness of cohomology for pseudorigid familiesof ( ϕ, Γ)-modules first, we can deduce the Euler characteristic formula by deforma-tion, without studying torsion objects.We then turn to ( ϕ, Γ)-modules with coefficients in finite extensions of F p (( u )), andwe prove Tate local duality: Theorem 1.0.3.
Tate local duality holds for every projective ( ϕ, Γ) -module D over Λ R, rig ,K . Our proof closely follows that of [Liu07, Theorem 4.7]; we compute the cohomologyof many rank-1 ( ϕ, Γ)-modules and then proceed by induction on the degree, usingthe Euler characteristic formula to produce non-split extensions. We are then ableto finish the computation of the cohomology of ( ϕ, Γ)-modules of character type.With this in hand, we are able to show that all rank-1 ( ϕ, Γ)-modules over pseu-dorigid spaces are of character type, following [KPX14], and we deduce that tri-angulations can be interpolated from a dense set of maximal points (in the senseof [JN19, Definition 2.2.7]):
Theorem 1.0.4.
Let X be a reduced pseudorigid space, let D be a projective ( ϕ, Γ K ) -module over X of rank d , and let δ , . . . , δ d : K × → Γ( X, O × X ) be a set ofcontinuous characters. Suppose there is a Zariski-dense set X alg ⊂ X of maximalpoints such that for every x ∈ X alg , D x is trianguline with parameters δ ,x , . . . , δ d,x .Then there exists a proper birational morphism f : X (cid:48) → X of reduced pseudorigidspaces and an open subspace U ⊂ X (cid:48) containing { p = 0 } such that f ∗ D | U is trian-guline with parameters f ∗ δ , . . . , f ∗ δ d . Unlike the situation in characteristic 0, the triangulation extends over every pointof characteristic p , and there are no critical points. This is again because there isno analogue of Fontaine’s element t in our positive characteristic analogue of theRobba ring.Finally, we turn to applications to the extended eigenvarieties constructed in [JN16].Adapting the Galois-theoretic argument of [DL16], we prove unconditionally thateach irreducible component of the extended eigencurve is proper at the boundary ofweight space, and that the Galois representations over characteristic p points of theextended eigencurve are trianguline at p . The latter answers a question of [AIP18].We actually prove these results under somewhat abstracted hypotheses, in order tofacilitate deducing analogous results for other extended eigenvarieties. In particular,our results apply to certain unitary and Hilbert eigenvarieties. However, for mostgroups the necessary results have not been proven even for Galois representationsattached to classical forms, nor have the required families of Galois representationsbeen constructed.In the appendices, we have collected several results on the geometry of pseudorigidspaces and Galois determinants over pseudorigid spaces necessary for our applica-tions. OHOMOLOGY OF ( ϕ, Γ)-MODULES OVER PSEUDORIGID SPACES 3
Remark 1.0.5.
We assume throughout that p (cid:54) = 2. It should be possible toremove this hypothesis without any real difficulty, but we would have had to worksystematically with R Γ (cid:0) Γ p − tors K , C • ϕ, Γ ( D ) (cid:1) , rather than the usual Fontaine–Herr–Liu complex. Acknowledgements.
I am grateful to Toby Gee, James Newton, and Lynnelle Yefor helpful conversations. 2.
Background
Rings of p -adic Hodge theory. Let R be a pseudorigid O E -algebra, forsome finite extension E/ Q p with uniformizer (cid:36) E , with ring of definition R ⊂ R ◦ and pseudo-uniformizer u ∈ R , and assume that p (cid:54)∈ R × . Let K/ Q p be afinite extension, let χ cyc : Gal K → Z × p be the cyclotomic character, let H K :=ker χ cyc , and let Γ K := Gal K /H K . Given an interval I ⊂ [0 , ∞ ], we defined rings( (cid:101) Λ R ,I,K , (cid:101) Λ + R ,I,K ) and (Λ R ,I,K , Λ + R ,I,K ) in [Bel20] which (when I = [0 , b ]) areanalogues of the characteristic 0 rings ( (cid:101) A (0 ,b ] K , (cid:101) A † ,s ( b ) K ) and ( A (0 ,r ] K , A † ,s ( r ) K ) definedin [Col08]. Here s : (0 , ∞ ) → (0 , ∞ ) is defined via s ( r ) := p − pr . We briefly recalltheir definitions here and state some of their properties.Let A inf := W ( O (cid:91) C K ), where O (cid:91) C K := lim ←− x (cid:55)→ x p O C K is the tilt of O C K . Let ε :=( ε (0) , ε (1) , . . . ) ∈ O (cid:91) C K be a choice of a compatible sequence of p -power roots ofunity, with ε (0) = 1 and ε (1) (cid:54) = 1, and let π := [ ε ] − ∈ A inf . Then if I = [ a, b ] for0 ≤ a ≤ b ≤ ∞ , we define ( (cid:101) Λ R ,I , (cid:101) Λ + R ,I ) such thatSpa( (cid:101) Λ R ,I , (cid:101) Λ + R ,I ) = (cid:0) Spa( R (cid:98) ⊗ A inf , R (cid:98) ⊗ A inf ) (cid:1) (cid:28) [ π ] s ( a ) u , u [ π ] s ( b ) (cid:29) If a = 0, we take [ π ] ∞ u = 0, and if b = 0, we take u [ π ] ∞ = π ] .The group H K acts on ( (cid:101) Λ R ,I , (cid:101) Λ + R ,I ), because Gal K acts on A inf and H K fixes [ π ].Then by [Bel20, Corollary 3.36],Spa( (cid:101) Λ H K R ,I , (cid:101) Λ + ,H K R ,I ) = (cid:16) Spa( R (cid:98) ⊗ A H K inf , R (cid:98) ⊗ A H K inf ) (cid:17) (cid:28) [ π ] s ( a ) u , u [ π ] s ( b ) (cid:29) If I ⊂ I (cid:48) , we have injective maps (cid:101) Λ R ,I (cid:48) → (cid:101) Λ R ,I and (cid:101) Λ H K R ,I (cid:48) → (cid:101) Λ H K R ,I . Thus, if I (cid:48) is an interval with an open endpoint, we may define( (cid:101) Λ R ,I (cid:48) , (cid:101) Λ + R ,I (cid:48) ) := ∩ I ⊂ I (cid:48) closed ( (cid:101) Λ R ,I , (cid:101) Λ + R ,I )and ( (cid:101) Λ H K R ,I (cid:48) , (cid:101) Λ + ,H K R ,I (cid:48) ) := ∩ I ⊂ I (cid:48) closed ( (cid:101) Λ H K R ,I , (cid:101) Λ + ,H K R ,I )The rings (Λ R ,I,K , Λ + R ,I,K ) are imperfect versions of these, defined when I ⊂ [0 , b ]with b sufficiently small. Given λ = m (cid:48) m ∈ Q > with gcd( m, m (cid:48) ) = 1, let ( D λ , D ◦ λ )denote the pair of rings corresponding to the localization ( O E [[ u ]] , O E [[ u ]]) (cid:68) (cid:36) mE u m (cid:48) (cid:69) .By [Lou17, Lemma 4.8], there is some sufficiently small λ such that R is topologi-cally of finite type over D λ , so we may assume that R is strictly topologically offinite type over D ◦ λ . R. BELLOVIN
For any unramified extension F/ Q p , the choice of ε gives us a natural map k F (( π )) → C (cid:91)K ; let E F denote its image, and let E ⊂ C (cid:91)K be its separable closure. ThenGal( E / E F ) ∼ = H F (by the theory of the field of norms), and for any extension K/F ,we set E K := E H K . Then E K is a discretely valued field, and we may choose auniformizer π K ; if we lift its minimal polynomial to characteristic 0, Hensel’s lemmaimplies that we have a lift π K ∈ W ( C (cid:91)K ) which is integral over O F [[ π ]][ π ] ∧ . We fixa choice π K for each K , and work with it throughout (when F/ Q p is unramified,we take π F to be π ).Assume that 0 ≤ a ≤ b < r K · λ , where r K is a constant defined in [Col08], and that a · v C (cid:91)K ( π K )) , b · v C (cid:91)K ( π K )) ∈ Z . Let F (cid:48) ⊂ K ∞ := K ( µ p ∞ ) be the maximal unramifiedsubfield. Then we define Λ R , [ a,b ] ,K to be the evaluation of O ( R ⊗ O F (cid:48) )[[ π K ]] on theaffinoid subspace of Spa( R ⊗ O F (cid:48) )[[ π K ]] defined by the conditions u ≤ π / ( b · v C (cid:91)K ( π K )) K and π / ( a · v C (cid:91)K ( π K )) K ≤ u (and similarly for Λ + R , [ a,b ] ,K ).If p = 0 in R , then we may take λ arbitrarily large, and hence b arbitrarily large.Thus, in this case we additionally define Λ R , [ a, ∞ ] ,K := ( R ⊗ O F (cid:48) )[[ π K ]].We further define Λ R, (0 ,b ] ,K := lim ←− a → Λ R, [ a,b ] ,K , and Λ R, rig ,K := lim −→ b → Λ R, (0 ,b ] ,K .The rings (cid:101) Λ H K R ,I and Λ R ,I,K are equipped with actions of Frobenius and Γ K . Moreprecisely, we have isomorphisms ϕ : (cid:101) Λ R ,I ∼ −→ (cid:101) Λ R , p I , ϕ : (cid:101) Λ H K R ,I ∼ −→ (cid:101) Λ H K R , p I and ring homomorphisms ϕ : Λ R , [ a,b ] ,K → Λ R , [ a/p,b/p ] ,K However, the latter are not isomorphisms; ϕ makes Λ R , [0 ,b/p ] ,K into a free ϕ (Λ R , [0 ,b ] ,K ) -module, with basis { , [ ε ] , . . . , [ ε ] p − } .If L/K is a Galois extension, (cid:101) Λ H L R ,I and Λ R ,I,L are also equipped with actions of H L/K := H K /H L . Lemma 2.1.1. If L/K is a finite Galois extension, then Λ R , [0 ,b ] ,L / Λ R , [0 ,b ] ,K isa finite free extension and Λ H K R ,I,L = Λ R ,I,K .Proof. Let F (cid:48) ⊂ K ∞ := K ( µ p ∞ ) , F (cid:48)(cid:48) ⊂ L ∞ := L ( µ p ∞ ) be the maximal unramifiedsubfields. A basis for O F (cid:48)(cid:48) over O F (cid:48) provides a basis for ( R (cid:98) ⊗ O F (cid:48)(cid:48) )[[ π K ]] over( R (cid:98) ⊗ O F (cid:48) )[[ π K ]], so we may assume that F (cid:48) = F (cid:48)(cid:48) . Then if e := e L ∞ /K ∞ = [ L ∞ : K ∞ ], the set { , π L , . . . , π e − L } is a basis for Λ R , [0 , ,L over Λ R , [0 , ,K .The trace map defines a perfect pairingΛ R , [0 , ,L × Λ R , [0 , ,L → Λ R , [0 , ,K ( x, y ) (cid:55)→ Tr( xy )The dual basis { f ∗ = 1 , . . . , f ∗ e } with respect to this pairing is the same as thatconstructed in [Col08, § R (cid:98) ⊗ Λ [0 ,b/λ ] ,L ) (cid:42) uπ / ( b · v C (cid:91)p ( πL )) K (cid:43) is a ring ofdefinition of Λ R , [0 ,b ] ,L by [Bel20, Proposition 3.38], [Col08, Corollaire 6.10] implies OHOMOLOGY OF ( ϕ, Γ)-MODULES OVER PSEUDORIGID SPACES 5 that f ∗ i ∈ Λ R , [0 ,b ] ,L for all i . Then for any x ∈ Λ R , [0 ,b ] ,L , we may uniquely write x = (cid:80) i Tr( xπ iL ) f ∗ i , as desired. (cid:3) By [Bel20, Proposition 3.10], the formation of (cid:101) Λ R,I behaves well with respect torational localization Spa R , and Λ R ,I,K does, as well, since it is sheafy. Thus, if X is a (not necessarily affinoid) pseudorigid space, we may let (cid:101) Λ H K X,I and Λ
X,I,K denote the corresponding sheaves of algebras.2.2. ( ϕ, Γ) -modules and cohomology. We briefly recall the theory of ( ϕ, Γ)-modules over pseudorigid spaces.
Definition 2.2.1. A ϕ -module over Λ R, (0 ,b ] ,K is a coherent sheaf D of modulesover the pseudorigid space (cid:83) a → Spa(Λ R, [ a,b ] ,K ) equipped with an isomorphism ϕ D : ϕ ∗ D ∼ −→ Λ R, (0 ,b/p ] ,K ⊗ Λ R, (0 ,b ] ,K D If a ∈ (0 , b/p ], a ϕ -module over Λ R, [ a,b ] ,K is a finite Λ R, [ a,b ] ,K -module D equippedwith an isomorphism ϕ D, [ a,b/p ] : Λ R, [ a,b/p ] ,K ⊗ Λ R, [ a/p,b/p ] ,K ϕ ∗ D ∼ −→ Λ R, [ a,b/p ] ,K ⊗ Λ R, [ a,b ] ,K D A ( ϕ, Γ K )-module over Λ R, (0 ,b ] ,K (resp. Λ R, [ a,b ] ,K ) is a ϕ -module over Λ R, (0 ,b ] ,K (resp. Λ R, [ a,b ] ,K ) equipped with a semi-linear action of Γ K which commutes with ϕ D (resp. ϕ D, [ a,b/p ] ).A ( ϕ, Γ K )-module over R is a module D over Λ R, rig ,K which arises via base changefrom a ( ϕ, Γ K )-module over Λ R, (0 ,b ] ,K for some b > γ ∈ Γ K . Then for a ( ϕ, Γ K )-module D , wedefine the Fontaine–Herr–Liu complex via C • ϕ, Γ : D ϕ D − ,γ − −−−−−−−→ D ⊕ D ( γ − ⊕ (1 − ϕ D ) −−−−−−−−−−→ D (concentrated in degrees 0, 1, and 2). We let H iϕ, Γ K ( D ) denote its cohomology indegree i .The main result of [Bel20] says that if M is a R -linear representation of Gal K ,there is an associated projective ( ϕ, Γ K )-module D rig ,K ( M ). Moreover, we have acanonical quasi-isomorphism R Γ(Gal K , M ) ∼ −→ C • ϕ, Γ between (continuous) Galoiscohomology and Fontaine–Herr–Liu cohomology. This extends similar results onfamilies of projective Galois representations with coefficients in classical Q p -affinoidalgebras [Pot13, Theorem 2.8] and earlier work in the setting of Q p -linear Galoisrepresentations [Liu07, Theorem 2.3].2.3. ( ϕ, Γ) -modules of character type. Let K/ Q p be a finite extension withramification degree e K and inertia degree f K , and let O K be its ring of integers, k K be its residue field, and (cid:36) K be a uniformizer. Let K ⊂ K be its maximal un-ramified subfield. Let R be a pseudoaffinoid algebra over Z p with ring of definition R ⊂ R and pseudo-uniformizer u ∈ R . By the structure theorem for pseudoaffi-noid algebras [Lou17], we may assume R is strictly topologically of finite type over D ◦ λ := Z p [[ u ]] (cid:68) p m u m (cid:48) (cid:69) , where m, m (cid:48) > λ := m (cid:48) m (we specifically rule out thesetting where R is a Q p -affinoid algebra, since that case was treated in [KPX14]). R. BELLOVIN
We begin by recalling the construction of ( ϕ, Γ K )-modules of character type from [KPX14]. Lemma 2.3.1.
Let α ∈ R × . Up to isomorphism, there is a unique rank- R ⊗ O K -module D f K ,α equipped with a ⊗ ϕ -semilinear operator ϕ α such that ϕ f K α = α ⊗ .Proof. This follows exactly as in [KPX14, Lemma 6.2.3]. (cid:3)
Definition 2.3.2.
Let δ : K × → R × , and write δ = δ δ , where δ , δ ⇒ R × arecontinuous characters such that δ is trivial on O × K and δ is trivial on (cid:104) (cid:36) K (cid:105) . Bylocal class field theory, δ corresponds to a continuous character δ (cid:48) : Gal K → R × .We let Λ R, rig ,K ( δ ) := D f K ,δ ( (cid:36) K ) ⊗ R ⊗ O K Λ R, rig ,K and Λ R, rig ,K ( δ ) := D rig ,K ( δ (cid:48) ),and we define Λ R, rig ,K ( δ ) := Λ R, rig ,K ( δ ) ⊗ Λ R, rig ,K ( δ ).If D is a ( ϕ, Γ K )-module and δ : K × → R × is a continuous character, we willlet D ( δ ) denote D ⊗ Λ R, rig ,K ( δ ). We will let C • ϕ, Γ K ( δ ) and H iϕ, Γ K ( δ ) denote theFontaine–Herr–Liu complex and the cohomology groups of Λ R, rig ,K ( δ ), respectively. Lemma 2.3.3.
Suppose
L/K is a finite extension, and (cid:36) L is a uniformizer of L with Nm L/K ( (cid:36) L ) = (cid:36) K . If δ : K × → R × is a continuous character, then Res LK Λ R, rig ,K ( δ ) is of character type, with associated character δ ◦ Nm L /K .Proof. We may consider separately the cases where δ is trivial on O × K and (cid:104) (cid:36) K (cid:105) .If δ is trivial on O × K , thenRes LK Λ R, rig ,K ( δ ) = D f K ,δ ( (cid:36) K ) ⊗ R ⊗ O K Λ R, rig ,L = (cid:0) D f K ,δ ( (cid:36) K ) ⊗ R ⊗ O K ( R ⊗ O L ) (cid:1) ⊗ R ⊗ O L Λ R, rig ,L But D f K ,δ ( (cid:36) K ) ⊗ R ⊗ O K ( R ⊗ O L ) is a rank-1 R ⊗ O L -module equipped with a 1 ⊗ ϕ -semilinear operator ϕ δ ( (cid:36) K ) such that ϕ f L δ ( (cid:36) K ) = δ ( (cid:36) K ) ⊗
1, so it is isomorphic to D f L ,δ ( (cid:36) K ) . By definition, D f L ,δ ( (cid:36) K ) ⊗ R ⊗ O L Λ R, rig ,L is equal to Λ R, rig ,L ( δ ◦ Nm L/K ).On the other hand, if δ is trivial on (cid:104) (cid:36) K (cid:105) , the statement follows from functorialityfor local class field theory. (cid:3) Continuous characters vary in analytic families, and hence ( ϕ, Γ)-modules do, aswell:
Proposition 2.3.4.
Let G be a topological group of the form G × Z ⊕ r × Z ⊕ r p ,where G is a finite discrete group. Then there is a pseudorigid space (cid:98) G and acontinuous character δ univ : G → O (cid:98) G such that every continuous character δ : G → R × , where R is a pseudoaffinoid algebra, arises via the pullback of δ univ along aunique morphism Spa R → (cid:98) G . This result is well-known in the Q p -affinoid setting (see e.g. [Buz, Lemma 8.2]or [KPX14, Proposition 6.1.1]), and the construction is identical in the pseudorigidsetting. In the case G = Q × p ∼ = µ p − × Z × Z p , the moduli space (cid:98) G has connectedcomponents indexed by the elements of µ p − , each of which is isomorphic to G an m × (Spa Z p [[ Z p ]]) an . OHOMOLOGY OF ( ϕ, Γ)-MODULES OVER PSEUDORIGID SPACES 7 Finiteness of cohomology
We wish to show that the cohomology of C • ϕ, Γ is R -finite. To do this, we willapply [KL, Lemma 1.10] to the morphisms of complexes D ∆ K [ a,b ] (cid:47) (cid:47) (cid:15) (cid:15) D ∆ K [ a,b/p ] ⊕ D ∆ K [ a,b ] (cid:47) (cid:47) (cid:15) (cid:15) D ∆ K [ a,b/p ] (cid:15) (cid:15) D ∆ K [ a (cid:48) ,b (cid:48) ] (cid:47) (cid:47) D ∆ K [ a (cid:48) ,b (cid:48) /p ] ⊕ D ∆ K [ a (cid:48) ,b (cid:48) ] (cid:47) (cid:47) D ∆ K [ a (cid:48) ,b (cid:48) /p ] induced by the natural homomorphisms Λ R, [ a,b ] ,K (cid:48) → Λ R, [ a (cid:48) ,b (cid:48) ] ,K (cid:48) (where [ a (cid:48) , b (cid:48) ] ⊂ ( a, b )). More precisely, Kedlaya–Liu show that if the morphisms D ∆ K [ a,b ] → D ∆ K [ a (cid:48) ,b (cid:48) ] are completely continuous and induce isomorphisms on cohomology groups, thenboth complexes have R -finite cohomology. Since C • ϕ, Γ is the direct limit (as b → a → C • ϕ, Γ has R -finite cohomology. Definition 3.0.1.
Let A be a Banach algebra, and let f : M → N be a morphismof Banach A -modules (equipped with norms |·| M and |·| N , respectively). We saythat f is completely continuous if there exists a sequence of finite A -submodules N i of N such that the operator norms of the compositions M → N → N/N i tend to 0(where N/N i is equipped with the quotient semi-norm) Definition 3.0.2.
Let f : ( A, A + ) → ( A (cid:48) , A (cid:48) + ) be a localization of complete Taterings over a complete Tate ring ( B, B + ). We say that f is inner if there is a strict B -linear surjection B (cid:104) X (cid:105) (cid:16) A such that each element of X maps to a topologicallynilpotent element of A (cid:48) . Here X is a (possibly infinite) collection of formal variables.If B is a nonarchimedean field of mixed characteristic and A and A (cid:48) are topologicallyof finite type over B , Kiehl proved that inner homomorphisms are completely con-tinuous. We prove the analogous result, using the definition of complete continuityfound in [KL]. Proposition 3.0.3. If [ a (cid:48) , b (cid:48) ] ⊂ ( a, b ) and [ a, b ] ⊂ (0 , ∞ ) , then the map Λ R, [ a,b ] ,K → Λ R, [ a (cid:48) ,b (cid:48) ] ,K induced by restriction is completely continuous.Proof. The pairs (Λ R, [ a,b ] ,K , Λ + R, [ a,b ] ,K ) and (Λ R, [ a (cid:48) ,b (cid:48) ] ,K , Λ + R, [ a (cid:48) ,b (cid:48) ] ,K ) are localiza-tions of ( R ⊗ O F (cid:48) [[ π K ]] , R ⊗ O F (cid:48) [[ π K ]]); since [ a (cid:48) , b (cid:48) ] , [ a, b ] ⊂ (0 , ∞ ), they are adicaffinoid algebras over ( R, R + ). Since [ a (cid:48) , b (cid:48) ] ⊂ ( a, b ), the natural restriction map isinner. Then [KL, Lemma 5.7] implies that it is completely continuous. (cid:3) Lemma 3.0.4.
Suppose a ∈ (0 , b/p ] . Then the functor D (cid:32) Λ R, [ a,b ] ,K ⊗ Λ R, (0 ,b ] ,K D =: D [ a,b ] induces an equivalence of categories between ϕ -modules over Λ R, (0 ,b ] ,K and ϕ -modules over Λ R, [ a,b ] ,K .Proof. Suppose we have a ϕ -module D [ a,b ] over Λ R, [ a,b ] ,K . Then the Frobenius pull-back ϕ ∗ D [ a,b ] is a finite module over Λ R, [ a/p,b/p ] ,K , and the isomorphism ϕ D, [ a,b/p ] :Λ R, [ a,b/p ] ,K ⊗ Λ R, [ a/p,b/p ] ,K ϕ ∗ D [ a,b ] ∼ −→ Λ R, [ a,b/p ] ,K ⊗ Λ R, [ a,b ] ,K D [ a,b ] (and the assump-tion that a ≤ b/p ) provides a descent datum. Thus, we may construct a finitemodule D [ a/p,b ] over Λ R, [ a/p,b ] ,K which restricts to D [ a,b ] . R. BELLOVIN
To show that D [ a/p,b ] is a ϕ -module over Λ R, [ a/p,b ] ,K , we need to construct anisomorphism ϕ D, [ a/p,b/p ] : Λ R, [ a/p,b/p ] ,K ⊗ Λ R, [ a/p ,b/p ] ,K ϕ ∗ D [ a/p,b ] ∼ −→ Λ R, [ a/p,b/p ] ,K ⊗ Λ R, [ a/p,b ] ,K D [ a/p,b ] By construction, we have an isomorphism ϕ D, [ a,b/p ] : ϕ ∗ D [ a,b ] ∼ −→ Λ R, [ a/p,b/p ] ,K ⊗ Λ R, [ a/p,b ] ,K D [ a/p,b ] and if we pull ϕ D, [ a,b/p ] back by Frobenius, we obtain an isomorphism ϕ D, [ a/p,b/p ] : Λ R, [ a/p,b/p ] ,K ⊗ Λ R, [ a/p ,b/p ,K ϕ ∗ D [ a/p,b/p ] ∼ −→ Λ R, [ a/p,b/p ] ,K ⊗ Λ R, [ a/p,b/p ] ,K D [ a/p,b/p ] On the overlap, they induce the same isomorphism ϕ ∗ D [ a,b/p ] → Λ R, [ a/p,b/p ] ,K ⊗ Λ R, [ a/p,b/p ] ,K D [ a/p,b/p ] (by construction), so we obtain the desired isomorphism ϕ D, [ a/p,b/p ] .Iterating this construction lets us construct a ϕ -module over Λ R, (0 ,b ] ,K .This proves essential surjectivity; full faithfulness follows because the natural mapsΛ R, (0 ,b ] ,K → Λ R, [ a,b ] ,K have dense image. (cid:3) Corollary 3.0.5. If D is a ϕ -module over Λ R, (0 ,b ] ,K , the morphism of complexes [ D (0 ,b ] ϕ − −−−→ D (0 ,b/p ] ] → [ D [ a,b ] ϕ − −−−→ D [ a,b/p ] ] is a quasi-isomorphism for any a ∈ (0 , b/p ] .Proof. This follows from the previous lemma because we may interpret the coho-mology groups as Yoneda Ext groups. (cid:3)
In order to prove that the restriction map D (0 ,b ] → D (0 ,b/p ] induces an isomorphismon cohomology, we will need the ψ operator. The isomorphism ϕ ∗ D (0 ,b ] ∼ −→ D (0 ,b/p ] induces an isomorphismΛ R, (0 ,b/p ] ,K ⊗ ϕ (Λ R, (0 ,b ] ,K ) ϕ ( D (0 ,b ] ) ∼ −→ D (0 ,b/p ] We therefore have a surjective homomorphism ψ : D (0 ,b/p ] → D (0 ,b ] defined bysetting ψ ( a ⊗ ϕ ( d )) = ψ ( a ) d , where a ∈ Λ R, (0 ,b/p ] ,K and d ∈ D (0 ,b ] . Lemma 3.0.6.
Let D be a ( ϕ, Γ) -module over Λ R, (0 ,b ] ,K for some b > . Thenthere is some < b (cid:48) ≤ b such that the action of γ − on ( D (0 ,b (cid:48) ] ) ψ =0 admits acontinuous inverse.Proof. We may replace D with Ind Q p K ( D ). Since ( D (0 ,b/p n ] ) ψ =0 = ⊕ j ∈ ( Z /p n ) × [ ε ] ˜ j ϕ n ( D ),it suffices to show that γ − ε ] j ϕ n ( D ) for j prime to p and sufficiently large n . Moreover, since γ n − γ − γ n − + · · · + 1), we mayreplace Γ Q p with a finite-index subgroup.If γ n ∈ Γ Q p is such that χ ( γ n ) = 1 + p n , then γ n (cid:0) [ ε ] j ϕ n ( x ) (cid:1) − [ ε ] j ϕ n ( x ) = [ ε ] j [ ε ] p n j ϕ n ( γ n ( x )) − [ ε ] j ϕ n ( x )= [ ε ] j ϕ n ([ ε ] j γ n ( x ) − x )= [ ε ] j ϕ n ( G γ n ( x )) OHOMOLOGY OF ( ϕ, Γ)-MODULES OVER PSEUDORIGID SPACES 9 where G γ n ( x ) := [ ε ] j γ n ( x ) − x = ([ ε ] j − · (cid:16) [ ε ] j [ ε ] j − ( γ n − (cid:17) ( x ). Thus, if wecan choose n such that (cid:80) ∞ k =0 (cid:16) − [ ε ] j [ ε ] j − ( γ n − (cid:17) k converges on D (0 ,b ] , we will bedone.The action of Γ Q p on D [ b/p,b ] is continuous, so we may choose n such that for n ≥ n , the sum above converges in End( D [ b/p,b ] ). But D [ b/p k +1 ,b/p k ] ∼ = ϕ ∗ D [ b/p k ,b/p k − ] for all k ≥ ϕ , so the sumconverges in End( D [ b/p k +1 ,b/p k ] ) for all k ≥ γ − D ψ =0(0 ,b/p n ] for n ≥ n . (cid:3) Proposition 3.0.7. If D is a ( ϕ, Γ) -module over R for some b > , then thecohomology of D is computed by D ∆ K [ a,b ] ϕ − ,γ − −−−−−−→ D ∆ K [ a,b/p ] ⊕ D ∆ K [ a,b ] ( γ − ⊕ (1 − ϕ ) −−−−−−−−−→ D ∆ K [ a,b/p ] for some sufficiently small b and any a ∈ (0 , b/p ] .Proof. We may assume that D is a ( ϕ, Γ)-module over Λ R, (0 ,b ] ,K for some b > D ∆ K (0 ,b ] ϕ − −−−→ D ∆ K (0 ,b/p ] ] → [ D ∆ K [ a,b ] ϕ − −−−→ D ∆ K [ a,b/p ] ] induces an isomorphism oncohomology, we see that the cohomology of D (0 ,b ] is computed by the above complex.Since the cohomology of C • ϕ, Γ ( D ) is computed by the direct limit of the cohomologygroups of C • ϕ, Γ ( D (0 ,b/p n ] ) as n → ∞ , it suffices to show that the natural morphism C • (0 ,b ] : D ∆ K (0 ,b ] (cid:47) (cid:47) (cid:15) (cid:15) D ∆ K (0 ,b/p ] ⊕ D ∆ K (0 ,b ] (cid:47) (cid:47) (cid:15) (cid:15) D ∆ K (0 ,b/p ] (cid:15) (cid:15) C • (0 ,b/p ] : D ∆ K (0 ,b/p ] (cid:47) (cid:47) D ∆ K (0 ,b/p ] ⊕ D ∆ K (0 ,b/p ] (cid:47) (cid:47) D ∆ K (0 ,b/p ] induces an isomorphism on cohomology groups for sufficiently small b .The maps ψ : D (0 ,b/p ] → D (0 ,b ] and ψ : D (0 ,b/p ] → D (0 ,b/p ] induce a surjection ofcomplexes Ψ : C • (0 ,b/p ] → C • (0 ,b ] such that the composition C • (0 ,b ] ϕ −→ C • (0 ,b/p ] Ψ −→ C • (0 ,b ] is the identity. By Lemma 3.0.6, if b is sufficiently small, γ − D ψ =0(0 ,b/p ] . Thus, we have a decomposition C • (0 ,b/p ] ∼ = ϕ ( C • (0 ,b ] ) ⊕ K • , where K • := ker Ψ, and K • has vanishing cohomology.In order to show that the base change map C • (0 ,b ] → C • (0 ,b/p ] is a quasi-isomorphism,it therefore suffices to show that id , ϕ : C • (0 ,b ] ⇒ C • (0 ,b/p ] are homotopic. But thisfollows by considering the diagram D ∆ K (0 ,b ] ( ϕ − ,γ − (cid:47) (cid:47) id (cid:15) (cid:15) ϕ (cid:15) (cid:15) D ∆ K (0 ,b/p ] ⊕ D ∆ K (0 ,b ] ( γ − ⊕ (1 − ϕ ) (cid:47) (cid:47) id (cid:15) (cid:15) ϕ (cid:15) (cid:15) pr (cid:117) (cid:117) D ∆ K (0 ,b/p ]id (cid:15) (cid:15) ϕ (cid:15) (cid:15) (0 , − id) (cid:117) (cid:117) D ∆ K (0 ,b/p ] ( ϕ − ,γ − (cid:47) (cid:47) D ∆ K (0 ,b/p ] ⊕ D ∆ K (0 ,b/p ] ( γ − ⊕ (1 − ϕ ) (cid:47) (cid:47) D ∆ K (0 ,b/p ] (cid:3) Corollary 3.0.8. If D is a ( ϕ, Γ) -module over Λ R, (0 ,b ] ,K and [ a (cid:48) , b (cid:48) ] ⊂ [ a, b ] , therestriction map D [ a,b ] (cid:47) (cid:47) (cid:15) (cid:15) D [ a,b/p ] ⊕ D [ a,b ] (cid:47) (cid:47) (cid:15) (cid:15) D [ a,b/p ] (cid:15) (cid:15) D [ a (cid:48) ,b (cid:48) ] (cid:47) (cid:47) D [ a (cid:48) ,b (cid:48) /p ] ⊕ D [ a (cid:48) ,b (cid:48) ] (cid:47) (cid:47) D [ a (cid:48) ,b (cid:48) /p ] induces an isomorphism on cohomology.Proof. We may assume that b (cid:48) ∈ [ b/p, b ], so that we have induced homomorphisms H iϕ, Γ ( D (0 ,b ] ) → H iϕ, Γ ( D (0 ,b (cid:48) ] ) → H iϕ, Γ ( D (0 ,b/p ] ) → H iϕ, Γ ( D (0 ,b (cid:48) /p ] )Since the compositions H iϕ, Γ ( D (0 ,b ] ) → H iϕ, Γ ( D (0 ,b/p ] ) and H iϕ, Γ ( D (0 ,b (cid:48) ] ) → H iϕ, Γ ( D (0 ,b (cid:48) /p ] )are isomorphisms, the homomorphism H iϕ, Γ ( D (0 ,b (cid:48) ] ) → H iϕ, Γ ( D (0 ,b/p ] ) is also an iso-morphism, and we are done. (cid:3) Now we can finally prove that ( ϕ, Γ)-modules have finite cohomology.
Theorem 3.0.9. If D is a ( ϕ, Γ) -module over Λ R, (0 ,b ] ,K , its cohomology is R -finite.Proof. If [ a (cid:48) , b (cid:48) ] ⊂ ( a, b ), the restriction map induces a quasi-isomorphism D [ a,b ] (cid:47) (cid:47) (cid:15) (cid:15) D [ a,b/p ] ⊕ D [ a,b ] (cid:47) (cid:47) (cid:15) (cid:15) D [ a,b/p ] (cid:15) (cid:15) D [ a (cid:48) ,b (cid:48) ] (cid:47) (cid:47) D [ a (cid:48) ,b (cid:48) /p ] ⊕ D [ a (cid:48) ,b (cid:48) ] (cid:47) (cid:47) D [ a (cid:48) ,b (cid:48) /p ] which is completely continuous. Then the result follows, by [KL, Lemma 1.10] (cid:3) Corollary 3.0.10. If D is a projective ( ϕ, Γ) -module over R , then C • ϕ, Γ K ( D ) ∈ D [0 , ( R ) .Proof. Finiteness of the cohomology of C • ϕ, Γ K ( D ) implies that C • ϕ, Γ K ( D ) ∈ D − perf ( R ),and by [Bel20, Proposition 3.47], the complex C • ϕ, Γ K ( D ) consists of flat A -modules.Then as in the proof of [KPX14, Theorem 4.4.5(1)], it follows that C • ϕ, Γ K ( D ) ∈ D [0 , ( R ). (cid:3) Corollary 3.0.11. If D is a projective ( ϕ, Γ) -module over R , then the cohomologygroups H iϕ, Γ ( D ) are coherent sheaves on Spa R .Proof. Since C • ϕ, Γ ( D ) ∈ D b coh ( R ), we have a quasi-isomorphism R (cid:48) ⊗ L R C • ϕ, Γ ( D ) ∼ −→ C • ϕ, Γ ( R (cid:48) ⊗ R D ) for any homomorphism R → R (cid:48) of pseudoaffinoid algebras. If R → R (cid:48) defines an affinoid subspace of Spa R , the morphism is flat and the derivedtensor product is an ordinary tensor product. On the other hand, we have a naturalhomomorphism C • ϕ, Γ ( R (cid:48) ⊗ R D ) → C • ϕ, Γ ( R (cid:48) (cid:98) ⊗ R D ), and it is a quasi-isomorphismafter every specialization R (cid:48) (cid:16) S to a finite-length algebra (since D is flat over R ).Since quasi-isomorphisms can be checked on finite-length specializations, the resultfollows. (cid:3) OHOMOLOGY OF ( ϕ, Γ)-MODULES OVER PSEUDORIGID SPACES 11
As a corollary, if R → R (cid:48) is a homomorphism of pseudoaffinoid algebras, there is anatural quasi-isomorphism R (cid:48) ⊗ L C • ϕ, Γ ( D ) ∼ −→ C • ϕ, Γ ( R (cid:48) ⊗ R D )and there is a corresponding second-quadrant base-change spectral sequence. Werecord the low-degree exact sequences of the base-change spectral sequence here: Corollary 3.0.12.
Let R → R (cid:48) be a morphism of pseudoaffinoid algebras and let D be a ( ϕ, Γ K ) -module over Λ R, rig ,K . Then (1) The natural morphism R (cid:48) ⊗ R H ϕ, Γ K ( D ) → H ϕ, Γ K ( R (cid:48) ⊗ R D ) is an isomor-phism. (2) The natural morphism R (cid:48) ⊗ R H ϕ, Γ K ( D ) → H ϕ, Γ K ( R (cid:48) ⊗ R D ) fits into anexact sequence → Tor R ( H ϕ, Γ K ( D ) , R (cid:48) ) → R (cid:48) ⊗ R H ϕ, Γ K ( D ) → H ϕ, Γ K ( R (cid:48) ⊗ R D ) → Tor R ( H ϕ, Γ K ( D ) , R (cid:48) ) → There is a filtration ⊂ F H ϕ, Γ K ( R (cid:48) ⊗ R D ) ⊂ F − H ϕ, Γ K ( R (cid:48) ⊗ R D ) ⊂ H ϕ, Γ K ( R (cid:48) ⊗ R D ) such that → F − H ϕ, Γ K ( R (cid:48) ⊗ R D ) → H ϕ, Γ K ( R (cid:48) ⊗ R D ) → ker (cid:0) Tor R ( H ϕ, Γ K ( D ) , R (cid:48) ) → R (cid:48) ⊗ R H ϕ, Γ K ( D ) (cid:1) → and → F H ϕ, Γ K ( R (cid:48) ⊗ R D ) → F − H ϕ, Γ K ( R (cid:48) ⊗ R D ) → (cid:0) Tor R ( H ϕ, Γ K ( D ) , R (cid:48) ) / Tor R ( H ϕ, Γ K ( D ) , R (cid:48) ) (cid:1) → are exact, and there is a natural surjection R (cid:48) ⊗ R H ϕ, Γ K ( D ) (cid:16) F H ϕ, Γ K ( R (cid:48) ⊗ R D ) , with kernel generated by Tor R ( H ϕ, Γ K ( D ) , R (cid:48) ) and ker (cid:0) Tor R ( H ϕ, Γ K ( D ) , R (cid:48) ) → Tor ( H ϕ, Γ K ( D ) , R (cid:48) ) (cid:1) .Proof. This follows from the convergence of the base-change spectral sequence. (cid:3)
We also deduce the Euler characteristic formula for all ( ϕ, Γ K )-modules: Corollary 3.0.13. If D is a projective ( ϕ, Γ K ) -module with coefficients in a pseu-doaffinoid algebra R , then χ ( D ) = − [ K : Q p ] .Proof. Euler characteristics are locally constant, so it suffices to compute χ ( D x )for a single maximal point x on each connected component of Spa R . Thus, we mayassume that R is a finite extension of either Q p or F p (( u )), and so we have accessto the slope filtration theorem of [Ked08].Since Euler characteristics are additive in exact sequences, we may assume that D is pure of slope s ; if necessary, replace R by an ´etale extension so that the slopeof D is in the value group of R . The moduli space (cid:92) (cid:104) (cid:36) K (cid:105) ∼ = G an m of continuouscharacters of (cid:104) (cid:36) K (cid:105) has a universal character δ univ : (cid:104) (cid:36) K (cid:105) → O × (cid:92) (cid:104) (cid:36) K (cid:105) , so we mayconsider the Fontaine–Herr–Liu complex C • ϕ, Γ of the ( ϕ, Γ K )-module D ( δ univ ) over (cid:92) (cid:104) (cid:36) K (cid:105) . Since C • ϕ, Γ ( D ( δ )) ∈ D [0 , ( R (cid:48) ) for every affinoid subdomain Spa( R (cid:48) ) ⊂ X , itsEuler characteristic is constant on connected components, and it suffices to verifythe statement at one point on each component. But each connected componentcontains a point x such that the slope of D ( δ ) at x is 0; then ( D ( δ ))( x ) is ´etale andwe may appeal to the Euler characteristic formula for Galois cohomology. (cid:3) In section 4.2, we will prove Tate local duality for ( ϕ, Γ)-modules when R is a finiteextension of F p (( u )). We deduce the corresponding result for families of ( ϕ, Γ)-modules over general pseudoaffinoid algebras here, and the reader may check thatthere is no circular dependence.
Theorem 3.0.14.
Let R be a pseudoaffinoid algebra and let D be a family of ( ϕ, Γ K ) -modules over R . Then the natural morphism C • ϕ, Γ K ( D ) → R Hom R ( C • ϕ, Γ K ( D ∨ ( χ cyc )) , R )[ − is a quasi-isomorphism.Proof. For every maximal point x ∈ Spa R , we have a quasi-isomorphism C • ϕ, Γ K ( R/ m x ⊗ R D ) ∼ −→ R Hom R ( C • ϕ, Γ K ( R/ m x ⊗ R D ∨ ( χ cyc )) , R/ m x ), by [Liu07, Theorem 4.7] (when R/ m x has characteristic 0) and Theorem 4.2.2 (when R/ m x has positive character-istic). Then by [KPX14, Lemma 4.1.5], the result follows. (cid:3) Positive characteristic function fields
In this section, we closely study overconvergent ( ϕ, Γ)-modules where the coef-ficients are finite extensions of F p (( u )). This is similar to the situation studiedby Hartl–Pink [HP04], but because we are interested in ( ϕ, Γ)-modules related torepresentations of characteristic-0 Galois groups, we may work with imperfect co-efficients. For this reason, we rely on the slope filtration theorem from [Ked08],rather than the Dieudonn´e–Manin classification theorem from [HP04]. We first cal-culate the cohomology of certain rank-1 ( ϕ, Γ Q p )-modules (using techniques similarto [Col08]), and then use those calculations to deduce the Tate local duality theoremfor all ( ϕ, Γ K )-modules (following the strategy of [Liu07]).4.1. Cohomology of rank- ϕ, Γ) -modules. We begin by computing the coho-mology of ( ϕ, Γ)-modules of character type.
Lemma 4.1.1.
Assume R is a finite extension of F p (( u )) , and let δ : K × → R × bea continuous character. Then H ϕ, Γ K ( δ ) = 0 unless δ is trivial, in which case it isa free R -module of rank .Proof. Write δ = δ δ , as above. We first show that the kernel of ϕ − R, rig ,K ( δ ) is trivial unless δ ( (cid:36) K ) = 1, in which case it is R , and then com-pute the elements of ker( ϕ −
1) fixed by Γ K .If f ( π K ) ∈ Λ R, rig ,K ( δ ), we may write f ( π K ) uniquely as f ( π K ) = (cid:80) i ∈ Z a i π iK ,where a i ∈ R ⊗ k (cid:48) (for some finite extension k (cid:48) /k K ). There is some integer f ≥ ϕ f K f fixes k (cid:48) . Using the fact that ϕ f K f ( π K ) = π f K fpK , a straightforwardcalculation shows that the kernel of ϕ f K f − δ ( (cid:36) K ) f = 1, in whichcase it is R ⊗ k (cid:48) . We now need to compute the kernel of ϕ f K − D f K ,δ ( (cid:36) K ) ⊗ k K k (cid:48) .But there is a basis { e , . . . , e f − } of k (cid:48) /k K such that ϕ f K acts via ϕ f K ( e i ) = e i − ,where the indices are taken modulo f , so the kernel of ϕ f K D − δ ( (cid:36) K ) = 1, in which case it is D f K ,α . We have reduced to computing the kernel of ϕ − D f K , , but the construction makes clear that this kernel is precisely R .Now suppose δ is trivial, so that H ϕ, Γ K (Λ R, rig ,K ( δ )) is R Γ K =1 . If γ ∈ Γ K is atopological generator of Γ K , it acts on R via multiplication by β for some β ∈ OHOMOLOGY OF ( ϕ, Γ)-MODULES OVER PSEUDORIGID SPACES 13 Λ R, (0 ,b ] ,K . This clearly fixes no elements unless β = 1, in which case it fixes all of R . (cid:3) Corollary 4.1.2. If R is a finite extension of F p (( u )) and D is a rank- ϕ, Γ) -module over Λ R, rig ,K of character type, then D has no proper non-trivial sub- ( ϕ, Γ) -module or quotient ( ϕ, Γ) -module. Lemma 4.1.3.
Suppose α ∈ R × satisfies v R ( α ) < . Then if f ∈ Λ R, [0 ,b ] , Q p isin the image of ϕ , there is some b (cid:48) > such that f is in the image of αϕ − R, [0 ,b (cid:48) ] , Q p → Λ R, [0 ,b (cid:48) /p ] , Q p .Proof. We are looking for a solution to the equation ( αϕ − g ) = ϕ ( f (cid:48) ); applying ψ to both sides, it suffices to show that the sum (cid:80) k ≥ ( α − ψ ) k converges on Λ R, [0 ,b (cid:48) ] ,K for b (cid:48) sufficiently small. But we may write ψ (cid:32)(cid:88) i ∈ Z α i π i (cid:33) = (cid:88) i ∈ Z p − (cid:88) j =0 α pi + j π i so for any b (cid:48) > v R,b (cid:48) (cid:32) ψ (cid:32)(cid:88) i ∈ Z α i π i (cid:33)(cid:33) ≥ v R,b (cid:48) /p (cid:32)(cid:88) i ∈ Z α i π i (cid:33) − pb (cid:48) ≥ v R,b (cid:48) (cid:32)(cid:88) i ∈ Z α i π i (cid:33) − pb (cid:48) where the second inequality follows from the maximum modulus principle. Thus,we may simply choose b (cid:48) < p v R ( α − ). (cid:3) Lemma 4.1.4.
Suppose α ∈ R × . Then αϕ − π K Λ R, (0 , ∞ ] ,K → π K Λ R, (0 , ∞ ] ,K issurjective. If α (cid:54) = 1 , then αϕ − R, (0 , ∞ ] ,K → Λ R, (0 , ∞ ] ,K is surjective.Proof. It suffices to show that (cid:80) k ≥ ( αϕ ) k converges on π K Λ R, [ a, ∞ ] ,K for all a > f = (cid:80) i ≥ α i π iK , we have v R, [ a, ∞ ] (( αϕ ) k ( f )) = k · v R ( α ) + inf i (cid:26) v R ( α i ) + p k +1 aip − (cid:27) ≥ k · v R ( α ) + a ( p k + · · · + p ) + inf i (cid:26) v R ( α i ) + paip − (cid:27) = v R, [ a, ∞ ] ( f ) + k · v R ( α ) + a ( p k + · · · + p )Thus, for any α ∈ R × and any a >
0, the sum (cid:80) k ≥ ( αϕ ) k ( f ) converges to anelement of π K Λ R, [ a, ∞ ] ,K , as desired.If α (cid:54) = 1, then ( αϕ − (cid:16) α − (cid:17) = 1, so R is also in the image of αϕ − (cid:3) Corollary 4.1.5.
Suppose α ∈ R × satisfies v R ( α ) < . If f ∈ Λ R, (0 ,b ] , Q p , then(possibly after shrinking b) there is some g ∈ Λ R, (0 ,b ] , Q p such that f − ( αϕ − g ∈ Λ ψ =0 R, [0 ,b ] , Q p .Proof. We have exact sequences0 → R [[ π ]] → Λ R, (0 , ∞ ] , Q p ⊕ Λ R, [0 ,b ] , Q p → Λ R, (0 ,b ] , Q p → b >
0, so we may write f = f + + f − , where f + ∈ π Λ R, (0 , ∞ ] , Q p and f − ∈ Λ R, [0 ,b ] , Q p . Then we can find g + ∈ π Λ R, (0 , ∞ ] , Q p and g − ∈ Λ R, [0 ,b (cid:48) ] , Q p for some b (cid:48) ≤ b such that f + = ( αϕ − g + ) and f − − ( αϕ − g − ) ∈ Λ ψ =0 R, [0 ,b (cid:48) ] , Q p , so f − ( αϕ − g + + g − ) ∈ Λ ψ =0 R, [0 ,b (cid:48) ] , Q p , as desired. (cid:3) Corollary 4.1.6. If δ : Q × p → R × is a continuous character trivial on p Z p ⊂ Z × p , such that v R ( δ ( p )) < , then H ϕ, Γ Q p ( δ ) = 0 .Proof. Lemma 4.1.5 implies that after subtracting an element of the form ( αϕ − g ), any cohomology class of H ϕ, Γ Q p ( δ ) has a representative f ∈ Λ ψ =0 R, [0 ,b ] , Q p , forsufficiently small b . But if γ is a topological generator of the procyclic part of Z × p , γ − ψ =0 R, [0 ,b ] , Q p , for sufficiently small b , and the result follows. (cid:3) Now we wish to compute H ϕ, Γ Q p ( δ ), where v R ( δ ( p )) < δ is trivial on 1 + p Z p . Lemma 4.1.7. If α ∈ R × satisfies v R ( α ) < and ( αϕ − f ) ∈ Λ ψ =0 R, (0 ,b ] , Q p , then f ∈ Λ R, (0 , ∞ ] , Q p .Proof. We may assume f = (cid:80) i ≤− α i π i with α i ∈ R , since ψ commutes withpassing to Laurent “tails”. Then0 = ψ (cid:0) ( ϕ − α − )( f ) (cid:1) = (1 − α − ψ )( f )so f = α − ψ ( f ). But for any b (cid:48) < p v R ( α − ), we have seen that v R,b (cid:48) ( α − ψ ( f )) >v R,b (cid:48) ( f ); since Λ R, [0 ,b (cid:48) ] , Q p is π -adically separated, this implies f = 0. (cid:3) Lemma 4.1.8. If γ is a topological generator of the procyclic part of Γ Q p , the actionof γ − defines a surjective map γ − ψ =0 R , [0 , ∞ ] , Q p → ⊕ j − j =1 ε j ϕ (cid:0) π Λ R , [0 , ∞ ] , Q p (cid:1) .Proof. We may assume χ ( γ ) = 1 + p ; then for any f ∈ Λ R , [0 , ∞ ] , Q p ,( γ −
1) ( εϕ ( f )) = εϕ ( G γ ( f ))where G γ ( f ) = εγ ( f ) − f = π (cid:0) − επ ( γ − (cid:1) ( f ). In addition, γ ( π ) = (1 + π ) p − π )(1 + π p ) − π + π p + π p +1 = π + ϕ ( π ) ε so γ − R , [0 , ∞ ] , Q p to ϕ ( π )Λ R , [0 , ∞ ] , Q p , and G γ ( f ) ∈ π Λ R , [0 , ∞ ] , Q p .It remains to show that π i is in the image of G γ ( f ) for all i ≥
1. To see this, itsuffices to show that (cid:80) k ≥ (cid:0) επ ( γ − (cid:1) k ( π ) i converges for all i ≥
1. But( γ − π i ) = i − (cid:88) j =0 (cid:18) ij (cid:19) π j ϕ ( π ) i − j ε i − j ∈ π i +( p − > (cid:80) k ≥ (cid:0) επ ( γ − (cid:1) k ( π ) i and G γ ( f ) is invertible. (cid:3) Lemma 4.1.9. If δ : Q × p → R × is a character with v R ( p ) < and δ | p Z p = 1 ,then H ϕ, Γ Q p ( δ ) is -dimensional.Proof. Let α := δ ( p ). For any cohomology class in H ϕ, Γ Q p ( δ ), we may choose arepresentative pair ( f, g ) ∈ Λ R, (0 ,b/p ] , Q p ⊕ Λ R, (0 ,b ] , Q p with ( γ − f ) = ( αϕ − g )(for some topological generator γ of the procyclic part of Z × p ). By Lemma 4.1.5, OHOMOLOGY OF ( ϕ, Γ)-MODULES OVER PSEUDORIGID SPACES 15 there is some h ∈ Λ R, (0 ,b ] , Q p such that f − ( αϕ − h ) ∈ Λ ψ =0 R, [0 ,b/p ] , Q p , so we mayreplace ( f, g ) with ( f − ( αϕ − h ) , g − ( γ − h )) and assume that f ∈ Λ ψ =0 R, [0 ,b/p ] , Q p .Since γ − ψ =0 R, [0 ,b ] , Q p , we have ( αϕ − g ) ∈ Λ ψ =0 R, [0 ,b ] , Q p , as well, so g ∈ Λ R, (0 , ∞ ] , Q p by Lemma 4.1.7. But then ( αϕ − g ) ∈ Λ R, (0 , ∞ ] , Q p , as well as inΛ ψ =0 R, [0 ,b ] , Q p , so ( αϕ − g ) ∈ Λ R , [0 , ∞ ] , Q p .Let c denote the image of ( αϕ − g ) in Λ R , [0 , ∞ ] , Q p /π p . We claim that ( f, g ) isa coboundary if and only if c = 0. Indeed, if ( f, g ) represents the 0 class, there issome h ∈ Λ R , (0 ,b ] , Q p such that f = ( αϕ − h ) and g = ( γ − h ) (possibly aftershrinking b ). Since f ∈ Λ ψ =0 R , [0 ,b ] , Q p , Lemma 4.1.7 implies that h ∈ Λ R, (0 , ∞ ] , Q p andhence f ∈ Λ ψ =0 R , [0 ,b ] , Q p ∩ Λ R, (0 , ∞ ] , Q p = Λ ψ =0 R , [0 , ∞ ] , Q p . Then Lemma 4.1.8 implies that c ≡ ( γ − f ) ≡ π p Λ ψ =0 R , [0 , ∞ ] , Q p .On the other hand, suppose c = 0. Then Lemma 4.1.8 implies that f ∈ Λ ψ =0 R , [0 , ∞ ] , Q p ⊂ Λ R , (0 , ∞ ] , Q p ; since αϕ − R, (0 , ∞ ] , Q p → Λ R, (0 , ∞ ] , Q p is surjective by Lemma 4.1.4( α (cid:54) = 1, by assumption), we may assume f = 0. It follows that ( αϕ − g ) = 0, soby Lemma 4.1.3, g = 0.Thus, there is an injective map H ϕ, Γ Q p ( δ ) → Λ ψ =0 R , [0 , ∞ ] , Q p /π p Λ ψ =0 R , [0 , ∞ ] , Q p . To com-pute its image, we consider the ∆-fixed subspace of the target (since we may assume( f, g ) is fixed by ∆). The subgroup ∆ ⊂ Z × p acts semi-linearly on Λ R , [0 , ∞ ] , Q p via δ | ∆ , and we claim it fixes a subspace of dimension 1 over R . It suffices to con-sider sums of the form (cid:80) j − j =1 ε j α j , where α j ∈ R ; we observe that (cid:80) p − j =1 ε j α j ∈ π Λ R , [0 , ∞ ] , Q p if and only if (cid:80) j α j = 0. Since ∆ is a cyclic group, we may choosea generator γ ∈ ∆ and compute γ (cid:16)(cid:80) j − j =1 ε j α j (cid:17) = δ ( γ ) (cid:80) p − j =1 γ ( ε ) j α j . Moreover, { γ ( ε ) j } = { ε j } since γ is a generator, so { α j } = { δ ( γ ) j α } . Thus, if δ ( γ ) = 1, wesee that (cid:88) j α j = 0 ⇔ α j = 0 for all j ⇔ p − (cid:88) j =1 ε j α j ∈ π p Λ ∆=1 ,ψ =0 R , [0 , ∞ ] , Q p and H ϕ, Γ Q p ( δ ) is 1-dimensional over R , with classes in bijection with ( αϕ − g )(0).If δ | ∆ is non-trivial, we have (cid:80) j α j = (cid:80) j δ ( γ ) j α = ( δ ( γ ) − − ( δ ( γ ) p − −
1) = 0automatically. Thus, the ∆-fixed subspace of Λ ψ =0 R , [0 , ∞ ] , Q p /π p Λ ψ =0 R , [0 , ∞ ] , Q p is a 1-dimensional R -vector space which lies in the image of (cid:0) π Λ R , [0 , ∞ ] , Q p (cid:1) ψ =0 . (cid:3) Tate local duality.Lemma 4.2.1. If δ : Q × p → R × is a continuous character such that v R ( δ ( p )) < and δ | p Z p is trivial, then Tate duality holds for Λ R, rig , Q p ( δ ) and Λ R, rig , Q p ( δ − χ cyc ) .Proof. Theorem 3.0.13 implies that dim R H ϕ, Γ Q p ( δ − χ cyc ) ≥
1, and so there is anon-split extension of ( ϕ, Γ Q p )-modules0 → Λ R, rig , Q p ( δ − χ cyc ) → D → Λ R, rig , Q p ( δ ) → Then an argument with slope filtrations shows that D is pure of slope 0, hencecomes from a Galois representation, so Tate local duality holds for D . The associ-ated long exact sequence in cohomology, combined with Lemma 4.1.1 shows that H ϕ, Γ Q p ( D ) = 0, so duality implies that H ϕ, Γ Q p ( D ∨ ( χ cyc )) = 0.Since δ − χ cyc and δ are non-trivial, H ϕ, Γ Q p ( δ − χ cyc ) = H ϕ, Γ Q p ( δ ) = 0, and since v R ( δ ( p )) < H ϕ, Γ Q p ( δ ) = 0 and H ϕ, Γ Q p ( δ ) is 1-dimensional.If we dualize our exact sequence and tensor with χ cyc , we get a second exact se-quence 0 → Λ R, rig , Q p ( δ − χ cyc ) → D ∨ ( χ cyc ) → Λ R, rig , Q p ( δ ) → H ϕ, Γ Q p ( δ − χ cyc ) H ϕ, Γ Q p ( D ∨ ( χ cyc )) H ϕ, Γ Q p ( δ ) H ϕ, Γ Q p ( δ − χ cyc ) H ϕ, Γ Q p ( δ − χ cyc ) ∨ H ϕ, Γ Q p ( δ ) ∨ H ϕ, Γ Q p ( D ) ∨ H ϕ, Γ Q p ( δ − χ cyc ) ∨ Since H ϕ, Γ Q p ( D ∨ ( χ cyc )) → H ϕ, Γ Q p ( D ) ∨ is an isomorphism (by the classical theo-rem), a diagram chase shows that H ϕ, Γ Q p ( δ − χ cyc ) → H ϕ, Γ Q p ( δ ) ∨ is injective, sodim R H ϕ, Γ Q p ( δ − χ cyc ) ≤ dim R H ϕ, Γ Q p ( δ ) = 1. But Theorem 3.0.13 implies thatdim R H ϕ, Γ Q p ( δ − χ cyc ) ≥
1, so dim R H ϕ, Γ Q p ( δ − χ cyc ) = 1 and the map H ϕ, Γ Q p ( δ − χ cyc ) → H ϕ, Γ Q p ( δ ) ∨ is an isomorphism. (cid:3) Theorem 4.2.2.
Tate local duality holds for every ( ϕ, Γ) -module D over Λ R, rig ,K .Proof. We may replace D by Ind Q p K D and treat the case of ( ϕ, Γ)-modules overΛ R, rig , Q p . We may also assume that D is pure of slope s , and by replacing it with D ∨ ( χ cyc ) if necessary, that s ≥ s = 0, D is ´etale and the result follows from the comparison with Galois cohomol-ogy. Otherwise, we proceed by induction on the degree of D , i.e. deg( D ) := (rk D ) s .Let δ : Q × p → R × be a continuous character with v R ( δ ( p )) = − δ | p Z p triv-ial. Since dim R H ϕ, Γ Q p ( D ( δ − )) ≥ rk D ≥ → D → D (cid:48) → Λ R, rig , Q p ( δ ) → D (cid:48) ; since it also holds for Λ R, rig , Q p ( δ ),we may deduce it for D .If D (cid:48) is pure, the result follows, since D (cid:48) has degree deg D − D − / (rk D +1) < s . Otherwise, D (cid:48) has a unique slope filtration 0 = D ⊂ D ⊂ · · · ⊂ D k = D (cid:48) by saturated ( ϕ, Γ)-submodules, such that the successive quotients arepure and µ ( D /D ) < µ ( D /D ) < · · · < µ ( D k /D k − ). Then µ ( D ) ≤ µ ( D (cid:48) ) <µ ( D ).We have an exact sequence0 → D ∩ D → D → D / ( D ∩ D ) → D / ( D ∩ D ) is the image of D in the quotient Λ R, rig , Q p ( δ ), so it is either 0or all of Λ R, rig , Q p ( δ ). If D ∩ D = 0, then D = Λ R, rig , Q p ( δ ) and the map D → D OHOMOLOGY OF ( ϕ, Γ)-MODULES OVER PSEUDORIGID SPACES 17 is a section of D → Λ R, rig , Q p ( δ ). But we constructed our extension to be non-split,so D ⊂ D and µ ( D ) > D is pure of positive slope).Thus, µ ( D i /D i − ) > i . Moreover, deg D (cid:48) = (cid:80) i deg( D i /D i − ) = (cid:80) i µ ( D i /D i − ) · rk( D i /D i − ), so deg( D i /D i − ) < deg D for all i . Then the inductive hypothesisimplies that Tate local duality holds for each D i /D i − , so it holds for D (cid:48) , and weare done. (cid:3) Now we can complete the computation of the cohomology of ( ϕ, Γ K )-modules ofcharacter type when the coefficients are a finite extension of F p (( u )). Corollary 4.2.3.
Let R be a finite extension of F p (( u )) and let δ : K × → R × be acontinuous character. Then (1) H ϕ, Γ K ( δ ) = 0 unless δ is the trivial character, in which case H ϕ, Γ K ( δ ) is a -dimensional R -vector space. (2) H ϕ, Γ K ( δ ) = 0 unless δ = χ cyc ◦ Nm K/ Q p , in which case H ϕ, Γ K ( δ ) is a -dimensional R -vector space. (3) H ϕ, Γ K ( δ ) is an R -vector space of dimension [ K : Q p ] unless either H ϕ, Γ K ( δ ) (cid:54) =0 or H ϕ, Γ K ( δ ) (cid:54) = 0 , in which case it is an R -vector space of dimension [ K : Q p ] + 1 . Triangulations
Classification of rank- ϕ, Γ) -modules. In this section, we show that rank-1 ( ϕ, Γ)-modules over a pseudorigid space X are free locally on X , and up totwisting by a line bundle on X , are of character type. The proof is largely the sameas in [KPX14, § R, (0 ,b ] ,K is B´ezout, and then deduce the case wherethe coefficients are artinian by a deformation argument. Proposition 5.1.1.
Suppose R is an artin local pseudoaffinoid algebra. If D is arank- ϕ, Γ) -module over Λ R , (0 ,b ] ,K , then there is a unique continuous character δ : K × → R × such that L := H ϕ, Γ K ( D ( δ − ) is free of rank over R . In addition, (1) The natural map Λ R, rig ,K ( δ ) ⊗ R L → D is an isomorphism. (2) H ϕ, Γ K ( D ( δ − )) is free over R of rank K : Q p ] . (3) H ϕ, Γ K ( D ( δ − )) = 0 .Proof. This proof is nearly identical to the proof of [KPX14, Lemma 6.2.13], so weonly give a sketch.When R is a field and D has slope s , we may choose α ∈ R with v R ( α ) = s . If δ : Q × p → R × be the character with δ ( p ) = α and δ | Z × p = 1, then δ ◦ Nm K/ Q p is a character K × → R × trivial on O × K and sending a uniformizer of K to α f ,and by construction, the associated ( ϕ, Γ)-module D ( δ ◦ Nm K/ Q p ) has slope s .Twisting D by its inverse, we reduce to the ´etale case. But when D is ´etale, M := (cid:16)(cid:101) Λ R, rig ⊗ D (cid:17) ϕ =1 is a Galois representation with D = D rig ( M ). Then localclass field theory and the construction of ( ϕ, Γ)-modules of character type implythat there is a unique character δ : K × with D = Λ R, rig ,K ( δ ). The calculation ofcohomology follows from Corollary 4.2.3. In order to bootstrap to the case where R is an artin local ring, we factor theextension R (cid:16) R/ m R as a sequence of small extensions, that is, extensions of theform 0 → I → R → R (cid:48) → m R ⊂ R is the maximal ideal of R and I ⊂ R is a principal ideal with I m R = 0.Then R → R (cid:48) is a square-zero thickening, and deformation theory (of charactersand of ( ϕ, Γ)-modules) implies that if D is a ( ϕ, Γ)-module over R of rank 1 with R (cid:48) ⊗ R D of character type, then D is of character type.If D = Λ R, rig ,K ( δ ), then H ϕ, Γ K ( D ( δ − )) contains R , and considerations on lengthsof R -modules imply that if H ϕ, Γ K ( R (cid:48) ⊗ R D ( δ − )) = R (cid:48) , then H ϕ, Γ K ( D ( δ − )) = R .Moreover, R/ m R ⊗ R H ϕ, Γ K ( D ( δ − )) = H ϕ, Γ K ( D R/ m R ( δ − )) = 0so H ϕ, Γ K ( D ( δ − )) = 0. Then the base change spectral sequence implies that theformation of H ϕ, Γ K ( D ( δ − )) commutes with base change on R , and the Euler char-acteristic formula implies that dim R/ m R H ϕ, Γ K ( D R/ m R ( δ − )) = 1 + [ K : Q p ]. Thenby Nakayama’s lemma, H ϕ, Γ K ( D ( δ − )) is free of the same rank, and we are done. (cid:3) In order to give a classification over a general base, we again follow the strategyof the proof of [KPX14, Theorem 6.2.14] and twist our rank-1 ( ϕ, Γ)-module bythe universal family of characters. Then we can use the settled case over artinlocal rings and cohomology and base change to cut out the appropriate character.The difficulty is in verifying that the slopes of a family of ( ϕ, Γ)-modules over apseudoaffinoid algebra are bounded; this is the essential content of the followingproposition, whose proof we do not duplicate here.
Proposition 5.1.2.
Let D be a ( ϕ, Γ Q p ) -module over Λ R, rig , Q p . Then (1) The quotient D/ ( ψ − is a finitely generated R -module (2) If n ∈ Z , let δ n : Q × p → R × be the character trivial on Z × p which sends p to u n . Then for all n (cid:29) , the map ψ − D ( δ − n ) → D ( δ − n ) is surjective.Proof. The proof of [KPX14, Proposition 3.3.2] carries over verbatim. One takes amodel of D as a finite projective module over Λ R, (0 ,b ] , Q p , considers it as a summandof a free module, and carefully analyzes the actions of ϕ and ψ . (cid:3) Now we give the desired general classification. The primary difference from theargument of [KPX14, Theorem 6.2.14] is in avoiding the use of Hodge–Tate weightsin the calculation.
Theorem 5.1.3.
Let R be a pseudoaffinoid algebra, and let D be a rank- ϕ, Γ) -module over Λ R, rig ,K . Then there exists a unique continuous character δ : K × → R × and a unique invertible sheaf L on Spa R such that D ∼ = Λ R, rig ,K ( δ ) ⊗ R L . Remark 5.1.4.
If such a δ and L exist, then L ( U ) = H ϕ, Γ K ( D ( δ − ) | U ) for everyopen subspace U ⊂ Spa R . Proof.
We first treat uniqueness. Since the formation of H ϕ, Γ K ( D ( δ − )) commuteswith flat base change on R , it suffices to show that if H ϕ, Γ K (Λ R, rig ,K ( δ )) is locally OHOMOLOGY OF ( ϕ, Γ)-MODULES OVER PSEUDORIGID SPACES 19 free of rank 1 over R , then δ is trivial. There is a Zariski-open dense subspace U ⊂ Spa R such that H iϕ, Γ K (Λ R, rig ,K ( δ ) | U ) is flat for all i ; if x ∈ U and m x ⊂ R isthe corresponding maximal ideal, then the base change spectral sequence impliesthat H iϕ, Γ K ( R/ m kx ⊗ R Λ R, rig ,K ( δ )) ∼ = R/ m kx ⊗ R H iϕ, Γ K ( δ ) for all i and all k ≥
1. Inparticular, H ϕ, Γ K (Λ R/ m kx , rig ,K ( δ )) is free of rank 1 over R/ m kx , which implies that δ : K × → ( R/ m kx ) × is trivial for all k ≥
1. It follows that δ : K × → ( R U (cid:48) ) × is trivial for all affinoid U (cid:48) ⊂ U . But the condition δ = 1 defines a Zariski-closedsubspace of Spa R ; since it contains a Zariski-open dense subspace, it is all of Spa R .To show existence, we follow [KPX14] and consider the twist of D by the inverse ofthe universal family of characters δ univ : K × → R × ; this is a ( ϕ, Γ K )-module overSpa R × (cid:100) K × , and we use Tate local duality to cut out a subspace correspondingto the desired character. More precisely, we let Γ (cid:48) D and Γ (cid:48)(cid:48) D be the support of H ϕ, Γ K ( D ∨ ( δ univ χ cyc )) and H ϕ, Γ K ( D ( δ − χ cyc )) in Spa R × (cid:100) K × , respectively, andlet Γ D := Γ (cid:48) D × Spa R × (cid:100) Γ K Γ (cid:48)(cid:48) D . Since the formation of H ϕ, Γ K commutes with arbitrarybase change on Spa R , the formation of Γ (cid:48) D and Γ (cid:48)(cid:48) D , and hence Γ D , commutes witharbitrary base change on Spa R .There is a natural projection map Γ D → Spa R ; a section induces a morphismSpa R → (cid:99) Γ K , or equivalently, a continuous character δ : K × → R × . We will showthat Γ D → Spa R is actually an isomorphism.Granting this, we may replace D with D ( δ − D ), where δ D : K × → R × is thecontinuous character corresponding to Spa R = Γ D → (cid:100) K × , so that Γ D corre-sponds to the trivial character. Then we need to show that H ϕ, Γ K ( D ) is a linebundle over Spa R , and D ∼ = Λ R, rig ,K ⊗ R H ϕ, Γ K ( D ) as a ( ϕ, Γ K )-module. If R (cid:48) is a pseudoaffinoid artin local ring and R → R (cid:48) is a homomorphism, there is aunique continuous character δ (cid:48) : K × → R (cid:48)× such that H ϕ, Γ K ( D R (cid:48) ( δ (cid:48)− )) is freeof rank 1 over R (cid:48) , and in addition, H ϕ, Γ K ( D R (cid:48) ( δ (cid:48)− )) is free of rank 1 + [ K : Q p ]and H ϕ, Γ K ( D R (cid:48) ( δ (cid:48)− )) = 0. Thus, the formation of H ϕ, Γ K ( D R (cid:48) ( δ (cid:48)− )) commuteswith arbitrary base change on R (cid:48) ; in particular, H ϕ, Γ K ( D R (cid:48) / m R (cid:48) ( δ (cid:48)− )) is non-zero. Since H ϕ, Γ K ( D ∨ R (cid:48) / m R (cid:48) ( δ (cid:48) χ cyc )) and H ϕ, Γ K ( D R (cid:48) / m R (cid:48) ( δ (cid:48)− χ cyc )) are dual to H ϕ, Γ K ( D R (cid:48) / m R (cid:48) ( δ (cid:48)− )) and H ϕ, Γ K ( D ∨ R (cid:48) / m R (cid:48) ( δ (cid:48) )), respectively, and the formation of H ϕ, Γ K commutes with arbitrary base change on R , we see that H ϕ, Γ K ( D ∨ R (cid:48) ( δ (cid:48) χ cyc ))and H ϕ, Γ K ( D R (cid:48) ( δ (cid:48)− χ cyc )) are both non-zero. Thus, the graph of the morphismSpa R (cid:48) → (cid:100) K × induced by δ (cid:48) is contained in Γ D ; since Γ D corresponds to the trivialcharacter, δ (cid:48) is trivial.In other words, for any homomorphism R → R (cid:48) with R (cid:48) a pseudoaffinoid artin localring, H ϕ, Γ K ( D R (cid:48) ) is free of rank 1 over R (cid:48) , H ϕ, Γ K ( D R (cid:48) ) is free of rank 1 + [ K : Q p ],and H ϕ, Γ K ( D R (cid:48) ) = 0; on residue fields, this implies that H ϕ, Γ K ( D R (cid:48) / m R (cid:48) ) = 0, soby Nakayama’s lemma, H ϕ, Γ K ( D R (cid:48) ) = 0, as well. This implies that H ϕ, Γ K ( D ) islocally free of rank 0, so by the base change spectral sequence, the formation of H ϕ, Γ K ( D ) commutes with arbitrary base change on R . It follows that H ϕ, Γ K ( D ) islocally free of rank 1 + [ K : Q p ], so the base change spectral sequence again impliesthat the formation of H ϕ, Γ K ( D ) commutes with arbitrary base change on R , andwe conclude that H ϕ, Γ K ( D ) is locally free of rank 1, as desired. We now prove that Γ D → Spa R is an isomorphism. In fact, it suffices to provethat Γ D is pseudoaffinoid. Indeed, since pseudoaffinoid rings are jacobson, an iso-morphism of pseudoaffinoid spaces can be detected on the level of completed localrings at maximal points, and this follows from the result on artin local rings.Since Γ D is a Zariski-closed subspace of the quasi-Stein space Spa R × (cid:100) K × , it isenough to show that that its image in (cid:100) K × is contained in an affinoid subspace, andsince (cid:100) K × ∼ = G an m × (cid:100) O × K , it is enough to show that its image in G an m is bounded.There is some N ≥ n ≥ N , ψ − D ∨ ( δ − n δ univ χ cyc ) → D ∨ ( δ − n δ univ χ cyc ) and ψ − D ( δ − n δ − χ cyc ) → D ( δ − n δ − χ cyc ) are surjective.Surjectivity is preserved under arbitrary base change R → R (cid:48) , and the isomorphism H • ϕ, Γ K → H • ψ, Γ K implies that H ( D ∨ ( δ − n δ univ δ (cid:48) χ cyc )) = H ϕ, Γ K ( D ( δ − n δ − δ (cid:48) χ cyc )) =0 for all continuous characters δ (cid:48) : O × K → R (cid:48)− . Thus, if T denotes the coordinateon G m , the image of Γ (cid:48) D is contained in the subspace T ≤ u N and the image of Γ (cid:48)(cid:48) D is contained in the subspace T − ≤ u N , and we are done. (cid:3) Interpolating triangulations.
Trianguline ( ϕ, Γ)-modules are those whichare extensions of ( ϕ, Γ)-modules of character type. More precisely,
Definition 5.2.1.
Let X be a pseudorigid space over O E for some finite exten-sion E/ Q p , let K/ Q p be a finite extension, and let δ = ( δ , . . . , δ d ) : ( K × ) d → Γ( X, O × X ) be a d -tuple of continuous characters. A ( ϕ, Γ K )-module D is trian-guline with parameter δ if (possibly after enlarging E ) there is an increasing fil-tration Fil • D by ( ϕ, Γ K )-modules and a set of line bundles L , . . . , L d such thatgr i D ∼ = Λ X, rig ,K ( δ i ) ⊗ L i for all i .If X = Spa R where R is a field, we say that D is strictly trianguline with parameter δ if for each i , Fil i +1 D is the unique sub-( ϕ, Γ K )-module of D containing Fil i D such that gr i +1 D ∼ = Λ R, rig ,K ( δ i +1 ). Lemma 5.2.2.
Let X = Spa R be a reduced pseudorigid space over Z p with p / ∈ R × ,let D be a ( ϕ, Γ) -module over Λ R, rig ,K , and let δ : K × → R × be a continuouscharacter such that H ϕ, Γ K ( D ∨ ( δ )) is free of rank over R and H iϕ, Γ K ( D ∨ ( δ )) has Tor-dimension at most for i = 1 , . Then the morphism D → Λ R, rig ,K ( δ ) corresponding to a basis of H ϕ, Γ K ( D ∨ ( δ )) is surjective over an open subspace U ⊂ X containing { p = 0 } ⊂ X .Proof. Choose a basis element of H ϕ, Γ K ( D ∨ ( δ )); there is some b > D → Λ R, rig ,K ( δ ) is defined over Λ R, (0 ,b ] ,K , and wemay view it as a morphism of coherent sheaves over the corresponding quasi-Steinspace. Moreover, ϕ -equivariance means that to check surjectivity, it suffices to checkthat Λ R, [ b/p,b ] ,K ⊗ Λ R, (0 ,b ] ,K D → Λ R, [ b/p,b ] ,K ⊗ Λ R, (0 ,b ] ,K Λ R, (0 ,b ] ,K ( δ ) is surjective.The morphism Λ R, [ b/p,b ] ,K ⊗ Λ R, (0 ,b ] ,K D → Λ R, [ b/p,b ] ,K ⊗ Λ R, (0 ,b ] ,K Λ R, (0 ,b ] ,K ( δ ) fails tobe surjective on a Zariski-closed subspace Z ⊂ Spa Λ R, [ b/p,b ] ,K . Since Spa Λ R, [ b/p,b ] ,K is affinoid, so is Z .Consider specializations at the characteristic p maximal points x ∈ Spa R . If H ϕ, Γ K ( D ∨ ( δ )) is flat of rank 1 over R , then k x ⊗ R H ϕ, Γ K ( D ∨ ( δ )) is a 1-dimensional k x -vector space. If H iϕ, Γ K ( D ∨ ( δ )) has Tor-dimension at most 1 for i = 1 ,
2, then
OHOMOLOGY OF ( ϕ, Γ)-MODULES OVER PSEUDORIGID SPACES 21 the specialization maps R → k x give us exact sequences0 → k x ⊗ R H ϕ, Γ K ( D ∨ ( δ )) → H ϕ, Γ K ( k x ⊗ R D ∨ ( δ )) → Tor R ( H ϕ, Γ K ( D ∨ ( δ )) , k x ) → k x ⊗ R D → k x ⊗ R Λ R, rig ,K ( δ ) are non-zero, and if k x haspositive characteristic, this implies that the corresponding map is surjective.Thus, p is a nowhere-vanishing function on Z , and since Z is affinoid, the maximummodulus principle discussed in Appendix A.1 implies that p | Z is bounded awayfrom 0. That is, there is some λ such that {| p | ≤ λ } ∩ Z is empty. Setting U := {| p | ≤ λ } ⊂ X yields the desired subspace. (cid:3) Theorem 5.2.3.
Let X be a reduced pseudorigid space over Z p , let D be a ( ϕ, Γ K ) -module over X of rank d , and let δ : K × → Γ( X, O × X ) be a continuous character.Suppose there is a Zariski-dense set X alg ⊂ X of maximal points such that for every x ∈ X alg , H ϕ, Γ K ( D ∨ x ( δ x )) is -dimensional and the image of Λ k x , rig ,K under anybasis of this space is saturated in D ∨ x ( δ x ) . Then there exists a proper birationalmorphism f : X (cid:48) → X of reduced pseudorigid spaces, a line bundle L on X (cid:48) , ahomomorphism λ : f ∗ D → Λ X (cid:48) , rig ,K ( δ ) ⊗ X (cid:48) L of ( ϕ, Γ K ) -modules, and an opensubspace U ⊂ X (cid:48) containing { p = 0 } such that (1) λ | U : f ∗ D | U → Λ U, rig ,K ( δ | U ) ⊗ U L | U is surjective (2) the kernel of λ | U is a ( ϕ, Γ K ) -module of rank d − Proof.
We may replace X with its normalization (using the theory of normalizationsof pseudorigid spaces developed in [JN19]), and we may consider the connectedcomponents of X separately.Using perfectness of C • ϕ, Γ K ( D ∨ ( δ )), we may use [KPX14, Corollary 6.3.6(2)] toconstruct a proper birational morphism f : X (cid:48) → X such that D (cid:48) := f ∗ ( D ∨ ( δ ))has H ϕ, Γ K ( D (cid:48) ) flat and H iϕ, Γ K ( D (cid:48) ) with Tor-dimension at most 1 for i = 1 ,
2. Thenfor any maximal point x ∈ X (cid:48) , the base change spectral sequence gives us a shortexact sequence0 → k x ⊗ R H ϕ, Γ K ( D (cid:48) ) → H ϕ, Γ K ( k x ⊗ R D (cid:48) ) → Tor R ( H ϕ, Γ K ( D (cid:48) ) , k x ) → x ∈ X (cid:48) such that the last term isnon-zero is a Zariski-closed subspace Z (cid:48) ⊂ X (cid:48) whose complement is open anddense. Thus, H ϕ, Γ K ( D (cid:48) ) is flat of rank 1. Letting L := H ϕ, Γ K ( D (cid:48) ) ∨ , we obtain ahomomorphism λ : f ∗ D → Λ X (cid:48) , rig ,K ( δ ) ⊗ X (cid:48) L .The formation of H ϕ, Γ K ( D (cid:48) ) commutes with flat base change on X ; we may finda collection { X (cid:48) i } of open pseudoaffinoid subspaces of X (cid:48) such that H ϕ, Γ K ( D (cid:48) ) | X (cid:48) i is free, { p = 0 } ⊂ ∪ i X (cid:48) i , and p is not invertible on X (cid:48) i . Then we may applyLemma 5.2.2 to conclude that λ | X (cid:48) i is surjective (possibly after shrinking X (cid:48) i ).Setting U := ∪ X (cid:48) i , we see that X (cid:48) , U ⊂ X (cid:48) , and λ satisfy the first of our desiredproperties.To check the second claim, observe that for some b > U → P → Λ U, (0 ,b ] ,K ⊗ D (cid:48) | U → Λ U, (0 ,b ] ,K ( δ ) ⊗ U L | U → U, (0 ,b ] ,K ( δ ) ⊗ X (cid:48) L is R (cid:48) -flat, this sequence remains exact after specializingat any point x ∈ U , so k x ⊗ P is a ( ϕ, Γ K )-module of rank d −
1. It follows that P is a vector bundle of rank d − U, (0 ,b ] ,K ,and hence is a ( ϕ, Γ K )-module of the correct rank. (cid:3) Remark 5.2.4.
The morphism f : X (cid:48) → X is, in general, not compatible with theanalogous morphism constructed in [KPX14, Theorem 6.3.9]; in that argument, theauthors make an additional blow-up, in order to control the cohomology groups of f ∗ M/t , which is what permits them to deduce that X alg ⊂ U . But Fontaine’s ele-ment t does not make sense in our mixed- or positive-characteristic overconvergentperiod rings, so we cannot deduce that X alg ⊂ U .As in [KPX14, Corollary 6.3.10], we may deduce the following: Corollary 5.2.5.
Let X be a reduced pseudorigid space over Z p , all of whoseconnected components are irreducible. Let M be a ( ϕ, Γ K ) -module over X of rank d and let δ := ( δ , . . . , δ d ) : ( K × ) d → Γ( X, O × X ) be a parameter such that D | x isstrictly trianguline with parameter δ at a Zariski-dense set X alg ⊂ X of maximalpoints x ∈ X . Then there exists a proper birational morphism f : X (cid:48) → X ofreduced pseudorigid spaces, an increasing filtration Fil • ( f ∗ D ) , and an open subspace U ⊂ X (cid:48) containing { p = 0 } such that (1) (Fil • ( f ∗ D )) | x is a strictly trianguline filtration on ( f ∗ D ) | x for all x ∈ U , (2) there are line bundles L i on X (cid:48) and injective maps of ( ϕ, Γ K ) -modules gr i ( f ∗ D ) → Λ X (cid:48) , rig ,K ( δ i ) ⊗ X (cid:48) L i which are isomorphisms over U , and (3) the first graded piece gr ( f ∗ D ) is isomorphic to Λ X (cid:48) , rig ,K ( δ ) ⊗ X (cid:48) L Proof.
We may apply Theorem 5.2.3 inductively to construct f : X (cid:48) → X , U ⊂ X (cid:48) , and { L i } satisfying the first two properties. It remains to check the thirdproperty. We may do this locally on X (cid:48) , so we may assume that X (cid:48) is connectedand affinoid and L is trivial. Now we have an injection of rank-1 ( ϕ, Γ K )-modules λ : gr ( f ∗ D ) → Λ X (cid:48) , rig ,K ( δ ) which has a model λ b : gr ( f ∗ D b ) → Λ X (cid:48) , (0 ,b ] ,K ( δ )over the pseudorigid space associated to Λ X (cid:48) , (0 ,b ] ,K for some b >
0. As in theproof of Lemma 5.2.2, it suffices to check surjectivity after extending scalars toΛ X (cid:48) , [ b/p,b ] ,K .We have an exact sequence0 → gr ( f ∗ D b ) → Λ X (cid:48) , [ b/p,b ] ,K ( δ ) → Q [ b/p,b ] → V ⊂ X (cid:48) (cid:114) { p = 0 } , the argument of [KPX14, Corol-lary 6.3.10] implies that Q [ b/p,b ] | V is O V -flat. For any affinoid subspace V (cid:48) ⊂ U , Q [ b/p,b ] | V (cid:48) = 0. Since X (cid:48) is connected by assumption and U and X (cid:48) (cid:114) { p = 0 } cover X (cid:48) , we conclude that Q [ b/p,b ] = 0. (cid:3) Applications to eigenvarieties
Set-up.
Extended eigenvarieties have been constructed by [AIP18], [JN16],and [Gul19] for various groups; these extended eigenvarieties are expected to (andin some cases known to) carry families of Galois representations such that localGalois-theoretic data matches certain Hecke-theoretic data. At places away from p and the level, this compatibility specifies that the local Galois representation isunramified and gives a characteristic polynomial for Frobenius. At places dividing OHOMOLOGY OF ( ϕ, Γ)-MODULES OVER PSEUDORIGID SPACES 23 p , this compatibility specifies that the local Galois representation is trianguline andgives the parameters of the triangulation.In this subsection, we use our results on trianguline ( ϕ, Γ)-modules to study ex-tended eigenvarieties at the boundary of weight space, in order to address twoquestions:(1) Are irreducible components proper at the boundary of weight space?(2) Are Galois representations at characteristic p points trianguline at p ?We will give partial affirmative answers to both questions.Before stating our assumptions more precisely, we recall the construction of [JN16].Let F be a number field, let H be a reductive group over F split at all placesabove p , and set G := Res F/ Q H . Fix a tame level by choosing a compact opensubgroup K (cid:96) ⊂ G ( Q (cid:96) ) for each prime (cid:96) (cid:54) = p , such that K (cid:96) is hyperspecial for allbut finitely many (cid:96) , and let K p ⊂ G ( Q p ) be an Iwahori subgroup. Let S (cid:48) denotethe set of places w of Q such that either w = ∞ , or K w is not hyperspecial, and let S denote the set of places of F lying above the places in S (cid:48) . Then [JN16] provedthe following: Theorem 6.1.1. [JN16, Theorems A and B]
The eigenvarieties for G constructedin [HN17] naturally extend to pseudorigid spaces X G equipped with a weight map wt : X G → W to extended weight space W := (Spa Z p [[ T (cid:48) ]]) an , where T (cid:48) is acertain quotient of the Z p -points of a (split) maximal torus of a model of G over Z p . Moreover, if F is totally real or CM and H = GL d , there is a continuous d -dimensional determinant D : O ( X G )[Gal F,S ] → O + ( X red G ) such that D (1 − X · Frob v ) = P v ( X ) for all v (cid:54)∈ S , where P v ( X ) is the Hecke polynomial. When F is totally real with p completely split, and H = GL , the characteristic 0eigenvariety X rig G contains a Zariski very dense set of “essentially classical” points(in the sense of [Che11], using [Che04, Lemme 6.2.10], [Che04, Lemme 6.2.8], anda “small slope implies classical” criterion). Furthermore, local-global compatibilityat places dividing p is known for classical Hilbert modular forms of motivic weightby [Ski09], [Liu12], [Sai09], [BR93], and so in this case X rig G contains a dense set ofpoints at which D corresponds to a trianguline Galois representation.When H is a totally definite quaternion algebra over a totally real field, split at p ,a similar argument shows that X rig G contains a Zariski very dense set of classicalpoints. Moreover, the p -adic Jacquet–Langlands correspondance of [Che05], [Bir19]can be extended to the pseudorigid setting. This identifies each irreducible compo-nent of a quaternionic eigenvariety with an irreducible component of an eigenvarietyfor Hilbert modular forms; it follows that a Galois determinant can be pulled backto X G , and it corresponds to a trianguline representation at a dense set of pointsof X rig G .There is a similar story when G is a definite unitary group over Q split at p . Char-acteristic 0 eigenvarieties have been constructed [Che04], [BC09] which interpolateclassical automorphic forms and carry a family of Galois determinants: Theorem 6.1.2. [BC09, Chapter 7]
Let F/ Q be an imaginary quadratic field andlet G be a definite unitary group associated to F , split at p . Then the characteristic eigenvariety X rig G contains a Zariski very dense set of classical points (corre-sponding to p -refined automorphic representations), and there is a continuous de-terminant D : O + ( X rig G )[Gal F,S ] → O + ( X rig G ) such that D (1 − X · Frob v ) = P v ( X ) for all v (cid:54)∈ S , where P v ( X ) is the Hecke polynomial. Moreover, the corresponding Galois representation is known to be trianguline atclassical points ; thus, there is a continuous Galois determinant D : O + ( X rig G )[Gal F,S ] → O + ( X rig G ) defined on the closure of X rig G in X G , and it corresponds to a triangulineGalois representation at a Zariski very dense of points.We will make more precise what kind of trianguline conditions we have (or hopefor) at places dividing p . If T is a split maximal torus of a model of G over Z p ,consider a splitting of the inclusion T ( Z p ) (cid:44) → T ( Q p ), and let Σ denote the kernel.There are two submonoids Σ cpt ⊂ Σ + ⊂ Σ; we refer the reader to [JN16, § G ( Z p ) ∼ = (cid:81) v | p GL d ( O F v ), we may take T to be the standard torus andΣ = (cid:89) v | p { diag( (cid:36) a v , . . . , (cid:36) a d v ) | a i ∈ Z } Σ + = (cid:89) v | p { diag( (cid:36) a v , . . . , (cid:36) a d v ) | a i +1 ≥ a i } Σ cpt = (cid:89) v | p { diag( (cid:36) a v , . . . , (cid:36) a d v ) | a i +1 > a i } The construction of X G depends on a choice of t ∈ Σ cpt , which in the above casewe take to be (cid:81) v | p diag(1 , . . . , (cid:36) d − v ); the authors construct a spectral variety Z ⊂W × G m using the Fredholm series of the corresponding controlling Hecke operator U t , and then construct X G → Z finite, such that there is a homomorphism ψ : T (∆ p , K p ) → O ( X G ). Here T (∆ p , K p ) is a Hecke algebra with no Hecke operatorsat places above p .However, it is possible to make the same construction using other choices of Heckealgebras, and we will need to do so (this is discussed in greater detail in [JN19, § A + p ⊂ Z p [ G ( Q p ) //I ] be the subring generated by thecharacteristic functions [ IsI ] for s ∈ Σ + . Then there is an extended eigenvariety X A + p G equipped with a homomorphism T (∆ p , K p ) ⊗ Z p A + p → O ( X A + p G ) and a finitemorphism X A + p G → X G . There is a surjective finite map X A + p G → X G , and weobtain a Galois determinant O ( X A + p , red G )[Gal F,S ] → O ( X A + p , red G ) by pulling backthe determinant on X red G .Let G m, W denote the pseudorigid space G an m ×W , so that we have finite morphisms X A + p G → X G → G m, W . By [JN19, Lemma 3.4.1], the image of [ IsI ] in O ( X A + p G )is invertible for all s ∈ Σ + , and so for s ∈ Σ, we can write s = s (cid:48) s (cid:48)(cid:48)− for s (cid:48) , s (cid:48)(cid:48) ∈ Σ + and obtain ψ ([ Is (cid:48) I ]) ψ ([ Is (cid:48)(cid:48) I ] − ) ∈ O ( X A + p G ) × . Thus, we have a morphism OHOMOLOGY OF ( ϕ, Γ)-MODULES OVER PSEUDORIGID SPACES 25 X A + p G → (cid:98) Σ := Hom(Σ , G m, W ) such that the diagram X A + p G (cid:98) Σ X G G m, W commutes and has finite horizontal maps. Here the right vertical map is inducedby evaluation at U t . Any choice of a basis of Σ will give us parameters δ i,v : F × v → O ( X A + p G ) × .When F is a number field and G = Res F/ Q GL d with the standard maximal torus,there is a natural ordered basis of Σ, namely { diag(1 , . . . , (cid:36) v , . . . , } i,v , where v | p and (cid:36) v is placed in the d − i slot. Then (restricting to non-critical pointsfor simplicity), [HN17, Conjecture 1.2.2(iii)] predicts that if x ∈ X A + p G is non-critical, the Galois representation corresponding to D x | Gal Fv is trianguline withparameters { δ i,v } such that { δ i,v ( (cid:36) v ) = ψ ( s i,v ) } (and δ i,v | O × Fv corresponds tothe automorphic weight). Similarly, in the unitary case sketched above, [BC09,Proposition 7.5.13] implies that at non-critical points x ∈ X A + p G , the Galois rep-resentation corresponding to D x | Gal Fv is trianguline with parameter { δ i,v } , where δ i,v ( p ) = ψ (diag(1 , . . . , p, . . . , Properness at the boundary.
We follow the strategy of [DL16] to showextended eigenvarieties are proper at the boundary. We assume we have a Ga-lois representation, and sufficiently many classical points where it is known to betrianguline at p , with parameters compatible with the Hecke algebra at p : Theorem 6.2.1.
Suppose X A + p G is an extended eigenvariety such that there is acontinuous determinant D : O ( X A + p red G )[Gal F,S ] → O ( X A + p , red G ) for some numberfield F . Suppose we have a commutative diagram Spa R (cid:114) { p = 0 } X A + p , red G Spa R W with Spa R → W corresponding to a weight κ : T ( Z p ) → R × and R a normalpseudoaffinoid algebra flat over Z p with p (cid:54)∈ R × . Suppose in addition that there isan ordered basis { s i,v } of Σ with (cid:81) ij =1 s j,v ∈ Σ + such that for a Zariski-dense setof points Z ⊂ Spa R (cid:114) { p = 0 } the Galois representation attached to the pull-back of D is trianguline at all places v | p , with parameters { δ i,v } induced by { s i,v } . Thenthe dashed arrow can be filled in. Remark 6.2.2.
For the eigenvarieties discussed in the previous section, we willbe able to check that Z exists so long as Spa R has the same dimension as weightspace, and classical weights are dense in κ . We can view this as saying that thelimit of a family of overconvergent automorphic forms exists, so long as it tends to { p = 0 } in a sufficiently regular manner. In particular, we may deduce thatevery irreducible component of the extended eigencurve is proper at the boundaryof weight space.We first treat the case of a finite morphism: Lemma 6.2.3.
Suppose X → Y is a finite morphism of pseudorigid spaces andwe have a commutative diagram Spa R (cid:114) { p = 0 } X Spa R Y where R is a Z p -flat normal pseudoaffinoid algebra with p (cid:54)∈ R × . Then the dashedarrow can be filled in.Proof. By [Hub13, 1.4.4], the pre-image in X of any affinoid subspace of Y isitself affinoid, so we may assume that X and Y are both affinoid. Then if X =Spa A , the morphism Spa R (cid:114) { p = 0 } → X is induced by a compatible sequenceof continuous homomorphisms ( A, A ◦ ) → ( R (cid:68) u k p (cid:69) (cid:2) u (cid:3) , R (cid:68) u k p (cid:69) (cid:2) u (cid:3) ◦ ) for somenoetherian ring of definition R ⊂ R and k ≥
1. By [Lou17, Theorem 5.1], R ◦ = ∩ k R (cid:68) u k p (cid:69) ◦ , so we have a continuous homomorphism A ◦ → R ◦ . Since A ◦ contains aring of definition of A , we obtain a continuous homomorphism A → R , as well. Sincethe composition Spa R → X → Y agrees with the specified morphism Spa R → PY after restricting to Spa R (cid:114) { p = 0 } and Y is separated, it agrees on all ofSpa R . (cid:3) Combined with the theory of determinants B, the assumption that a determinant D : O ( X G )[Gal F,S ] → O ( X G ) exists implies that there is a natural map X red G → X an p , where X p is the adic space associated to the deformation rings of all of thedeterminants attached to isomorphism classes of d -dimensional modular residualrepresentations of Gal F,S . Lemma 6.2.4.
Let R be an integral normal pseudoaffinoid algebra flat over Z p with pseudouniformizer u . Then any morphism Spa R (cid:114) { p = 0 } → X an p extendsuniquely to a morphism Spa R → X an p .Proof. We use the Hebbarkeitss¨atze of [Lou17]. The pseudorigid space Spa R (cid:114) { p =0 } is connected, so its image in X an p has constant residual determinant D . Thus,the morphism Spa R (cid:114) { p = 0 } → X an p is induced by a series of homomorphisms R D → R (cid:68) u k p (cid:69) ◦ for k ≥
1, where R D is the pseudodeformation ring parametrizinglifts of D . But by [Lou17, Theorem 5.1], R ◦ = ∩ k R (cid:68) u k p (cid:69) ◦ , so we get a continuoushomomorphism R D → R ◦ and a morphism Spa R → (Spa R D ) an . (cid:3) Lemma 6.2.5.
For any s ∈ Σ + , ψ ( s ) ∈ O ( X A + p , red G ) is power-bounded.Proof. This follows from the construction of [JN16] and we use the notation of thatpaper freely. By [JN16, Corollary 3.3.10], the action of s on D rκ is norm-decreasing, OHOMOLOGY OF ( ϕ, Γ)-MODULES OVER PSEUDORIGID SPACES 27 for any weight κ : T → R × and any r (cid:29) /p (depending on κ ). It follows that theaction of s is power-bounded, hence power-bounded on C ∗ ( K, D rκ ), hence power-bounded on K • := ker • Q ∗ ( U t ), and hence power-bounded on H ∗ K • . (cid:3) Now we are in a position to prove Theorem 6.2.1:
Proof of Theorem 6.2.1.
We first assume that
R/p is an integral domain. Since X A + p , red G → (cid:98) Σ is finite, Lemma 6.2.3 implies that it therefore suffices to lift κ to amorphism (cid:101) κ : Spa R → (cid:98) Σ (compatibly with the given map Spa R (cid:114) { p = 0 } → (cid:98) Σ).In other words, we need to show that the image of ψ ( s i,v ) in ∩ k R (cid:68) u k p (cid:69) is an elementof R × for all i and all v | p .Let U i,v := (cid:81) ij =1 s i,v ∈ Σ + ; by Lemma 6.2.5, ψ ( U i,v ) is power-bounded for all i and all v | p , and by [Lou17, Theorem 5.1], the image of ψ ( U i,v ) in ∩ k R (cid:68) u k p (cid:69) landsin R + ⊂ R .It suffices to show that the image of ψ ( U i,v ) is a unit of R , i.e. that it does notvanish at any point in the locus { p = 0 } . We proceed by induction on i . Let F ,v denote the image of ψ ( s ,v ) = ψ ( U ,v ) in R . By Lemma 6.2.4, the determinant D : O ( X A + p , red G )[Gal F,S ] → O ( X A + p , red G ) extends to a determinant R [Gal F,S ] → R .By B.0.4 there is a morphism f : X (cid:48) → Spa R and a family of Galois representations M (cid:48) over X (cid:48) such that M (cid:48) induces the pullback f ∗ D R of D R to X (cid:48) , and we mayassume that X (cid:48) → Spa R is the composition of a blow-up and a finite morphism.We may make a further blow-up, and assume that H ϕ, Γ ( M (cid:48) ) and H ϕ, Γ ( M (cid:48) ) haveTor-dimension at most 1.Let M (cid:48) v denote the restriction of M (cid:48) to the local Galois group at v | p . For each point x ∈ Spa R , the morphism f − ( x ) → Spa k ( x ) is flat, and so H iϕ, Γ ( M (cid:48) v | f − ( x ) ) = H iϕ, Γ ( M v | x ), where by abuse of notation, we let M v | x refer to the Galois represen-tation at x (possibly after extending scalars). In particular, there is a Zariski-denseset of points x (cid:48) ∈ X (cid:48) such that D rig ( M (cid:48) v ⊗ k ( x (cid:48) )) Γ=1 ,ϕ fv = f (cid:93) ( F ,v ) is free of rank 1 over k ( x (cid:48) ) ⊗ F v, . It follows that there is a Zariski-dense open subspace U (cid:48) ⊂ X (cid:48) (cid:114) { p = 0 } such that D rig ( M v | U (cid:48) ) Γ=1 ,ϕ fv = f (cid:93) ( F ,v ) is a rank-1 vector bundle over O ( U (cid:48) ) ⊗ F v, .There is a finite affinoid cover { U (cid:48) j } of X (cid:48) and a finite extension L v /F v such that D L v rig ( M (cid:48) v | U (cid:48) j ) is a free Λ U (cid:48) j , rig ,L V -module. Fix an open affinoid Spa R (cid:48) j ⊂ U (cid:48) j (cid:114) { p = 0 } such that D rig ( M v | Spa R (cid:48) j ) Γ=1 ,ϕ fv = f (cid:93) ( F ,v ) is free of rank 1 and fix a generator e v,j of D rig ( M v | Spa R (cid:48) j ) Γ=1 ,ϕ fv = f (cid:93) ( F ,v ) ; after dividing by a suitable power of p , we mayassume that e v,j does not vanish on the entire locus { p = 0 } . Since ϕ f v ( e v,j ) = f (cid:93) ( F ,v ) e v,j holds on a Zariski-dense subset of U (cid:48) j , it holds on all of U (cid:48) j .By construction, H ϕ, Γ ( M (cid:48) ) and H ϕ, Γ ( M (cid:48) ) have Tor-dimension at most 1, so Corol-lary 3.0.12 implies that for every maximal point x (cid:48) ∈ U (cid:48) j , we have an injection k ( x (cid:48) ) ⊗ D L rig ( M (cid:48) ) (cid:44) → D L rig ( k ( x (cid:48) ) ⊗ M (cid:48) ). If e v,j is non-zero at x (cid:48) , injectivity of ϕ on D L rig ( k ( x (cid:48) ) ⊗ M (cid:48) ) implies that the image of F ,v does not vanish at x (cid:48) . But F ,v van-ishes at a maximal point x ∈ Spa R if and only if its image vanishes at every pointof f − ( x ). Its vanishing locus is therefore a proper closed subspace of Spa R/p ; since we assumed Spec R/p irreducible, Krull’s Hauptidealsatz implies that F ,v isactually a unit of R .Let δ ,v : F × v → R × be the character which sends the uniformizer (cid:36) v to F ,v and is defined by κ on O × F v . We have a morphism f ∗ Λ X (cid:48) , rig ,L ( δ ,v ) → D L rig ( M (cid:48) )over a Zariski-open subspace containing the locus { p = 0 } which is injective withprojetive quotient. Then we may repeat this argument with ψ ( s ,v ) = ψ ( U ,v ) /F ,v ,and eventually conclude that ψ ( s i,v ) is a unit of R for all i , as desired.Now we remove the hypothesis on the integrality of R/p . In the general case, byworking locally on Spa R , the previous arguments yield a morphism Spa R (cid:114) Z → (cid:98) Σ,where Z ⊂ Spa(
R/p ) denotes the union of intersections of at least two irreduciblecomponents. The codimension of Z in X is therefore at least 2, and the secondHebbarkeitsatz of [Lou17, Theorem 5.1] implies that our morphism extends to allof Spa R . (cid:3) Trianguline points.
In this section, we show that the Galois representationsattached to certain characteristic p points of X G in the closure of the characteristic0 eigenvariety are trianguline, partially answering a question of [AIP18] and [JN16].Our setup is similar to the previous section: Theorem 6.3.1.
Suppose X A + p G is an extended eigenvariety such that there is acontinuous determinant D : O ( X A + p red G )[Gal F,S ] → O ( X A + p , red G ) for some numberfield F , and let X (cid:44) → X A + p , red G be an irreducible Zariski-closed subspace. Supposein addition there is an ordered basis { s i,v } of Σ with (cid:81) ij =1 s j,v ∈ Σ + for all i such that for a very Zariski-dense set of points Z ⊂ X the Galois representationattached to D is trianguline at all places v | p , with parameters { δ i,v } induced by { s i,v } . If x ∈ X is a maximal point whose residue field has positive characteristic,then the Galois representation associated to the restriction D | x is also triangulineat all places v | p , with parameters { δ i,v } induced by { s i,v } .Proof. Let U = Spa R ⊂ X red G be an irreducible affinoid pseudorigid subspacecontaining x , with U (cid:114) { p = 0 } non-empty. By [WE18, Theorem 3.8], there is atopologically finite-type cover f (cid:48) : U (cid:48) := Spa R (cid:48) → U and a Galois representation ρ (cid:48) : Gal F,S → GL n ( R (cid:48)◦ ) such that the determinant associated to ρ (cid:48) is equal to R (cid:48)◦ ⊗ R ◦ D . By [Bel20, Theorem 1.1], for each place v | p of F , there is a projective( ϕ, Γ F v )-module D rig ( ρ (cid:48) v ) associated to ρ (cid:48) v .By assumption, there is a Zariski-dense set of points { x i } ⊂ U (cid:114) { p = 0 } andcontinuous characters δ v,j : E v → R × such that the ( ϕ, Γ K v , Gal K v /E v )-moduleattached to D x i | Gal Ev is trianguline with parameters ( δ v,j ) x i . Thus, after passingto a further cover f (cid:48)(cid:48) : U (cid:48)(cid:48) → U , there is an open subspace V ⊂ U (cid:48)(cid:48) containing { p = 0 } ⊂ U (cid:48)(cid:48) such that f (cid:48)(cid:48)∗ ρ (cid:48) | V is trianguline with parameters δ v,j .In particular, f (cid:48)(cid:48)∗ ρ (cid:48) | ( f (cid:48) ◦ f (cid:48)(cid:48) ) − ( x ) is trianguline. Since ( f (cid:48) ◦ f (cid:48)(cid:48) ) − ( x ) → { x } is faith-fully flat, the triangulation descends to a triangulation on D rig ( ρ x ) with the desiredparameters. (cid:3) This argument applies strictly to positive characteristic points lying in the closureof points where the Galois representation is already known to be trianguline. In
OHOMOLOGY OF ( ϕ, Γ)-MODULES OVER PSEUDORIGID SPACES 29 particular, if there are irreducible components supported entirely in positive char-acteristic, we can say nothing at all. However, [JN19, Lemma 4.2.2] implies that theextended Coleman–Mazur eigencurve does not contain any strictly characteristic p components, and so the Galois representations associated to its boundary pointsare all trianguline at p . Appendix A. Complements on pseudorigid spaces
Let E be a complete discretely valued field with ring of integers O E , uniformizer (cid:36) E , and residue field k E . We briefly recall the definition of a pseudorigid spaceover O E , before discussing pseudorigid generalizations of the maximum modulusprinciple and the generic fiber constructions of Bosch–L¨utkebohmert [BL93] andBerthelot [dJ95]. Definition A.0.1.
Let R be a Tate ring. We say that R is a pseudoaffinoid O E -algebra if it has a noetherian ring of definition R ⊂ R which is formally of finitetype over O E . If X is an adic space over Spa O E , we say that X is pseudorigid if itis locally of the form Spa R := Spa( R, R ◦ ), where R is a pseudoaffinoid O E -algebra. Example A.0.2.
Let λ = nm ∈ Q > be a positive rational number with ( n, m ) = 1,and set D ◦ λ := O E [[ u ]] (cid:68) (cid:36) mE u n (cid:69) and D λ := D ◦ λ (cid:2) u (cid:3) . Then D λ is a pseudoaffinoidalgebra.Every pseudoaffinoid algebra R is a topologically finite type D λ -algebra for somesufficiently small λ >
0, by [Lou17, Lemma 4.8].A.1.
Maximum modulus principle.
In classical rigid analytic geometry, themaximum modulus principle states roughly that every function on an affinoid do-main attains its supremum at some closed point. We wish to give an analogousresult for affinoid pseudorigid spaces. We note that we have chosen to present itusing the language of valuations, rather than norms, since there is no longer anatural exponential base, so what we prove might better be called a “minimumvaluation principle”.Let E and D λ be as above, and let D λ,r := D λ (cid:104) X , . . . , X r (cid:105) , which is a pseudoaffi-noid algebra corresponding to a closed ball over D λ . Let D ◦◦ λ denote the ideal oftopologically nilpotent elements of D ◦ λ .We begin by defining valuations on D λ and on Tate algebras over it. Each elementof D λ may be written uniquely in the form (cid:80) i ∈ Z a i u i with a i ∈ O E , which permitsthe following definition: Definition A.1.1.
We define analogues of the Gauss norm on D λ and D λ,r , via v D λ ( (cid:88) i ∈ Z a i u i ) := inf i (cid:26) v E ( a i ) + iλ (cid:27) and v D λ,r (cid:88) j ∈ Z ⊕ r a j X j := inf j v D λ ( a j ) respectively. For any Tate ring R with ring of definition R and pseudo-uniformizer u ∈ R , we also define the spectral semi-valuation v R, sp ( f ) := − inf { ( a,b ) ∈ Z ⊕ N | u a f b ∈ R } ab for f ∈ R .Note that v R, sp depends on a choice of pseudouniformizer, but we suppress this inthe notation.Then it is clear from the definition that f ∈ D ◦ λ,r if and only if v D λ,r ( f ) ≥ f is topologically nilpotent if and only if v D λ,r ( f ) >
0. Moreover, for all f ∈ D λ,r , v D λ,r ( uf ) = λ − + v D λ,r ( f ), and v D λ,r ( af ) = v E ( a ) + v D λ,r ( f ) for all a ∈ O E . Similarly, for a general pseudoaffinoid algebra R and f ∈ R , v D λ , sp ( uf ) =1 + v D λ , sp ( f ) and v D λ , sp ( f k ) = k · v D λ , sp ( f ) for all integers k ≥ Lemma A.1.2. If f ∈ R and v R, sp ( f ) > , then f is topologically nilpotent.Proof. By assumption, there is some a ∈ Z < and some b ∈ N such that u a f b ∈ R ,so f b ∈ u − a R and f is topologically nilpotent. (cid:3) Corollary A.1.3. If f ∈ D λ,r and v D λ,r ( f ) = 0 , then v D λ,r , sp ( f ) = 0 , as well.Proof. The hypothesis implies that f ∈ D ◦ λ,r (cid:114) D ◦◦ λ,r , so v D λ,r , sp ( f ) ≥ v D λ,r , sp ( f ) ≤
0, as desired. (cid:3)
Lemma A.1.4. If f ∈ D λ,r , then v D λ,r , sp ( af ) = λv E ( a ) + v D λ,r , sp ( f ) for all a ∈ O E .Proof. We may assume that f ∈ D ◦ λ,r and a = (cid:36) E . Then v D λ,r , sp ( (cid:36) E f ) ≥ v D λ,r , sp ( (cid:36) E ) + v D λ,r , sp ( f ) ≥ nm + v D λ,r , sp ( f ) = λ + v D λ,r , sp ( f )since u − n (cid:36) mE ∈ D ◦ λ,r .On the other hand, we need to show that if u a ( u − n (cid:36) mE f m ) b ∈ D ◦ λ,r , then u a f bm ∈ D ◦ λ,r . Writing D ◦ λ,r ∼ = O E [[ u ]] (cid:104) X, X , . . . , X r (cid:105) / ( u n X − (cid:36) mE ), it suffices to show thatif f (cid:48) ∈ D λ,r satisfies Xf (cid:48) ∈ D ◦ λ,r , then f (cid:48) ∈ D ◦ λ,r . Writing f (cid:48) = u − N f (cid:48)(cid:48) for some f (cid:48)(cid:48) ∈ D ◦ λ,r , we see that Xf (cid:48)(cid:48) ≡ D ◦ λ,r /u ∼ = ( O E /(cid:36) mE )[ X, X , . . . , X r ], whichimplies that f (cid:48)(cid:48) ≡ D ◦ λ,r /u . Replacing f (cid:48)(cid:48) with f (cid:48)(cid:48) u and N with N − f (cid:48) ∈ D ◦ λ,r , as desired. (cid:3) Corollary A.1.5. If f ∈ D λ,r , then f ∈ D ◦ λ,r if and only if v D λ,r , sp ( f ) ≥ . Corollary A.1.6.
For f ∈ D λ,r , v D λ,r ( f ) = λ − v D λ,r , sp ( f ) . The maximal points of Spa D λ consist of the points of a classical half-open annulus(which is not quasi-compact), together with a positive characteristic “limit point”Spa( D λ /(cid:36) E ) = Spa k E (( u )). Similarly, the maximal points of Spa D λ,r consist ofthe product of a closed r -dimensional unit ball with a classical half-open annulusand a closed r -dimensional unit ball over Spa k E (( u )). OHOMOLOGY OF ( ϕ, Γ)-MODULES OVER PSEUDORIGID SPACES 31
The advantage of working with v R, sp for a pseudoaffinoid algebra R is that it makessense on the residue fields of maximal points, letting us compare v R, sp ( f ) and v R/ m x , sp ( f ( x )) for f ∈ R and x a maximal point of Spa R (here we use the image of u in R to compute the spectral semi-valuation on residue fields). It was demonstratedin the proof of [JN19, Lemma 2.2.5] that if m ⊂ R is a maximal ideal, then Spa( R/ m )is a singleton. Then for a maximal point x ∈ Spa R , the corresponding equivalienceclass of valuations contains the composition of the specialization map R → R/ m x with v R/ m x , sp ( f ( x )).Then we have the following analogue of [BGR84, Lemma 3.8.2/1]: Lemma A.1.7.
Let R be a pseudoaffinoid algebra over O E and let x ∈ MaxSpec R be a maximal point, corresponding to a maximal ideal m x ⊂ R . Then for any f ∈ R , v R/ m x , sp ( f ) ≥ v R, sp ( f ) Proof.
The quotient map R → R/ m x carries R to the ring of definition of R/ m x .Thus, if u a f b ∈ R then u a f b ∈ ( R/ m x ) ◦ . (cid:3) Thus, the spectral semi-valuation is a lower bound for the residual spectral semi-valuations. In fact, it is the minimum. Analogously to the classical setting, we firstprove this for R = D λ,r before deducing the general result. Lemma A.1.8.
For any f ∈ D λ,r , the function x (cid:55)→ v D λ,r / m x , sp ( f ) on MaxSpec D λ r attains its infimum, and its minimum is equal to v D λ,r , sp ( f ) .Proof. Fix f ∈ D λ,r . After scaling by an element of the form au k for a ∈ O E , wemay assume that v D λ,r ( f ) = v D λ,r , sp ( f ) = 0, so f ∈ D ◦ λ,r and we need to find amaximal point x ∈ Spa D λ,r such that v D λ,r , sp ( f ( x )) = 0. Let f = (cid:80) j a j X j denotethe image of f in D ◦ λ,r /D ◦◦ λ ∼ = k E [ X, X , . . . , X r ]. Since f (cid:54) = 0 by assumption, thereis some closed point ( x, x , . . . , x r ) ∈ k r +1 E such that f ( x, x , . . . , x r ) (cid:54) = 0. Thenfor any maximal point x ∈ Spa D λ,r whose kernel m x reduces to ( x, x , . . . , x r ), v D λ,r , sp ( f ( x )) = 0, as desired. (cid:3) In order to deduce the same result for more general pseudoaffinoid algebras, we usethe Noether normalization result of [Lou17, Proposition 4.14]:
Proposition A.1.9.
Let R be an O E -flat pseudoaffinoid algebra such that (cid:36) E / ∈ R × and let f ∈ R . Then inf x ∈ MaxSpec R v R/ m , sp ( f ) = min x ∈ MaxSpec R v R/ m , sp ( f ) = v R, sp ( f ) .Proof. If p , . . . , p s are the minimal prime ideals of R , it suffices to prove the resultfor each R/ p j . Thus, we may assume that R is an integral domain. The algebra R/(cid:36) E is a k E (( u ))-affinoid algebra, so Noether normalization for affinoid algebrasprovides us with a finite injective map k E (( u )) (cid:104) X , . . . , X r (cid:105) → R/(cid:36) E for some r ≥
0. Then [Lou17, Proposition 4.14] implies that it lifts to a finite injective map D n,r → R (cid:10) (cid:36) E u n (cid:11) for some sufficiently large integer n . Then we may argue as in the proof of [BGR84, Proposition 3.8.1/7] to see thatinf x ∈ MaxSpec R (cid:104) (cid:36)Eun (cid:105) v R/ m , sp ( f ) = inf y ∈ MaxSpec D n,r min x ∈ MaxSpec R (cid:104) (cid:36)Eun (cid:105) ⊗ D n,r / m y v R/ m , sp ( f )= inf y ∈ MaxSpec D n,r min j d − j v D n,r / m y , sp ( b j ( y ))= min j v D n,r , sp ( b j )where Y d + b d − Y d − + . . . + b is the minimal polynomial for f over D n,r . Since v D n,r , sp ( b j ) attains its infimum on MaxSpec D n,r by Lemma A.1.8 and the fibers ofMaxSpec R (cid:10) (cid:36) E u n (cid:11) over MaxSpec D n,r are finite, v R/ m , sp ( f ) also attains its infimum.Since v R, sp is “power-additive” for all R , [BGR84, Proposition 3.1.2/1] implies inaddition that v R (cid:104) (cid:36)Eun (cid:105) , sp ( f ) ≥ min j d − j v D n,r , sp ( b j ). Since the right-side is equal toinf x ∈ MaxSpec R (cid:104) (cid:36)Eun (cid:105) v R/ m , sp ( f ), Lemma A.1.7 implies that min x ∈ MaxSpec R (cid:104) (cid:36)Eun (cid:105) v R/ m , sp ( f ) = v R (cid:104) (cid:36)Eun (cid:105) , sp ( f ).To conclude, we use the result for classical affinoid algebras on Spa R (cid:68) u n (cid:36) E (cid:69) . SinceSpa R (cid:10) (cid:36) E u n (cid:11) ∪ Spa R (cid:68) u n (cid:36) E (cid:69) is a cover of Spa R , the result follows. (cid:3) A.2.
Analytic loci of formal schemes.
Recall that a point of a pre-adic spaceSpa(
A, A + ) is said to be analytic if the kernel of the corresponding valuation is notopen. We can describe the analytic locus in certain O E -formal schemes as explicitpseudorigid spaces, following [dJ95]. Let X = Spf A be a noetherian affine formalscheme over O E , and let X = Spa A be to corresponding adic space; let I ⊂ A bethe ideal of topologically nilpoent elements, and assume in addition that A/I is afinitely generated k E -algebra.If I = ( f , . . . , f r ), then X an = ∪ i X (cid:68) If i (cid:69) . In the special case when A = R is thering of definition of a O E -pseudoaffinoid algebra R , we have X an = Spa R .As in [dJ95, Proposition 7.1.7], we can give a functor-of-points characterization of X an : Proposition A.2.1.
Let X and X be as above, and let Y := Spa R be an affinoidpseudorigid space. Then (A.2.1) lim −→ R ⊂ R ring of definition Hom FS / O E (Spf R , X ) ∼ −→ Hom(
Y, X an ) is an isomorphism.Proof. Given any two rings of definition of R , there is a third which contains bothof them. Moreover, suppose R ⊂ R is a ring of definition, and g : R → R (cid:48) isa continuous homomorphism of pseudoaffinoid algebras. By [Hub93, Proposition1.10], g is adic; it therefore carries R ⊂ R to a ring of definition of R (cid:48) .Thus, for a fixed X , we can view lim −→ R ⊂ R ring of definition Hom FS / O E (Spf R , X ) as acovariant functor evaluated on R , and equation A.2.1 as a natural transformation.We will construct an inverse. Suppose we have a morphism Y → X an ; it is inducedby a continuous ring homomorphism g : A → R . Using the description of the OHOMOLOGY OF ( ϕ, Γ)-MODULES OVER PSEUDORIGID SPACES 33 analytic locus of X and the quasi-compactness of Y , we see that the image of Y iscontained in U := ∪ i U i , where U i := X (cid:68) If i (cid:69) , for some finite set { f i } ⊂ I .Let V i ⊂ Y denote the rational subset Spa R (cid:68) g ( I ) g ( f i ) (cid:69) , and let R i denote its coordi-nate ring. Each morphism V i → U i is induced by a continuous ring homomorphism A (cid:68) If i (cid:69) → R ,i , for some ring of definition R ,i ⊂ R i .Let R ⊂ R ◦ be the equalizer of (cid:81) i R ,i ⇒ (cid:81) i,j R ◦ i,j ; we claim that the map A → Γ( U, O X ) → R factors through R , and R is a ring of definition of R . Forthe first claim, we consider the diagram0 Γ( U, O X ) (cid:81) i Γ( U i , O X ) (cid:81) i,j Γ( U i,j , O X )0 R (cid:81) i R ,i R ◦ (cid:81) i R ◦ i (cid:81) i,j R ◦ i,j It is commutative with exact rows, and a diagram chase shows that the dottedarrow exists.For the second claim, let u ∈ R be a pseudouniformizer; we check that R [ u − ] = R and R is a bounded subring of R . Given r ∈ R , we may write (cid:81) i r i for its imagein (cid:81) i R i . Since R ,i ⊂ R i is a ring of definition, there is some n ∈ N such that u n ( (cid:81) i r i ) ∈ (cid:81) i R ,i ; by construction, u n ( (cid:81) i r i ) is in the kernel of (cid:81) i R ,i ⇒ (cid:81) i,j R ◦ i,j , so it defines the desired element of R .To see that R ⊂ R is bounded, we let R (cid:48) ⊂ R be a ring of definition. It inducesrings of definition R (cid:48) ,i ⊂ R i ; since R ,i ⊂ R i is bounded, there is some n (cid:48) ∈ N suchthat u n (cid:48) (cid:81) i R ,i ⊂ (cid:81) i R (cid:48) ,i , and a diagram chase shows that u n (cid:48) R ⊂ R (cid:48) .We have constructed a continuous homomorphism A → R inducing the morphism Y → X an , where R ⊂ R is a ring of definition. The corresponding morphismSpf R → Spf A is the desired element of the left side of equation A.2.1. It isstraightforward to verify that this defines a natural transformation. (cid:3) Appendix B. Pseudorigid determinants
We need to extend some of the results of [Che14] on moduli spaces of Galois deter-minants from the rigid analytic setting to the pseudorigid setting. Recall that forany topological group G and d ∈ N , [Che14] defines functors (cid:101) E d : FS / Z p → Setand (cid:101) E d,z : FS / Z p → Set on the category of formal schemes over Z p , where z is a d -dimensional determinant G → k for some finite field k . More precisely, (cid:101) E d ( X ) := { continuous determinants O ( X )[ G ] → O ( X ) of dimension d } and (cid:101) E d,z ( X ) ⊂ (cid:101) E d ( X ) is the subset of continuous determinants which are residuallyconstant and equal to z .Suppose that G is a topological group satisfying the following property:For any open subgroup H ⊂ G , there are only finitely many continuous grouphomomorphisms H → Z /p Z . Under this condition (which is satisfied by absolute Galois groups of characteristic0 local fields, and by groups Gal
F,S , where F is a number field and S is a finite setof places of F ), [Che14, Corollary 3.14] implies that (cid:101) E d and (cid:101) E d,z are representable.Moreover, every continuous determinant is residually locally constant, and so (cid:101) E d = (cid:96) z (cid:101) E d,z .We may define an analogous functor (cid:101) E an d on the category of pseudorigid spaces,and we wish to prove the following: Theorem B.0.1.
The functor (cid:101) E an d is representable by a pseudorigid space X d ,and X d is canonically isomorphic to the analytic locus of (cid:101) E d . The functor (cid:101) E an d,z isrepresentable by a pseudorigid space X d,z , and X d,z is canonically isomorphic tothe analytic locus of (cid:101) E d,z . Moreover, (cid:101) E an d is the disjoint union of the (cid:101) E an d,z . Remark B.0.2.
This is a direct analogue of [Che14, Theorem 3.17], and the proofis virtually identical. However, we sketch it here for the convenience of the reader.If R is a pseudoaffinoid algebra, it contains a noetherian ring of definition R ⊂ R ,and we have R ◦ = lim −→ R , where R ◦ ⊂ R is the subring of power-bounded elementsof R and the colimit is taken over all rings of definition of R . We have an injectivemap ι : lim −→ R ⊂ R ring of definition (cid:101) E (Spf R ) → (cid:101) E an ( R )Exactly as in [Che14], we have the following: Lemma B.0.3.
Let R be a pseudoaffinoid algebra, and let D ∈ (cid:101) E an d ( R ) . Then (1) For all g ∈ G , the coefficients of D (1 + gt ) ∈ R [ t ] lie in R ◦ . (2) The map ι : lim −→ R ⊂ R (cid:101) E (Spf R ) → (cid:101) E an ( R ) is bijective. (3) If R is reduced, then (cid:101) E d (Spf R ◦ ) = (cid:101) E an d ( R ) . In particular, if L is the residue field at a maximal point of Spa R and O L is its ringof integers, (cid:101) E d (Spf O L ) = (cid:101) E an d ( L ). Every L -valued point of Spa R therefore definesa map (cid:101) E d (Spa R ) → (cid:101) E d ( L ) = (cid:101) E d (Spf O L ) → (cid:101) E d ( k L )where k L is the residue field of O L .Thus, we may talk about residual determinants of determinants R [ G ] → R , anddefine (cid:101) E an d,z for any continuous determinant z valued in a finite field.Now the proof of Theorem B.0.1 follows by combining Proposition A.2.1 andLemma B.0.3.For the convenience of the reader, we record the following analogue of [BC09,Lemma 7.8.11]; its proof carries over verbatim to the setting of pseudorigid familiesof determinants. Lemma B.0.4.
Let D : G → O ( X ) be a continuous d -dimension determinant of atopological group on a reduced pseudorigid space X . Let U ⊂ X be an open affinoid. (1) There is a normal affinoid Y , a finite dominant map g : Y → U , anda finite-type torsion-free O ( Y ) -module M ( Y ) of generic rank d equipped OHOMOLOGY OF ( ϕ, Γ)-MODULES OVER PSEUDORIGID SPACES 35 with a continuous representation ρ Y : G → GL O ( Y ) , whose determinant atgeneric points of Y agrees with g ∗ D .Moreover, ρ Y is generically semisimple and the sum of absolutely ir-reducible representations. For y in a dense Zariski-open subset Y (cid:48) ⊂ Y , M ( Y ) y is free of rank- d over O y , and M ( Y ) y ⊗ k ( y ) is semisimple andisomorphic to ρ g ( y ) . (2) There is a blow-up g (cid:48) : Y → Y of a closed subset of Y (cid:114) Y (cid:48) such thatthe strict transform M Y of the coherent sheaf on Y associated to M ( Y ) is a locally free O Y -module of rank d . That sheaf M Y is equipped with acontinuous O Y -representation of G with determinant ( g (cid:48) g ) (cid:93) ◦ D , and for all y ∈ Y , ( M Y,y ⊗ k ( y )) ss is isomorphic to ρ g (cid:48) g ( y ) . References [AIP18] Fabrizio Andreatta, Adrian Iovita, and Vincent Pilloni,
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