Degenerating products of flag varieties and applications to the Breuil--Mezard conjecture

We consider closed subschemes in the affine grassmannian obtained by degenerating $e$-fold products of flag varieties, embedded via a tuple of dominant cocharacters. For $G= \operatorname{GL}_2$, and cocharacters small relative to the characteristic, we relate the cycles of these degenerations to the representation theory of $G$. We then show that these degenerations smoothly model the geometry of (the special fibre of) low weight crystalline subspaces inside the Emerton--Gee stack classifying $p$-adic representations of the Galois group of a finite extension of $\mathbb{Q}_p$. As an application we prove new cases of the Breuil--M\'ezard conjecture in dimension two.


Introduction
Overview.Let K be a finite extension of Q p with residue field k and let X d denote the Emerton-Gee stack classifying d-dimensional p-adic representations of G K .Inside X d there are closed substacks X µ,τ d classifying potentially crystalline representations of type (µ, τ ), for µ and τ respectively Hodge and inertial types.When µ is regular (i.e.consists of distinct integers) these closed substacks have • m(λ, µ, τ ) denotes the multiplicity with which λ appears in an explicit Frepresentation V (µ, τ ) of GL d (k) attached to µ and τ .
(there is also a version of the conjecture for substacks of potentially semistable representations; the conjecture has the same shape but with altered V (µ, τ )).These identities have been verified in only a small number of cases: (1) When K = Q p and d = 2, using the p-adic Langlands correspondence.See [Kis09a, Paš15, HT15, San14, Tun21].(2) When d = 2 and µ = (1, 0), as consequence of certain modularity lifting theorems.See [GK14].(3) When K is unramified over Q p , d is arbitrary, and both p and τ are generic relative to µ. See [LLHLM20].Again modularity lifting technique play an important role.
In this paper we construct Breuil-Mézard identities in a fourth setting: we are interested in the two dimensional case where τ = 1 (i.e.we consider only crystalline rather than potentially crystalline representations) and µ is bounded so that the representation theory of GL 2 (k) in the conjecture behaves as it does in characteristic zero.We do this by constructing analogous identities involving certain degenerations of products of flag varieties embedded in the affine grassmannian, and then relating the geometry of these degenerations to the geometry of the X λ,1 d .
Main result.First we describe a bound on the Hodge types considered above, which is natural in the sense that the GL d (k)-representation theory appearing in the conjecture changes markedly once the bound is passed.Recall that a Hodge type µ consists of a d-tuple of integers for each embedding κ ∶ K → Q p .If one assumes that (1.2) for each embedding κ 0 ∶ k ↪ F p then: • V (µ, 1) is a tensor product over the embeddings κ of representations of highest weight µ κ and the Jordan-Holder factors of this tensor product are computed in characteristic p just as they are in characteristic zero, by Littlewood-Richardson coefficients.• Each Jordan-Holder factor λ of V (µ, 1) can be written as V ( λ, 1) for some Hodge type λ uniquely determined up to an ordering of the embeddings κ.In particular, the cycles C λ appearing in (1.1) for these small µ are uniquely determined by the conjectured identity for µ = λ; one has [X λ d ] = C λ .Thus, the following theorem establishes new cases of the conjecture: Theorem 1.3.Assume that d = 2, p > 2, µ is regular, and that There are some comments to make before we discuss what goes into the proof.Firstly, the theorem has two clear limitations: the assumption that d = 2 and the fact that the bound on µ is stronger than that in (1.2) (we have no expectation whatever that the methods in this paper apply beyond (1.2)).
As we explain in more detail below, the proof of the theorem has two key inputs.The first involves relating the X µ,1 d with certain local models we define inside the affine grassmannian.This can be done without any restriction on d but our current argument requires the stronger bound on µ.The second key input is a lower bound on the Breuil-Mézard identities which has been established when d = 2 using global techniques from [GK14] (this is also where the assumption p > 2 appears).
Finally, taking λ = µ shows that the cycle [X λ 2 ] is independent of the choice of "lift" λ of λ.We also show that each of these cycles consists of a single irreducible component occurring with multiplicity one.
Method.The proof of the theorem divides into three parts: Part 1: Local models in the affine grassmannian.The starting point of the proof is the construction of certain projective schemes whose special fibres give upper bounds on the multiplicities appearing in Theorem 1.3.To explain their construction we fix a sufficiently large extension E of Q p , with ring of integers O and residue field F, and consider a mixed characteristic version of the affine grassmannian Gr O over O whose special and generic fibres are given by Here κ 0 and κ are embeddings and Gr is the affine grassmannian over O K whose A points, for A a p-adically complete O K -algebra, classify rank d-projective A[[u]]modules satisfying for some a ∈ Z ≥0 and π ∈ K a fixed choice of uniformiser.For each dominant cocharacter λ of G = GL d there is a closed immersion of the flag variety G P λ → Gr (P λ ⊂ G being the parabolic corresponding to λ).This allows us to define, for any Hodge type µ = (µ κ ), an O-flat closed subscheme M µ in Gr O by taking the closure in Gr O of κ The following summarises the key results we prove regarding these M µ 's Proposition 1.5.Assume that µ is regular.
(1) If µ satisfies (1.2) then there exist n(λ, µ) ∈ Z such that in the group of with the sum running those irreducible GL d (k)-representations for which the Hodge type λ also satisfies (1.2).
The first part is proved by constructing an explicit closed locus Gr ∇ O ⊂ Gr O defined in terms of a differential operator ∇ (this is a variant of locus considered in [LLHLM20]).A direct computation shows that if we bound the height according to (1.2) then the resulting closed subscheme of Gr ∇ O ⊗ O F consists of irreducible components of dimension ≤ dim M µ .Furthermore, those components with maximal dimension are labelled by the λ's appearing in (1).One can also show that M µ ⊗ O F is contained in this closed subscheme.From these observations we are able to prove (1).
Remark 1.6.Unfortunately, this explicit moduli interpretation is only a good topological approximation of M µ ⊗ O F; typically the components appear with much too high multiplicity.
Part (2) is proved by constructing an explicit resolution of X → M λ with X smooth and which is an isomorphism on the generic fibre.Unfortunately, we do not know how to construct such resolutions when d > 2 (or whether they are likely to exist).
For part (3) we consider the restriction of the determinant line bundle on Gr O to M µ .Since the generic fibre of M µ is a product of flag varieties it is easy to compute that for where H 0 (µ κ ) denotes the algebraic representation of G over K of highest weight µ κ .We point out that this tensor product differs from the V (µ, 1) appearing in the Breuil-Mézard conjecture in that V (µ, 1) is obtained as the reduction modulo p of such a tensor product, but in which µ κ is replaced by µ κ − ρ for ρ = (d − 1, d − 2, . . ., 1, 0).Nevertheless, these multiplicities are approximately the same, in the sense that if, in the Grothendieck group of E-representations, one has equals the value at n of a polynomial of degree < dim M µ .Since the representations ⊗ κ H 0 (nµ κ ) ⊗ O K ,κ E can be obtained by replacing L det with L ⊗n det in (1.7), the identity of cycles in part (1) implies that is also the value of a polynomial in n of degree < dim M µ , at least for n >> 0. Taking the difference shows that To prove (1) it suffices to show this factorisation for A-points for every finite Falgebra A. For simplicity, we sketch the argument only in the case where A = F.The general case requires only minor technical changes.We also assume k = F p as this greatly simplifies the notation.If e = 1 then M µ = G P µ is just a single flag variety and the claimed factorisation comes down to showing that for any M ∈ Y µ,1 d (A) and any basis β the module M is generated by for some g ∈ GL d (A).This follows from results in [GLS14] where it is shown, for any lift of M to M ∈ Y µ d (A), with A the ring of integers in a finite extension of E, and any basis β that (u − π) p M is generated by for a matrix X err divisible by a power of π p−µ1+µ d +1 and g ∈ GL d (A).Here π ∈ K is a fixed uniformiser.This result does not directly extend to the case e > 1.However, a variant of the method is able to show that, for each embedding κ ∶ K → E, the module M ϕ ∩ (u − κ(π)) p M can be generated by for some g κ ∈ GL d (A) and X err,κ a matrix divisible by π p−µκ,1+µ κ,d +1 .This was done in [GLS15] (actually they only consider the case d = 2 but it is straightforward to extend their arguments to higher dimensions).If the X err,κ 's are divisible by a high enough power of π then it follows that is congruent modulo π to the intersection of the submodules generated by This sufficient divisibility is ensured by the bound on µ from Theorem 1.3 and this congruence is precisely what it means of M to be mapped onto an element of For the proof of (2) we factor the morphism Ψ as Ỹd → Zd → Gr O where Zd denotes the moduli stack of Breuil-Kisin modules (without a crystalline Galois action) and Ỹd → Zd forgets the Galois action.An easy calculation shows that over the special fibre Zd → Gr O is smooth with irreducible fibres of dimension equal the relative dimension of Ỹd → Y d .Part (2) therefore reduces to understanding when Ỹd → Zd is an isomorphism.To address this we note that for any Breuil-Kisin module M with basis β we can define a naive Galois action σ naive,β on M by semilinearly extending the trivial G K -action on ϕ(β).Usually σ naive,β will not be ϕ-equivariant or crystalline.However, we show that if σ naive,β − 1 is suitably divisible and if M satisfies height conditions imposed by (1.2) (actually a very slight strengthening of this bound is required to avoid certain "Steinberg" situations) then lim n→∞ ϕ n ○ σ naive,β ○ ϕ −n converges to a unique ϕ-equivariant crystalline G K -action.It turns out that the locus of Zd on which σ naive,β − 1 is sufficiently divisible is closed, and obtained as the preimage of a closed subscheme in Gr O .Part (2) is then proved by showing that M µ ⊗ O F is contained in this closed subscheme.
Part 3: Upper and lower multiplicity bounds.The final ingredient which goes into the proof of Theorem 1.3 is a lower bound on the multiplicities appearing in the Breuil-Mézard conjecture.The is the most critical place where we require d = 2.It is also where we use that p > 2. Under these assumptions it is shown in [EG23, 8.6] (using global automorphy lifting techniques from [GK14]) that one always has This holds without any assumption on µ or τ .Combining [GK14] with the potential diagaonalisability established in [Bar19] one also obtains that 2 ] so long as λ is not Steinberg (for d = 2 this means λ is not a twist of ⊗ κ0∶k→F Sym p−1 F 2 ).The bounds on µ ensure Steinberg λ do not appear in Theorem 1.3 (except if K Q p is unramified, but in this case the theorem is trivial).Therefore, for µ as in Theorem 1.3, we have To finish the proof we have to show that the results from parts 1. and 2. can be combined to give equality.First we consider the identity Using part (2) of Proposition 1.5 we are even able to deduce this is an equality when Since Ỹd → Y d is smooth and surjective it follows that also Pushing this identity forward along the proper morphism Y 2 → X 2 gives an in- Combining this with the lower bound we obtain n(λ, µ) ≥ m(λ, µ, 1).By part (3) of Proposition 1.5 this must be an equality, which proves Theorem 1.3.Actually, in the paper we follow the same argument, but for the final step we prefer to work with deformation rings rather than X 2 .This allows us to avoid dealing with stacks.As explained in [EG23, 8.3], Theorem 1.3 is implied by its analogue in the setting of deformation rings.
Acknowledgements.I would like to thank Toby Gee, Eugen Hellmann, Brandon Levin and Timo Richarz for helpful conversations and correspondences.I would also like to thank the Max Plank Institute for Mathematics in Bonn, where parts of this work were done, for providing an excellent working environment.

Notation
2.1.We fix a finite extension K of Q p with residue field k of degree f over F p and ramification degree e.Let C denote the completed algebraic closure of K, with ring of integers O C , and fix a compatible system π 1 p ∞ of p-th power roots of a fixed choice of uniformiser for the minimal polynomial of π.Thus E(u) is Eisenstein of degree equal to the ramification degree e of K over Q p .We also fix another finite extension E of Q p with ring of integers O and residue field F. We assume that E contains a Galois closure of K. We typically use κ and κ 0 respectively to denote embeddings K → E and k → F. For each κ 0 we fix an embedding κ0 ∶ K → E with κ0 k = κ 0 .

For any Z p -algebra A we write S
].This comes equipped with the A-linear endomorphism ϕ which on W (k) acts as the lift of the p-th power map on k and sends u ↦ u p .We also consider We view A inf,A as an S A -algebra via u.Note that the lift of Frobenius on W (O C ♭ ) induces a Frobenius ϕ on A inf,A which is compatible with that on S A .The natural G Kaction on O C also induces a continuous (for the (u, p)-adic topology) G K -action on A inf,A commuting with ϕ.Write If A is topologically of finite type (i.e.A ⊗ Zp F p is of finite type) then S A → A inf,A is faithfully flat (in particular injective) [EG23, 2.2.13].
We also fix a compatible system (1, ǫ 1 , ǫ 2 , . ..) of p-th power roots of unity in O C which we view as an element ǫ ∈ O C ♭ .We write µ = Proof.By choosing a Z p -basis of O this follows immediately from the assertion that 2.4.We frequently consider modules as in 2.2 defined over O ⊗ Zp W (k) for an O-algebra A. Using the isomorphism given by a ⊗ b ↦ (aκ 0 (b)) κ0 (here we write κ 0 to its extension to an embedding W (k) → O) we see that any such module M can be expressed as a product where M κ0 can be identifies with the submodule of M on which the two actions of W (k) given by (1 ⊗ a)m and a ↦ (κ 0 (a) ⊗ 1)m coincide.Similarly, there is an isomorphism given by a ⊗ b ↦ (κ(b)a) κ which allows us to write an where again M κ can be identified with the submodule consisting of m ∈ M with (1 ⊗ a)m = (κ(a) ⊗ 1)m for all a ∈ O K .We warn the reader that the idempotents in (2.5) will not be contained in O ⊗ Zp O K whenever K Q p is ramifies and so the product decomposition M = ∏ κ M κ is not valid integrally, i.e. when M is an 2.6.Applying the previous discussion to Using this identification we define where π κ ∶= κ(π) and the u − π κ appears in the κ k -th factor in the product.Notice that 3. Cycles 3.1.For a Noetherian scheme X let Z m (X) denote the free abelian group generated by integral closed subschemes Z ⊂ X of dimension m.If F is a coherent sheaf on X with support of dimension ≤ m then we define Lemma 3.2.Let X be a projective scheme over k equipped with an ample line bundle L. Suppose that Y, Y 1 , . . ., Y s are m-dimensional closed subschemes in X and that Proof.This follows from [Sta17,0BEN] and the fact that, since L is ample, the higher cohomologies of L ⊗n vanish for n >> 0 [Sta17, 0B5U].
Lemma 3.3.Suppose that f ∶ X → Y is a proper morphism between equidimensional flat O-schemes which becomes an isomorphism after applying Therefore, the two stated properties of the specialisation map give

We begin by defining an ind-scheme Gr over O
for some a ≥ 0. For each κ 0 ∶ k → F, which we extend to an embedding W (k) → O, we also define Gr κ0 as the ind-scheme over O whose A-points classify rank dprojective A[u]-modules satisfying In particular, this illustrates the ind-representability of the functor; the locus of E as in (4.2) identifies with a closed subscheme of the usual grassmannian classifying submodules of (u−π) 4.4.Write X(T ) for the group of characters of GL d relative to T , the diagonal torus, and identify X(T ) = Z d as usual.We say an element µ for (e 1 , . . ., e d ) the standard basis in O K [u] d .There is an obvious action of G = GL d on Gr and, since the stabiliser of E µ under this action is a parabolic subgroup P µ ⊂ G, the orbit map induces a proper monomorphism of type µ on A d (which means the n-th graded piece is A-projective of rank equal to the multiplicity of −n in µ) then on A-points this closed immersion is given by where we view A d as a submodule of A[u] d in the obvious way.
Lemma 4.5.If A is a p-adically complete O-algebra then Gr-identifies with the set of rank for some a ≥ 0. Similarly, for each Gr κ0 .
Lemma 4.6.For each κ 0 ∶ k → F there is an isomorphism with inverse given by A denote the open obtained by inverting κ 0 (E(u)) and write A for the open obtained by inverting A .Note that an A-valued point of Gr κ0 is the same thing as a rank d vector bundle on The map in the lemma can therefore be expressed as E ↦ (E κ ) where E κ is the vector bundle obtained by glueing E Uκ with the trivial bundle on ⋃ κ ′ ≠κ U κ ′ .The inverse of this map sends (E κ ) onto the vector bundle obtained by glueing the E κ Uκ .Concretely, this glueing corresponds to taking the intersection of each of the E κ 's which gives the lemma.

We define one last ind-scheme Gr
for some a ≥ 0. From 2.4 we see that Lemma 4.6 implies that the generic fibre of Gr O identifies with ∏ κ (Gr ⊗ O K ,κ E) with the product running over all embeddings κ ∶ K → E. Note also that the analogue of Lemma 4.5 applies to Gr O and identifies its points valued in p-adically complete O-algebras A with rank d projective S A -modules satisfying Lemma 4.9.Let µ be a Hodge type and suppose n κ ≥ 0 so that µ κ,d ≥ −n κ for every κ.
(1) Let A be an E-algebra.
(recall the elements E κ (u) from 2.4).(2) Let A be a p-adically complete Noetherian flat O-algebra and suppose there are A-submodules for each κ such that Fil i κ Fil i+1 κ is p-torsionfree and becomes A[ 1 p ]-projective of constant rank after inverting p.If E ∈ Gr(A) can be expressed as Proof.Note that multiplication by Thus we can assume n κ = 0 throughout.
For (1) we first decompose E = ∏ κ0 E κ0 ∈ ∏ κ0 Gr κ0 according to the action of W (k). Then Lemma 4.6 and the description of G P µκ ↪ Gr from 4.4 implies E ∈ M µ if and only if, for each κ 0 , Thus, to prove (1) we just need to identify the κ 0 -th term inside this intersection with For part (2) we use that A is Noetherian to ensure commutes with finite intersections and so As a consequence of (1) it follows that E[ 1 p ] ∈ M µ and so E ∈ M µ also.Part (3) relies on the fact that, for A as in the proposition, being A-projective is equivalent to being p-torsion free and finitely generated.Applying (1) to E[ 1 p ] produces filtrations on A[ 1 p ] d for each κ.If Fil i κ denotes the intersection of this filtration with A d then the graded pieces are p-torsionfree.This is equivalent to asking that each is also p-torstionfree, and so the valuative criterion for properness implies E ′ = E.

5.1.
Recall G = GL d viewed as an algebraic group over O K .Let λ ∈ X(T ) be dominant and set is a G-equivariant line bundle on G P λ and H 0 (G P λ , L(λ) ⊗n ) can be viewed as an algebraic representation of G on a flat O K -module whose generic fibre identifies with H 0 (nλ), the algebraic representation over K of highest weight nλ.See for example [Ful97, p.143-144].
5.2.On Gr there is an ample G-equivariant line bundle L det whose fibre over any We also write L det for the G-equivariant ample line bundle constructed analogously on Gr O .
Lemma 5.3.The restriction of L det to G P λ inside Gr identifies G-equivariantly with L(λ).
and so the fibre of In either case, the second factor in these tensor products identifies with ⊗ i≥λ d det(Fil −i Fil −i+1 ) λ d which finishes the proof.
Corollary 5.4.For any n > 0, there is an identification where L det here denotes the restriction to G P µκ of the determinant line bundle on Gr.
The Kunneth formula [Sta17,0BED] gives as G-representations.Therefore, we just have to show H 0 (G P µκ , L ⊗n det ) ⊗ O K K = H 0 (nµ κ ) as G-representations, and this follows from Lemma 5.3.

Multiplicity bounds
6.1.The formal character of an algebraic representation V of G on a finite dimensional vector space is defined as where V λ is the λ-weight space of V and e(λ) denotes λ viewed as an element of the group ring Z[X(T )].This induces an isomorphism between the Grothendieck group of such representations and Z[X(T )] W where W denotes the Weyl group of G [Jan03, II.5.7].
where A(λ) ∶= ∑ w∈W det(w)e(wλ) [Jan03, II.5.10].If we write dim ∶ Z[X(T )] → Z for the map ∑ λ a λ e(λ) ↦ ∑ a λ then one also has Though here H 0 (λ) is defined over a field of characteristic zero, all the above goes through with H 0 (λ) replaced by the representation over a field of characteristic p of highest weight λ.What differs in characteristic p is that this highest weight representation may not be irreducible.
Lemma 6.3.Let µ 1 , . . ., µ e ∈ X(T ) with µ i − ρ dominant for each i and suppose that is a polynomial in n of degree < ∑ i dim G P µi .
Proof.Using Weyls character formula from 6.2 and multiplying by A(ρ) e gives Taking the image of this identity under the endomorphism of Z[X(T )] induced by multiplication by n on X(T ) (because the formation of A commutes with this endomorphism).Dividing by A(ρ) e then gives The lemma therefore follows by taking the dimension and observing that dim H 0 (nλ)− dim H 0 (nλ−ρ) is a polynomial in n of degree < dim G P λ for any dominant λ ∈ X(T ) (use the last equation from 6.2).
Remark 6.4.Since K has characteristic zero each H 0 (λ) is irreducible.Moreover, every irreducible G-representation is isomorphic to one such H 0 (λ).The observation from 6.2 that ch induces an identification between Z[X(T )] and the Grothendieck group of G-representations shows that the integers m(λ, µ) in the previous lemma are ≥ 0 and are uniquely determined.
Proof.We apply Lemma 3.2 to the line bundle L det .This gives that is, for large n, equal the value at n of a polynomial of degree < ∑ κ dim G P µκ .Applying Corollary 5.4 implies the same is true for is a polynomial of degree < ∑ κ dim G P µκ in n.We conclude that the dimension of is also polynomial of degree < ∑ κ dim G P µκ in n for n >> 0. Since n(λ, µ) − m(λ, µ) ≥ 0, each term in the above sum is a polynomial in n of degree ∑ κ dim G P µκ with non-negative leading term.We must therefore have n(λ, µ) = m(λ, µ) for each λ.

Topological descriptions
7.1.Recall that for λ, λ ′ ∈ X(T ) we write . .+ λ i for each i with equality when i = 1.Proposition 7.2.Let µ be a Hodge type with µ κ − ρ dominant for each κ.Assume that (1) There are integers n(λ, µ) with the sum running over tuples λ = (λ κ0 ) with each λ κ0 ∈ X(T ) dominant and satisfying 7.3.To prove the proposition we will approximate M µ ⊗ O F via explicit moduli conditions.In fact we give two such moduli interpretations, based on the following two operators: We also write ∇ for the coordinate-wise extension to S A [ 1 E(u) ] d .• If A is a p-adically complete O-algebra of topologically finite type then for each σ ∈ G K we can also define Note this is well defined because σ(µ) ∈ µA inf .We also note that and E(u) generate the same ideal inside A inf (as follows from Lemma 2.3) and so we can also view ∇ σ as an operator on Again write ∇ σ also for the coordinate-wise extension to A inf,A [ 1 E(u) ] d .The advantage of ∇ is that it is easier to compute with.The advantage of ∇ σ is that it is more directly related to Galois representations.

Lemma 7.4. There exist closed subfunctors
Proof.That Gr ∇ O is a closed subfunctor if clear, so we focus on Gr ∇σ O .Since Gr O is an inductive system of proper Noetherian O-schemes it suffices to show that for any E ∈ Gr O (A) the condition is closed on Spec A whenever A is a p-adically complete topologically finite type O-algebra.This follows from an application of [EG23, B.29].
Remark 7.5.Since E(u) and µ ϕ −1 (µ) generate the same ideal in A inf the condition defining Gr ∇σ O can also be expressed as This description may be more familiar from the point of view of crystalline Breuil-Kisin modules.
Proposition 7.6.For every Hodge type µ one has for any E-algebra A. Lemma 4.9 allows us to write E as an intersection of It is therefore enough to show that E(u)∇(E κ ) ⊂ uE κ and this follows since There is a slight difficulty in giving an identical argument to show M µ ⊂ Gr ∇σ O because the moduli description for Gr ∇σ O does not apply when A is an E-algebra.To address this we first note that the generic fibre of M µ is reduced so to show O it suffices to show this on A-points whenever A is a finite extension of E. By the valuative criterion for properness, any such A-valued point is induced from a point valued in the ring of integers of A. Thus we are reduced to showing M µ (A) ⊂ Gr ∇σ O (A) whenever A is the ring of integers in a finite extension of E. Using part (3) of Lemma 4.9 this comes down to proving that This would follow from the claim that To prove the claim first note that σ(E κ (u)) and arguing by induction on i then gives that the claim follows.
Remark 7.7.After possibly replacing the compatible system of primitive p-th power roots of unity ǫ we can choose σ ∈ G K so that σ(u) u = ǫ.Then Thus ∇ σ = u∇ q where ∇ q is the q-derivation for q = [ǫ].In particular ∇ σ ≡ u d du = ∇ modulo [ǫ] − 1.This illustrates the close relationship between the Gr ∇σ O and the locus Gr ∇ O .

For the rest of this section we focus on Gr
where Gr ∇ κ0 is defined similarly.Let us write Gr = Gr κ0 ⊗ O F (note this is independent of κ 0 ) and Gr ∇ = Gr ∇ κ0 ⊗ O F. The description from Lemma 4.5 shows that the group scheme LG acts on Gr.For λ ∈ X(T ) dominant we set Gr λ equal to the LG + -orbit of E λ ∈ Gr (recall E λ is defined in 4.4) and we set Gr ≤λ equal to its reduced closure.Then Gr λ ′ Lemma 7.9.Suppose λ ∈ X(T ) is dominant with Proof.We begin by giving an open cover of Gr λ : let U λ ⊂ L + G denote the subfunctor whose A-points consist of unipotent upper triangular matrices where ).Therefore the lemma reduces to showing U λ ∩ Gr ∇ is an affine space of the claimed dimension.Observe that g ∈ U λ ∩ Gr ∇ if and only if (7.10) u e−1 g −1 ∇(g) ∈ If we write g −1 = (b ij ) ij then, using that b jj = 1, b lj = 0 for l < j, and ∇(a ii ) = 0, we see that (7.10) is equivalent to asking that for every i > j.By assumption λ j − λ i − e + 1 ≤ p and so ∑ j<l<i ∇(a il )b lj modulo u λj −λi−e+1 admits an antiderivative; in other words, there exists a unique Since a ij has degree < λ j − λ i it follows that for some a This shows, by an inductive argument, that b ij is a function of a lk for l < k with k − l ≤ i − j.Therefore the element X ∈ uA[u] considered above depends on a lk with k − l < i − j.As a consequence the morphism given by (a ij ) ↦ (a ij ) has a well-defined inverse which finishes the proof.

Proof of Proposition. First observe that under the identification Gr
∏ Gr λ where the product runs over λ = (λ κ0 ) with λ κ0 ≤ ∑ κ k =κ0 µ κ .Since M µ ⊂ Gr ∇ O and each µ κ − ρ is dominant the dimension calculations from Lemma 7.9 imply that where the union now runs over λ = (λ κ0 ) with λ κ0 + eρ ≤ ∑ κ k =κ0 µ κ and where we write C λ+eρ = ∏ κ0 C λκ 0 +eρ .Thus, one can write as cycles, for integers n(λ, µ) ≥ 0. Furthermore, since Gr Since this intersection is clearly non-empty it follows that n(λ, µ) = 1 for this particular λ.This shows that Arguing by induction we conclude that we can always write ) satisfying λ ′ κ0 ≤ λ κ0 and some n λ ′ ∈ Z.This proves the first part of the proposition.
The second part follows from the first provided we can show M λ is irreducible whenever λ κ0,1 − λ κ0,d ≤ p − 1.To establish this irreducibility we require d = 2. Choose an indexing κ 0,1 , . . ., κ 0,e of those κ with κ k = κ 0 so that κ 0,1 = κ0 .Then construct a scheme X which classifies tuples (E e ⊂ . . . for each i.Then the map (E i ) ↦ E e produces a proper morphism X → Gr O which on the generic fibre identifies X ⊗ O E with M λ ⊗ O E. In particular, this shows that X ⊗ O F → M λ ⊗ O F is surjective.On the other hand, X is a successive extension of (products of) grassmannians over a (product of) flag varieties.Thus X is O-smooth, and so X ⊗ O F is irreducible.We conclude the same is true of M λ ⊗ O F.

If E corresponds to an
Proof.It suffices to prove this on the generic fibre.Thus, one is reduced to prove that for any λ ∈ X(T ), G P λ ⊂ Gr is identified with G P −w0λ by the version of E ↦ E * on Gr.But this follows easily from the fact that if E is generated by (e 1 , . . ., e d )X then E * is generated by (e 1 , . . ., e d )(X −1 ) t for (X −1 ) t the conjugate transpose.In particular, the G-orbit of any E is mapped onto the G-orbit of E * .Since E * λ = w 0 E −w0λ the lemma follows.9. Breuil-Kisin modules 9.1.Let A be a p-adically complete O-algebra.Then a Breuil-Kisin module M over A is a finite projective S A -module equipped with an S A -linear homomorphism whose cokernel is killed by a power of E(u).We say M has height ≤ h if the cokernel is killed by E(u) h .We write M ϕ for the image of ϕ M and ϕ(M) for the image of  where Γ forgets the choice of basis and, with Gr O viewed as a formal scheme over Spf O, the morphism Ψ sends Writing L + GL d for the group scheme given by A ↦ GL d (S A ) we see that Γ is an L + GL d -torsor for the action on Z≤h d given by g ⋅ (M, β) = (M, βg).A second action of L + GL d on Z≤h can be given by where M g = M as an S A -module, with Frobenius given by ϕ g (β) = βgC for C the matrix determined by ϕ(β) = βC.It is easy to see that this action makes Ψ into an L + GL d -torsor over its image in Gr O .9.4.An alternative viewpoint on 9.3 is as follows.Let L ≤h GL d denote the group scheme over O given by given by (M, β) ↦ C, for C defined by ϕ(β) = βC, is an isomorphism (here we view L ≤ h GL d as a formal scheme over Spf O).Under this isomorphism the action g⋅(M, β) corresponds to (right) action of L + GL d by ϕ-conjugation: C ↦ g −1 Cϕ(g).The action g * (M, β) corresponds to the (left) multiplication action: C ↦ gC.Thus (9.6) identifies with the diagram Here L ≤h GL d ϕ L + GL d indicates the quotient by Frobenius conjugation The issue with the construction in 9.3 is that Z≤h d ≅ L ≤h GL d is not of finite type over Spf O.To address this we instead consider: (the quotient being by Frobenius conjugation).The following is a essentially [PR09, 2.1].
Proposition 9.6.Fix n ≥ 1.Then, for N sufficiently large, (9.3) induces a diagram Z≤h,N and Ψ N is smooth of relative dimension dim O G N with irreducible fibres.More precisely, the action g * (M, β) from 9.3 induces an action of We will see in the proof that N is required large enough that E(u) h divides u (p−1)N −1 in S O π n (this can be made explicit using e.g.[EG21, 5.2.6]).
Proof.The crucial observation is that if E(u) h divides u (p−1)N −1 in S O π n then, as stacks over Spec O π n , the identity map (the quotient on the left being by Frobenius conjugation and that on the right being by left multiplication).This immediately shows that the morphism Ψ induces a morphism Ψ N as claimed, and that Ψ N is a Concretely, the claimed isomorphism follows from the two assertions: • For each g 0 ∈ U N (A) we have g −1 0 Cϕ(g 0 ) = gC for a unique g ∈ U N (A).• For each g ∈ U N (A) there exists a unique g 0 ∈ U N (A) with g −1 0 Cϕ(g 0 ) = gC.Note these are exactly the same assertions as (1) and (2) in [PR09, 2.2].The first point is easy because g 0 ∈ U N (A) implies Therefore, the fact that E(u) h C −1 ∈ Mat(S A ) and that E(u) h divides u pN ensures g ∈ U N (A).For the second point notice that, if J n = (gC)ϕ(gC) . . .ϕ n (gC) and I n = Cϕ(C) . . .ϕ n−1 (C), then any such g 0 satisfies for every n ≥ 1.We claim that for any g 1 ∈ U N (A) we have I n (ϕ n (g 1 ) − 1)J −1 n ∈ u N Mat(S A ) and that this element conveges u-adically to zero as n → ∞.This ensures that I n J −1 n converges in U N (A) as n → ∞ and that this limit is the unique Therefore the claim will follow if u (p n −1)N J −1 n−1 ∈ Mat(S A ) converges u-adically to zero.Since ϕ i (E(u) h ) divides u ((p−1)N −1)p i and equipped with the G K -action induced by σ is a crystalline representation of G K on a finite projective A-module.Furthermore, the Hodge type of (M, σ) coincides with that attached to V via the filtered module The theorem as stated is taken from [Bar20, 2.1.12],but the result originates from a combination of ideas appearing in [Kis06,GLS14,Oze14].
Remark 10.6.These conventions mean that the Hodge type of the cyclotomic character is −1.
A so that ϕ (induced semilinearly from that on M) is the identity on ρ and so that the G K -action (induced semilinearly from that on ρ) satisfies given by (M, ρ) ↦ (M, σ) with σ the G K -action induced by ρ is easily seen to be formally smooth.Therefore, the preimage Proof.We saw in the proof of Lemma 10.8 that L µ ρ → Y µ d is formally smooth with fibres classifying framings of the corresponding Galois representation, and so of relative dimension d 2 .Hence Y µ d has dimension (in the sense of e.g.

Naive Galois actions
In this section we consider the morphism Y ≤h induced by ϕ(β).Thus, σ naive,β is uniquely determined as the semilinear G K -action fixing ϕ(β).
11.2.Let us fix integers 0 ≤ r κ ≤ h for each κ.Then we consider the closed subfunctor Gr ∇σ,r O ⊂ Gr ∇σ O defined by requiring that Proposition 11.3.Assume that ∑ κ k =κ0 r κ ≤ e + p − 1 for each κ 0 with at least one inequality strict.Then Proof.By Lemma 15.1 is enough to show that this morphism induces equivalences on groupoids of A-valued points for any local finite type F-algebra A. Equivalently, we must show that for any (M, β) ∈ Z∇σ,r d (A) there exists a unique action σ of G K making (M, σ) into an object of Y ≤h d .Existence implies essential surjectivity on A-valued points and full-faithfullness follows from the uniqueness.
Write Hom(M, M) for the S A -module of S A -linear endomorphisms of M and equip Hom(M, M) with the Frobenius ϕ Hom given by h ↦ ϕ ○ h ○ ϕ −1 .If we identify Hom(M, M) with matrices in S A using the basis β then ϕ Hom acts by where ϕ acts entrywise on the matrix M and C is such that ϕ(β) = βC.The following claim shows that, after extending scalars to A inf,A , this operator is topologically nilpotent on matrices with entries divisible by [π ♭ ]ϕ −1 (µ).
Then H is ϕ Hom -stable and ϕ Hom is topologically nilpotent on H.

Proof of claim.
Recall that M κ0 ⊂ M is the submodule on which W (k) acts via κ 0 and ϕ on M restricts to a semilinear map (the last equality uses that A is an F-algebra) and so, as ∑ κ κ k =κ 0 r κ ≤ e + p − 1, it follows that H is ϕ Hom -stable.Since the inequality is strict at least once we have ϕ Hom (H κ ) ⊂ uH κ0○ϕ −1 for at least one κ 0 .In particular ϕ Hom is topologically nilpotent.
Set M inf = M ⊗ S A A inf,A .Note that for each σ ∈ G K the endomorphism σ naive,β is σ-semilinear and so defines an element of Any such semilinear map is determined by where β is sent; thus we obtain an (additive) identification Hom(M inf , inf ) and H. Via this identification we view ϕ Hom as a Frobenius on Hom(M inf , M σ inf ).We claim that ϕ n Hom (σ naive,β ) ∈ Hom(M inf , M σ inf ) for all n ≥ 0 and that this sequence converges to a σ-semilinear endomorphism which we simply write as σ.By construction this endomorphism is ϕ-equivariant.To see this claim note that by assumption where 1 σ the σ-semilinear extension of the map β ↦ β.Since we can write the claimed convergence follows from the topological nilpotence of the operator ϕ Hom on Hom(M inf , [π ♭ ]ϕ −1 (µ)M σ inf ) = H.This formula also shows that σ(x) − x ∈ M ⊗ S A [π ♭ ]ϕ −1 (µ)A inf,A for each x ∈ M. Since σ naive,β defines a G K -action so to does σ.Similarly, continuity of σ naive,β implies continuity of σ.
Finally, to see uniqueness suppose that σ ′ was another such G K -action.Then for each σ ∈ G K one has

Comparison with local models
The following proposition is the key technical result which goes into the proof of the theorem.It is a reworking of techniques originally developed in [GLS14,GLS15].
Proposition 12.3.Let A be a finite flat O-algebra and suppose Use the ϕ(S A )-basis ϕ(β) to define a section s of ϕ(M) → M ϕ → M κ .Then there exists a filtration Fil • κ on M κ by A-submodules with p-torsionfree graded pieces such that when Fil n κ is viewed as a submodule of M ϕ via s and M ϕ err,κ ∶= ∑ p−1 l=1 π p−l E κ (u) l M ϕ .Note that, by construction, the image of the section s generates M ϕ over S A .
Proof.First we define the filtration Fil • κ .Recall that we equipped M ∶= M ϕ E(u) with the filtration whose n-th piece is the image of M ϕ ∩ E(u) n M. Then D K ∶= M [ 1 p ] is a filtered A ⊗ Zp K-module and can be written as given by u ↦ π κ .Therefore, its kernel is E κ (u)M ϕ .This means M κ can be viewed as a submodule of D K,κ and M κ [ The filtered pieces of D K,κ are Q p -vector spaces so the graded pieces of Fil • κ are ptorsionfree.This also shows that Fil n κ [ 1 p ] = Fil n (D K,κ ) which proves Corollary 12.4 below.
Next we use: Claim.For x ∈ Fil n κ with n ≤ p there exists x 1 , . . ., x p−1 ∈ M ϕ such that s(x) Proof of Claim.This follows from results in [Bar19,§5].To apply these first note that in loc.cit. the embeddings K → E are indexed by integers 1 ≤ i ≤ f and 1 ≤ j ≤ e so that κ ij k depends only on i.This labelling can be chosen so that κ from the proposition equals κ i1 for some i.In [Bar19, 5.2.5] it is shown that for any x ∈ ϕ(M) there exist x 1 , . . ., x p−1 ∈ M ϕ so that We apply this to x = s(x).Then the image of x in D K,κ is contained in Fil n (D K,κ ) and so [Bar19, 5.2.2] implies x (n) is contained in a submodule of M ϕ ⊗ S S[ 1 p ] denoted Fil {n,0,...,0} .In [ Bar19,5.1.3]it is shown that Fil {n,0,...,0} ∩M ϕ = M ϕ ∩ E 1 (u) n M. Therefore Therefore, in the above identity we can replace each E 1 (u) with E κ (u), and the claim follows.
The claim shows that for any 0 ≤ m ≤ r κ and we want to prove the opposite inclusion for 0 ≤ m ≤ r κ by induction on m.When m = 0 this is clear since both sides equal M ϕ (recall that the section s was chosen so that s(M κ ) generates M ϕ over S A ).For m > 0 note that the image of M ϕ ∩ E κ (u) m M in M κ is contained in Fil m κ , while Fil m κ equals the image of Y m .The above inclusion therefore shows these images are equal.As a consequence, if x ∈ M ϕ ∩ E κ (u) m M then there exists x ′ ∈ Y m so that The second equality uses that M ϕ err,κ ⊂ E κ (u)M ϕ .The inductive hypothesis therefore gives that err,κ as desired.
Corollary 12.4.The graded pieces of Fil As in Proposition 12.3 the Fil n κ 's are viewed as submodules of M ϕ using the basis ϕ(β).Corollary 12.4 together with part (2) of Lemma 4.9 (taking n κ = r κ ) shows that E[ 1 p ] defines an A[ 1 p ] point of M −w0µ under the identification M ϕ = S d A induced by ϕ(β).Note, however, that it is not a priori clear E defines an A-valued point.We will be done if we can show this is the case, and if we can show that We begin with the second assertion.Take z ∈ (∏ κ E κ (u) rκ ) M. Then z ∈ M ϕ ∩ E κ (u) rκ M for each κ and so Proposition 12.3 ensures the existence of We claim there then exists m ∈ πM ϕ such that m ≡ m κ mod E κ (u) rκ M ϕ for each κ.Since M ϕ is S A -free this claim follows from Lemma 12.5 below.This is where we use the bound on the r κ .Since for each κ (due to the filtration Fil n κ being concentrated in degrees [0, r κ ] we have Fil This establishes the two required conditions mentioned in the second paragraph, and therefore finishes the proof.
Lemma 12.5.Let A be a finite flat O-algebra and suppose are given with m κ,l ∈ A. Then there exists m ∈ πS A with m ≡ m κ modulo E κ (u) rκ for each κ.
Proof.Firstly, by choosing an O-basis of A we can reduce to the case A = O.Secondly, we can fix κ and assume that m κ ′ = 0 for all κ ′ ≠ κ.Using the identification satisfies the two bullet points above.To finish it suffices to show that N , and hence For this view N as a polynomial in (u − π κ ).By assumption the coefficient of (u − π κ ) n in m has valuation ≥ p − n.On the other hand, the coefficient of (u − π κ ) in X κ ′ has valuation ≥ −(n + r κ ′ )ν for ν ∶= ν κ k .Since ν ≥ 1 we have p − n ≥ p − nν and as such the coefficient of (u − π κ ) n in m ∏ κ ′ ≠κ X j has valuation ν + 1 so we are done.

Lower bounds
In this section we recall from [GK14,EG23] the lower bound on the cycles appearing in the Breuil-Mézard conjecture attained when d = 2.We do this in the context of cycles in deformation rings.We also give a minor improvement using the potential diagaonalisability established in [Bar19].13.2.We also recall from the introduction the F-representation V (µ, τ ) of GL d (k) attached to any pair (µ, τ ) with µ a Hodge type with each µ κ − ρ dominant and τ an inertial type.Taking τ = 1 we obtain V (µ, 1) by evaluating the algebraic representation (here we write κ also for its composite with the surjection O → F) on k-points.Since the exact definition of V (µ, τ ) will not be needed for τ ≠ 1 we refer to [EG23, 8.2] for the general construction.
Notation 13.4.If λ denotes an isomorphism class of absolutely irreducible Frepresentation of GL d (k) then we write λ for the Hodge type obtained as in 6.5 for (λ κ0 ) the tuple corresponding to λ in 13.1.
13.5.In the next proposition we fix a continuous homomorphism ρ ∶ G K → GL d (F) and, as in the proof of Lemma 10.8, we write R ρ for the O-framed deformation ring of ρ.If (µ, τ ) is a pair consisting of a Hodge type µ and an inertial type τ then we also write R µ,τ ρ for the unique reduced O-flat quotient of R ρ whose points valued in a finite extension of E are correspond to potentially crystalline representations of type (µ, τ ).

Then
(1) There are cycles C ρ,λ in Spec R ρ , indexed by isomorphism classes of absolutely irreducible F-representations λ of GL d (k), such that, for any pair (µ, τ ), one has an inequality . This is the single point where the assumption p > 2 arises.
Proof.In [EG23, 8.6.6] it is shown that if X µ,τ 2 denotes the closed algebraic substack of the Emerton-Gee stack X 2 , whose A-points (for A a finite flat O-algebra) correspond to potentially crystalline G K -representations of type (µ, τ ), then there are top dimensional cycles C λ ⊂ X 2 such that [X It remains to prove that e(R λ,1 ρ ⊗ O F) ≤ 1 for ρ maximally non-split of niveau 1 and weight λ.For this we recall [GK14, 3.5.5]which asserts that for any ρ there is a unique set of integers e λ (ρ) ≥ 0 such that, if every potentially crystalline lift of ρ of type µ, τ is potentially diagonalisable, then The integers e λ (ρ) are the Hilbert-Samuel multiplicities of the cycles C λ,ρ and the above discussion implies these are 1 when λ is non-Steinberg.Therefore, part (2) will follow if we can show every crystalline lift of ρ with Hodge type λ is potentially diagaonalisable.This is the main result of [Bar19] (which applies since ρ is not an unramified twist of χ −1 cyc by 1).

Main result
We can now prove our main result.
Proof.First, by a simple twisting argument, we can assume µ κ,2 = 0 for each κ.Since µ κ,1 ≥ 1 the bound ∑ κ k =κ0 µ κ,1 ≤ p implies that either e < p or e = p and µ κ,1 = 1 for each κ.In the latter case µ = λ for λ the trivial representation and in this case there is nothing to prove.Thus, we can assume e < p.This means K is tamely ramified over Q p and so Remark 12.2 indicates that Theorem 12.1 applies.We can also assume that e > 1, since if e = 1 then again µ = λ and the theorem is again trivial.This means that ∑ κ k =κ0 µ κ,1 < p + e − 1 for each κ and so Proposition 11.3 also applies, with r κ = µ κ,1 .Proposition 7. The next goal is to descend this identity to an inequality of cycles in Spec R ρ ⊗ O F. For this we recall the projective R ρ -scheme L ≤h ρ introduced in the proof of Lemma 10.8.There is a formally smooth morphism L ≤h ρ → Y ≤h shows that this must be an equality.The theorem follows.

Miscellany
Let X and Y be algebraic stacks of finite type over a field k and let f ∶ X → Y be a morphism of stacks.
Lemma 15.1.If, for A any local finite k-algebra, the induced functor X (A) → Y(A):

Here
for n >> 0. For µ satisfying (1.2) the multiplicities m(λ, µ) computed in characteristic zero coincide with the m(λ, µ, 1) computed in characteristic p.Thus, the assumption in (3) is that n(λ, µ)−m(λ, µ) ≥ 0. Each term in (1.8) is a polynomial in n of degree dim M µ and positive leading term.Therefore we must have n(λ, µ) = m(λ, µ).Part 2: From local models to moduli of crystalline Galois representations.The second step is to relate the M µ 's to the geometry of X d .The basic strategy is to study the geometry of X d via a resolutionY d → X d with Y d a stack whose A-points classify Breuil-Kisin modules with A-coefficients (i.e.projective (W (k) ⊗ Zp A)[[u]]-modules equipped with a semilinear endomorphism ϕ).A local version of this construction was first made in[Kis09b] (with X d replaced by Spec of a deformation ring) and its globalisation to stacks first appeared in[PR09], before being built upon in[EG23].In our case, we takeYd as the stack classifying pairs (M, σ) with M a rank d Breuil-Kisin module and σ a ϕ-equivariant action of G K on M ⊗ W (k)[[u]] A inf satisfying a "crystalline" condition (which means that σ − 1 is sufficiently divisible).Inside Y d there are Z p -flat closed substacks Y µ d whose O-valued points correspond to Breuil-Kisin modules associated to crystalline representations of Hodge type µ whenever O is the ring of integers in a finite extension of Q p .Then X µ,1 d is, by definition, the scheme theoretic image of the morphism Y µ d → X d .To relate Y d to the affine grassmannian we use the following diagram: Ỹd classifies Breuil-Kisin modules in Y d together with a choice of basis (to stay in the world of finite type stacks this basis is taken modulo u N for N >> 0) and, using this choice of basis, the morphism Ψ takes a Breuil-Kisin module M to the relative position of M and its image of Frobenius.The morphism Γ forgets this choice of basis.The key result we prove is then Proposition 1.10.(1) If µ satisfies the bound from Theorem 1.3 then the restriction of Ỹd → Gr O to Ỹ µ d ⊗ O F (for Ỹ µ d the preimage of Y µ d in Ỹd ) factors through M −w0µ ⊗ O F for w 0 ∈ W the longest element.(2) For such µ, the morphism Ỹd → Gr O is smooth over M µ ⊗ O F with irreducible fibres of dimension equal the relative dimension of Ỹd → Y d .
Proposition 1.5.Applying an involution of Gr O which sends a lattice onto its dual allows us to replace µ and each λ in this identity with −w 0 µ and −w 0 λ.Thus [M −w0µ ⊗ O F] = λ n(λ, µ)[M −w0 λ ⊗ O F] Part (2) of Proposition 1.10 ensures Ỹ2 → Gr O is smooth over the closed subschemes appearing in this identity of cycles.This allows us to pull the identity back to Ỹd to obtain [Y µ,flag 2 ] = λ n(λ, µ)[Y λ,flag 2 ] where Y µ,flag 2 equals the preimage of M −w0µ ⊗ O F under this map.Part (1) of Proposition 1.10 (together with a dimension comparison) implies [ Ỹ µ d ⊗ O F] ≤ [Y µ,flag 2 ].
appearing in this sum is irreducible and generically reduced.Since the [M λ⊗ O F] are pairwise distinct this implies n(λ, µ) ≥ 0.
defines a morphism Gr λ → G P λ .Since the parabolic P λ is contained in the Borel of lower triangular matrices B − ⊂ G we can compose this map with G P λ → G B − .Then the morphism U → Gr λ identities U λ with the preimage under this composite of the open U ⊂ G B − consisting of upper triangular unipotent matrices.In particular, U λ → Gr λ is an open immersion and Gr λ = ⋃ w wU λ with w running over the permutation matrices in G (as follows by considering the open cover G B − = ⋃ wU ).Since ∇(w) = 0 we have wU λ ∩ Gr ∇ = w(U λ ∩ Gr ∇

the composite M m↦m⊗1 →
M ⊗ ϕ,S A S A → S. Thus ϕ(M) is an ϕ(S A )-submodule of M ϕ which generates M ϕ over S A .Definition 9.2.For any p-adically complete O-algebra A write • Z ≤h d (A) for the groupoid of rank d Breuil-Kisin modules over A with height ≤ h.Morphisms are S A -linear isomorphisms compatible with the Frobenius.• Z≤h d (A) for the groupoid of pairs (M, β) with M ∈ Z ≤h d (A) and β = (β 1 , . . ., β d ) an S A -basis of M. Morphisms are S A -linear isomorphisms compatible with the Frobenius and identifying the bases.With pull-backs defined by base-change these categories form an fpqc stacks Z ≤h , Z≤h d over Spf O.

Definition 9. 5 .
For N ≥ 1 let Z≤h,N d (A) denote the category of pairs (M, β) ∈ Z≤h d (A) whose morphisms are ϕ-equivariant morphisms of S A -modules which identify the bases modulo u N .Equivalently, Z≤h,N d is the quotient of Z≤h d under the action of the group scheme Corollary 9.7.Z≤h,N d × O O π n is a finite type O-scheme for N >> 0 and Z ≤h d is a p-adic formal algebraic stack (in the sense of [EG23, A7]) of finite type over Spf O. Proof.The first part follows since we've just seen that Z≤h,N d × O O π n is a torsor for a finite type group scheme over a finite type O π n -scheme.The second part follows from the first and the definition of a p-adic formal algebraic stack.10.Crystalline Breuil-Kisin modules 10.1.If A is a p-adically complete O-algebra which is of topologically finite type then a crystalline Breuil-Kisin module over A is a pair (M, σ) with M a Breuil-Kisin module over A and σ a continuous ϕ-equivariant A inf,A -semilinear action of Definition 10.2.Write Y ≤h d (A) for the groupoid consisting of rank d crystalline Breuil-Kisin modules over A with height ≤ h.10.3.One can attach a Hodge type to crystalline Breuil-Kisin modules (at least over coefficient rings which are O-flat): For n ∈ Z one defines Fil n (M ϕ ) = M ϕ ∩ E(u) n M and equips the finite projective O K ⊗ Zp A-module M ϕ E(u) with the filtration whose n-th filtered piece is the image of Fil n (M ϕ ).The graded pieces of this filtration become (A ⊗ Zp W (k)[ 1 p ]-projective after inverting p.This allows us to say that (M, σ) has Hodge type µ = (µ κ ) if the part of gr n M ϕ [ 1 p ] E(u) on which K acts via κ has E-dimension equal the multiplicity of n in µ κ .Remark 10.4.In other words, M has Hodge type µ if the part of M ϕ [ 1 p ] E(u) on which K acts via κ is a filtration of type −w 0 µ κ = (−µ κ,d , . . ., −µ κ,1 ) in the sense of 4.4.Theorem 10.5.If (M, σ) ∈ Y ≤h d (A) with A a finite flat O-algebra then

Proposition 10. 7 .
There exists a limit preserving p-adic algebraic formal stack Y ≤h d of topologically finite type over O whose groupoid of A-valued points, for any p-adically complete O-algebra topologically of finite type, is canonically equivalent to Y ≤h d (A).

For
each Hodge type µ there exists a unique O-flat closed substack Y µ d of Y ≤h d with the property that the full subcategory Y µ d (A) of Y ≤h d (A) consists of all crystalline Breuil-Kisin modules with height ≤ h and Hodge type µ.Proof.The first part follows from [EG23, §4.5].There, algebraic stacks C a π ♭ ,s,d,h over Spec O π a are constructed [EG23, 4.5.8] with π ♭ = (π, π 1 p , . ..) and s some sufficiently large integer.In the proof of [EG23, 4.5.15] it is explained how Y ≤h µ × O O π a can be realised as a closed substack of C a π ♭ ,s,d,h .The second part is [EG23, 4.8.2].We conclude with a useful lemma giving a description of the points of Y µ d valued in a finite local F-algebra: Lemma 10.8.Assume that µ κ ⊂ [0, h] for each κ and suppose that (M, σ) corresponds to an A-valued point of Y µ d for A some finite local F-algebra.Then there exists a local finite flat O-algebra A with A = A ⊗ O F and an A-valued point (M, σ) of Y µ d with special fibre (M, σ).Proof.Let F ′ F be a finite extension and write R ρ for the framed O-deformation ring corresponding to some ρ crystalline representations of Hodge type µ [Kis08, 3.3.8].In particular, L µ ρ is reduced.Now apply the above construction with ρ = (M ⊗ S A W (C ♭ ) A ) ϕ=1 ⊗ A F ′ where F ′ denotes the residue field of A. Then (M, σ) induces an A-valued point of L µ ρ .Applying [Bar20, 4.1.2]to the local ring of L µ ρ at this points produces a finite flat O-algebra A with A → A and an A-valued point of L ≤µ ρ pulling back to our Avalued point.The image of this A-valued point in Y µ d then corresponds to (M, σ) as desired.Corollary 10.9.Assume µ κ ⊂ [0, h] for each κ.Then Y µ d has relative dimension ∑ κ dim G P µκ over O.

d.
→ Z ≤h d which forgets the Galois action.More precisely, we consider its base-change Y ≤h d × Z ≤h d 0, and show this is an isomorphism over certain closed subschemes in the special fibre of Z≤h,N d Construction 11.1.The aim is to establish conditions which allow the following "naive" crystalline G K -action on (M, β) ∈ Z≤h,N d (A) to be perturbed into one

d→
as the preimage of Gr ∇σ ,r O under the morphism Φ ∶ Z≤h,N d Gr O (for the moment N is arbitrary).Then the A-points of Z∇σ,r d for A a p-adically complete O-algebra of topologically finite type are precisely those (M, β) ∈ Z≤h,N d For every σ • κ become A[ 1 p ]-projective after inverting p and Fil • κ [ 1 p ] has type −w 0 µ κ = (−µ κ,d ≥ . . .≥ −µ κ,1 ).Proof of Theorem 12.1.By Corollary 15.2 it suffices to prove the factorisation on the level of A-valued points for A any finite local F-algebra.Let (M, σ, β) be such a point.Applying Lemma 10.8 we obtain a local finite flat O-algebra A with a map A → A and (M, σ) ∈ Y µ d (A) lifting (M, σ).Additionally, choose an S A -basis β lifting β.We will then be done if we can show that the special fibre of (M, σ, β) is mapped into M −w0µ by Ψ.We can assume that A = A ⊗ O F. Applying Proposition 12.3 for each κ we obtain filtrations Fil • κ .Define E by κ E κ (u) rκ E = ⋂ κ rκ n=0 E κ (u) rκ−n S A Fil n κ
, µ, τ )C λ Furthermore, the cycles C λ are described explicitly in[EG23, 8.6.2](using results from[CEGS19]).In particular, if λ is non-Steinberg then C λ is the irreducible component of X 2 labelled by λ as in[EG23, 5.5.11].As explained in [EG23, 8.3] the above inequality implies part (1) of the proposition by pulling back along the formally smooth morphism SpfR ρ ⊗ O F → X 2 .Likewise, part (2) will follow if we can show [X λ,1 2 ] = C λ for λ non-Steinberg.Furthermore,[EG23, 8.3]  explains that this equality is implied by the assertion that e(R λ,1 ρ ⊗ O F) = 1 for every ρ contained in a dense open subset of C λ , where e(R) denotes the Hilbert-Samuel multiplicity of a local ring R. We do this by considering the dense open consisting of ρ maximally non-split of niveau 1 and weight λ, which is indicated in part (2) of[EG23, 5.5.11].By[EG23, 5.5.4]we know e(R λ,1 ρ ⊗ O F) ≥ 1 for such ρ.From the definition given in[EG23, 5.5.1]we also know that if λ is non-Steinberg then any such ρ is not of the form ψ ⊗ 1 * 0 χ −1 cyc for ψ an unramified character and χ cyc the mod p cyclotomic character.

( Z∇σ,r 2 ⊗ O F) → Z∇σ,r 2 ⊗]]](
2 gives an identity of cycles[M µ ⊗ O F] = λ n(λ, µ)[M λ ⊗ O F]for integers n(λ, µ) ≥ 0 and λ running over tuples (λ κ0 ) with λ κ0 ≤ ∑ κ k =κ0 (µ κ − ρ).Each such λ κ0 then satisfies λ κ0,1 − λ κ0,2 ≤ p − 1 so we can also view the sum as running over absolutely irreducible F-representations of GL 2 (k) by 13.1.Applying the automorphism from Section 8 gives[M −w0µ ⊗ O F] = λ n(λ, µ)[M −w0 λ ⊗ O F]Proposition 7.6 allows us to view this identity of cycles as occurring within the closed subscheme Gr ∇σ ,r O ⊗ O F from 11.2 for r = (r κ ).We want to consider its preimage under the composite(14.2) Y ≤h 2 × Z ≤h 2 O F → Gr ∇σ ,r O ⊗ O F(here the auxilliary integer N is chosen sufficiently large so that Proposition 9.6 applies).To do this we need to show the composite is flat.As the first map is an isomorphism by Proposition 11.3, and G N is a smooth and irreducible group scheme, this composite is smooth with irreducible fibres.Smooth morphisms are flat so the pull-back of cycles is well defined and we obtain[Y of M −w0µ ⊗ O F. These are identities of dim G N + ∑ κ dim G P µκ -dimensionalcycles.Theorem 12.1 shows that Y µ 2 × Z ≤h 2 as cycles.We also point out that since each M λ ⊗ O F is irreducible and generically reduced (see Proposition 7.2) the same is true of M −w0 λ ⊗ O F. The same is then also true of Y λ,flag 2 since (14.2) is smooth with irreducible fibres.In particular, this implies the inequality [Y is an equality when µ = λ.As a consequence as dim G N +∑ κ dim G P µκ -dimensional cycles inside the scheme Y ≤h 2 × Z ≤h 2

(( Z≤h,N 2 ⊗
d 2 = 4. Pulling back the previous inequality along the special fibre of L ≤h 2 × Z ≤h 2 being a formally smooth morphism between Noetherian schemes, this map is flat and so the pull-back is defined) gives an inequality µ ρ ⊗ O F for L µ ρ the preimage of Y µ 2 in L ≤h ρ (defined just as in the proof of Lemma 10.8).This is an identity ofd 2 +dim G N +∑ κ dim G P µκ -dimensional cycles.Since the morphism L ≤h ρ × Z ≤h 2 O F) → L ≤h ρ is a G N -torsor (in particular smooth, surjective, and of relative dimension dim G N ) it follows that [L µ ρ ] ≤ λ n(λ, µ)[L λ ρ ] as d 2 + ∑ κ dim G P κ -dimensional cycles inside L ≤h ρ .Recall that the projective morphism Θ ∶ L ≤h ρ → Spec R ρ becomes a closed immersion after inverting p and this closed immersion identifiesL µ ρ [ 1 p ] = Spec R µ,1 ρ [ 1 p ].This was discussed in the proof of Lemma 10.8.Since the R µ ρ are O-flat an application of Lemma 3.3 shows that Θ * [L µ ρ ] = [Spec R µ,1 ρ ⊗ O F] Therefore, pushing forward the previous inequality of cycles gives [Spec R µ,1 ρ ⊗ O F] ≤ λ n(λ, µ)[Spec R λ,1 ρ ⊗ O F] now as d 2 + ∑ κ dim G P κ -dimensional cycles inside Spec R ρ ⊗ O F.Proposition 13.6 then gives that n(λ, µ) ≥ m(λ, µ, 1).Combining Lemma 13.3 and Proposition 6.6 we can view E * as an A-valued point of Gr O .Since E * * = E the endomorphism of Gr O induced by E ↦ E * is an automorphism.Lemma 8.2.The above automorphism identifies M λ with M −w0λ where w 0 ∈ W denotes the longest element.
Remark 12.2.If K is tamely ramified, i.e. if e is not divisible by p, then π κ − π κ ′ generates the same ideal of O K as π whenever κ ′ ≠ κ.Therefore v κ0 = 0 in this case.To see this consider the π-adic valuation of d du κ 0