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.


Overview
Mazur [35] 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 0-dimensional. 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 and Wiles [51] and Bellaïche and Chenevier [3]. The space Ext 1 G (ρ 2 ,ρ 1 ) of extensions of two distinct irreducible residual representationsρ 1 ,ρ 2 of G ρ 1 * 0ρ 2 (1.1) 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 [51] and [3] 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 [9] 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 pseudorepresentationD.
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 is multiplicity-free if ρ ss D has no multiplicity among its simple factors. Theorem A (Theorem 3. 8

) If G satisfies Mazur's finiteness condition 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) 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 [26]. 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 Sect. 3).  [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 algebraizable, 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 One may also produce the RD-algebraic closed substack Rep τ,v D without any condition onD assuming an algebraization conjecture for ψ (see Sect. 3.3).
We emphasize that the methods to cut out these loci of representations are due to Kisin [26,Sects. 1,2] 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 the loci follow from results on the ring-theoretic properties of equi-characteristic zero potentially semi-stable deformation rings, principally [5].
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). (2) When we replace "semi-stable" by "crystalline," R τ,v is pseudo-rational (see Definition 7.4); in particular, it is normal and Cohen-Macaulay.
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 have not been directly applicable to study pseudodeformation rings R τ,v D or R τ,v . The author intends to report on this in future work.
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 Sect. 7.3. These ordinary pseudodeformation rings are compared to Hecke algebras in [57,58].
As a final point, we emphasize that Theorem A is based on a study of the moduli of representations of a finitely generated associative algebra over a Noetherian ring in Sect. 2. Theorem A is deduced from this study by "removing the topology" from the representation theory of profinite groups. The conclusions of Theorem A may be viewed as generalizations, allowing for the profinite topology and non-zero characteristic, of parts of the investigations of Le Bruyn [30,31] (building on [39]) in non-commutative algebraic geometry.

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 [9]. 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 Sect. 2, we augment the theory of generalized matrix algebras of [3,Sect. 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 Sect. 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 3.6, namely that the universal Cayley-Hamilton quotient E(G)D, which factors the representations of G with residual pseudorepresentationD, 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 pseudorepresentationD. Adding the results of Sect. 2, we get Theorem A. We then discuss how Theorem A implies that formal GAGA holds for ψ in certain cases.
In Sect. 4 we begin our study of potentially semi-stable representations of G = G K . We adapt the methods of Kisin [26,Sect. 1] to cut out a locus of representations with E-height ≤ h within the universal families Rep D . The point of the 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 in the theory of Fontaine [13,Sect. 1.2], drawing an equivalence between continuous representations of G K ∞ with arbitrary discrete coefficients, and étale ϕ-modules. The work of Emerton and Gee [12] will interpolate these families, as there exist families of étale 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 the étale 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 [26,Sect. 2], descending, in families, the comparison of a Kisin module to a G K ∞ -representation down to a comparison of a (ϕ, N )-module to a G K -representation. This comparison is valid over a certain locus, and Sect. 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 potential semi-stability with a certain Galois type. Theorem B follows from applying these constructions to a universal family of representations and algebraizing these closed subschemes using formal GAGA for ψ. Finally, geometric properties of these spaces in characteristic 0 are then deduced from existing results on their local rings at closed points.
In Sect. 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 [1,2] and Schoutens [47] allow us to deduce the ring-theoretic properties of potentially crystalline pseudodeformation rings using invariant theory. We 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.

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 (cf. [36,Sect. 1.6.2]), e.g. the main result Theorem 2.20. 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 ddimensional pseudorepresentations of R, ultimately showing in Theorem 2.20 that they almost form an adequate moduli space when R is finitely generated. Later, in Sect. 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.
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. [30,31,33,39] 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 [9], following previous notions due to Wiles [56] and Taylor [52]. He uses the notion of a multiplicative polynomial law due to Roby [43,44].
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), (2) D is functorial on A-algebras, i.e. for any commutative A-algebras B → B , the diagram 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 χ(r, t) ∈ A[t] is given by D A [t] (t − r ) and is written (2.1) Indeed, the χ(r, t) for r ∈ R characterize the pseudorepresentation [9, Lem. 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 2.3 With R and A as above and any
We denote the resulting Spec B-groupoid by Rep R,D and note that Rep Chenevier, following the work of Roby [43,44], proved that the functor of d- When R is finitely generated over A, d A (R) ab is also finitely generated over A [9,Prop. 2.38].
The notion of a kernel of a pseudorepresentation provides a first step toward our goal of understanding ψ.

Definition 2.4
The kernel ker(D) of a pseudorepresentation D : R → A is a twosided ideal of elements r ∈ R such that for all A-algebras B and all r ∈ R ⊗ A B, the characteristic polynomial See [9,Sect. 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 following In Sect. 2.3, we will refine Corollary 2.6 using geometric invariant theory. This will rely in part upon understanding what base extensions make a pseudorepresentation become realizable as the determinant of a representation. That is, we are seeking a version of Theorem 2.5 where k =k.
Toward this goal, we first recall the following theorem of Chenevier. To state it, we need the following definitions. For any algebraic field extension K /k, we write K s for the maximal separable extension of k in K . Let ( f i , q i ) be the exponent of a field extension k i /k, i.e. f i = [k s i : k] and q i is the least power of p = char k such that k q i i ⊂ k s i . Given any simple k-algebra S i with center k i , there is a canonical n i f i q idimensional pseudorepresentation, denoted det S i : S i → k and defined as follows: det S i is the composition of the standard reduced norm S i → k i followed by the q i -Frobenius map F q i : k i → k s i followed by the standard field-theoretic norm k s i → k. (1) k is a perfect field, (2)  We consider our refinement to be a corollary of Chenevier's result, and we use the same notation. Proof The question of realizing det S i as the induced pseudorepresentation of a representation after some scalar extension may be addressed separately for each of the three factors composing det S i . First we address the reduced norm θ i : S i → k i . It is well-known that there exists a minimal finite separable extension k i /k i such that [22,Prop. 4.5.5]). This representation of S i over k i realizes θ i ⊗ k i k i . Later in the proof, it will be useful to draw this isomorphism as α i : Next we address the field-theoretic norm N i : k s i → k, which is a f i -dimensional k-linear pseudorepresentation. By definition, N i is the pseudorepresentation induced by the determinant of the regular representation ρ i : k s i → M f i (k) of k s i over k. We claim that the q i -Frobenius map F q i : k i → k s i , which is a q i -dimensional pseudorepresentation, is realized as the determinant of some representation after a finite scalar extension if and only if k i /k s i is finite. Secondly, when k i /k s i is finite we claim that this scalar extension may be taken to be separable if and only if it may be taken to be trivial if and only if k i /k s i is a simple extension. To prove the first claim, observe that the minimal dimensional k s i -linear representation of k i is the the regular representation Such an L i exists since any purely inseparable extension may be realized as a sequence of extensions of degree p achieved by adjoining pth roots. The ring k i ⊗ k s i L i is a local ring with residue field k i . The action of k i ⊗ k s i L i on k i realized by projection of the regular action to the residue field, which we will call β i below, is then a q i -dimensional L i -linear representation. One can check that its induced q i -dimensional pseudorepresentation is F q i ⊗ k s i L i . Indeed, the characteristic polynomial of the L i -linear action of α ∈ k i on k i is X q i − α q i .
Having proved the first claim, the second claim follows from the following observations. If K i /k s i is any finite separable extension, then k i ⊗ k s i K i is a field, and so the minimal dimension of a K i -linear representation of Also, we observe that q i = [k i : k s i ] if and only if k i /k s i is simple if and only if k i /k is simple.
Taking the fields L i and k i as above, let L i be the composite field L i k i s ∼ = k i s ⊗ k s i L i , which has degree over k satisfying the bound [L i : k] ≤ [k i : k]n i /q i . We claim that there is a L i -linear representation of S i realizing det S i ⊗ k L i . First we note that we can draw an isomorphism The rightmost ordering of tensor factors makes it clear that we can apply the composition of appropriate scalar extensions of α i and β i , followed by ρ i on the k s i factor to obtain a L i -linear n i q i f i -dimensional representation The composite field k of all of these field extensions L i of k then satisfies the properties sought after in the statement, including the degree bound. In particular, the pseudorepresentation D from the statement of the corollary is realized, after a base change to D ⊗ k k , by the d-dimensional k -linear representation Moreover, (1) follows from the observation that when k i /k is simple, L i may be taken to be separable with [L i : k] bounded by n i f i . This results on the desired restrictions on k in the case that k i /k is simple for all i. Under the assumptions of (2), the arguments above allow us to take L i /k i to be the trivial extension, so that k /k is also the trivial extension. In particular, the assumption about the Brauer group guarantees that k i = k i .

Cayley-Hamilton algebras are polynomial identity rings
The notion of a Cayley-Hamilton pseudorepresentation will be critical in what follows.
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 [39]. When we take the Cayley-Hamilton algebra produced out of the universal d-dimensional pseudorepre- we can get a "universal Cayley-Hamilton algebra" (E(R) d , D u ) and "universal Cayley-Hamilton representation" ρ u : [9,Sect. 1.22]). The consequence of this universality that we are concerned with is the following where the map d A (R) ab → B is induced by the pseudorepresentation det •ρ B : R ⊗ A B → B. In partulcar, for ρ B = ρ u , there is a canonical isomorphism to the moduli of representations of E(R) d compatible with D u (see Definition 2.3). Now we will discuss polynomial identity rings, written PI-rings; we refer to the book [38] 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 x y − yx = 0.

Proposition 2.13 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 [38,Prop. 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 for all r ∈ R. By [38, Ch. VI, Thm. 2.7], any integral, finitely generated noncommutative 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 [36, Sects. 1.1. 3, 9.1.3]. Remarkably, this proposition along with Proposition 2.12 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. In particular, we have this strengthening of Chenevier's Theorem 2.7. Corollary 2.14 With the assumptions of Theorem 2.7, dim k R/ ker(D) is finite when R is finitely generated as a k-algebra.
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 2.15
Let A = k be field and let R be an associative (non-unital) kalgebra 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 [46] 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 [46] is the key input needed to loosen conditions guaranteeing that a deformation ring of pseudorepresentations of a profinite group is Noetherian (see Proposition 3.2). It will be used in the form of the following Proof The definition of ker(D) implies that 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 2.15 implies the result.

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 2.17
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 [1,2] provides a useful perspective on geometric 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.
Example 2.18 (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) Now G be a linearly reductive S-group scheme acting on X ; see e.g. [1,Sect. 12] for a definition. Then [X/G] → X//G is an example of a good moduli space. We will only require the fact that a torus is linearly reductive over any S. If S = Spec k and char k = 0, reductive is equivalent to linearly reductive; if char k > 0, linearly reductive means that the connected component of the identity in G is a torus, and the group of components has order prime to char k.
We will be interested in the particular case of the adequate moduli space Here are the main results of Alper's work.  19, we find that ν induces an isomorphism on geometric points. It is the same to say that ν is surjective and radicial [16, 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. an integral universal homeomorphism that is an isomorphism in characteristic zero. In the affine Noetherian case, this means that the kernel and cokernel of a ring map consists of finite modules of p-torsion nilpotents. It is possible to eliminate this difference in certain cases (see Theorem 2.27).

Theorem 2.20
If R is finitely generated over A, the difference ν between ψ : Rep d R → PsR d R and an adequate moduli space is an adequate homeomorphism.
We emphasize that the isomorphism in characteristic zero is due to Chenevier, using ideas of Procesi [39].
Proof The proof that ψ is precisely an adequate moduli space in equi-characteristic zero is due to Chenevier [8,Prop. 2.3]. We know that ν is surjective and radicial by the comments above, so in light of [21,Cor. 18.12.11], it remains to show that ν is finite. It will suffice to prove that ν is universally closed, since it is clearly affine, hence separated, and proper affine morphisms are finite. We will prove this by verifying the valuative criterion for universally closed morphisms given in [ The kernel of the action of R ⊗ A K on this representation factors through (R ⊗ A K )/ ker(D), which is finite-dimensional over K by Corollary 2.14. Then Corollary 2.9 tells us that this representation is, in fact, realizable as a representation ρ : where K /K is some 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 , which is a DVR [49,Prop. II.3]. We claim that ρ is isomorphic to ρ B ⊗ B K , where ρ B : R ⊗ A B → End B (L ) and L is a rank d projective B -module, which will complete the proof. Choose a ddimensional 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 Bmodule because the action of R ⊗ A B factors through its Cayley-Hamilton quotient (R ⊗ A B)/CH(D ⊗ A B) by Proposition 2.12, and this quotient is B-module-finite by Proposition 2.13. Therefore L is a B -lattice because it is finite and torsion-free, and the induced ρ B : R ⊗ A B → End B (L ) yields ρ after applying ⊗ B K .

Generalized matrix algebras
The concept of a generalized matrix algebra (GMA) with respect to a pseudocharacter has been carefully studied in [3, Sect. 1]. It will be helpful in the sequel 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 [55,Remark 2.3.3.6].
A pseudocharacter is the data of a trace coefficient function 1 satisfying identities expected of a trace function coming from a representation (see [3,Sect. 1.2], [52] for the definition), while a d-dimensional pseudorepresentation D keeps track of all characteristic polynomial coefficients (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 : e i Re i such that the trace map Tr = Tr E : R → A defined by is a central function, i.e. Tr(x y) = 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 [3, Sects. 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 [3, Sect. 1] on the structure of a GMA, which we record here in order to introduce notation.
• Write 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 We recall the definition of an adapted representation of a GMA.

Definition 2.22 ([3, Definition 1.3.6]) Let B be a commutative A-algebra and let
We define Rep Ad (R, E) to be the functor associating to an A-algebra B the set of adapted representations of (R, E) over B.
is represented by the affine scheme corresponding to the quotient of the A-algebra These are precisely the relations required to ensure that the 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.

Proposition 2.23 Given a GMA A-algebra
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 . We also have Tr = D E 1 .
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 : 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 = D E 1 by computing the characteristic polynomial χ D E (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 [3, Lem. 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,D E 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,D E via the immersion above. This means that the morphism (2.4) exists, and, furthermore, we show the following.

Proposition 2.24 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 Indeed, the base change of V X to G from X is a free rank d O G -vector bundle with a canonical basis adapted to (R, E). This defines a map G → Rep Ad (R, E), equivariant for the action of Z (E). We have therefore established a morphism We observe that this provides a quasi-inverse to (2.4).
In the case of a generalized matrix algebra, we can improve on Theorem 2.20.

Corollary 2.25 Let (R, E) be a generalized matrix A-algebra with canonical pseudorepresentation D
Proof We will argue that the invariant ring (T /J ) Z (E) of the Z (E)-action on the coordinate ring T /J of Rep Ad (R, E), given in (2.3), is equal to A. In light of Example 2.18, Rep Ad (R, E) → Spec A is a good moduli space because tori are linearly reductive over any base, and because A = (T /J ) Z (E) . Then the statement of the Corollary follows from Proposition 2.24.

It is clear that A is contained in (T /J ) Z (E) , and we will show any Z (E)-invariant in T /J is in A.
Indeed, Z (E) acts on A i, j as the torus in GL r acts by roots on the (i, j)coordinate of M r , so that the invariant subring of T is generated by tensors of the form [1,Rem. 4.11]. By considering the generators of J and the property (ASSO), we conclude that all of the invariant tensors are equivalent to elements of A.
The following conditions will be useful to show that certain Cayley-Hamilton algebras are GMAs. (1) We denote byD the residual pseudorepresentation D ⊗ A F : R ⊗ A F → F, and callD 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 Recall from Theorem 2.7 that (R ⊗ A F)/ ker(D) is a semi-simple F-algebra, so it is split after at base change by a finite extension of F.
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 D 1 of D [9, Thm. 2.22(ii)]. We will now remark that his proof also shows that the pseudorepresentation D E is equal to D, generalizing [3, Cor. 1.3.16] to any characteristic.

Theorem 2.27 Let (R, D) be a finitely generated Cayley-Hamilton A-algebra where A is a Noetherian Henselian local ring. Assume that D is residually multiplicityfree. Then (R, D) admits a structure E of a generalized matrix A-algebra such that the pseudorepresentation induced by (R, E) is equal to D. Moreover, there is an isomorphism
Proof It will suffice to prove the theorem after replacing the Cayley-Hamilton A-algebra (R, D) with the universal Cayley-Hamilton with residual pseudorep-resentationD. This is the Cayley-Hamilton RD-algebra (D u ) and D u is the universal pseudorepresentation deformingD, D u : R u → RD. Indeed, the good moduli space property claimed in the theorem will follow because it is stable under base change (Theorem 2.19 (7)).
Using the assumptions of the statement, Chenevier [9, Thm. 2.22(ii)] shows that there exists a data of idempotents E of R u inducing a generalized matrix algebra (R u , E) over RD. Both D u ⊗ RD F and D E ⊗ RD F are equal toD, as they each arise from the product of matrix algebras Consequently, D E is a deformation ofD with coefficients in RD, and the universal property of RD induces a map f : RD → RD induced by D E . The desired equality D u = D E will follow from this map being the identity. This follows immediately from the fact that D E is RD-linear by construction. One concrete way to observe this is to use the GMA structure to restrict D E and D u to the matrix subalgebra Both restrictions equal the determinant pseudorepresentations (see e.g. the proof of [9,Thm. 2.22]). In particular, for x ∈ RD, we can compute the traces of matrices with one non-zero entry: , so f is the identity. The rest of the theorem now follows from 2.24 and 2.25.

Formal moduli of representations and pseudorepresentations of a profinite group
Let G be a profinite group; we will often impose the p -finiteness condition of Mazur [35] 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 Z p , which is anti-equivalent to the category of affine Noetherian Spf Z p -formal schemes [16, Sect. 10.1]. We will use the category of Spf Z p -formal schemes FS Z p as coefficient spaces.
As in the previous section, we will use A as a base coefficient ring, and will let A ∈ Adm Z p 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 Definition 3.1 Let A and R be as specified above, and let d be a positive integer.
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 Aformal scheme when R satisfies a finiteness condition equivalent to the 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 [9]. Firstly, he shows in [9, Prop. 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 loosen Chenevier's criteria for RD to be Noetherian [9, Prop. 3.7].

Proposition 3.2 Assume that the continuous cohomology group H
Then RD is Noetherian. Proof Let F represent the coefficient field of ρ ss D . We apply Chenevier's strategy to prove [9, Prop. 3.7] via arguing from [9, 2.7, 2.26, 2.35]. The one change we make is that in [9, Lem. 2.7(iii)], we produce N (d) such that ker(D) N (d) = 0 using Corollary 2.16. This removes the condition that d! is invertible in F. [9,Remark 2.29]. It fulfills his suggestion that some result in the spirit of Shirshov's height theorem would allow for the proof of the Noetherianness of RD in terms of the finiteness of cohomology at ρ ss D alone, instead of the stronger condition that G satisfies the p finiteness condition. This result was provided by Samoilov [46] (see Corollary 2.16). On the other hand, the finiteness of cohomology for every ρ ss D implies that G satisfies condition p [9, Example 3.6].

Remark 3.3 Proposition 3.2 answers a question of Chenevier
The deformation theory of variousD will suffice to describe the entire moduli functor of d-dimensional pseudorepresentations on formal schemes X ∈ FS Z p , It will be helpful to establish notation about residual pseudorepresentation 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 pseudorepre-sentationsD : Definition 3. 4 We will letD : R ⊗ A FD → FD represent a residual pseudorepresentation, which is the unique representative of each equivalence class with smallest field of definition FD. By Corollary 2.9(2), there exists a semi-simple representation Notice that the irreducible factors of ρ ss D need not be absolutely irreducible, i.e. R/ ker(D) may not be a product of matrix algebras over F (cf. Definition 2.26). Rather, it may be a product of matrix algebras over finite extensions of F.
Chenevier has shown that the entire moduli of pseudorepresentations is simply a disjoint union of deformation functors of residual representations. 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) 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 to PsR d R , for exampleψ Using the decomposition of Theorem 3.5, we may study this morphism over one component of the base at the time. Fixing a residual pseudorepresentationD : PsRD to be the groupoid of representations with residual pseudorepresentationD, and writeψ for the base changê We may analogously define Rep D and RepD.
It is well-known thatψ is an isomorphism whenD is absolutely irreducible, i.e. when ρ ss D is absolutely irreducible (see [37,45] in the case of pseudocharacters, and [9, Thm. 2.22(i)] for pseudorepresentations). In this case, RepD is purely formal.

Algebraization of moduli of representations over moduli of pseudorepresentations
Our goal is to draw a conclusion aboutψ similar to Theorem 2.20. As before, our principal tool will be the universal Cayley-Hamilton algebra of Sect. 2.2. As nothing about its construction was particular to the finitely generated case, the universal continuous pseudodeformation ofD induces the universal Cayley-Hamilton algebra As a consequence of the following important properties of the universal Cayley-Hamilton algebra, the two definitions are identical.

(1) The natural map E(G)D →Ẽ(G)D is a topological isomorphism. (2) The quotient map RD[[G]] E(G)D is continuous. (3) E(G)D is module-finite as an RD-algebra, and therefore Noetherian. (4) On E(G)D, the profinite topology, the mD-adic topology, and the quotient topology from the surjection RD[[G]] E(G)D are equivalent. (5) WhenD is multiplicity-free, E(G)D is a generalized matrix algebra with canon-
ical pseudorepresentation equivalent to D uD . In particular, whenD is absolutely The image of this map is precisely the two-sided ideal generated by the image of the χ [α] , i.e. CH(D uD ), 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 finite-dimensional over FD. Firstly, Theorem 2.7(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 2.15 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 [9,Prop. 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 The algebra structure results of part (5) come from [9,Thm. 2.22], and the equivalence of the pseudorepresentations was proved in Theorem 2.27.
Using these results on E(G)D, we get a result for continuous representations analogous to Proposition 2.12.
There is a functorial equivalence of categories between continuous representations of G with coefficients in Adm Z p with induced residual pseudorepresentationD and continuous representa- There exists an inverse functor, associating to an admissible RD-algebra B with The following important result shows thatψ is algebraizable. This allows us to deduce further properties ofψ from the study of Sect. 2. We write PsRD for Spec RD.
Following [16,Sect. 10.13], we will call a homomorphism A → B in Adm Z p "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.
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 := The natural morphism ψ : RepD → PsRD has mD-adic completionψ and satisfies 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 3.2 and 3.6 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 uD whenD is multiplicityfree. Theorem 3.7 allows us to study representations of E(G)D in place of those of G. In order to see that the mD-adic completion of (RepD, ψ) is (RepD,ψ) (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 3.9
The condition "constant residual pseudorepresentationD" is no real restriction to the scope of Theorem 3.8 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 algebraization This algebraization result implies that the topology on an integral p-adic family V A of representations of G with coefficients in A ∈ Adm Z p and residual pseudorep-resentationD can always be strengthened to the mD-adic topology, and that there are subrings of A that are finitely generated over RD over which a model for V A exists.

Corollary 3.10 Let A ∈ Adm Z p and let (ρ A , V A ) ∈ RepD(A). Then with assumptions as in Theorem 3.8, (1) the G-action on V A remains continuous for the possibly stronger mD A-adic topology on A, where A has the structure of a continuous RD-algebra induced by the pseudorepresentation det •ρ A , (2) there exists a minimal formally finitely generated sub-RD-algebra
there exists a minimal finitely generated mD-adically separated 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 3.6. 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 uD -compatible representation of G valued in an arbitrary RD-algebra, i.e. not just those in Adm Z p . 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 will never be mD-adically separated because p ∈ mD is a unit in Q p .
The moduli functor Rep D parameterizes these continuous representations. As the image of G under ρ lies in a module-finite RD-subalgebra of M d (A), the mD-adic topology on this subalgebra is separated and ρ is continuous with respect to this topology.  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 3.14 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 3.8.

Consequences of formal GAGA for ψ
In order to descend closed loci under ψ, it will be helpful to know formal GAGA for ψ. By "formal GAGA for a morphism f " we mean that the completion functor on coherent sheaves on the domain of f is an equivalence of categories. The classical case is a proper morphism [18,Cor. 5.1.3]. We also know formal GAGA for ψ when ψ a good moduli space by [23]. Even when ψ is not a good moduli space, it satisfies formal GAGA under the following hypothesis.

(FGAMS)
Formal GAGA holds for adequate moduli spaces realized as quotient stacks by GL d .
Consequently, we have Proof IfD is multiplicity free, ψ is a good moduli space by Theorem 3.8, and [23] 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 3.17
According to the authors of [23], 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 6.9.) Lemma 3.18 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 morphismsŶ →X, and (c) closed immersions Y → X and closed immersionsŶ →X.

Remark 3.19
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 O S 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 Zariskilocal on the base provided that the data of an ample line bundle is included with the morphism (cf. [53,Sect. 17.3.4]).
Proof To prove (1), one observes that each of the possible types of schematic morphisms over X orX is controlled by coherent sheaves, whence the statements follow from formal GAGA. The cases (b) and (c) are covered by [18,Prop. 5.4.4]. The case (a) is also controlled by coherent sheaves, as a projective morphism Y → X is by definition where the quasi-coherent O X -algebra B = i≥0 B i is a direct sum of coherent sheaves with multiplication law composed of morphisms of coherent sheaves. The corresponding projective morphism toX is given byŶ = Proj OX i≥0B i .
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. [34, Cor. 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).

Families of Étale ϕ-modules and Kisin modules
After introducing notation in Sect. 4.1, we will describe the main point of this section in Sect. 4.2.

Background for families of G K ∞ -representations of bounded E-height
For a reference to the following fundamental definitions in p-adic Hodge theory, see e.g. [4].
Let k be a finite field of characteristic p > 0 and W := W (k) its ring of ptypical Witt vectors. W is the ring of integers of the finite unramified extension K 0 := W (k)[1/ p] of Q p . Let K /K 0 be a totally ramified extension of degree e. Fix an algebraic closureK of K , and a completion C p ofK and let G K := Gal(K /K ).
We recall the definitions of some p-adic period rings. Let OK be the ring of integers ofK and O C p 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 is k and is also canonically ak-algebra. The fraction field Fr R of R is a complete nonarchimedean algebraically closed characteristic p field. The elements x of R are in natural bijection with sequences of elements (x (n) ) n≥0 of O C p 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) 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 p n x n, (n) 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). 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), 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.
Then O E is a discrete valuation ring with residue field k((u)) and maximal ideal generated by 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 completion of S⊗ Z p A with respect to a defining system of ideals for the topology of A. It will be important to know that the following such rings are Noetherian.
There is also an isomorphism of rings S R ∼ → S⊗ Z p R, but it is not necessarily a topological isomorphism.
By [16,Prop. 10.13.5(ii)], the rings O E,A and S A are Noetherian because Spf A → Spf R is a formally finite type morphism of formal schemes.

Algebraic families of Étale 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. In this section, we also fix R to be an arbitrary Artinian (and, therefore, finite cardinality) subring of A. In Sect. 4.4, we apply these results where R is Z p -subalgebra of A generated by characteristic polynomial coefficients of the G K or G K ∞ -action.
Our goal is to compare these families of Galois representations to étale ϕ modules. These étale ϕ-modules are finite modules M over certain of the Noetherian rings of  (5) Let V be the covariant 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. The proof will be given below after stating one more result.

(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 .

In particular, (a) V A is projective as an A-module of constant rank d if and only if M(V A ) is a projective O E,A -module of constant rank d. (b) V A is free as an A-module with rank d if and only if M(V A ) is a free O E,Amodule of rank d.
First we assemble useful facts about limits. We will append (−) ∞ to categories defined in Definition/Lemma 4.2 to indicate that the A-module finiteness condition has been dropped.   In order to substantiate Definition/Lemma 4.2 and Proposition 4.4, 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) Let V A ∈ ob Mod G K∞ (A). Because the action of G K ∞ has a finite index kernel, we have a canonical isomorphism as R[G K ∞ ]-modules of
We note that the functor M (resp. V ) commutes with injective direct limits in Mod G K∞ (R) (resp. M (R)), using Facts 4.5 and 4.6 above along with the fact that the tensor product ⊗ Z p O E ur (resp. ⊗ O E O E ur ) preserves injective maps.
Therefore there are canonical isomorphisms in ∞ M (R) of colimits of objects of M (R), and the fact that M is an equivalence of categories out of Mod G K∞ (R) commuting with the necessary colimits implies that there is a canonical isomorphism respecting all structures (4. 2) 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 Amodule; moreover, it is an O E,A -module with a Frobenius semi-linear endomorphism. To complete the proof that M is well-defined in Definition/Lemma 4.2, we must show that M(V A ) is finite as an O E,A -module.
Let H be the open kernel of the action of G K ∞ on V A . Since H acts trivially on V A , the canonical isomorphism above induces a canonical isomorphism

Functors of lattices and affine grassmannians
The assumptions on A, R, and V A remain the same as in the previous section. We will study lattices in the étale ϕ-module and we may consider S A -lattices within M A which are stable under the Frobenius semi-linear endomorphism ϕ on M A . We will show in this section that the functor associating to an A-algebra B the ϕ-stable S B -sublattices of M A ⊗ A B satisfying the condition "E-height ≤ h" is represented by a projective A-scheme. We will use the affine Grassmannian for this, generalizing the result of [28, Sect. 2.1] and [26, Sect. 1], which was done in the case that A is Artinian.
First we will briefly review the theory of the affine Grassmannian; see [42, Sect. 2] for thorough and general treatment of affine Grassmannians. Affine Grassmannians for GL d and related groups (see below) are functors of sublattices of projective, constant rank modules. The local affine Grassmannian parameterizes these vector bundles over the formal one-dimensional disk D which are trivialized on the punctured disk. The global affine Grassmannian parameterizes these vector bundles over the affine line A 1 which are trivialized on the punctured line.

Definition 4.8 Let V A be a projective rank d A-module. Write
Then the affine Grassmannians we will require are the following functors.
(1) The local affine Grassmannian Gr loc G for G is the functor associating to a A-algebra B the set of pairs (P D , η) where P D is a projective rank d W B [[t]]-module and η is an isomorphism (2) The global affine Grassmannian Gr glob G for G is the functor assigning to an Aalgebra B the set of pairs (P A 1 , η), where P A 1 is a projective rank d W B [t]-module and η is an isomorphism We observe that there is a natural functor We also want to know that Gr G is ind-projective over Spec A with a canonical ample line bundle. Like before, we will reduce to the case A = Z p by replacing Spec A with a Zariski cover trivializing V A . The fact that Gr Res W/Z p GL d is ind-proper over Spec Z p is given in e.g. [ We summarize what we have proved in the following

Theorem 4.9 Let S be a locally Noetherian scheme, and let V be a projective, coherent, constant rank O S -module. Then Gr Res W/Z p GL W A (V ⊗ Z p W ) is an ind-projective S-scheme with a canonical ample line bundle.
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. In what follows, we writeŜ B for the u-adic completion of S B ; they are both Noetherian (Lemma 4.1).

glob G for G/A. (2) The functor F V A associating to a finitely generated A-algebra B the S Bsublattices of M B := M A ⊗ A B (3) The local affine Grassmannian Gr loc G for G/A. induced by tensoring
Gr loc G (4.5) which factors the composite isomorphism (4.4).

Remark 4.11
We will see in the proof that the natural isomorphisms between the affine Grassmannians and F V A is not canonical. This is not a new phenomenon that arises when A is no longer Artinian as it was in [26]; bases were implicitly chosen there as well. However, because of the nature of its construction, the canonical ample line bundle on F V A does not depend on the choice of basis.
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 4.4 (4). We observe that the two morphisms in (4.5) factor (4.4), and therefore it will suffice to show that the latter morphism ⊗ S B B [[u]] is a monomorphism of functors.
Choose a basis for M A . For any finitely generated A-algebra B, let M B ∈ F V A (B) denote the S B -lattice generated by the induced basis of M B = M A ⊗ A B. As remarked in [32, Example 2.2.7], we note that F V A (B) is a direct limit over n ≥ 0 of functors of lattices N B ⊂ M B satisfying u −n M B ⊇ N B ⊇ u n M B . This filtered direct limit exists for each of the global/local affine Grassmannian functors as well as F V A , and is compatible with and unchanged by the tensor maps of (4.5). Therefore the latter map of (4.5) 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 4.4(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.

7]) The functor L ≤h V A is represented by a projective A-scheme L ≤h V A with a canonical ample line bundle. If A → A is finitely generated and V
and applying (−) ϕ=1 to the canonical isomorphism (4.2), we have a canonical isomorphism It remains to show that the rightmost factor of (4.9) and the rightmost factor of (4.6) are canonically G K ∞ -equivariantly isomorphic.
Then tensor-Hom adjunction results in an isomorphism

A universal family of Kisin modules in characteristic 0
While the previous parts of Sect. 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 )| G K∞ for each i ≥ 1 and extend to the limit, where R above may be set to be the image of RD in A/m iD A. For example, the functor M generalizes to this setting naturally from the above, since the map of limits is an isomorphism by Fact 4.7 and the fact that the ideal ( p ⊗ 1) + O E ur ⊗ mD A (for which the left side is the completion) is equal to O E ur ⊗ Z p mD A (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 4.15
The functor L ≤h V A on A-algebras B such that m iD · B = 0 for some i ≥ 1 is represented by a projective A-scheme L ≤h V A . Proof By Proposition 4.13 and Remark 3.19, this functor is represented by a projective formal scheme with a ample line bundle compatible with its limit structure. By applying formal GAGA 3.18(1a), we conclude that L ≤h V A is the mD-adic completion of a projective A-scheme.
We now study the the map A : L ≤h A → Spec A, showing that it is a closed immersion in equi-characteristic zero.

4]) Let A and V A be as specified above. Then (1) The map A : L ≤h V A → Spec A is a closed immersion after inverting p. (2) If A ≤h is the quotient of A corresponding to the scheme-theoretic image of A , then for any finite W (F)[1/ p]-algebra B, a continuous A → B factors through A ≤h if and only if V B = V A ⊗ A B is of E-height ≤ h.
Part (1)

a quotient A of A[1/ p] is characterized by those homomorphisms from A[1/ p] to finite Q p -algebras which factor through A , (4) a finite A[1/ p]-module V is projective if and only if V ⊗ A B is projective for every finite Q p -algebra B receiving a homomorphism A → B, (5) the image of A in a residue field from part (2) is an order in its ring of integers, and (6) the image of A in a finite local Q p -algebra B with residue field E lies in some
Statements (1), (2), (3), and (4) (3). 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).
To prove (1) and (2) for A alg , consider that (1) and (2)  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 [26] needs to be modified.

) 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 ⊗ Z p B-modules (4) There is a canonical isomorphism Proof LetL ≤h V A be the mD-adic completion of L ≤h V A . Then is a projective morphism of Spf(A)-formal schemes, and its base is Noetherian by Lemma 4.1. By Corollary 4.15, there exists a universal Kisin moduleM on the mDadic completionL ≤h it is a locally free coherent sheaf. Applying formal GAGA for S A ,M is the completion of a finite locally free module M on the projective S A -scheme The scheme theoretic image of S A is S A ≤h . We set With this work done, the proofs of part (1), (2), (3) and (4) may be repeated from [26,Cor. 1.7]. For part (4), we remark that just as in loc. cit., there is a canonical isomorphism produced by combining Lemma 4.14, which gives this isomorphism whenÃ is replaced byÃ/m iD A for each i ≥ 1, and the theorem on formal functions [18,Thm. 4.1.5]. Then the right side is de-completed by formal GAGA, and (4) then follows from the fact that the kernel and cokernel of A ≤h →Ã are p-torsion.

Period maps and (ϕ, N)-modules in families
So far, we have cut out loci of G K ∞ -representations with E-height ≤ h. In this section and in Sect. 6, we refine this locus to cut out semi-stable G K -representations with Hodge-Tate weights in [0, h]. In order to do this, we will construct a family of (ϕ, N )modules from the family of Kisin modules already produced. The locus will be cut out in characteristic 0, but the construction relies on the family having an integral model. In Sect. 5.1 we will set up notation and prove lemmas so that we can use the required period rings in families. In Sect. 5.2 we will carry out the constructions needed to cut out the semi-stable locus; the result will appear in Sect. 6.

Background and notation
We now change notation, writing A • for what we called A above, and writing A for what we called A[1/ p] above. Also, assume that A • is p-torsion free. We also make analogous notation changes to RD as follows.  [1/ p], and therefore to O as well.
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.  [1/ p], 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  (ker θ, p) for A cris . Define B + dR to be the ker(θ )adic completion of W (R) [1/ p], where θ is extended to a map θ : W (R) [1/ p] C p , and let B dR be its fraction field.
Recall from Sect. 4.1 the definition of [π ], the image of u in W (R), and [ε] of (4.1). Write 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 u is transcendental over the fraction field of B cris . One computes that Frobenius ϕ acts as ϕ( u ) = p u and ϕ(t) = pt.
Equip B + st with an endomorphism N by formal differentiation d/d u of the variable u with coefficients in B + cris , i.e. so that N (B + cris ) = 0. Extend ϕ to B + st as well, with ϕ( u ) = p u . We note that ϕ and N define endomorphisms of the polynomial subring K 0 [ u ] ⊂ B + st , and that pϕ N = N ϕ on B st . There is an exhaustive, decreasing filtration on each of A cris , B + cris , written induced by their inclusion in the filtered ring B + dR , such that Fil 0 A cris = A cris and Fil 0 B + cris = B + cris . The filtration on B + dR is given by In fact, t ∈ Fil 1 B + dR and t / ∈ Fil 2 B + dR , 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. It is also well-known that B + st is stable under G K ; this also follows from the following calculation (see [55,Lem. 4.6.4]), which we will need later. (1) with respect to the cyclotomic character, belonging to the cohomology class associated to π by Kummer theory. When β(σ ) = 0, it generates the maximal ideal of B + dR . As this maximal ideal generates the filtration on B + dR , if β(σ ) = 0 then

Lemma 5.2 The map β given by
We will use the following families of period rings over A.
The map ϕ extends to each of these rings B-linearly, with N again acting as formal differentiation with respect to 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 mD A • -adic completion of Fil i A cris ⊗ Z p 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 (5.4) 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, 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 by showing, for large enough n, the existence of maps Indeed, these maps exist for n ≥ e because the eth power of u maps to a divided power ideal for A cris relative to W (R), as mentioned above. With the construction complete, Lemma 5.3 implies that this map will remain injective after tensoring with A • and completing with respect to the mD A • -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 5.3 ([26, Lem. 2.3.1]) Let R be a Noetherian ring that is I -adically complete and separated for some ideal I ⊂ R. For any R-module M, denote by M its I -adic completion. If M is a flat R-module, then
(1) For any finite R-module N , the natural map Lemma 5.3 is given in [26] 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.

Lemma 5.4 (Generalizing [26, Lem. 2.3.2])
Let R be a admissible Z p -algebra, Iadically complete and separated. Also, assume that R is p-torsion free.
(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 ( To prove part (5), consider that if 0 = f ∈ A cris,R , we may fix some n such that 0 = f ∈ A cris,R/I n = A cris ⊗ Z p R/I n . There is an injective map with m varying over the maximal ideals of R/I n , which are in natural bijective correspondence with the maximal ideals of R. Because A cris is Z p -flat, there exists some maximal ideal m ⊂ R such that the projection of f to A cris,(R/I n ) ∧ m is non-zero.
Then there exists a positive integer a such that the projection of f to A cris,R/(I n +m a ) is non-zero. Notice that f naturally projects from A cris,R to A cris,(R/I n ) ∧ m , which then projects to A cris,R/(I n +m a ) . Therefore the images of f in A cris,(R/I n ) ∧ m and A cris,R ∧ m are non-zero. Because the statement of (5) is known in the case that R is local [26, Lem. 2.3.2(5)], we apply this case to R ∧ m to produce an ideal q ⊂ R ∧ m with the desired property. Then the kernel of the surjective composite R → R ∧ m → R ∧ m /q is an ideal of R with the desired property.
The following lemma will be useful to construct loci cutting out conditions realized over a family of period rings.

Lemma 5.5 (Generalizing [26, Lem. 2.3.3]) Let M be an A • -module and x ∈
Proof Assume that m nD · M = 0 for some n ≥ 1, and choose some Because the statement was shown to be true when A • is local in [26,Lem. 2.3.3], we may apply the statement, with RD in place of A • , to the left hand side. Here we are using the assumption that n exists as above, since this implies that A cris,RD ⊗ RD M ∼ = A cris,A • ⊗ A • M. 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 [26,Lem. 2.3.3] deduces the Lemma in the case that A • is local from the case that M has finite length. The same proof works in this setting, so we briefly sketch it. Assume that M is a finitely generated A • -module, as the general case will follow. Denote by x n the image of x in A cris,A • ⊗ A • M/mD n M, and denote by N (x n ) the submodule of M/mD n M obtained from the argument above. One then observes that the image of N (x n+1 ) in M/mD n M is N (x n ), and that the limit N (x) := lim ← −n N (x n ) is the desired submodule of M (using Lemma 5.3(1)).

Period maps in families
Suppose now that V A • has an A • -linear action of G K such that its restriction to G K ∞ has E-height ≤ h, in the sense that (A • ) ≤h = A • (cf. Proposition 4.16 (2)). Write We will now follow [26,Sect. 2.4] in constructing a period map (5.4) comparing the family V A of G K -representations to a family of (ϕ, N )-modules. Using the results of Sect. 4, we can compare V A | G K∞ to a family of Kisin modules. We will produce a family of (ϕ, N )-modules from Kisin modules, and add additional structure needed to descend G K ∞ -representations to G K -representations. Our goal is to produce A st , the maximal quotient of A over which V A is semi-stable with Hodge-Tate weights in [0, h].
has cokernel killed by E(u) h . One part of the period map comes from Proposition 4.18 (4), which provides a canonical, 3) The following lemma supplies the other half of the period map, comparing V A with the candidate family of (ϕ, N )-modules D A . Indeed, we write each of which have a natural induced action of ϕ. Denote by S 0,A the completion of We may now produce the period map from ξ and ι as follows. The map ι induces a S A -linear, ϕ-equivariant map

Lemma 5.6 ([26], Lem. 2.2) There is a unique, ϕ-compatible, W A -linear map ξ : D A → M A , whose reduction modulo u is the identity on D A . The induced map D A ⊗ W A O A → M A has cokernel killed by λ h , and the image of the map D
Applying ⊗ S A O A to it and composing it with ξ , we have a ϕ-equivariant map Tensoring the composition of these maps by ⊗ A B, where B is any A-algebra, there is a B + cris,B -linear 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 because ι above is G K ∞ -equivariant. In order to extend the action of G K ∞ on the left hand side of (5.4) to an action of G K , we suppose that there is a W B -linear map which satisfies the identity pϕ N = 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 u to B + cris,B to get B + st, with a B-linear action of N and ϕ. Consider the composite of the isomorphisms where the first map is the inverse to the natural isomorphism We claim that (5.4) is G K -equivariant if and only if (5.7) is equivariant when G K is regarded as acting trivially on D B . A key observation is that the an inverse to the 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 (5.4), and for B = A. Indeed, the case for general B arises from the case B = A by applying ⊗ A B, and this map will remain injective after ⊗ A B if the cokernel for B = A is a flat A-module. Lemma 5.4(5) immediately reduces the injectivity claim to the case that A • is finite over Z p , which was proved in [26,Lem. 2.4.6].
To show that the cokernel of (5.4) is flat, it suffices to show that (5.4) 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 Proof We may freely assume that V A is a free rank d A-module. In this case, the construction of A st in [26,Prop. 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 5.5 and 5.3 and the map (5.4), 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 pϕ N = N ϕ is representable by a finitely generated A-algebra A N .
Write η B for the map of (5.4), 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 N -basis for Q and let x 1 , . . . , x r be the coordinates of δ σ (d) with respect to the basis. Applying Lemma 5.5 with M = A N and x = x i for x i varying over a B + cris,A N -basis 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 5.3(1)

Algebraic families of potentially semi-stable Galois representations
We will maintain the notation for A, A • , etc. established in Sect. 5.1, but we drop the assumption of Sect. 5.2 that the family V A • of representations of G K has E-height ≤ h. We have studied the period map 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 semistable 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 equicharacteristic 0.

Families of semi-stable Galois representations with bounded Hodge-Tate weight
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 have followed the techniques of Kisin [26]; see Hartl and Hellmann [24] for another approach.
respecting the action of ϕ and N .  (4).
We must fully explain the proof of part (3). By part (2) and Lemma 5.7, (6.1) 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 • [9,Prop. 3.13]; this is the case because 1-dimensional representations are identical to 1dimensional pseudorepresentations. It is then evident that (6.1) arises by ⊗ B st,R B st,A from the same map where A is replaced by R, and that it suffices to prove that (6.1) is an isomorphism when A = R. This was done in [26,Prop. 2.7.2]. Then (6.2) is an isomorphism by Lemma 6.2.
We need the following lemma in order to find the G K -invariants in B st,A .
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 r i 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.

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 equicharacteristic 0). First we will address the Hodge type, following [26,Sect. 2.6] and [27,Sect. A.4]. For this, we fix an finite extension field E of Q p and suppose that A admits the structure of an E-algebra.
realizes the ith part of the Hodge filtration of D B ⊗ K 0 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 4.17(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 := A st,h ⊗ A A v .

Galois type
Next we will study families of potentially semi-stable G K -representations, following [26,Sect. 2.7.5]. We stipulate that B is an Artinian local E-algebra with residue field E. Let V B ∈ Rep d G K (B). Following [14], set where K runs over finite field extensions of K . LetK 0 ⊂K denote the maximal unramified extension of K 0 , and let G K 0 ⊂ G K be the inertia group of G K . Then D * pst (V B ) is a B ⊗ Q pK 0 -module with a Frobenius semilinear Frobenius automorphism ϕ, a nilpotent endomorphism N such that pϕ N = N ϕ, and a B ⊗ Q pK 0 -linear action of G K 0 which has open kernel and commutes with ϕ and N .
Following [26,Sect. 2.7] along the line of reasoning of [28, Lem. 1.2.2(4)], we see that D * pst (V B ) is finite and free as a B ⊗ Q pK 0 -module. Since the action of G K 0 commutes with the action ϕ, the traces of elements of G K 0 are contained in B, and D * pst descends to a representation of G K 0 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 K 0 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 .
Definition 6.5 Let T : G K 0 → GL d (Q p ) be a representation with open kernel. We say that V B is potentially 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 K 0 , 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 6.3; fix a representation We will give a sketch.
Proof Let L/K be a finite Galois extension such that I L ⊆ ker T . Theorem 6.4 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 | G L is semi-stable with Hodge type v. One then applies Theorem 6.1 (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(3).

Universal families of representations, and algebraization
We will summarize what we have proved by producing universal spaces of potentially semi-stable 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 .
We will use the following notation for the universal spaces, in analogy with Sect. 5.1.  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

Formerly Rep
Proof 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 6.7 directly. That is, using formal GAGA for ψ freely, it is possible to carry out all of the work of Sect. 4.4 and Sect. 5 with the mD-adically separated finitely generated RD-subalgebra A • alg ⊂ A • of Corollary 3.10 in place of A • . For example, even the Cauchy sequence used to construct the map ξ of Lemma 5.6 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 6.7 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 [19, Ch. 0, Thm. 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 [5] 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 [26,Thm. 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 [15,Cor. 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)  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. [26,Sect. 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 [28, Sect. 2.3], we observe that there is an isomorphism of AR W (F),

Potentially semi-stable pseudorepresentations
We must clarify what it means to ask if a pseudorepresentation satisfies some property which, a priori, only applies to representations. Definition 7.1 Let K be a full subcategory of the category of perfect fields which is closed under finite 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 finite base change V ⊗ K K are each in P.
Then a pseudorepresentation D ∈ D over K ∈ K has property P if, given a finite 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 2.9), 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 Q p over p-adic fields, and let the property P be "crystalline," or any of the conditions of Sect. 6.4.
While it seems possible to emulate Definition 7.1 over non-fields or non-perfect fields if appropriate conditions on P are imposed, we do not pursue 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.
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 3.8(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 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 [1,Thm. 4.16(viii)]. Then, the uniqueness of R C follows from Lemma 4.17 and the fact that it is reduced. 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 pseudo-rational when C is a crystalline condition.  (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 [47] of Schoutens is a generalization of the Hochster-Roberts theorem to this setting.

Corollary 7.5
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. [1,Remark 4.13]. The main theorem [47,Thm. 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 [47,Sect. 4]. Formally unramified is equivalent to reduced, since R C is finitely generated over the excellent ring RD. Remark 7.6 Reducedness and normality of R C are clear from the regularity of S without resorting to Schoutens' result. Cf. also [1,Thm. 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. [1,Thm. 4.16(vii)], Theorem 3.8(2)), the analysis of the geometrically connected components of R C amounts to analysis of the geometrically connected components of the affine scheme Rep D .

Corollary 7.7
There is a natural bijective correspondence between the geometrically connected components of each of Rep ,C D , Rep ,C D , Rep C D , and Spec R C .

Global potentially semi-stable pseudodeformation rings
In this section, we will assume that all algebraizations of stacks of potentially semistable 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 p finiteness condition, the universal ramified-only-at-S pseudodeformation ring RD of global Galois representations is Noetherian (Theorem 3.5). 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| G v is not irreducible), and the quotients R v ρ → R C v ρ corresponding to the condition C v on representations of G v deformingρ| G v . Then one sets in order to obtain a deformation ring parameterizing representations of G F,S deforminḡ ρ 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 vD → RD and R vD → R C v D exist. The reason is that ifρ is irreducible butρ| G v is not, then a deformation D : G F,S → E ofD (where E/Q p is a finite extension) such that D| G v is reducible may have information about extensions between the Jordan-Hölder factors of D| G v , while D| G v 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 pseu-dorepresentationD; it is algebraizable of finite type over Spec RD via ψ. There are also analogous local spaces Rep Proof After possibly allowing a finite extension of the coefficient field E /E, there exists a unique representation V ss z of G F,S over E inducing D z . Recall that V ss z induces the unique closed point ζ of RepD lying over z ∈ Spec RD. Therefore, because ψ : RepD → Spec RD is universally closed and Rep C D → RepD is a closed immersion, we must have ζ ∈ Rep C D if and only if z ∈ Spec R C D . This means that V ss z satisfies C, i.e. its restriction to G v satisfies C v for all v ∈ S.

Example: ordinary pseudodeformation rings
The first local deformation conditions commonly dealt with were "ordinary" [35] and "flat" [41]. The constructions of this paper result in a "flat pseudodeformation ring" since flat is equivalent to "crystalline with Hodge-Tate weights in [0, 1]" [7,25]. However, because the ordinary condition allows for arbitrary Hodge-Tate weights, it is not included in these constructions. Here, we will construct an ordinary pseudodeformation ring in the 2-dimensional case; this should be compared with the ordinary pseudodeformation ring of [10,Sect. 3]. There are several notions of "ordinary;" we use the following one. Definition 7.10 Let K be a p-adic local field. We call a 2-dimensional representation of G K ordinary when it is reducible and there exists an unramified 1-dimensional quotient.
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 a 2-dimensional pseudorepresentation D of G Q,S such that D| G p det •(ψ ⊕ χ) where ψ is unramified is not necessarily ordinary.
We letD arise from the sum of two charactersψ,χ valued in F × , writingD = det(ψ ⊕χ) and stipulating thatψ| I p = 1 so that the set of ordinary pseudodeformations ofD is not empty. We also assume thatψ| G p =χ | G p .

Remark 7.11
One may naturally ask if there is a reasonable generalization of the ordinary condition to n-dimensional representations. For simplicity, assume thatD splits into a sum of characters of G Q,S . In this setting, ordinary will mean totally reducible when restricted to G p (no condition on the inertia action). By observing the following construction, one can see that this will be possible if the residual characters are pairwise distinct after restriction to G p , generalizing the "residually p-distinguished" conditionψ| G p =χ| G p . Lemma 7.12 With the assumptions above, there exists a closed substack Rep ord G p ,D of Rep G p ,D parameterizing representations of G p with induced residual pseudorepre-sentationD that are ordinary.
Proof We will use Theorem 3.7 and consider representations of E(G p )D. AsD is multiplicity free, Theorem 2.27 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 just as in (7.1), and then construct the global ordinary pseudodeformation ring R ord D by Definition 7.8. BecauseD is multiplicity-free, ψ is a good moduli space (Theorem 3.8(4)). The restriction of ψ to Rep ord D → Spec R ord D is a good moduli space as well (Theorem 2.19 (7)), and R ord D is precisely the associated GIT quotient ring. Proof Combine Lemma 7.12 and Theorem 7.9.
See [57] for a deformation-theoretic definition of ordinary pseudorepresentation.