Algebraic families of Galois representations and potentially semi-stable pseudodeformation rings

We construct and study the moduli of continuous representations of a profinite group with integral $p$-adic coefficients. We present this moduli space over the moduli space of continuous pseudorepresentations and show that this morphism is algebraizable. When this profinite group is the absolute Galois group of a $p$-adic local field, we show that these moduli spaces admit Zariski-closed loci cutting out Galois representations that are potentially semi-stable with bounded Hodge-Tate weights and a given Hodge and Galois type. As a consequence, we show that these loci descend to the universal deformation ring of the corresponding pseudorepresentation.

1. Introduction 1.1. Overview. Mazur [Maz89] initiated the systematic study of the moduli of representations of a Galois group G in terms of complete local deformation rings. For a fixed residual representationρ with coefficients in the finite residue field F, which admits a universal deformation ring Rρ, the resulting moduli space Spf Rρ is "purely formal" in the sense that the underlying algebraic scheme Spec F is 0dimensional. These deformation rings have been studied extensively in recent years, playing a significant role in automorphy lifting theorems.
In contrast, the moduli of Galois representations that are not purely formal, i.e. positive-dimensional algebraic families of residual representations, have been somewhat neglected. They do appear implicitly in the work of Skinner-Wiles [SW99] and Bellaïche-Chenevier [BC09]. The space Ext 1 G (ρ 2 ,ρ 1 ) of extensions of two distinct irreducible residual representationsρ 1 ,ρ 2 of G (1.1.1) ρ 1 * 0ρ 2 is the most basic example of such a residual family. The goal of this paper is to set up a general theory of families of Galois representations and to show that conditions from p-adic Hodge theory may be sensibly imposed on them. The ad hoc use of these ideas in [SW99] and [BC09] suggests that these spaces should have applications to modularity lifting theorems, and the study of Selmer groups.
To state our first main result, recall that a pseudorepresentation of G is the data of a polynomial for each element of G, satisfying coherence and continuity conditions one expects from characteristic polynomials of a representation. Chenevier [Che14] has shown that a residual pseudorepresentationD of G admits only formal deformations, and that these are parameterized by a universal deformation ring (RD, mD). Let RepD denote the groupoid which attaches to any quotient B of Z p [[t 1 , . . . , t n ]] z 1 , . . . , z m , the category of locally free B-modules V B equipped with a continuous linear action ρ B : G → Aut B (V B ) having residual pseudorepresenta-tionD.
Writeψ(ρ B ) for the pseudorepresentation induced by a representation (V B , ρ B ). Denote by ρ ss D the unique semi-simple representation such thatψ(ρ ss D ) =D. We say thatD multiplicity-free if ρ ss D has no multiplicity among its simple factors. Theorem A (Theorem 3.2.4). If G satisfies Mazur's finiteness condition Φ p , then RD is Noetherian andψ : RepD −→ Spf RD is a formally finite type Spf RD-formal algebraic stack. Moreover, (1) RepD arises as the mD-adic completion of a finite type Spec RD-algebraic stack ψ : RepD −→ Spec RD.
(2) The defect ν : Spec ψ * (O RepD ) → Spec RD between the GIT quotient and the pseudodeformation space is a finite universal homeomorphism which is an isomorphism in characteristic zero. In particular, ψ is universally closed.
(3) IfD is multiplicity-free, then RD is precisely the GIT quotient ring.
The theorem should be compared with the result thatψ is an isomorphism when ρ ss D is absolutely irreducible. The fact thatψ is algebraizable may be thought of as an interpolation of the algebraicity of each fiber ofψ over a pseudorepresentation D. This fiber consists of the representations with semi-simplification isomorphic to ρ ss D , and is naturally algebraic as in (1.1.1) above. The proof of Theorem A uses the existence of a "universal Cayley-Hamilton algebra" whose representations naturally factor the continuous representations of RepD. The theorem then follows from results on the moduli of representations of finitely generated algebras.
Having constructed and algebraized these families, we prove that it is possible to impose conditions from p-adic Hodge theory on them, namely that they are potentially semi-stable with a given Hodge type v and Galois type τ in the sense of [Kis08]. Let K be a finite extension of Q p , with Galois group G = G K . We will refer to Rep D , which is the framed version of RepD (see §3).
Theorem B ( §6.4). Let τ and v be a fixed Galois and Hodge type. There exists a closed formal substack Rep τ,v D ֒→ RepD such that for any finite Q p -algebra B and point ζ : Spec B → RepD, ζ factors through Rep τ,v D if and only if the corresponding representation V B of G K is potentially semi-stable of Hodge and Galois type (τ, v). Moreover, (1) Rep ,τ,v D [1/p] is reduced, locally complete intersection, equi-dimensional, and generically formally smooth over Q p . If we replace "semi-stable" with "crystalline," it is everywhere formally smooth over Q p .
(2) IfD is multiplicity-free, then Rep τ,v D is algebraizible, i.e. Rep τ,v D is the completion of a closed substack Rep τ,v D of RepD. The geometric properties of (1) also apply to Rep τ,v D [1/p], except equi-dimensionality, which applies to its framed version Rep ,τ,v D [1/p].
One may also produce the RD-algebraic closed substack Rep τ,v D without any condition onD assuming an algebraization conjecture for ψ (see §3.3).
We emphasize that the methods to cut out these loci of representations are due to Kisin [Kis08,§1] in the case of complete local coefficient rings, and that we adapt his arguments to hold over more general coefficient schemes. The geometric properties of this space are deduced from existing results on the ring-theoretic properties of equi-characteristic zero potentially semi-stable deformation rings, principally [Kis08,§3] and [Bel14].
The fact that ψ is algebraic of finite type and universally closed can be used to produce a potentially semi-stable pseudodeformation ring. A pseudorepresentation D valued in a p-adic field E will be said to satisfy a condition applying to representations when the associated semi-simple representation ρ ss D satisfies this condition (see Definition 7.1.1).
Theorem C ( §7.1). Assuming that the algebraization Rep τ,v D of Rep τ,v D exists, the scheme-theoretic image of Rep τ,v D under ψ corresponds to a quotient of R τ,v D of RD which satisfies the following property: for any finite extension E/Q p and any point z : Spec E → Spec RD, the corresponding pseudorepresentation D z : G K → E is potentially semi-stable of Galois and Hodge type (τ, v) if and only if z factors through R τ,v D . Moreover, (1) R τ,v := R τ,v D [1/p] is reduced for any (τ, v) and does not depend on the choice of R τ,v D .
We remark that the ring-theoretic properties of the potentially semi-stable pseudodeformation rings in Theorem C are deduced from the geometric properties of the families of potentially semi-stable representations in Theorem B using invariant theory: Theorem A tells us that R τ,v is a GIT quotient ring. The conventional techniques used to study ring-theoretic properties of Galois deformation rings in terms of Galois cohomology do not seem directly applicable to study the pseudodeformation rings R τ,v D or R τ,v . Having shown that conditions from p-adic Hodge theory cut out a Zariski-closed condition on algebraic families of local Galois representations, we end the paper with a discussion of the corresponding constructions for families of global Galois representations, and pseudorepresentations. We remark that the correct notion of "a global pseudorepresentation that is locally potentially semi-stable" is more restrictive than "a global pseudorepresentation such that its restriction to each decomposition group over p is potentially semi-stable" This is well-illustrated through the explicit example of a 2-dimensional global ordinary pseudodeformation ring, which we discuss in §7.3.
1.2. Summary Outline. Section 2 discusses the geometry of the moduli spaces of d-dimensional representations Rep d R and pseudorepresentations PsR d R of an associative algebra R, especially with reference to the natural map ψ : Rep d R → PsR d R associating a representation to its induced pseudorepresentation. The main idea pursued is that the adjoint action of GL d on the scheme of framed representations Rep ,d R , whose associated quotient stack is Rep d R , has GIT quotient nearly equal to PsR d R . In order to establish this, we draw geometric and algebraic consequences of Chenevier's work on pseudorepresentations [Che14]. First, we establish that the GIT quotient and PsR d R naturally have identical geometric points because each set of geometric points naturally corresponds to isomorphism classes of semi-simple representations. We then introduce the notion of universal Cayley-Hamilton quotient, which factors the representations of R. Using the theory of polynomial identity rings to derive finiteness properties of the Cayley-Hamilton quotient, we show that the discrepancy between the GIT quotient and PsR d R is finite. To conclude section 2, we augment the theory of generalized matrix algebras of [BC09,§1.3] so that it functions well in arbitrarily small characteristic, attaching a canonical pseudorepresentation to a generalized matrix algebra. We also discover that when R is a generalized matrix algebra, the associated space of pseudorepresentations is precisely the GIT quotient.
In Section 3 we study the map ψ in the setting of continuous representation theory of a profinite group G, so that we take coefficients in formal schemes over Z p . The key result is Proposition 3.2.2, namely that the universal Cayley-Hamilton quotient E(G)D, which factors the representations of G with residual pseudorep-resentationD, is finite as a module over RD and that its adic topology as an RD-module is equivalent to the topology induced by G. Consequently, the moduli space of representations of E(G)D overD is a finite-type RD-algebraic model RepD for the formal moduli space RepD of representations of G with residual pseudorep-resentationD. Adding the results of section 2, we get Theorem A. We then discuss how Theorem A implies that formal GAGA holds for ψ in certain cases.
In Section 4 we begin our study of potentially semi-stable representations of G = G K . We adapt the methods of Kisin [Kis08,§1] to cut out a locus of representations with E-height ≤ h within the universal families Rep D . The the point of generalization is that coefficients must now be allowed to be quotients of RD z 1 , . . . , z a , while the coefficients rings were taken to be local in loc. cit. Along the way, we expand the allowable coefficients the theory of Fontaine [Fon90,§1.2], drawing an equivalence between continuous representations of G K∞ with arbitrary discrete coefficients, andétale ϕ-modules. The forthcoming work of Emerton and Gee will interpolate these families, as there exist families ofétale ϕ-modules larger than those that admit a Galois representation. We then construct a projective subscheme of an affine Grassmannian parameterizing lattices of E-height ≤ h (i.e. Kisin modules) in theétale ϕ-module, and produce a characteristic zero period map relating a family of G K∞ -representations to an family of Kisin modules.
Section 5 continues with the next part of Kisin's method [Kis08,§2], adding the data of an monodromy operator on the Kisin module side of the map, so that on the other side, we may descend from a G K∞ -action to a G K -action. The result is a closed locus cut out by the condition that the period map interpolates the usual period relation between semi-stable G K -representations and (ϕ, N )-modules.
Section 6 begins with the conclusion that this locus consists of exactly those G K -representations that are semi-stable with Hodge-Tate weights in [0, h]. Then, we cut out connected components corresponding to a given p-adic Hodge type or potentially semi-stability with a certain Galois type. The constructions required for Theorem B follow from applying the ideas above to universal families of representations and algebraizing these closed subschemes using formal GAGA for ψ. We finish with arguments deducing geometric properties of these spaces in characteristic 0 from existing results on their local rings at closed points, which are equi-characteristic zero potentially semi-stable Galois deformation rings.
In section 7, we apply Theorems A and B to cut out potentially semi-stable pseudodeformation rings as the scheme-theoretic image of the potentially semi-stable locus in RepD under ψ, proving Theorem C. Works of Alper [Alp08,Alp10] and Schoutens [Sch08] allow us to deduce the ring-theoretic properties of potentially crystalline pseudodeformation rings using invariant theory. We then discuss representations of the Galois group G F,S of a number field and cut out loci of representations and pseudorepresentations which are potentially semi-stable at decomposition groups over p. There are subtleties in this definition, which we illustrate through an example of ordinary pseudorepresentations.
1.3. Acknowledgements. We wish to thank Mark Kisin and recognize his influence in two capacities, firstly as the originator of the p-adic Hodge theoretic methods and ideas in this paper, and secondly for his suggestion to examine the geometric relationship between moduli spaces of Galois representations and moduli spaces of pseudorepresentations. This was begun in the author's Ph.D. thesis under his supervision. The influence of Gaëtan Chenevier will also be clear to the reader. It is also a pleasure to thank Brian Conrad, Barry Mazur, Gaëtan Chenevier, Joël Bellaïche, Preston Wake, Rebecca Bellovin, and David Zureick-Brown for helpful discussions related to this work. Part of this work was completed with support from the National Science Foundation in the form of a graduate research fellowship. They have our thanks. Finally, we are grateful for the support and hospitality of the mathematics departments at Harvard University and Brandeis University.

Moduli of Representations of a Finitely Generated Group or Algebra
Let A be a commutative Noetherian ring, let R be an associative but not necessarily commutative A-algebra, and let d ≥ 1 be an integer. We will often assume that R is finitely generated over A. For example, we may have R = A[G] for some finitely generated group G. We will study the moduli of d-dimensional representations of R relative to the space of d-dimensional pseudorepresentations of R, ultimately showing in Theorem 2.3.4 that they almost form an adequate moduli space when R is finitely generated. Later, in §3, we will apply this study to continuous representations of a profinite group.

Moduli Spaces of Representations and Pseudorepresentations. With
A, R, and d as above and S = Spec A, here are the moduli groupoids we will consider.
(1) Define the functor on S-schemes Rep ,d R by setting with the natural O X -linear, R-equivariant isomorphisms of such objects.
One can check that Rep ,d R is representable by an affine scheme which is finite type over S if R is finitely generated over A. It has been studied extensively, especially when A is an algebraically closed field of characteristic zero (see e.g. [LM85,Pro87] We will be interested in the geometry of Rep d R relative to the moduli space of d-dimensional pseudorepresentations of R. We will use the notion of pseudorepresentation due to Chenevier [Che14], following previous notions due to Wiles [Wil88] and Taylor [Tay91]. He uses the notion of a multiplicative polynomial law due to Roby [Rob63,Rob80].
A is a homogenous multiplicative polynomial law D : R → A, i.e. an association of each commutative A-algebra B to a function satisfying the following conditions: (1) D B is multiplicative and unit-preserving (but not necessarily additive), (3) D is functorial on A-algebras, i.e. for any commutative A-algebras B → B ′ , the diagram R is a functor on A-algebras. A pseudorepresentation may be thought of as an ensemble of characteristic polynomials, one for each element of R, satisfying compatibility properties as if they came from a representation of R. For r ∈ R, its characteristic polynomial Indeed, the χ(r, t) for r ∈ R characterize the pseudorepresentation [Che14, Lemma 1.12(ii)]. Any B-valued representation (V, ρ) ∈ Rep d R (B) of R induces a pseudorepresentation, denoted ψ(V ), given by composition of ρ : R ⊗ A B → End B (V ) with the determinant map det : End B (V ) → B. This is easily checked to be functorial in A-algebras and therefore defines a morphism There also exist analogous maps to PsR d R from Rep ,d R and Rep d R . The usual notion of characteristic polynomial of a representation coincides with the characteristic polynomial of the representation's induced pseudorepresentation.
Base changes of ψ have a natural interpretation as follows.
Definition 2.1.4. With R and A as above and any B-valued d-dimensional pseu- When R is finitely generated over A, Γ d A (R) ab is also finitely generated over A [Che14, Proposition 2.38].
The notion of a kernel of a pseudorepresentation provides a first step toward our goal of understanding ψ.
Definition 2.1.5. The kernel ker(D) of a pseudorepresentation D : R → A is a two-sided ideal of elements r ∈ R such that for all A-algebras B and all r ′ ∈ R⊗ A B, the characteristic polynomial See [Che14,§1.17] for further properties of the kernel, among them being the fact that the quotient algebra R/ ker(D) is the minimal quotient through which D factors. Moreover, in the case that A is an algebraically closed field, the surjection R → R/ ker(D) realizes the representation ρ ss D of the foliowing Theorem 2.1.6 ([Che14, Theorem A]). Letk be an algebraically closed field and let D : R →k be a d-dimensional pseudorepresentation. Then there exists a unique (up to isomorphism) d-dimensional semi-simple representation ρ ss D : R → M d (k) such that ψ(ρ ss D ) = D. From this theorem, we know that there is a unique semi-simple representation in each geometric fiber of ψ. Moreover, it is precisely the orbits (under the adjoint action of GL d ) of geometric points corresponding to semi-simple representations in Rep ,d R (k) which are closed orbits (cf. [Kra82, §II.4.5, Prop.]). It is equivalent to say that each fiber ψ −1 (D) of a geometric point D ∈ PsR d R (k) has a unique closed point corresponding to the representation ρ ss D . We summarize these facts: Corollary 2.1.7. If R is finitely generated over A, the morphism ψ : Rep d R → PsR d R induces an isomorphism of sets from the closed geometric points of Rep d R to the geometric points of PsR d R . The inverse map sends D ∈ PsR d R (k) to the semisimple representation ρ ss D , where the geometric points of the fiber ψ −1 (D) correspond to representations with Jordan-Hölder factors identical to ρ ss D . Next, we will refine this result past the level of geometric points using geometric invariant theory. Before that, we record this more precise version of Theorem 2.1.6, which is a corollary of Cheneiver's work, bounding the degree of field extension over which a field-valued pseudorepresentation splits, i.e. becomes realizable as the determinant of a representation.
Corollary 2.1.8 (of [Che14, Theorem 2.16]). Let k be a field and let R be a kalgebra with a d-dimensional pseudorepresentation D : R → k. Then R/ ker(D) is a semi-simple k-algebra and there exists a finite separable field extension k ′ /k of degree at most d such that there is a representation ρ : R/ ker(D) → M d×d (k ′ ) such that ψ(ρ) = D ⊗ k k ′ . Also, R/ ker(D) is finite-dimensional over k if any of the following conditions are satisfied.
The conditions (1) to (4) under which R/ ker(D) is finite-dimensional come from [Che14, Theorem 2.16], along with the fact that R/ ker(D) is semi-simple. We prove the remainder of the corollary.
Proof. For any field extension K/k, we write K s for the maximal separable subextension of k. We will adopt the notation and notions of [Che14, Theorem 2.16]. Namely, we know that R/ ker(D) ≃ s i=1 S i , where S i is a simple k-algebra with center k i and dimension n 2 i over k i , where (f i , q i ) is the exponent of the field extension k ′ /k, i.e. f i = [k ′s : k]. Given any simple k-algebra S i , Chenevier shows that any pseudorepresentation S i → k is an integer power of a canonical n i f i q i -dimensional pseudorepresentation denoted det Si : S i → k. In fact, S i is the reduced norm S i → k i followed by the q i -Frobenius map k i → k s i followed by the Galois norm Let k ′ /k denote the maximal separable subfield extension of the composite field of all of the k ′ i . Because each k i /k has finite separable degree f i , k ′ /k is a finite separable field extension. Then we have and it follows that One can check that det Si ⊗ k k ′ i s on (2.1.9) is the product over each of the Galois factors of the usual determinant on the matrix algebra followed by the q i -Frobenius map. Recalling the decomposition R/ ker(D) ≃ s i=1 S i , we observe that the natural product of these maps evaluated on R/ ker(D) is It can be useful to take the perspective that a Cayley-Hamilton algebra is a generalization of a matrix algebra, and to consider Cayley-Hamilton algebra-valued representations. For instance, Procesi proved that in equi-characteristic 0, any Cayley-Hamilton algebra admits an embedding into a matrix algebra [Pro87]. When we take the Cayley-Hamilton algebra produced out of the universal d-dimensional , we can get a "universal Cayley-Hamilton algebra" (E(R) d , D u ) and "universal Cayley-Hamilton representation" ρ u : to the moduli of representations of E(R) d compatible with D u (see Definition 2.1.4). Now we will discuss polynomial identity rings, written PI-rings; we refer to the book [Pro73] for the precise definition of a polynomial identity ring. It will suffice to say that an associative ring R is called a polynomial identity ring when there exists some non-commutative polynomial in n variables that every n-tuple in R ×n satisfies. For example, every commutative ring R is a polynomial identity ring because any x, y ∈ R satisfy the equation xy − yx = 0.
Proposition 2.2.4. If (R, D) is a Cayley-Hamilton A-algebra, it is a PI-A-algebra with polynomial identity dependent only on the dimension of D. If, in addition, R is finitely generated over the Noetherian ring A, R is finite as an A-module.
Proof. By [Pro73, Proposition 3.22], given any d ∈ Z ≥1 , there is an explicit polynomial identity with coefficients in Z such that any associative A-algebra R that is integral over A with degree bounded by d is a PI-A-algebra with this particular polynomial identity. Consequently, any Cayley-Hamilton A-algebra (R, D) is a PI-A-algebra because any element of R is integral over A of degree bounded by d = dim(D); indeed, χ(r, r) = 0. By [Pro73, Theorem 2.7], any integral, finitely generated non-commutative PI-algebra over a commutative Noetherian ring is module-finite.
Consequently, such R is Noetherian, finite as a module over its center, and Jacobson when A is Jacobson. Remarkably, this proposition along with Proposition 2.2.3 implies that the study of d-dimensional representation theory of a finitely generated non-commutative A-algebra R amounts to the study of representations of a certain module-finite algebra over a Noetherian ring.
Here are some results from PI-theory that will be useful even in the infinitely generated cases we will study later, namely group algebras of profinite groups.
Proposition 2.2.5. Let A = k be field and let R be an associative (non-unital) k-algebra satisfying the polynomial identity x d , i.e. every element of R is nilpotent of degree at most d ∈ Z ≥1 . Then there exists some N = N (d) ∈ Z ≥1 depending only upon d such that R is nilpotent of degree N , i.e. R N = 0.
Proof. When char(k) = 0 or char(k) > d, the Nagata-Higman theorem states that On the other hand, the main theorem of [Sam09] states that if char(k) = p > 0, then there exists an integer N = N (p, d) depending only on p and d such that R N = 0. Combining these two results, we may set The work of Samoilov is the key input needed to loosen conditions needed to guarantee that a deformation ring of pseudorepresentations of a profinite group is Noetherian (see Proposition 3.1.3). It will be used in the form of the following Corollary 2.2.6. Given a positive integer d, there exists a positive integer N (d) with the following property. Let k be a field and let (R, D) be a Cayley-Hamilton k-algebra of degree d (which may not be finitely generated over k). Then the kernel ker(D) ⊂ R is nilpotent of order N (d).
Proof. The definition of ker(D) implies that its every element r ∈ ker(D) has characteristic polynomial χ(r, t) = t d , and because (R, D) is Cayley-Hamilton we have that χ(r, r) = r d = 0. Then Proposition 2.2.5 implies the result.
2.3. Invariant Theory. For this paragraph, we will assume that R is finitely generated over the Noetherian commutative ring A so that Rep d R and PsR d R are finite type over S = Spec A. The fact that ψ : Rep d R → PsR d R is a bijection on closed geometric points suggests a comparison between PsR d R and the geometric invariant theoretic (GIT) quotient. Definition 2.3.1. The GIT quotient of the action of an affine algebraic S-group scheme G on an affine S-scheme X = Spec B, written X//G, is given by X//G := Spec B G .
The work of Alper [Alp08,Alp10], generalizing the classical geometric invariant theory of Mumford, provides a useful perspective on invariant theory that is appropriate for our use. We will refer to loc. cit. for the definitions of adequate and good moduli spaces, since for our purposes, the following examples of adequate and good moduli spaces suffice.
(1) Let G be a reductive S-group scheme acting on an affine S-scheme X. Then the natural morphism from the quotient stack to the GIT quotient [X/G] → X//G is an example of an adequate moduli space.
(2) When G is linearly reductive, then [X/G] → X//G is an example of a good moduli space.
We will be interested in the particular case of the adequate moduli space Here are the main results of Alper's work. Following Alper, we call an integral universal homeomorphism that is an isomorphism in characteristic zero an adequate homeomorphism. Alp08]). Let φ : X → Y be an adequate moduli space.
(2) Two geometric points x 1 , x 2 ∈ X (k) are identified in Y if and only if their closures {x 1 } and {x 2 } intersect in X × Zk . (3) If X is finite type over a Noetherian scheme S, then Y is finite type over S and for every coherent O X -module F , φ * F is coherent. (4) φ is universal for maps from X to algebraic spaces which are either locally separated or Zariski-locally have affine diagonal.
Combining Corollary 2.1.7 with part (2) of Theorem 2.3.3, we find that ν induces an isomorphism on geometric points. It is the same to say that ν is surjective and radicial [Gro60, 3.5.5]. What we will show is that PsR d R differs from the GIT quotient by at most an adequate homeomorphism, i.e. the difference between the two rings are finitely many p-torsion nilpotents. It is possible to eliminate this difference in certain cases (see Theorem 2.4.10).
Theorem 2.3.4. If R is finitely generated over A, ν is an adequate homeomorphism, i.e. is an integral universal homeomorphism which is an isomorphism in characteristic zero. That is, ψ : Rep d R → PsR d R differs from an adequate moduli space by at most an adequate homeomorphism.
We emphasize that the isomorphism in characteristic zero is due to Chenevier, using ideas of Procesi [Pro87].
Proof. The proof that ψ is precisely an adequate moduli space in equi-characteristic zero is due to Chenevier [Che13, Proposition 2.3]. We know that ν is surjective and radicial by the comments above, so in light of [Gro67, Corollaire 18.12.11], it remains to show that ν is finite. We will prove this by verifying the valuative criterion for universally closed morphisms given in [LMB00, Theorem 7.10] (see also [Gro61a, Remark 7.3.9(i)]).
Let B represent a complete discrete valuation ring with an algebraically closed residue field and fraction field K. Given a diagram of A-schemes is a finite extension of fields, and whose induced pseudorepresentation det •ρ is identical to D ⊗ K K ′ .
Let B ′ be the integral closure of B in K ′ . We claim that ρ is isomorphic to and L ′ is a rank d projective B ′module, which will complete the proof. Choose a d-dimensional K ′ -vector space V ′ realizing ρ, and let L be a B ′ -lattice L ⊂ V ′ . Now let L ′ be the B ′ -linear span of the translates of L by R ⊗ A B. This is a finite B ′ -module because the action of R ⊗ A B factors through its Cayley-Hamilton quotient (R ⊗ A B)/CH(D ⊗ A B) by Proposition 2.2.3, and this quotient is B-module-finite by Proposition 2.2.4. Therefore L ′ is a B ′ -lattice because it is finite and torsion-free, and the induced 2.4. Generalized Matrix Algebras. The concept of a generalized matrix algebra (GMA) with respect to a pseudocharacter has been carefully studied in [BC09,§1]. It will be helpful in what follows to develop the notion of GMA relative to a pseudorepresentation in order to eliminate complications with pseudocharacters arising in small characteristic. In particular, this will allow us to adapt the theory of GMAs to characteristic smaller than the dimension. However, no change to the definition of the GMA is necessary: we will show that a GMA admits a canonical pseudorepresentation. This was also shown independently by Ann-Kristin Juschka, following the suggestion of [WE13, Remark 2.3.3.6].
A pseudocharacter is the data of a trace coefficient function Λ 1 satisfying identities expected of a trace function coming from a representation (see [Tay91], [ (1) A set of r orthogonal idempotents e 1 , . . . , e r with sum 1, and (2) A set of isomorphisms of A-algebras φ i : such that the trace map Tr = Tr E : R → A defined by is a central function, i.e. Tr(xy) = Tr(yx) for all x, y ∈ R. We call E the data of idempotents of R and write (R, E) for a GMA.
We note that all of the arguments of [BC09, §1.3.1- §1.3.6] have no dependence on the characteristic of A or the invertibility of d! in A, except the proof that the trace map Tr associated to E is a pseudocharacter. Therefore, we have access to these results of [BC09,§1] on the structure of a GMA, which we record here in order to introduce notation.
• We fix notation δ j,k ∈ M d (A) for the matrix with entries 0 except in the (j, k)th entry, where the value is 1.
• we also write E l for the lth primitive idempotent of R given by the order down the diagonal of the idempotents E 1 • Likewise, write A i,j := E i RE j , and write ϕ i,j,k for the map A i,j ⊗ A A j,k → A i,k induced by multiplication, where 1 ≤ i, j, k ≤ d, which will also satisfy (UNIT), (COM), and (ASSO).
We define Rep Ad (R, E) to be the functor associating an A-algebra B to the set of adapted representations of (R, E) over B.
By [BC09, Proposition 1.3.9], Rep Ad (R, E) is represented by the affine scheme corresponding to the quotient of the A-algebra These are precisely the relations required to ensure that the morphism of A-modules R → M d (B) induced by the A-algebra homomorphism T → B along with (2.4.2) is actually a morphism of A-algebras. Then, the universal adapted representation f : With the above notions in place, we are equipped to introduce a canonical pseudorepresentation associated to a GMA.
1 , namely the pseudorepresentation given by, for any commutative A-algebra B, the formula Here, the product is first over the cycles γ of σ and then over the elements l of the cycle taken in the order that they appear in the cycle, where k is a choice of initial element of γ.
Proof. It is clear that we have a homogenous degree d polynomial law D E : R → A, and it will be a pseudorepresentation if it is multiplicative. This follows from the fact that, by inspection of the definition of D E , the injection f : R ֒→ M d (T /J) satisfies D E = det •f . These maps remain injective after any base extension ⊗ A B because the injections A i,j → T /J are split. Therefore, the determinant is a multiplicative homogenous degree d polynomial law, i.e. a pseudorepresentation. One may check that Tr E = Λ DE 1 by computing the characteristic polynomial χ DE (r, t). We now verify that D E does not depend upon the choice of initial element k in each cycle γ composing σ. This follows from the property (COM) of the multiplication maps ϕ deduced from the centrality of Tr E in [BC09, Lemma 1.3.5], which reads as follows: Therefore, for any σ ∈ S d , cycle γ of σ, and k ∈ γ at which we will begin the multiplication, we have that where we apply (COM) in the central equality.
The proof shows that the determinant of the universal adapted representation R → M d (T /J) is compatible with the pseudorepresentation D E : R → A induced by the GMA structure (R, E). Consequently, we have a monomorphism Rep Ad (R, E) ֒→ Rep R,DE induced by forgetting the adaptation; it may be easily checked to be a closed immersion.
Considering the adjoint action of GL d on framed representations, the stabilizer subgroup of an adaptation is the center Z(E) of the diagonally embedded subgroup compatibly with the action of GL d on Rep R,DE via the immersion above. This means that the morphism (2.4.7) exists, and, furthermore, we show the following.
Proposition 2.4.6. Given a GMA (R, E) over A, the natural morphism and the fact that the pseudorepresentation induced by V X lies over D E , we see that the action of e i Re i on V i is faithful and induces an isomorphism We observe that this provides a quasi-inverse to (2.4.7).
In the case of a generalized matrix algebra, we can improve on Theorem 2.3.4.
Corollary 2.4.8. Let (R, E) be a generalized matrix A-algebra with canonical pseudorepresentation D E : R → A. Then Rep R,DE → Spec A is a good moduli space. In particular, it is an adequate moduli space.
Proof. It is clear that the action of Z(β) on the coordinate ring of Rep Ad (R, E) leaves A as its invariant subring. Indeed, consider its form as described in (2.4.4), and that Z(β) acts on A i,j as the torus in GL r acts on its roots. In light of Example 2.3.2, Rep Ad (R, E) → Spec A is a good moduli space because tori are linearly reductive over any base. Then apply Proposition 2.4.6.
The following conditions will be useful to show that certain Cayley-Hamilton algebras are GMAs.
Definition 2.4.9. Let (A, m A ) be a local ring with the usual data D : R → A and residue field F := A/m A .
(1) We denote byD the residual representation D ⊗ A F : R ⊗ A F → F, and call D split and D residually split if (R ⊗ A F)/ ker(D) is a product of matrix algebras.
(2) We callD multiplicity-free and call D residually multiplicity-free whenD is split and the semi-simple representation ρ ss D : R ⊗ A F → M d (F) has distinct Jordan-Hölder factors.
Recall from Corollary 2.1.8 that (R ⊗ A F)/ ker(D) is a semi-simple F-algebra and is split after at most a finiteétale extension.
Chenevier has shown that in a certain case, a Cayley-Hamilton algebra (R, D) may be endowed with the structure of a GMA (R, E) such that the pseudocharacter induced by E is equal to the trace Λ D 1 of D [Che14, Theorem 2.22(ii)]. We will now remark that his proof also shows that the pseudorepresentation D E is equal to D, generalizing [BC09, Corollary 1.3.16] to any characteristic.
Theorem 2.4.10. Let (R, D) be a finitely generated Cayley-Hamilton A-algebra where A is a Noetherian Henselian local ring. Assume that D is residually split and multiplicity-free. Then (R, D) admits a structure E of a generalized matrix Aalgebra such that the pseudorepresentation induced by (R, E) is equal to D. Moreover, there is an isomorphism Proof. Chenevier's proof [Che14, Theorem 2.22] shows that there exists a data of idempotents E of R inducing a generalized matrix algebra (R, E) over A. As in the proof of Proposition 2.4.6, to any object Because By Proposition 2.2.4, R is finite as an A-module, and by construction C is finite as an A-module as well. Therefore the idempotent of C/m A C corresponding tof 1 (andf 2 ) has a unique lift, which must induce the maps f 1 and f 2 , showing that The theorem now follows directly from Proposition 2.4.6 and Corollary 2.4.8.

Formal Moduli of Representations and Pseudorepresentations of a
Profinite Group. Let G be a profinite group; we will often impose the Φ pfiniteness condition of Mazur [Maz89] on G. We wish to understand the moduli space of continuous representations of G and how it relates to the moduli space of continuous pseudorepresentations. We will study these representations over integral p-adic coefficient rings for some prime p which we fix. We will not insist that these rings are local because of the existence of positive-dimensional algebraic families of residual representations such as the one-dimensional family whereẽ 1 ,ẽ 2 are representatives of linearly-independent extension classes e 1 , e 2 ∈ Ext 1 G (ρ 2 ,ρ 1 ). Namely, we will let our category of coefficient rings be admissible continuous Z p -algebras Adm Zp , which is anti-equivalent to the category of affine Noetherian Spf Z p -formal schemes [Gro60, §10.1]. We will use the category of Spf Z p -formal schemes F S Zp as coefficient spaces.
As in the previous section, we will use A as a base coefficient ring, and will let A ∈ Adm Zp be a local ring, so that A is profinite and, in particular, has a finite residue field. Let R be a profinite continuous (non-commutative) A-algebra. When we wish to consider the case of group representations, we may set A = Z p and Definition 3.1.2. Let A and R be as specified above, and let d be a positive integer.
(1) Define the functor and morphisms being isomorphisms of this data.
and morphisms being isomorphisms of this data.
It is not difficult to show that Rep ,d R is representable by an affine Noetherian Spf A-formal scheme when R satisfies a finiteness condition equivalent to the Φ p finiteness condition on profinite groups. But we will first show that all of these groupoids are algebraizable, from which their representability by formal schemes or formal algebraic stacks follows.
The moduli functor of continuous pseudorepresentations of a profinite algebra has been defined and studied by Chenevier [Che14]. Firstly, he shows in [Che14, Proposition 3.3] that given any finite field-valued pseudorepresentationD : R ⊗ A F → F, the natural deformation functor PsRD to complete local A-algebras with residue field F is representable by a complete local A-algebra (RD, mD), i.e. PsRD ∼ = Spf RD. We call the objects of PsRD "pseudodeformations." When certain finiteness conditions are satisfied, RD is Noetherian. Here we slightly generalize Chenevier's criterion [Che14, Proposition 3.7].
Proposition 3.1.3. Assume that the continuous cohomology group H 1 c (G, ad ρ ss D ) is finite dimensional over the coefficient field of ρ ss D . Then RD is Noetherian. Proof. Let F represent the coefficient field of ρ ss D . We apply the proof strategy of Chenevier [Che14, 2.7, 2.26, 2.35], except that we remove the condition that d! is invertible in F using Corollary 2.2.6. This deformation theory will suffice to describe the entire moduli functor of d-dimensional pseudorepresentations on formal schemes X ∈ F S Zp , will be helpful to establish notation about residual representations and their fields of definition. There is a natural equivalence relation on continuous d-dimensional finite field-valued pseudorepresentations of R, namely thatD ∼D ′ for pseudorep-resentationsD : There is a unique representative of each equivalence class with smallest field of definition.
Definition 3.1.5. We will letD ′ : R ⊗ A FD′ → FD′ represent a residual representation, which is the special representative of each equivalence class of finite field-valued pseudorepresentations as described above. We also write FD for the minimal extension of FD′ over which a semi-simple representation (V ss D , ρ ss D ) induc-ingD ′ exists, and callD :=D ′ ⊗ FD′ FD a split residual pseudorepresentation.
Chenevier has shown that the entire moduli of pseudorepresentations is simply a disjoint union of deformation functors of residual representations.
Theorem 3.1.6 ([Che14, Corollary 3.14]). Assuming that G satisfies the Φ p finiteness condition and of local Noetherian formal schemes over the set of d-dimensional residual pseudorepresentations.
That is, the moduli of continuous pseudorepresentations of a profinite group is "purely formal," i.e. semi-local, unlike the moduli of its representations. This means that all non-trivial positive-dimensional algebraic families of residual representations consist of varying extension classes with a fixed semi-simplification, with (3.1.1) as the archetypal example.
Because continuous representations induce continuous pseudorepresentations, there exists a natural morphism from each of the moduli spaces of Definition 3.1.2 to PsR d R , for exampleψ : Using the decomposition of Theorem 3.1.6, we may study this morphism over one component of the base at the time. Fixing a split residual pseudorepresentationD : to be the groupoid of representations with residual pseudorepresentationD, and writeψ for the base changê We may analogously define Rep D and RepD.
Remark 3.1.7. We will carry out our computations and results using a split residual representation, since this is often required in order to state conditions and carry out our work. However, many of the results we will prove descend under the finité etale cover PsRD → PsRD′ although we will not remark on this.
It is well-known thatψ is an isomorphism whenD is absolutely irreducible, i.e. when ρ ss D is absolutely irreducible (see [Nys96,Rou96] in the case of pseudocharacters, and [Che14, Theorem 2.22(i)] for pseudorepresentations). In this case, RepD is purely formal.
3.2. Algebraization of Moduli of Representations over Moduli of Pseudorepresentations. Our goal is to draw a conclusion aboutψ similar to Theorem 2.3.4. As before, our principal tool will be the universal Cayley-Hamilton algebra. As nothing about its construction was particular to the finitely generated case, the universal continuous pseudodeformation ofD The image of this map is precisely the two-sided ideal generated by the image of the χ [α] , i.e. CH(D ū D ), and it is closed by the closed map lemma, proving (2). Consequently, the quotient map RD[[G]] ։Ẽ(G)D induces a quotient topology equivalent to the profinite topology.
We now work withĒ :=Ẽ(G)D ⊗ RD FD. We wish to show thatĒ is finitedimensional over FD. Firstly, Corollary 2.1.8(1) gives us thatĒ/ ker(D) is finitedimensional as a FD-vector space. BecauseĒ is a Cayley-Hamilton algebra over a field, we may also apply Proposition 2.2.5 to conclude that ker(D) ⊂Ē is nilpotent. Because of the natural surjection ker(D)/ ker(D) 2 ⊗i ։ ker(D) i / ker(D) i+1 induced by multiplication, it will suffice to show that ker(D)/ ker(D) 2 is finitedimensional. We now invoke the finiteness of H 1 c (G, ad ρ ss D ), which, by the argument of [Che14, Proposition 3.35], contains ker(D)/ ker(D) 2 as a sub-vector space.
The finiteness of the FD-dimension ofĒ along with the fact that E(G)D is clearly mD-adically separated implies that E(G)D is finite as a RD-module. This completes (3), from which (4) follows.
Observe that the image of the composite map Using these results on E(G)D, we get a result for continuous representations analogous to Proposition 2.2.3.
There is a functorial equivalence of categories between continuous representations of G with coefficients in Adm Zp with induced split residual pseudorepresentationD and continuous rep- There exists an inverse functor, associating to an admissible RD-algebra B with Now we have our main result, showing thatψ is algebraizable and then deducing its further properties from the study of §2. We write PsRD for Spec RD.
Following [Gro60, §10.13], we will call a homomorphism A → B in Adm Zp "formally finitely generated" if this map is compatible with a presentation of B as a quotient of a restricted power series ring A x 1 , . . . , x n , and use the term "formally finite type" to describe the corresponding morphisms of formal schemes.
Theorem 3.2.4. Assume that dim FD H 1 c (G, ad ρ ss D ) is finite. The groupoids RepD and RepD (resp. the functor Rep D ) over Spf Z p are representable by formally finite type formal algebraic stacks (resp. formally finite type formal scheme) over Spf RD which are algebraizable of finite type over Spec RD with algebraizations RepD : Consequently, we have a finite type morphism ψ : RepD → PsRD with the following properties.
(1) ψ is universally closed, (2) ψ has connected geometric fibers with a unique closed point corresponding to the unique semi-simple representation inducing the pseudorepresentation corresponding to the base of the fiber, (3) ψ consists of an adequate moduli space RepD → ψ * (O RepD ) followed by an adequate homeomorphism ν : ψ * (O RepD ) → Spec RD, and (4) ifD is multiplicity free, then ψ is precisely an adequate moduli space; moreover, it is a good moduli space.
The condition on ν in (3) means that there exist finitely many p-power torsion nilpotents Proof. Under the assumption that dim FD H 1 c (G, ad ρ ss D ) is finite, Propositions 3.1.3 and 3.2.2 tell us that RD is a Noetherian ring and that the RD-algebra E(G)D is finite as a RD-module and has a GMA structure compatible with D ū D whenD is multiplicity-free. Theorem 3.2.3 allows us to study representations of E(G)D in place of those of G. Then the existence and properties of each of the representation functors Rep D , RepD, RepD and ψ are the content of §2. More precisely, we have these statements for Rep d E(G)D over PsR d E(G)D , and the condition that the induced pseudorepresentation lies over D ū D simply cuts out a connected component (namely Spec RD) of the base PsR d E(G)D . In particular, we apply Theorem 2.4.10 to obtain part (4).
In order to see that the mD-adic completion of RepD is RepD (and similarly for RepD, Rep D ), we observe that all (non-topological) homomorphisms from RD to admissible Z p -algebras B or from E(G)D to End B (V B ) are automatically continuous, as E(G)D is a finite RD-module with the mD-adic topology.
Remark 3.2.5. The condition "constant residual pseudorepresentationD" is no real restriction to the scope of Theorem 3.2.4 in view of the bijective correspondence between d-dimensional residual pseudorepresentationsD and connected components of Rep d G . Putting together the connected components, we can say that there is an This algebraization result implies that the topology on an integral p-adic family V A of representations of G with coefficients in A ∈ Adm Zp and residual pseudorep-resentationD can always be strengthened to the mD-adic topology, and that there are may be proper subrings of A over which a model for V A exists. where A has the structure of a continuous RD-algebra induced by the pseudorepresentation det •ρ A , (2) there exists a canonical formally finitely generated sub-RD-algebra The corollary follows directly from the fact thatψ is formally finite type and algebraizable by the finite type morphism ψ, or, alternatively, directly from Proposition 3.2.2. In particular, this means that the matrix coefficients of a family of representations of G with residual pseudorepresentationD generate a finite type algebra over RD.
The algebraization theorem also suggests that there exists a notion of continuous D ū D -compatible representation of G valued in an arbitrary RD-algebra, i.e. not just those in Adm Zp . When such an algebra A is mD-adically separated, the mD-adic topology may be used, and the usual notions of continuity may be applied. On the other hand, there are common cases of concern where A is not mD-adically separated. For example, one often wants to consider continuous Galois representations with coefficients in Q p . However, Q p is will never be mD-adically separated because p ∈ mD is a unit in Q p , and the only topologies on Q p induced by ideals are the discrete topology and the trivial topology.  Also, notice that whenD is not absolutely irreducible, RepD is generally nonseparated. If O E is the ring of integers of a p-adic field E, then there may be multiple O E points inducing a single E-point, reflecting the existence of multiple isomorphism classes of G-stable O E -lattices in an E-valued representation.
Example 3.2.9. Letρ : G → GL d (F) be a residual representation with induced split residual pseudorepresentationD := det •ρ. Write Rρ for the versal deformation ring ofρ with versal representation Vρ. There exists a canonical map RD → Rρ, and Vρ is continuous with respect to the mD-adic topology on Rρ, which is often a strictly stronger topology than its native topology. Also, there exists a canonical, finite-type, mD-adically separated RD-subalgebra Rρ ,alg of Rρ with a canonical mD-adically continuous representation Vρ ,alg such that Vρ ≃ Vρ ,alg ⊗ Rρ ,alg Rρ.
Remark 3.2.10. The influence that Galois cohomology exerts on the structure of Rρ is well-understood. Analogously, appropriate Galois cohomology groups control the structure of RepD, which will be explained in forthcoming work. See the following example for a basic case.
Let us give an explicit example of a fiber of ψ, illustrating how ψ satisfies the properties of Theorem 3.2.4.
Example 3.2.11. Let G = G Qp where p > 3 and letD = detχ ⊕ 1 over F p , wherē χ is the mod p cyclotomic character. Then, using local Tate duality, we calculate that the fiber ψ −1 (D) in RepD consists of • extensions of 1 byχ, parameterized by P 1 3.3. Consequences of Formal GAGA for ψ. In order to descend closed loci under ψ, it will be helpful to know formal GAGA for ψ. We know this is true in the case that ψ is a good moduli space by [GZB12]. It also follows from the following hypothesis.

(FGAMS)
Formal GAGA holds for adequate moduli spaces realized as quotient stacks by GL d .

Consequently, we have
Theorem 3.3.1. IfD is multiplicity free, or if the assumption (FGAMS) holds, then formal GAGA holds for the morphism ψ : RepD → RD.
Proof. IfD is multiplicity free, ψ is a good moduli space by Theorem 3.2.4, and [GZB12] shows that formal GAGA holds for good moduli spaces. Otherwise, ψ is an adequate moduli space followed by a finite morphism, and (FGAMS) implies that formal GAGA holds along ψ.
Remark 3.3.2. According to the authors of [GZB12], it is unclear whether to expect that formal GAGA holds for adequate moduli spaces. However, they can prove formal GAGA for adequate moduli spaces such as BG for G a reductive algebraic group.
The following lemma shows how we can apply formal GAGA. The foremost use will be to algebraize loci of Galois representations that we initially produce only formally. (See, however, Remark 6.4.3).
Lemma 3.3.3. Let (R, m R ) be a complete Noetherian local Z p -algebra, and let X be an algebraic stack of finite type over Spec R. WriteX for its m R -adic completion. We assume that formal GAGA holds for X over Spec R.
(1) There is a natural bijective correspondence between (a) projective morphisms Y → X and projective morphismsŶ →X, (b) finite schematic morphisms Y → X and finite schematic morphismŝ Y →X, and (c) closed immersions Y ֒→ X and closed immersionsŶ ֒→X.
(2) If Y is a finite type Spec R-scheme that is a presentation Y → X of X, then (a)Ŷ →X is a fppf cover ofX as a map of formal algebraic stacks.
(b)Ŷ → X is a fpqc cover of X as a map of algebraic stacks.
Remark 3.3.4. It is important to specify the notion of "projective morphism," as there are definitions which differ over non-local bases. A projective morphism over a scheme S is a morphism of the form Proj OS B for some quasi-coherent sheaf B = i≥0 B i of graded algebras which is generated by B 1 and where B 1 is finite type. As we will work in the case of a Noetherian base, we note that this notion of projectivity is Zariski-local on the base provided that the data of an ample line bundle is included with the morphism (cf. Part (2a) is clear. Because Y is locally Noetherian,Ŷ → Y is flat. Since Y → X is smooth,Ŷ → X is then flat as well. It is also clearly quasi-compact. The surjectivity ofŶ → X may be deduced as follows: For any point z ∈ X, its closure in the Zariski topology is realized by a closed substack Z ֒→ X (cf. [LMB00, Corollarie 5.6.1(ii)]). Then there is a point ofŶ lying over z, namely, a generic point of the base changeẐ ×XŶ of the closed substackẐ ֒→X corresponding to Z ֒→ X by (1c), completing the proof of (2b). Finally, (2b) implies (2c). We recall the definitions of some p-adic period rings. Let OK be the ring of integers ofK and O Cp the ring of integers of C p . Let R = lim ← − OK/p, where each transition map is the Frobenius endomorphism of the characteristic p ring OK/p. This is a complete valuation ring which is perfect of characteristic p and whose residue field isk and is also canonically ak-algebra [FO,Proposition 4.6]. The fraction field Fr R of R is a complete non-archimedean algebraically closed characteristic p field. The elements x of R are in natural bijection with sequences of elements (x (n) ) n≥0 of O Cp such that x p (n+1) = x (n) for all n ≥ 0. A canonical valuation on R is given by taking the valuation v on C p normalized so that v(p) = 1 and setting v R ((x (n) ) n≥0 ) = v(x (0) ). Consider the ring W (R), and write an element of W (R) as (x 0 , x 1 , . . . , x n , . . . ). There is a unique continuous surjective W -algebra map

Families ofÉtale ϕ-modules and Kisin Modules
lifting the projection to the first factor R → OK/p onto the 0th truncation W 0 (R) of the limit of truncated Witt vectors defining W (R) (cf. [FO,Remark 5.10]). The natural Frobenius action on R induces a Frobenius map ϕ on W (R) which sends (x 0 , x 1 , . . . ) to (x p 0 , x p 1 , . . . ). We fix the notation S := W [[u]], the power series ring in the variable u. We equip S with a Frobenius map denoted ϕ, which acts by the usual Frobenius map on W and sends u to u p . We think of S as the functions bounded by 1 on the open analytic unit disk over K 0 , and S[1/p] as the ring of bounded functions on the open unit disk. Fix a uniformizer π ∈ K, and elements π n 1 for n ≥ 0 such that π 0 = π and π p n+1 = π n . Write E(u) ∈ W [u] for the minimal, Eisenstein polynomial of π over K 0 .
Write π := (π n ) n≥0 ∈ R, and let [π] ∈ W (R) be its Teichmüller lift (π, 0, 0, . . . ). Because R is canonically ak-algebra, we have a canonical embedding W ֒→ W (k) ֒→ W (R). We consider W (R) as a W [u]-algebra by sending u to [π]. Since θ([π]) = π, this embedding extends to an embedding of S into W (R) (cf. the formulation of W (R) in [FO, §5.2.1]), and we will consider W (R) and rings derived from W (R) as S-algebras via this map from now on. From the discussion above, this map is visibly ϕ-equivariant.
Let O E be the p-adic completion of S[1/u]. Then O E is a discrete valuation ring with residue field k((u)) and maximal ideal generated by p. Write E for its fraction field and O E ur its ring of integers. Since Fr R is algebraically closed, the residue field O E ur /pO E ur is a separable closure of k((u)). If O E ur is the p-adic completion of O E ur , or, equivalently, the closure of O E ur in W (Fr R) with respect to its p-adic topology, set S ur := O E ur ∩ W (R) ⊂ W (Fr R). All of these rings are subrings of W (Fr R)[1/p], and are equipped with a Frobenius operator coming from W (Fr R)[1/p].
Let K ∞ = ∪ n≥0 K(π n ) and G K∞ := Gal(K/K ∞ ). Clearly the action of G K∞ on W (R) fixes the subring S, since it fixes both W and π n ∀n ≥ 0. Therefore G K∞ has an action on S ur and E ur .
Recall that for any Z p -algebra S, S A denotes the mDA-adic completion of S ⊗ Zp A.
It will be important to know that the following such rings are Noetherian.

4.2.
Algebraic Families ofÉtale ϕ-modules. In this section, we will work with representations V A of G K∞ with coefficients in admissible Z p -algebras A with the discrete topology, which are quotients of Z/p i [x 1 , . . . , x j ] for some integers i, j. Unlike the previous sections, we will not study the most general moduli space of these families, but simply fix such an A and study the category of A-linear representations. Later, these results will be applied to a family of G K -representations in RepD, considered as a G K∞ -representation. We will often take R to be an arbitrary Artinian (and, therefore, finite cardinailty) subring of A such that A is a finitely generated R-algebra; R can be taken to be the image of Our goal is to compare these families of Galois representations toétale ϕ modules.  (5) Let V be the functor It remains to be confirmed that parts (3) and (5) above are valid, e.g. that M (V A ) is finite as a O E,A -module when V A is finite as an A-module.
(3) If A ′ is a finitely generated A-algebra, then there is a commutative diagram of functors where the downward functors are induced by − ⊗ A A ′ . (4) M restricts to an equivalence of categories

In particular, (a) V A is projective as an A-module of constant-rank d if and only if
First we assemble useful facts about limits. We will append (−) ∞ to various categories to indicate that the A-module finiteness condition has been dropped; however, in this setting we insist that that the linear action of G K∞ has open kernel instead of merely being continuous.   In order to substantiate Definition/Lemma 4.2.1 and Proposition 4.2.3, A-linear structure on the objects will be forgotten down to R-linear structure. Then, the objects are direct limits of finite R-submodules for which the statements are known, and we establish appropriate compatibility with the limits.
Proof (Definition/Lemma 4.2.1). Let V A ∈ ob Mod GK ∞ (A). Because the action of G K∞ has a finite index kernel, we have a canonical isomorphism as R We note that the functor M (resp. V ) commutes with injective direct limits in Mod GK ∞ (R) (resp. Φ ′ M (R)), using Facts 4.2.4 and 4.2.5 above along with the fact that the tensor product ⊗ Zp O E ur (resp. ⊗ OE O E ur ) preserves injective maps.
Therefore there are canonical isomorphisms in Φ and the fact that M is an equivalence of categories out of Mod GK ∞ (R) commuting with the necessary colimits implies that there is a canonical isomorphism respecting all structures The A-linear structure on the left hand side then provides a canonical A-linear structure on the right hand side, commuting with the action of O E ur , G K∞ , and ϕ. Therefore, M (V A ), being the G K∞ -invariants of the right hand side, has the structure of an A-module, and also  . We will show in this section that the functor associating to B the ϕ-stable S B -sublattices of M (V A ) ⊗ A B satisfying the condition "E-height ≤ h," for B a commutative A-algebra, is represented by a projective A-scheme. We will use the affine Grassmannian for this, generalizing the result of [Kis09b, §2.1] and [Kis08, §1], which was done in the case that A is Artinian.
We writeŜ B for the u-adic completion of the S B ; they are both Noetherian (Lemma 4.1.2).  (1) The local affine Grassmanian Gr loc GL(VA) for GL(V A ) is the functor associating to a A-algebra B the set of pairs (P D , η) where P D is a projective rank d B[[t]]-module and η is an isomorphism (2) The global affine Grassmannian Gr glob GL(VA) for GL(V A ) is the functor assigning to an A-algebra B the set of pairs (P A 1 , η), where P A 1 is a projective rank d B[t]-module and η is an isomorphism We observe that there is a natural functor The functor of sublattices that arises in our study is not identical to the global nor the local affine Grassmannian, but it is scheme-theoretically isomorphic.  Remark 4.3.7. We will see in the proof that the equivalence between the affine Grassmannians and F VA is not canonical. This is not a new phenomenon that arises when A is no longer Artinian as it was in [Kis08]; bases were implicitly chosen there as well.
Proof. We will prove the case W = Z p . First let us assume that V A is free of rank d, so that M A is as well, by Proposition 4.2.3(4). We observe that the two morphisms in (4.3.6) factor (4.3.3), and therefore it will suffice to show that the latter morphism ⊗ This filtered direct limit exists for each of the global/local affine Grassmannian functors as well as F VA , and is compatible with and unchanged by the tensor maps of (4.3.6). Therefore the latter map of (4.3.6) is an isomorphism.
In the case that V A is a projective, rank d A-module trivialized by a Zariski cover SpecÃ → Spec A, Proposition 4.2.3(2,4) implies that the same cover trivializes M A . Therefore the argument above applies after base change to SpecÃ, and by descent we have the statement of the proposition.  In the case that A is Artinian, L ≤h VA is represented by a projective A-scheme [Kis09b, Proposition 2.1.7] (see also [Kis08, Proposition 1.3]). The same proof will apply in the non-Artinian case.     VÃ It remains to show that the rightmost factor of (4.3.14) and the rightmost factor of (4.3.11) are canonically G K∞ -equivariantly isomorphic. The arguments given in [Kis08, Lemma 1.4.1] apply verbatim in the present case which we observe is a S R -submodule of MÃ. We have the natural surjection N ⊗ R A ։ Θ A * (M). Upon applying ⊗ RÃ , the isomorphism (4.3.13) induces anÃ-linear isomorphism Then tensor-Hom adjunction results in an isomorphism Finally, because the map S ur A → O E,Ã inducing this isomorphism may be checked to be an injection, an element of the left hand side factors through the quotient Θ A * (M) if and only if its image on the right hand side factors through Θ A * (M).

A Universal Family of Kisin Modules in Characteristic 0.
While the previous parts of §4 have been carried out over a fixed discrete coefficient ring A, we now fix a residual pseudorepresentationD of G K and let A be a formally finitely generated RD-algebra with a G K -representation V A with induced pseudorepresentation compatible with the RD-algebra structure of A.
The results above can be applied to (V A ⊗ RD RD/m iD )| GK ∞ for each i ≥ 1 and extend to the limit. For example, the functor M generalizes to this setting naturally from the above, since the map of limits is an isomorphism by Fact 4.2.6 and the fact that the ideal (p ⊗ 1) + O E ur ⊗ mDA (for which the left side is the completion) is equal to O E ur ⊗ Zp mDA (for which the right side is the completion). This means that M A is a projective O E,A -module of rank d as expected.
For B an A-algebra such that m iD · B = 0 for some i ≥ 1, set L ≤h Corollary 4.4.2. The functor L ≤h VA on A-algebras B such that m iD · B = 0 for some i ≥ 1 is represented by a projective A-scheme L ≤h VA . Proof. By Proposition 4.3.9 and Remark 3.3.4, this functor is represented by a projective formal scheme with a ample line bundle compatible with its limit structure. By applying formal GAGA, we conclude that L ≤h VA is the mD-adic completion of a projective A-scheme.
We now study the the map Θ A : L ≤h A → Spec A, showing that it is a closed immersion in equi-characteristic zero.
Part (1) expresses the uniqueness of S-lattices of E-height ≤ h in characteristic zero. According to part (2), scheme-theoretic image of Θ A has the property we expect.
Proposition 4.4.3 follows from the arguments in the case of local A, done in [Kis08, Prop. 1.6.4]. All that must be done to apply those arguments -which are based on the behavior of Θ A on B-points where B is an Artinian Q p -algebra -is to prove the following lemma. We establish the following notation: let B be an Artinian local Q p -algebra with residue field E. Let Int B denote the set of finitely generated O E -subalgebras of B 0 , the preimage of O E via B ։ E.  Because faithfully flat morphisms are descent morphisms for the flat property, and flatness is equivalent to projectivity for finite modules over Noetherian rings, we also have (4).
We now show that there exists a family of S-lattices of E-height ≤ h with coefficients in A ≤h which are universal in characteristic 0 in the sense of part (4) below. Only the construction of [Kis08] needs to be modified.
Proposition 4.4.5 (Generalizing [Kis08, Corollary 1.7]). There exists a finite S A ≤h -module M A ≤h with the following structures and properties. ( (3) For any finite W (FD)[1/p]-algebra B, any map f : A ≤h → B and any C ∈ Int B through which f factors, there is a canonical, ϕ-compatible isomorphism of S ⊗ Zp B-modules (4) There is a canonical isomorphism Proof. LetL ≤h VA be the mD-adic completion of L ≤h VA . Then For S a Z p -algebra we write S A := S A • [1/p], where we recall that S A • is the mDA-adic completion of S ⊗ Zp A • . We will use the canonical isomorphism In order to study families over A of ϕ-modules over O, we need to define the correct notion of the ring of coefficients. Two candidate definitions end up being the identical: While it is clear that these rings are isomorphic when A • is local, we prove the isomorphism here in the general case. , with the canonical map B n ֒→ C n that we get from considering an element of B n as a power series in u. Since the maps making up these limits are injective, it will suffice to show for f ∈ C 2n that its image in C n under the inclusions making up the limit lies in the image of B n in C n . With f ∈ C 2n chosen, write it as Because p ∈ mDA • , the coefficient f m p ⌊m/n⌋−⌊m/2n⌋ of u n /p ⌊m/n⌋ lies in (mDA • ) i(n) where lim n→+∞ i(n) = +∞. This means that f lies in the image of B n in C n , which is what we wanted to prove.
We observe that O A has an A-linear Frobenius endomorphisms compatible with the natural map S A → O A . Write S 0,A for the completion of K 0 [u] ⊗ Qp A at the ideal (E(u)).
There are natural maps from S A and O A to other period rings in families. First we recall the period rings and some properties (cf. [FO,§6]). Let A cris be the p-adic completion of the divided power envelope of W (R) (see §4.1) with respect to ker(θ), and let B , the image of u in W (R), and [ε] of (4.1.1). Write ℓ u , t ∈ B + dR for the elements defined by one can check that these series converge in B + dR . We may now define several more period rings: We can and will think of B + st as a polynomial ring over B + cris , for ℓ u is transcendental over the fraction field of B cris . As both ℓ u and t are "logarithms," Frobenius ϕ acts as ϕ(ℓ u ) = pℓ u and ϕ(t) = pt.
Equip B + st with an endomorphism N by formal differentiation d/dℓ u of the variable ℓ u with coefficients in B + cris , i.e. so that N (B + cris ) = 0. Extend ϕ to B + st as well, with ϕ(ℓ u ) = pℓ u . We note that ϕ and N define endomorphisms of the polynomial subring K 0 [ℓ u ] ⊂ B + st , and that pϕN = N ϕ on B st . There is an exhaustive, decreasing filtration on each of A cris , B + cris , written Fil i A cris , Fil i B + cris , induced by their inclusion in the filtered ring B + dR , such that where Fil 0 A cris = A cris and Fil 0 B + cris = B + cris . The filtration on B + dR is given by Fil i B + dR := (ker θ) i , i ≥ 0. In fact, t ∈ Fil 1 B + dR and t ∈ Fil 2 B + dR [FO,Proposition 5.19], so also t ∈ Fil 1 A cris , and t is a generator for the maximal ideal of B + dR . There is an action of G K on these rings arising from its action on OK/p to a continuous action on R, W (R), and the derivative rings above. In particular, it will be useful to know the action of G K on t is given by σ(t) = χ(σ) · t where χ represents the p-adic cyclotomic character. That B + st is stable under G K follows from the following calculation (see [WE13,Lemma 4.6.4]).
The map ϕ extends to each of these rings B-linearly, with N again acting as formal differentiation with respect to ℓ u here. In particular, N (B + cris,B ) = 0. Analogous notation is used for the elements of the filtration on these rings: denote by Fil i A cris,A • the mDA • -adic completion of Fil i A cris ⊗ Zp A • , and for any A-algebra B let Fil i B + cris,B := Fil i A cris,A • ⊗ A • B.
It will be important to know in the construction of (5.2.3) that there is a canonical inclusion O A ֒→ B + cris,A extending the map O ֒→ B + cris discussed above, and also a map S ur A ֒→ B + cris,A . By Lemma 5.1.2, it will suffice to show that for large enough n, In order to construct the map, it will suffice to draw, for sufficiently large n, maps for each j ≥ 1. We will get such maps if we show, for large enough n, the existence of maps W [[u, u n /p]] ֒→ A cris .
Then Lemma 5.1.5 implies that this map will remain injective after tensoring with A • and completing with respect to the mDA • -adic topology. This same construction gives us a canonical map S ur A ֒→ B + cris,A . We will now record some lemmata to ensure that the large rings B + cris , A cris , and so forth behave well in families. Lemma 5.1.5 is given in [Kis08] in the case that R is local, but its proof is valid for any adically complete Noetherian ring.
The following lemma requires some generalization from the local case.
(1) For i ≥ 0, the ideal Fil i A cris,R of A cris,R is a faithfully flat R-module.
(2) For i ≥ 0, Fil i A cris,R / Fil i+1 A cris,R is a faithfully flat R-module, which is isomorphic to the I-adic completion of (Fil i A cris / Fil i+1 A cris ) ⊗ Zp R. is injective, where q runs over ideals of R such that R/q is a finite flat Z p -algebra. (6) If 0 = f ∈ A cris , then f is not a zero divisor in A cris,R .
Proof. Parts (1), (2), (3), (4), and (6) are proved by the same arguments as the corresponding parts of [Kis08, Lemma 2.3.2], where for part (4) the the fact that f : R → B • is finite and continuous implies that f is adic, i.e. that f (I) · B • is an ideal of definition for the J-adic topology of B • .
To prove part (5), consider that if 0 = f ∈ A cris,R , then there exists some n such that 0 = f ∈ A cris,R/I n = A cris ⊗ Zp R/I n . The ring R/I n satisfies the Jacobson condition (it is finitely generated over Z, for example) so that there is an injection for a = 1, where N denotes the nilradical of R/I n and m varies over the maximal ideals of R/I n . One then observes that this injection exists for any positive integer a. Because N is nilpotent, we then have an injection R/I n ֒→ m (R/I)/m a for some a depending only on I and n. Since A cris is Z p -flat, we then have that the image of f in A cris,(R/I n )/m a is non-zero for some choice of m. It follows that there exists maximal ideal m ′ ⊂ R such that the image of f in A cris,R/m ′a is non-zero. Therefore the image of f under the natural map A cris,R → A cris,R ∧ m ′ is non-zero, where R ∧ m ′ denotes the completion of R at m ′ . Since the statement of (5) is known in the case that R is local [Kis08, Lemma 2.3.2(5)], we are done.
The following lemma will be useful to construct loci cutting out conditions realized over a family of period rings. Proof. Assume that m n D · M = 0 for some n ≥ 1, and choose some x ∈ A cris,A • ⊗ A • M . Therefore there is a natural isomorphism of A • -modules Because the statement was shown to be true when A • is local in [Kis08, Lemma 2.3.3], we may apply the statement to the left hand side. As a result, there exists a smallest RD-submodule P of M such that x ∈ A cris,RD ⊗ RD P . We claim that the image N (x) of the natural map is the smallest A • -submodule of M with the required property. Clearly it contains x. If there were a A • submodule N with the property, then N ⊃ P since N is also a RD-module with the property. But then N must contain N (x), which is the A • -span of P . This shows that N (x) is the smallest A • -submodule of M with the property.
The proof of [Kis08, Lemma 2.3.3] deduces the Lemma in the case that A • is local from the case that M has finite length using Lemma 5.1.5. That same deductive argument may be used to prove the lemma from the case m n D · M = 0 proved above.

Period Maps in
. We will now follow [Kis08, §2.4] in constructing a period map comparing G Krepresentations to (ϕ, N )-modules from the data above, adding additional data needed to descend G K∞ -representations to G K -representations. In what follows, B is an arbitrary A-algebra.
We derive from the map ι a S A -linear, ϕ-equivariant map ). Tensoring this map by ⊗ SA O A and using the map ξ : D A → M A from Lemma 5.2.1, we have a ϕ-equivariant map . We see that the right hand side has an action of G K , and the left hand side has an action of G K∞ through the action on B + cris,B . This map is G K∞ -equivariant because G K∞ acts equivariantly on the inclusions S ֒→ O ֒→ B + cris and that ι above is G K∞ -equivariant. In order to extend the action of G K∞ on the left hand side of (5.2.3) to an action of G K , we suppose that there is a W B -linear map which satisfies the identity pϕN = N ϕ. Then the action of for σ ∈ G K . One can check that this action of G K commutes with ϕ.
In order to parameterize semi-stable representations, we must work with B + st . Recall that we adjoin ℓ u to B + cris,B to get B + st,B = B + cris,B ⊗ K0 K 0 [ℓ u ] with a Blinear action of N and ϕ. Consider the composite of the isomorphisms where the first map is the inverse to the natural isomorphism The following lemma is an important step toward the comparison of semi-stable Galois representations and filtered (ϕ, N )-modules in families. Proof. First we note that it suffices to prove the assertions only for (5.2.3), and for B = A. Lemma 5.1.6(5) immediately reduces the injectivity claim to the case that A • is finite over Z p , which was proved in [Kis08, Lemma 2.4.6].
To show that the cokernel of (5.2.3) is flat, it suffices to show that (5.2.3) remains injective after applying ⊗ A A/I for any finitely generated ideal I of A. If we had started our proof with A/I in the place of A, we would still have the injectivity statement for A/I, just as we proved it for A above. Therefore it only remains to show that is injective. This is precisely what Lemma 5.1.6(4) tells us -indeed, the sources and targets are isomorphic, respectively -completing the proof.
Our goal is to produce A st , the maximal quotient of A over which V A is semistable with Hodge-Tate weights in [0, h]. This means that for any A-algebra B which is finite as a Q p -algebra, the representation V A ⊗ A B semi-stable with Hodge- Tate  Proof. We may freely assume that V A is a free rank d A-module. In this case, the construction of A st in [Kis08, Proposition 2.4.7] of a finitely generated A-algebra A st representing the functor of the statement generalizes to this setting, since the ingredients, Lemmas 5.1.7 and 5.1.5 and the map (5.2.3), generalize. We sketch the argument to demonstrate these dependencies.
Firstly, the functor assigning to B the set of W B -linear maps N : D B → D B satisfying pϕN = N ϕ is representable by a finitely generated A-algebra A N .
Write η B for the map of (5.2.3), and for d ∈ D A N and σ ∈ G K set which are elements of Q := Hom A N (V A N , B + cris,A N ). We wish to show that the vanishing of this map for all σ, d is cut out by an ideal of A. Choose a B + cris,A Nbasis for Q and let x 1 , . . . , x r be the coordinates of δ σ (d) with respect to the basis. Applying Lemma 5.1.7 with M = A N and x = x i for x i varying over a B + cris,A Nbasis for Q, the span of the resulting ideals of A N is the kernel of the quotient A st of A N . Then, because Q is a faithfully flat A N -module by Lemma 5.1.5(1), A N → B will factor through A st if and only if η B is compatible with the action of G K .

Algebraic Families of Potentially Semi-Stable Galois Representations
In the previous section, we studied the period map D A ⊗ WA B + cris,A −→ Hom A (V A , B + cris,A ) of a family of G K -represetations with bounded E-height and demonstrated that it is injective with flat cokernel, and also G K -equivariant over a Zariski-closed locus. In this section, we will show that these properties allow for the construction of Zariski-closed loci of crystalline and semi-stable Galois representations with bounded Hodge-Tate weights. In addition, loci corresponding to a given Hodge type or potentially semi-stable Galois type will be cut out. We will conclude by stating these results for the universal spaces of Galois representations Rep D and RepD, producing algebraic versions of these spaces, and drawing conclusions about their geometry in equi-characteristic 0.
6.1. Families of Semi-stable Galois Representations with Bounded Hodge-Tate Weight. We now drop the assumption that the family V A • of representations of G K has E-height ≤ h, but maintain the other assumptions given in §5.1. The following theorem shows that a semi-stability condition with bounded Hodge-Tate weights cuts out a closed locus and that the corresponding period maps interpolate along this locus.   We must fully explain the proof of part (3). By part (2) and Lemma 5.2.7, (6.1.2) is an injective map of projective B st,A -modules of rank d. Therefore it will suffice to show that this map induces an isomorphism on top exterior powers, and we may freely restrict ourselves to the case that d = 1.
In the one-dimensional case, V A • arises by extension of scalars ⊗ RD A • [Che14, Proposition 3.13]; this is the case because 1-dimensional representations are identical to 1-dimensional pseudorepresentations. It is then evident that (6.1.2) arises by ⊗ Bst,R B st,A from the same map where A is replaced by R, and that it suffices to prove that (6.1.2) is an isomorphism when A = R. This was done in [Kis08, Proposition 2.7.2]. Then (6.1.3) is an isomorphism by Lemma 6.1.4.
We need the following lemma in order to find the G K -invariants in B st,A .
Lemma 6.1.4 (Generalizing [Kis08, Lemma 2.7.1]). For i ≥ 0 there is an isomorphism (A(i), B + st,A ) induced by multiplication by p −ri for r i defined below, where A(i) denotes A with G K acting via the ith power of the p-adic cyclotomic character χ. In particular, if The key part of the proof of the local case in [Kis08, Lemma 2.7.1] is that the χ i -isotypic part of A cris,A • is given by W A • · t i /p ri where r i is the greatest nonnegative integer such that t i /p ri is in A cris . Applying this to A cris,RD , which we may do because RD is local, it follows that the Because the action of G K on A cris,A • is continuous (where the topology is the mD-adic topology), the closure of W A • ·t i /p ri in A cris,A • is the χ i -isotypic part. However, this module is already closed.
With this fact in place, the proof of loc. cit. supplies the rest of the argument.
6.2. p-adic Hodge Type. Our remaining goal is to find loci corresponding to more refined p-adic Hodge theoretic conditions, namely, a certain Hodge type or being potentially semi-stable of a certain Galois type. In fact, these conditions will cut out connected components (in equi-characteristic 0). First we will address the Hodge type, following [Kis08, §2.6] and [Kis09a, §A.4]. First we recall the notion of p-adic Hodge type. For this, we fix an finite extension field E of Q p and suppose that A admits the structure of an E-algebra.
The proof from [Kis08, Corollary 2.6.2] does not require any generalization to account for A • being non-local, in light of Lemma 4.4.4. However, we will sketch the proof in order to incorporate the erratum [Kis09a, §A.4].
Proof. By applying parts (1) and (2) of Theorem 6.1.1, the later parts of the proof of [Kis08, Corollary 2.6.2] explains that the finite A-module realizes the ith part of the Hodge filtration of D B ⊗ K0 K when specialized to any finite Q p -algebra B. Therefore, because these pieces of the filtration are projective B-modules, the A-module is projective by Lemma 4.4.4(4). Because the rank of a finite projective module is locally constant, Spec A v is a union of connected components of Spec A. One may then set A st,v : 6.3. Galois Type. Next we will study families of potentially semi-stable G Krepresentations, following [Kis08, §2.7.5]. We stipulate that B is an Artinian local E-algebra with residue field E. Let V B ∈ Rep d GK (B). Following [Fon94], set where K ′ runs over finite field extensions of K. LetK 0 ⊂K denote the maximal unramified extension of K 0 , and let G K0 ⊂ G K be the inertia group of G K . Then D * pst (V B ) is a B ⊗ QpK0 -module with a Frobenius semi-linear Frobenius automorphism ϕ, a nilpotent endomorphsm N such that pϕN = N ϕ, and a B ⊗ QpK0 -linear action of G K0 which has open kernel and commutes with ϕ and N .
Following [Kis08, §2.7] along the line of reasoning of [Kis09b, Lemma 1.2.2(4)], we see that D * pst (V B ) is finite and free as a B ⊗ QpK0 -module. Since the action of G K0 commutes with the action ϕ, the traces of elements of G K0 are contained in B, and D * pst descends to a representation of G K0 on a finite free B-moduleP B . Because characteristic zero representations of finite groups are rigid, this representation must be an extension of scalars from a representation P B of G K0 over E.
We have associated to a potentially semi-stable d-dimensional representation V B of G K over B a representation of the inertia group of K over E which reflects the failure of V B to be semi-stable. We will call this the "Galois type" of V B , as follows.
Fix an algebraic closureQ p of Q p . We say that V B is potetnially semi-stable of Galois type T provided that P B defined above is isomorphic to T overQ p .
It is equivalent to say that for any γ ∈ G K0 , the trace of T (γ) is equal to the trace of γ on D * pst (V B ). Let v be a p-adic Hodge type as in Definition 6.2.1; fix a representation Theorem 6.3.2 (Generalizing [Kis08, Theorem 2.7.6 and Corollary 2.7.7]).
(1) There exists a quotient A T,v of A such that for any finite E-algebra B, a map of E-algebras ζ : These constructions may be repeated verbatim from [Kis08, Theorem 2.7.6 and Corollary 2.7.7]. We will give a sketch.
Proof. Let L/K be a finite Galois extension such that I L ⊆ ker T . Theorem 6.2.2 gives the existence of a quotient A pst,v of A such that ζ factors through A pst,v if and only if V B | GL is semi-stable with Hodge type v. One then applies Theorem 6.1.1(3) and studies the action of the inertia subgroup I L/K , which is L 0 -linear and commutes with ϕ, and therefore has trace function in A pst,v . As this inertia group is finite, its trace function is locally constant on Spec A pst,v . The condition that its trace is T therefore cuts out a union of connected components of A pst,v , as desired.
For the second result, first produce A T,v as above, and then take the quotient corresponding to the equation N = 0, where N is the endomorphism of D A st,h defined in Theorem 6.1.1(3).
6.4. Universal Families of Representations, and Algebraization. We will summarize what we have proved by producing universal spaces of potentially semistable Galois representations with bounded Hodge-Tate weights. These can then be algebraized using Theorem A, under some assumptions. In particular, let C be one of the following conditions on representations of G K over a finite Q p -algebra.
(1) Crystalline with Hodge-Tate weights in the range [a, b].
(2) Semi-stable with Hodge-Tate weights in the range [a, b].
(3) Any of the above two conditions, with fixed Hodge type v.
(4) Any of the above three conditions after restriction to G L , for some finite field extension L/K. (5) Condition (4) with L/K a Galois extension, and in addition, a particular Galois type T .
Consider the case where A • is the coordinate ring of Rep •, D , the universal formal moduli scheme of framed representations of G K with residual pseudorepresentation D. This admits an action of GL d , the mD-adic completion of GL d ⊗ Z RD. The results from the previous sections produce a quotient of A C of A = A • [1/p] of representations satisfying condition C. The ideal I C ⊂ A such that A/I C = A C is stable under the action of GL d ; this is the case because the B-valued points in the C-locus of Spec A, where B is a finite Q p -algebra, are obviously preserved by the action of GL d , and these points characterize the C-locus by Lemma 4.4.4(3).
The kernel I •,C ⊂ A • of the natural map A • → A C cuts out a quotient A •,C := A • /I •,C such that A •,C [1/p] ∼ → A C and therefore A •,C has the same property on B-points. Moreover, I •,C is GL d -stable. We summarize our discussion in this Theorem 6.4.1. Given any of the conditions C above, there is a closed substack Rep •,C D of Rep • D , formally of finite type over Spf RD, such that for any finite Q palgebra B and representation V B of G K with residual pseudorepresentationD, there exists a model V B ′ for V B , where B ′ ∈ Int B , such that the corresponding map ζ : Spf B ′ → Rep • D factors through Rep •,C D if and only if V B has property C. The statement of the theorem is an example of the way that we think of the "generic fiber" over Q p of a Spf Z p -formal stack while considering the formal stack only as a limit of algebraic stacks over Spec RD/m iD . The theorem expresses that the locus of such points has an integral model and is GL d -stable, as RepD is a quotient stack of Rep •, Proof. Without loss of generality we may assume that B is local. Choose a basis for V B , so that we have a corresponding map A → B. If V B satisfies C, it factors through A C . By Lemma 4.4.4(4), there exists some B ′ ∈ Int B and a natural map Proof. In either case, formal GAGA holds for ψ : RepD → RD by Theorem 3.3.1. We apply Lemma 3.3.3(1c) to find a natural corresponding closed immersion Remark 6.4.3. In the corollary above, we have invoked formal GAGA for ψ produce an RD-algebraic universal family of potentially semi-stable representations Rep •,C D after first producing a formal version. However, if one freely invokes formal GAGA from the start, it is possible to carry out the construction of algebraic universal families of potentially semi-stable Galois representations in Theorem 6.4.1 directly. That is, using formal GAGA for ψ freely, it is possible to carry out all of the work of §4.4 and §5 with the mD-adically separated finitely generated RD-subalgebra A • alg ⊂ A • of Corollary 3.2.6 in place of A • . For example, even the Cauchy sequence used to construct the map ξ of Lemma 5.2.1 can be shown to be have algebraic coefficients, i.e. defining a map In this sense, once we know formal GAGA, the construction of Rep •,C D is not merely algebro-geometric, but is natural in that all of the semi-linear algebraic data and period maps exist algebraically relative to RD. However, we have constructed potentially semi-stable loci in the formal setting first, so that Theorem 6.4.1 is not conditional on assumption (FGAMS).
Here are some geometric properties of the generic fiber over Z p of these algebraic stacks of representations, deduced from established ring-theoretic properties of equicharacteristic zero deformation rings of Galois representations. (1) When C is a potentially crystalline condition, each of these spaces is formally smooth over Q p . In view of [Gro64, Ch. 0, Théorème 22.5.8], the formal smoothness of these spaces over Q p is equivalent to their being regular. We will work with the latter condition in the proof.
Proof. Bellovin [Bel14] proves that for any p-adic field-valued representation ρ satisfying C, the complete local ring ring parameterizing liftings of ρ with property C is complete intersection and reduced. It is also equi-dimensional of the dimension given in the statement of part (2) by [Kis08, Theorem 3.3.4] when C has a fixed Hodge type v. As these rings are the complete local rings of the closed points of the excellent Jacobson scheme Rep ,C D , we know that Rep ,C D is reduced and locally complete intersection. Indeed, see [GM78, Corollary 3.3] for the openness of the complete intersection locus of an excellent ring. We also know that Rep ,C D is equi-dimensional as in statement (2) when C implies a fixed p-adic Hodge type. Because being complete intersection and reduced is local in the smooth topology, these properties hold for Rep C D as well. By the main result of [Val76], the coordinate ring of Rep ,•,C D is excellent, and therefore so is the coordinate ring of Rep ,C D . The arguments above may then be applied in this case as well.
Kisin in [Kis08, Theorems 3.3.4 and 3.3.8] and [Kis09a, Theorem A.2] proves the generic regularity statement and part (1) of the proposition above, but with the generic fiber of a framed deformation ring R V F of a residual representation V F in place of RepD. We will deduce our claim from this case. First, let us prove generic regularity in Spec The arguments for generic regularity in [Kis08] are statements 3.1.6, 3.2.1, and 3.3.1 of loc. cit. Their validity and their application in the proof of [Kis08, Theorem 3.3.4] generalize verbatim from the case that A • is a complete Noetherian local Z p -algebra to the case that A • is topologically finite type over Z p , with the exception of Proposition 3.3.1 of loc. cit. We deduce the non-local case of this statement in Lemma 6.4.5 below. This gives us the generic regularity of Rep ,C D . In particular, Kisin's arguments identify the singular locus with the support of a finite A-module H 2 (D A ) produced out of D A and its structure maps ϕ, N . This construction may be carried out over Rep C D to produce a coherent sheaf H 2 (DD). As the support of H 2 (DD) is nowhere dense after changing base to the fpqc cover be formally smooth and let C be one of the conditions above, so that A C is a quotient of A parameterizing representations with condition C. Then for a closed point x ∈ A C corresponding to a maximal ideal m = m x ⊂ A C , the morphism SpfÂ C m → Mod F,ϕ,N is formally smooth. Proof. Let V A • be the rank d G K -representation corresponding to f with specialization V x at x to a representation with coefficients in the p-adic residue field E = A/m. By Lemma 4.4.4(3), the map A • → E factors through its ring of integers O E ⊂ E, giving a choice of G K -stable lattice V • x ⊂ V x . Let V F denote V • x ⊗ OE F, where F is the residue field of O E . Let D V F denote its deformation groupoid as in [Kis08,§3]. Note that D V F → RepD, as a morphism of groupoids on complete Noetherian W (F)-algebras with residue field F, is schematic. Then, using the notation of [Kis09b, §2.3], we observe that there is an isomorphism of AR W (F),(OE ) -groupoids . Following the arguments of [Kis09b, §2.3], one may check that the complete local Z p -algebra A ′• given by Spf A ′• = Spf A • × RepD D V F has a map x ′ : A ′ = A ′• [1/p] → E factoring x : A → E and thatÂ mx →Â ′ m x ′ is an isomorphism. This is all we need to reduce the proof to the case that A • is local, which then follows by [Kis08, Proposition 3.3.1].
7. Potentially Semi-stable Pseudodeformation Rings 7.1. Potentially Semi-Stable Pseudorepresentations. We must be clear regarding what it means to ask if a pseudorepresentation satisfies some property which, a priori, only applies to representations.
Definition 7.1.1. Let K be a full subcategory of the category of fields which is closed under finite separable extensions, let D be a setoid of pseudorepresentations fibered over K, 2 and let Rep be a groupoid of representations fibered over K. Let P be a full subcategory of Rep of representations with property P such that if V ∈ P(K), then its semisimplification V ss and any separable base change V ⊗ K K ′ are each in P.
Then a pseudorepresentation D ∈ D over K ∈ K is said to have property P if, given a finite separable extension K ′ /K such that there exists a semi-simple representation V ss D ∈ Rep(K ′ ) such that ψ(V ss D ) = D ⊗ K K ′ (which exists by Corollary 2.1.8), V has property P .
For example, one can let K be the category of p-adic fields, let D and Rep be the continuous pseudorepresentations and representations of G Qp over p-adic fields, and let the property P be "crystalline," or any of the conditions of §6.4.
While it also seems possible to define such a notion for pseudorepresentations valued in non-fields, we will not require this here.
We now return to the case of C being a potentially semi-stable condition as in the previous section. Recall that R = RD[1/p] and that Rep C D exists unconditionally whenD is multiplicity-free.
Theorem 7.1.2. If Rep C D exists, there exists a canonical quotient R C of R with the property that for any finite field extension E of W (F)[1/p], the map z : R → E factors through R C if and only if the semi-simple representation associated to the pseudorepresentation corresponding to z satisfies the condition. This quotient R C is reduced.
Proof. We have from Corollary 6.4.2 that there is a closed subscheme Rep C D of RepD. Because ψ is universally closed, the scheme-theoretic image Spec R C of Rep C D under ψ defines a closed subscheme of Spec R, and ψ restricted to Rep C D is a good moduli space over Spec R because Rep C D is realizable as a quotient stack Having constructed the quotient R C of R, we show that it has the desired property. Choose a closed point ζ : Spec E → R C . By Theorem 3.2.4(2), there exists a unique closed point z in the fiber of ψ in RepD over ζ, with residue field some finite extension E ′ /E, corresponding to the unique (up to isomorphism) semi-simple representation inducing ζ. Because Rep C D ֒→ RepD is a closed immersion, we must have z ∈ Rep C D . When Rep C D is reduced, then R C is also [Alp08,Theorem 4.16(viii)]. Then, the uniqueness of R C follows from Lemma 4.4.4 and the fact that it is reduced.
Corollary 7.1.3. If Rep C D exists, there exists a quotient R C D of RD with the property that for any finite field extension E of W (F)[1/p], the map z : RD → E factors through R C D if and only if the semi-simple representation associated to the pseudorepresentation corresponding to z satisfies the condition. There is a unique such quotient which is reduced, namely the image of RD in R C .
Proof. One may take R C D to be any quotient of RD such that it realizes R C after inverting p.
The generic fiber Spec R C is in fact pseudo-rational when C has a crystalline condition.
Definition 7.1.4 ([Sch08, §6.1]). A Noetherian local ring (R, m) is called pseudorational if it is analytically unramified, normal, Cohen-Macaulay, and for any projective birational map f : Y → Spec R with Y normal, the canonical epimorphism between the top cohomology groups δ : H d m (R) → H d Z (Y ) is injective, where Z is the closed fiber f −1 (m) and d the dimension of R. A Noetherian ring A is called pseudo-rational if A p is pseudo-rational for every prime ideal p in A.
The notion of pseudo-rational is a generalization, to rings over which no resolution of singularities exists, of the notion of rational singularities for finite type algebras over a characteristic zero field. The work of [Sch08] Schoutens is a generalization of the Hochster-Roberts theorem to this setting.
Corollary 7.1.5. If the condition C implies potentially crystalline, R C is pseudorational. In particular, it is reduced, normal, and Cohen-Macaulay.
Proof. Write S for the coordinate ring of the regular affine scheme Rep ,C D , so that R C = S GL d ֒→ S is an inclusion of the invariant subring by the action of basis change. Therefore the map R C ֒→ S is cyclically pure (also known as ideally closed), cf. [Alp08, Remark 4.13]. The main theorem [Sch08, Theorem A] states that a cyclically pure subring of a regular Noetherian equi-characteristic zero ring is pseudo-rational. Therefore R C is pseudo-rational, and hence also formally unramified, normal, and Cohen-Macaulay [Sch08,§4]. Formally unramified is equivalent to reduced, since R C is finitely generated over the excellent ring RD.
Remark 7.1.6. Reducedness and normality of R C are clear from the regularity of S without resorting to Schoutens' result. Cf. also [Alp08,Theorem 4.16(viii)].
There is often interest in understanding the connected components of potentially semi-stable deformation rings. It is no more complicated to study the connected components of potentially semi-stable pseudodeformation rings. Using the fact that each of the maps Rep D → RepD → Spec R and Rep D → RepD is surjective with connected geometric fibers (cf. [Alp08, Theorem 4.16(vii)], Theorem 3.2.4(2)), the analysis of the connected components of R C amounts to analysis of the connected components of the affine scheme Rep D .
Corollary 7.1.7. There is a natural bijective correspondence between the connected components of each of Rep ,C D , Rep ,C D , Rep C D , and Spec R C .
7.2. Global Potentially Semi-stable Pseudodeformation Rings. In this section, we will assume that all algebraizations of stacks of potentially semi-stable representations exist. Let F/Q be a number field, let S be a finite set of places of F containing those over p, and takeD : G F,S → F to be a global Galois pseudorepresentation ramified only at places in S. As G F,S satisfies Mazur's Φ p finiteness condition, the universal ramified-only-at-S pseudodeformation ring RD of global Galois representations is Noetherian [Che14, Proposition 3.7]. Fix decomposition subgroups G v ⊂ G F,S for places v ∈ S. In analogy to a common construction in the case of deformations of Galois representations, we want to find a quotient R C D of RD, C = (C v ) v∈S , parameterizing pseudodeformations which satisfy certain conditions C v at each v ∈ S, such as a condition C v coming from p-adic Hodge theory when v | p.
In the case of deformations of a irreducible Galois representationρ : G F,S → GL d (F), one may accomplish this construction using the natural maps R v ρ → Rρ from a local deformation ring to a global deformation ring (usually discussed as a "deformation condition" to avoid unnecessary technical complications whenρ| Gv is not irreducible), and the quotients R v ρ → R Cv ρ corresponding to the condition C v on representations of G v deformingρ| Gv . Then one sets in order to obtain a deformation ring parameterizing representations of G F,S deformingρ with conditions C v upon restriction to G v . In contrast, one does not want to do the same construction with pseudodeformation rings (as ifD replacedρ in each place in the line above), even though the corresponding maps R v D → RD and R v D → R Cv D exist. The reason is that ifρ is irreducible butρ| Gv is not, then a deformation D : G F,S → E ofD (where E/Q p is a finite extension) such that D| Gv is reducible may have information about extensions between the Jordan-Hölder factors of D| Gv , while D| Gv lacks this information. If a condition C is sensitive to the extension classes in a representation, then we may get too large of a quotient in this way.
Instead, the following construction is appropriate: fixingD as above, we have the Noetherian moduli stack Rep • D of representations of G F,S inducing residual pseudorepresentationD; it is algebraizable of finite type over Spec RD via ψ. There are also analogous local spaces Having constructed this space, we may construct the pseudodeformation ring parameterizing pseudodeformations with property C.
For simplicity we will address representations of G Q,S where p ∈ S, and cut out a locus of representations satisfying ordinariness with respect to a choice of decomposition group with its inertia group at p, G Q,S ⊃ G p ⊃ I p . We will find a quotient of RD parameterizing ordinary pseudorepresentations. Obviously, the ordinary condition is sensitive to extension classes, so that an 2-dimensional pseudorepresentation D of G Q,S such that D| Gp ≃ det •(ψ ⊕ χ) where ψ is unramified is not necessarily ordinary.
We letD arise from the sum of two charactersψ,χ valued in F × , writinḡ D = det(ψ ⊕χ) and stipulating thatψ| Ip = 1 so that the set of ordinary pseudodeformations ofD is not empty. We also assume thatψ| Gp =χ| Gp . Proof. We will use Theorem 3.2.3 and consider representations of E(G p )D. AsD is multiplicity free, Theorem 2.4.10 gives us a generalized matrix algebra structure on E(G p )D: the data of two idempotents, e 1 associated to the factorχ ofρ ss D =ψ ⊕χ and e 2 associated toψ. We write the generalized matrix algebra in the form We may now define the moduli stack Rep ord D of G Q,S -representations that are ordinary at p by setting Rep ord D := RepD × Rep Gp ,D Rep ord Gp,D , just as in (7.2.1), and then construct the global ordinary pseudodeformation ring R ord D by Definition 7.2.2. BecauseD is multiplicity-free, ψ is a good moduli space (Theorem 3.2.4(4)). The restriction of ψ to Rep ord D → Spec R ord D is a good moduli space as well (Theorem 2.3.3(7)), and R ord D is precisely the associated GIT quotient ring.
Corollary 7.3.3. Let E be a p-adic field with ring of integers O. With the datā D,ψ,χ as above, choose a pseudorepresentation D z : G Q,S → O ⊂ E deforminḡ D, so that there is a corresponding morphism z : Spec E → Spec RD. Then z factors through R ord D if and only if D z is ordinary in the sense that the associated semi-simple representation V ss z is ordinary. Proof. Combine Lemma 7.3.2 and Theorem 7.2.4.