Harder-Narasimhan filtrations for Breuil-Kisin-Fargues modules

We define and study Harder-Narasimhan filtrations on Breuil-Kisin-Fargues modules and related objects relevant to p-adic Hodge theory.


C H R IST O P H E C O R N U T M A C A R E N A P E C H E IR IS S A R R Y H A R D E R -N A R A SI M H A N F ILT R ATIO N S F O R B R E UIL -K ISIN -FA R G U E S M O D U L E S F ILT R ATIO N S D E H A R D E R -N A R A SI M H
universal such functor, and thus also a universal realisation category, which he called the category of motives. He also worked out an elementary bottom-up construction of this universal functor and its target category, assuming a short list of hard conjectures (the so-called standard conjectures) on which little progress has been made. A top-down approach to Grothendieck's conjecture aims to cut down the elusive category of motives from the various realisation categories of existing cohomology theories, and this first requires assembling them in some ways.
Over an algebraically closed complete extension C of Q p , Bhatt, Morrow and Scholze [BMS16] have recently defined a new (integral) p-adic cohomology theory, which specializes to all other known such theories and nicely explains their relations and pathologies. It takes values in the category of Breuil-Kisin-Fargues modules (hereafter named BKF-modules), a variant of Breuil-Kisin modules due to Fargues [Far15]. This new realisation category has various, surprisingly different but nevertheless equivalent incarnations, see [SW17,14.1.1], [Sch17,7.5] or Section 3; beyond its obvious relevance for p-adic motives, it is also expected to play a role in the reformulation of the p-adic Langlands program proposed by Fargues [Far16].
In this paper, we mostly investigate an hidden but implicit structure of these BKF-modules: they are equipped with some sort of Harder-Narasimhan formalism, adapted from either [Iri16] or [LWE16], which both expanded the original constructions of Fargues [Far19] from p-divisible groups over O C to Breuil-Kisin modules.

Overview
In Section 2, we define our categories of BKF-modules, review what Bhatt, Morrow and Scholze had to say about them, exhibit the HN-filtrations (which we call Fargues filtrations) and work out their basic properties. In Section 3, we turn our attention to the curvy avatar of BKF-modules up to isogenies, namely admissible modifications of vector bundles on the curve, and to their Hodge-Tate realizations. The link between all three incarnations of sthukas with one paw was established by Fargues, according to Scholze who sketched a proof in his lectures at Berkeley (1) . We redo Scholze's proof in slow motion and investigate the Fargues filtration on the curve and Hodge-Tate side, where it tends to be more tractable. We also clarify various issues pertaining to exactness, and introduce some full subcategories where the Fargues filtration is particularly well-behaved. In a subsequent work, we will show that ordinary BKFmodules with G-structures factor through these subcategories and compute the corresponding reduction maps, from lattices in the étale realization to lattices in the crystalline realization.

Results
We refer to the main body of the paper for all notations.
(1) Between [SW17] and [BMS16], the paw was twisted from Aξ to Aξ . We follow the latter convention. No sthukas were harmed in the making of our paper, but our valuations have lame normalizations.
We define Fargues (3.2). We define opposed Newton filtrations F N and F ι N and their types t N and t ι N on Mod ϕ L,f (2.6.2), and a Newton (or slope) filtration F N with type t N on Bun X (3.1.3) and Modif ad X (3.1.6). The Hodge and Newton filtrations are compatible with ⊗-product constructions and satisfy some exactness properties. If K = C is algebraically closed, then for a finite free BKF-module M ∈ Mod ϕ A,f mapping to the admissible modification E ∈ Modif ad X , we establish the following inequalities:

References
Here are some overall references for the material covered in this paper: [Cor] or [Cor18] for filtrations, lattices and related invariants, [And09] and [Cor18] for the Harder-Narasimhan formalism, respectively with categories and buildings, [Far10], [Far19] and [Iri16] or [LWE16] for Fargues filtrations on respectively finite flat group schemes, p-divisible groups and Breuil-Kisin modules, [FF18] and [Far15] for the Fargues-Fontaine curve, [SW17] and [BMS16] for BKF-modules, aka sthukas with one paw. Quasi-abelian categories abound in our paper. These are additive categories with well-behaved kernels and cokernels, which differ from abelian categories by the fact that the canonical morphism from the coimage (cokernel of the kernel) to the image (kernel of the cokernel) may fail to be an isomorphism. They usually show up as the torsion-free part of a cotilting torsion theory on an abelian category [BvdB03,App. B]. The Harder-Narasimhan formalism takes as input a quasi-abelian category C equipped with two additive functions rank : sk C → N and deg : sk C → R subject to various conditions, and outputs a canonical slope or Harder-Narasimhan filtration on C.

Filtrations
In [Cor], we defined a notion of Γ-filtrations for finite free quasi-coherent sheaves (aka vector bundles) over schemes, and in [Cor18] we investigated a notion of Rfiltrations on bounded modular lattices of finite length. Here is a common simple framework for Γ-filtrations and their types. If (X, ) is a bounded partially ordered set with smallest element 0 X and largest element 1 X , then a Γ-filtration on X is a function F : Γ → X which is non-increasing, exhaustive, separated and leftcontinuous: F(γ 1 ) F(γ 2 ) for γ 1 γ 2 , F(γ) = 1 X for γ 0, F(γ) = 0 X for γ 0 and for every γ ∈ Γ, there is a γ < γ such that F is constant on ]γ , γ] := {η ∈ Γ|γ < η γ}. If all chains of X are finite, the formula yields a bijection between the set F Γ (X) of all Γ-filtrations on X and the set of all pairs (c • , γ • ) where c • = F(Γ) = (c 0 < · · · < c s ) is a (finite) chain of length s in X with c 0 = 0 X and c s = 1 X , while γ • = Jump(F) = (γ 1 > · · · > γ s ) is a decreasing sequence in Γ. We then set F + (γ) := max {F(η) : η > γ}. If rank : X → N is an increasing function and r = rank(1 X ), then all chains of X are finite of length s r and any Γ-filtration F ∈ F Γ (X) has a well-defined type t(F) ∈ Γ r : for any γ ∈ Γ, the multiplicity of γ in t(F) is equal to rank(F(γ)) − rank(F + (γ)).
If C is an essentially small quasi-abelian category equipped with a rank function rank : sk C → N, as defined in [Cor18,3.1], then for every object X of C, the partially ordered set Sub(X) of all strict subobjects of X is a bounded modular lattice of finite length. A Γ-filtration on X is then a Γ-filtration on Sub(X), and we denote by F Γ (X) the set of all Γ-filtrations on X. For F ∈ F Γ (X), we typically write F γ = F γ = F(γ), F γ + = F >γ = F + (γ) and Gr γ F = F γ /F γ + . If r = rank(X), the type map t : F Γ (X) → Γ r is given by t(F) = (γ 1 · · · γ r ) ⇐⇒ ∀ γ ∈ Γ : rank Gr γ F = #{i : γ i = γ} and the degree map deg : F Γ (X) → Γ is given by deg(F) = deg(t(F)) = γ∈Γ rank Gr γ F ·γ.
and F y ∈ F Γ (y), and we have We denote by Gr Γ C and Fil Γ C the quasi-abelian categories of Γ-graded and Γ-filtered objects in C. For finite dimensional vector spaces over a field k, we set When Γ = R, we simplify our notations to F(X) := F R (X).

Invariants
Let O be a valuation ring with fraction field K, maximal ideal m and residue field k. We denote by (Γ, +, ) the totally ordered commutative group (K × /O × , ·, ), when we want to view it as an additive group. We extend the total orders to K/O × = K × /O × ∪ {0} and Γ ∪ {−∞}, by declaring that the added elements are smaller than everyone else. We denote by | · | : K → K/O × the projection. Thus for every λ 1 , λ 2 ∈ K, |λ 1 | |λ 2 | ⇐⇒ Oλ 1 ⊂ Oλ 2 . We write for the corresponding isomorphisms. When the valuation has height 1, i.e. when it is given by a genuine absolute value | · | : K → R + , we will identify K × /O × with the corresponding subgroup |K × | ⊂ R × + , and Γ with a subgroup of R, using genuine logarithms and exponential maps in some base b > 1. In this case, we will always normalize the related choices of the absolute value | · | and the base b by requiring that log b |π| = −1 for some specified nonzero element π of m, taking π to be a uniformizer whenever O is a discrete valuation ring. For every element γ ∈ Γ, is a finitely presented torsion O-module. These modules are the building blocks of the category of finitely presented torsion O-modules, which we denote by C.
Remark 1.4. -When O is a discrete valuation ring with uniformizer π, I(γ) = Oπ n for γ = n in Γ = Z according to our various conventions, and we thus retrieve the usual invariants of finitely generated torsion O-modules.
Proof. -We just need to establish the second claim when N is either a submodule or a quotient of M . For X ∈ C, set X ∨ := Hom O (X, K/O). One checks using the previous lemma that this defines an exact duality on C, with inv(X) = inv(X ∨ ). We may thus even assume that N is a quotient of M . Our claim now follows from the previous lemma and the surjectivity of Being a valuation ring, O is a coherent ring. Thus any finitely generated submodule of a finitely presented O-module is also finitely presented, and the finitely generated submodules of any M ∈ C also belong to C. Lemma 1.6. -For M ∈ C and any positive integer r, Proof. -It is plainly sufficient to establish that for every submodule N of M generated by r elements, length(N ) Suppose that two out of {M 1 , M 2 , M 3 } belong to C. Then so does the third one and if and only if the exact sequence splits.
For every M ∈ C, there is a canonical Γ-filtration F(M ) on M ⊗ k defined by It depends functorially upon M and one checks easily that we have where r = r(M ). In particular, length(M ) = − deg(F(M )).
Lemma 1.11. -Let 0 → V 1 → V 2 → V 3 → 0 be an exact sequence of K-vector spaces. For any pair of O-lattices L 2 , L 2 ∈ L(V 2 ), their inverse and direct images in V 1 and V 3 are O-lattices L 1 , L 1 ∈ L(V 1 ) and L 3 , L 3 ∈ L(V 3 ), and we have is an exact sequence.
Proof. -Plainly 0 → L 1 → L 2 → L 3 → 0 and 0 → L 1 → L 2 → L 3 → 0 are exact; thus L 3 and L 3 are finitely generated over O, in particular they are both O-lattices in V 3 and free over O; it follows that both exact sequences split, which implies that L 1 and L 1 are also (finite free) O-lattices in V 1 . For the remaining claims, we may as above replace L 2 by xL 2 for some x ∈ K × (which replaces L i by xL i for i ∈ {1, 3}) to reduce to the case where L i ⊂ mL i ⊂ L i for all i ∈ {1, 2, 3}. Applying Lemma 1.7 to the resulting exact sequence of finitely presented torsion O-modules . Moreover by Lemma 1.8, equality holds in either one of them if and only if for every γ ∈ Γ, is an exact sequence of k-vector spaces. This proves the lemma. For L 1 , L 2 ∈ L(V ), we denote by ν(L 1 , L 2 ) ∈ Γ the degree of d(L 1 , L 2 ).

Tensor products
There are also compatible notions of tensor products, symmetric and exterior powers for types, objects and Γ-filtered objects in arbitrary quasi-tannakian categories, and O-lattices in K-vector spaces. All of these notions are fairly classical, and their various compatibilities easily checked. For instance if L and L are O-

The rings
Let p be a prime number, E a finite extension of Q p , K a perfectoid field extension of E, K the tilt of K. We denote by O E , O K and O K the ring of integers in E, K and K , with maximal ideals m E , m K and m K , and perfect residue fields F q := O E /m E (finite with q elements) and F := O K /m K = O K /m K . We fix once and for all a uniformizer π of E. We denote by W O E ( · ) the Witt vector functor with values in O E -algebras, as defined in [FF18, 1.2]. We set Thus A(K) and O L are complete discrete valuation rings with uniformizer π and residue fields respectively equal to K and F, while our main player A := A(O K ) is a non-noetherian complete local ring with maximal ideal m and residue field F. We denote by ϕ the Frobenius x → x q in characteristic p or its extension to O E -Witt vectors. The ring homomorphisms ,n for all i 0. We fix a generator ξ of ker(θ) and set ξ := ϕ(ξ). We write for the image of ξ in O K = A 1 , where more generally A n := A/π n A for n ∈ N. Thus is a pseudo-uniformizer of K , i.e. a non-zero element of m K . For an A-module M and n ∈ N, we define In particular, M 1 = M ⊗ A O K . We normalize the absolute value of K by requiring that q | q | = 1. This weird normalization is meant to simplify some formulas, under the conventions borrowed from [BMS16] for sthukas with one paw, which differ from those of [SW17]; the latter would have lead us to the normalization q | | = 1. We normalize the discrete absolute value on E and L by requiring that q |π| = 1, and we similarly normalize the discrete absolute values on the fraction fields B dR and B dR of the completed local rings B + dR and B + dR of A at (ξ) and (ξ ) by requiring that respectively q |ξ| = 1 and q |ξ | = 1.
Remark 2.1. -In the sequel, we will often refer to [BMS16] or [SW17] where E = Q p . We have carefully checked that the results we quote from either of these references also hold when E is an arbitrary finite extension of Q p , with essentially identical proofs.

Categories of
is a finitely generated torsion module over the complete discrete valuation ring A(K), there is a unique sequence of integers Chasing denominators, we may modify any such isomorphism into one that fits in a commutative diagram of π ∞ -torsion A-modules If π n M [π ∞ ] = 0, the cokernel of the top map is a finitely generated This is a finitely presented A-module killed by π n for n 0.

2.2.3.
We will consider the following strictly full subcategories of Mod A, * : Then any M ∈ Mod A, * has a canonical and functorial dévissage with everyone in the relevant subcategory. The projective dimension of the nonzero A-modules in Mod A,f , Mod A,t and Mod A,m ∞ are respectively 0, 1 and 2.

2.2.4.
For n 0, π n kills M [π ∞ ] and M , thus for any m ∈ M f , π n m is the image of some m ∈ M and π n m ∈ M only depends upon m. This defines an embedding M f → M whose cokernel Q is a finitely presented A-module killed by π 2n : By construction, the composition M f → M → M f is multiplication by π 2n and since M [π 2n ] = M [π ∞ ], we obtain an exact sequence of A-modules

2.2.5.
Any A-module M in Mod A,π ∞ has yet another canonical and functorial dévissage, the finite non-decreasing filtration by the finitely presented A-submodules M [π n ] of M whose successive quotients M [π n ]/M [π n−1 ] π n−1 M [π n ] are finitely presented O K -modules. If M belongs to Mod A,t , these subquotients are torsion free, thus finite free over O K . If M belongs to Mod A,m ∞ , they are finitely presented torsion O Kmodules, thus themselves non-canonically isomorphic to direct sums of modules of

2.2.6.
For every A-module N and any nonzero x ∈ O K , the exact sequences , and an exact sequence

2.2.7.
The category Mod A, * is stable under extensions in Mod A . The next proposition implies that it inherits from Mod A the structure of a closed symmetric monoidal category, which just says that Mod A, * is a ⊗-category with internal Homs. Proof. -Fix a finite resolution P • of M 1 by finite free A-modules. Then , and similarly So all of these A[ 1 π ]-modules are indeed finite and free.

2.2.8.
The categories Mod A,π ∞ and Mod A,m ∞ are weak Serre subcategories of Mod A : they are stable under kernels, cokernels and extensions. In particular, they are both abelian. The category Mod A,t is also stable by extensions and kernels in Mod A , but it is only quasi-abelian. In fact, the exact sequence (for yields a cotilting torsion theory [BvdB03] on the abelian category Mod A,π ∞ with torsion class Mod A,m ∞ and torsion-free class Mod A,t : any M ∈ Mod A,π ∞ is a quotient of A r n ∈ Mod A,t for some n, r ∈ N (being finitely generated over A and killed by a power of π), and there is no nonzero morphism from an object in Mod A,m ∞ to an object in Mod A,t . The kernel and coimage of a morphism in Mod A,t are the corresponding kernel and coimage in the abelian category Mod A,π ∞ or Mod A . The image and cokernel of f : M → N in Mod A,t are given by

2.2.9.
The categories Mod A,f , Mod A,π ∞ and Mod A,m ∞ are stable under the usual Ext's and Tor's in Mod A , and so they are also ⊗-categories with internal Homs (but only Mod A,f has a neutral object). They are also stable under symmetric and exterior powers (of rank k 1 for the torsion categories).
The category Mod A,t is stable under the internal Hom of Mod A , but it is not stable under the ⊗-product of Mod A . For instance, if x = 0 belongs to m K , then is a finitely generated ideal of A 2 , so it belongs to Mod A,t , but the image of π in is a nonzero element killed by [x] ∈ m \ πA. We can nevertheless equip Mod A,t with a tensor product compatible with the usual internal Hom, given by With this definition, Mod A,t becomes yet another ⊗-category with internal Homs.

2.2.10.
As explained in 2.2.2 or 1.6.3, there is an invariant Alternatively, inv t (M ) is the unique element (n 1 · · · n s ) of N ∞ such that ∀ n 1 : This follows from 2.2.6, which indeed implies that for every n 1, This invariant yields a function rank t : sk Moreover by 2.2.6 and Lemma 1.7, for every exact sequence with equality if and only if the exact sequence is a rank function on Mod A,t in the sense of [Cor18]: it is additive on short exact sequences, nonzero on nonzero objects, and constant on mono-epis in Mod A,t .

2.2.11.
For M ∈ Mod A, * , let I be the image of M → M f . For any n 1, recall that M n = M/π n M , which is a finitely presented A n -module. The dévissage of M from 2.2.3 yields exact sequences of finitely presented A n -modules with n → rank t M t,n non-decreasing and equal to rank t M t for n 0.

2.2.12.
A good filtration on a module M in Mod A,m ∞ is a sequence We have seen in 2.2.5 that any M in Mod A,m ∞ has such a good filtration. We claim that the principal ideal It is therefore sufficient to treat the case where M = O K (x) for some nonzero x ∈ O K , which follows from Lemma 1.7. We thus obtain a generalized length function, we have the following formula: Since log q ξ = log q | q | = −1, we obtain another formula for the rank on Mod A,t : 2.2.14.
The functor M → M [ 1 π ] extends to the isogeny categories,

Categories of ϕ-A-modules
Its kernel and cokernels are given by (ker(f ), ϕ 1 ) and (coker(f ), ϕ 2 ) with commutative. This makes sense since M → M [ξ −1 ] and M → ϕ * M are exact. The category Mod ϕ A is abelian, and it is a ⊗-category: using the isomorphisms the tensor product, symmetric and exterior powers, and neutral object are A, the strictly full subcategory of Mod ϕ A, * of all BKF-modules (M, ϕ M ) whose underlying A-module M lies in the strictly full subcategory with everyone in the relevant strictly full subcategory.

2.3.3.
The categories Mod ϕ A,π ∞ and Mod ϕ A,m ∞ are weak Serre subcategories of Mod ϕ A : they are stable under kernels, cokernels and extensions. In particular, they are both abelian. The category Mod ϕ A,t is also stable under extensions and kernels in Mod ϕ A , but it is only quasi-abelian. This last statement now requires some argument, given below: for every M ∈ Mod ϕ A,π ∞ , the exact sequence yields a torsion theory on the abelian category Mod ϕ A,π ∞ with torsion class Mod ϕ A,m ∞ and torsion-free class Mod ϕ A,t , but we do not know whether this torsion theory is cotilting (is every object M of Mod ϕ A,π ∞ a quotient of some N in Mod ϕ A,t ?), and thus we can not appeal to the criterion of [BvdB03, B.3] for quasi-abelian categories, as we did for Mod A,t in 2.2.8. Plainly, kernels and coimages in Mod ϕ A,t are the corresponding kernels and coimages in Mod ϕ A . The image and cokernel of a morphism f : where the Frobeniuses are induced by ϕ N : which is valid for any finitely presented M 1 also yields an internal Hom, , ϕ 2 on any of these categories. The subcategory Mod ϕ A,t of Mod ϕ A, * is stable under this internal Hom, but it is not stable under the tensor product. As for Mod A,t , there is a modified tensor product (M 1 , ϕ 1 ) ⊗ t (M 2 , ϕ 2 ) := (M 1 ⊗ t M 2 , ϕ 1 ⊗ t ϕ 2 ) which turns Mod ϕ A,t into a genuine ⊗-category with internal Hom's.

2.3.6.
There is a Tate object

The Fargues filtration on Mod
The rank function on Mod A,t yields a rank function on Mod ϕ A,t , In addition, the length function on Mod A,m ∞ yields a degree function on Mod ϕ A,t , From the short exact sequences and 2.2.13, we thus obtain that does not depend upon n 0. Plainly, for every n ∈ Z. A short exact sequence in Mod ϕ A,t yields, for every n 0, a commutative diagram with exact rows and columns, from which easily follows that with equality if and only if f is an isomorphism.

2.4.2.
We may now apply the Harder-Narasimhan formalism of [And09] or [Cor18] to the quasi-abelian category Mod ϕ A,t , equipped with the rank and degree functions that we have just defined, for the slope function µ = deg t / rank t . Specializing the general theory to the case at hand, we obtain the following definitions and results.
The semi-stable BKF-modules of slope µ form an abelian full subcategory of Mod ϕ A,t , and every BKF-module M in Mod ϕ A,t has a unique decreasing R-filtration F by strict subobjects F γ with Gr γ with equality on the left (resp. right) for s = 0 (resp. s = r). In particular, with equality for s = 0 and s = r 2 . In particular, Proof. -These are standard properties of Harder-Narasimhan filtrations on quasiabelian categories, see for instance [Cor18,Proposition 21] or [And09,4.4.4].
Proposition 2.5. -For every M ∈ Mod ϕ A,t of rank r ∈ N and any n ∈ Z, Proof. -This is obvious: the map N → N {n} induces a bijection between strict subobjects of M and strict subobjects of We thus have the following relations: Moreover, we have We obtain the following inequalities: for 0 s r 1 , and for r 1 s r 2 , which also implies that for 0 s r 3 ,
For any M ∈ Mod ϕ A and n 1, consider the exact sequence Suppose that M is a BKF-module, i.e. belongs to Mod ϕ A, * . Then M n and M [π n ] both belong to Mod ϕ A,π ∞ . Moreover, rank t M n n rank A M by 2.2.11. Viewing t F (M n ) as a concave function on [0, rank t M n ], we may thus define Proposition 2.6. -There is a constant C(M ) such that the functions t F,n (M ) are C(M )-Lipschitzian. They converge uniformly to a continuous concave function Suppose first that M is free. Then for every n, m 1, the exact sequence It follows that for every n, k 1 and 0 s r, f nk (s) f n (s) with equality for s ∈ {0, r}.
In particular, f n (s) f 1 (s) with equality for s ∈ {0, r}, and the slopes of the continuous piecewise linear functions f n are uniformly bounded by the constant Fix n 0 , n 1. For n = n 0 q n + r n with q n 0 and 0 r n < n 0 , we have from which we obtain that for 0 s r, for some s , s ∈ [0, r] with n 0 q n s + r n s = ns. But then s − s = rn n (s − s ), thus Therefore lim sup f n (s) f n 0 (s) and this being true for all n 0 1, lim sup f n (s) lim inf f n (s) i.e. f n (s) converges to some limit f ∞ (s) ∈ R. Since all the f n 's are C-Lipschitzian concave, so is f ∞ = f ∞ (M ) and the convergence is uniform. Suppose next that M is torsion free, so that 0 → M → M f → M → 0 is exact and for n 0 (such that π n M = 0), we obtain an exact sequence It remains to establish that M → t F,∞ (M ) is constant on isogeny classes, and we already know that t F,∞ (M ) = t F,∞ (M f ). We thus have to show that if is an exact sequence in Mod ϕ A with M 1 , M 2 finite free and Q torsion, then t F,∞ (M 1 ) equals t F,∞ (M 2 ). For n 0 (such that π n Q = 0), we obtain exact sequences Splitting them in two short exact sequences and using again the computations of Section 2.4.3 yields the desired equality. (1) For every γ ∈ R and n, m 1, the exact sequence induces an exact sequence defines an R-filtration on M by finite free BKF-submodules whose underlying A-submodules are direct summands: the quotient Gr γ F (M ) = F γ F (M )/F >γ F (M ) is a finite free BKF-module. (3) For every γ ∈ R and n 1, In particular, the type of the R-filtration F • F (M ) is given by Proof. -(1) Since t F,∞ (M ) = t F,1 (M ), also t F,n (M ) = t F,1 (M ) for every n 1, thus for every n, m 1, from which (1) immediately follows by Proposition 2.4.
(2) and (3): This follows from (1) by a standard argument: consider for n, m 1 and γ ∈ R the commutative diagram with exact rows and columns Taking the projective limit over n, and since every one is Mittag-Leffler surjective, we obtain a commutative diagram of A-modules with exact rows and columns , the first row tells us that F γ F (M ) is separated and complete in the π-adic topology, with F γ F (M ) 1 F γ F (M 1 ) finite free over A 1 = O K , say of rank s ∈ N. Pick a morphism α : A s → F γ F (M ) reducing to an isomorphism modulo π. By the topological version of Nakayama's lemma, α is surjective, and F γ F (M ) is finitely generated over A. Playing the same game with the third row, we obtain a surjective morphism β : A s G γ F (M ) reducing to an isomorphism modulo π. But now the kernel N of β has to be finitely generated over A since G γ F (M ) is finitely presented over A by the second column. Applying Tor A , which is trivial by the third row, thus N = 0 by the classical version of Nakayama's lemma. It follows that β is an isomorphism, G γ F (M ) is free, the middle column is split (in Mod A ), and F γ F (M ) is also free, being finite projective over the local ring A. The remaining assertions of (2) and (3)  Definition 2.12. -We say that a finite free BKF-module M is semi-stable (of slope γ ∈ R) if M 1 is semi-stable (of slope γ ∈ R).
Example 2.13. -Any finite free BKF-module M of rank 1 is semi-stable of slope deg t (M 1 ), thus A is semi-stable of slope 0 and A{1} is semi-stable of slope 1.
By Proposition 2.8, a finite free BKF-module M is semi-stable (of slope γ) if and only if M n is semi-stable (of slope γ) for every n 1, in which case M is of HN-type and t F,∞ (M ) = t F,1 (M ) = t F (M 1 ) is isoclinic (of slope γ). By Proposition 2.10, a finite free BKF-module M of HN-type has a canonical filtration F F (M ) whose graded pieces are finite free semi-stable BKF-modules with decreasing slopes. Conversely, any finite free BKF-module which has such a filtration is of HN-type (by uniqueness of the Fargues filtration on Mod ϕ A,t ) and its filtration is the canonical one.

2.5.3.
We denote by Mod ϕ, * A,f the strictly full subcategory of Mod ϕ A,f whose objects are the finite free BKF-modules of HN-type. The functoriality of the Fargues filtration on

Categories of ϕ-R-modules
For any A-algebra R equipped with a ring isomorphism ϕ : R → R compatible with ϕ : A → A, we may analogously define the abelian ⊗-category Mod ϕ R and its full ⊗-subcategories Mod ϕ R, * and Mod ϕ R,f . They come equipped with ⊗-functors Mod ϕ A,? → Mod ϕ R,? for ? ∈ {∅, * , f }, which are exact when A → R is flat. In this section, we discuss the following cases: In this case, Mod ϕ R,f is the full subcategory of Mod ϕ A,t made of all BKF-modules killed by π. This is the quasi-abelian category of all finite free O K -modules M equipped with an isomorphism ϕ M : ϕ * M ⊗ K → M ⊗ K , or equivalently, with a ϕ-semilinear isomorphism φ M : M ⊗ K → M ⊗ K . As a subcategory of Mod ϕ A, * , it is stable under tensor products, internal Homs, symmetric and exterior powers, and it has a neutral object of its own. Using the isomorphisms the tensor products, internal Homs and neutral object in Mod ϕ R,f are given by The rank and degree functions on with equality if and only if for every γ ∈ R, the complex of F-vector spaces Proof. -This follows from 1.6.5 and Lemma 1.11.
Proof. -Let X • = γ X γ be the R-graded object of Mod ϕ R,f attached to the Fargues filtration of X = (M, ϕ). Then by Propositions 2.4 and 2.16, We may thus assume that X is semi-stable, in which case the result is obvious since the concave polygons t F (X) and t H (X) have the same terminal points.
where I γ = x ∈ K : |x| q −γ , and ξ mod π = q , i.e. For λ = d h with d ∈ Z and h ∈ N * relatively prime, D λ is the union of the finitely generated O L -submodules X of D such that φ h D (X) = π d X. This Newton decomposition is functorial, compatible with all tensor product constructions, thus is an exact ⊗-functor, and so are the corresponding opposed Newton Q-filtrations which are given by the usual formulas We denote by t N (D, ϕ D ) and t ι N (D, ϕ D ) the corresponding opposed types. Both Newton filtrations are Harder-Narasimhan filtrations, for the obvious rank function on Mod ϕ L,f and for the opposed degree functions which are respectively given by . These degree functions are Z-valued! If the residue field F is algebraically closed, the category Mod ϕ L,f is even semi-simple, with one simple object D • λ for each slope λ ∈ Q. If λ = d h as above, then rank(D  O L ,f → Fil Z F is compatible with tensor products, duals, symmetric and exterior powers. For every exact sequence with equality if and only if for every γ ∈ Z, the complex of F-vector spaces Proof. -We first show that F ι N (X ⊗ L) and F H (X) have the same degree. Since both filtrations are compatible with exterior powers, we may assume that the rank of X = (M, ϕ M ) equals 1.
Returning to the general case, both polygons thus have the same terminal points. We now follow the proof of Corollary 2.17. Let X • = γ X γ be the Q-graded object of Mod ϕ O L ,f attached to the filtration on X induced by F ι N (X ⊗ L). Then by exactness of F ι N and the previous proposition . We may thus assume that X ⊗ L is semi-stable (i.e. isoclinic), in which case the result is obvious since t ι N (X ⊗ L) and t H (X) have the same terminal points.  The category Mod ϕ R, extends to a commutative diagram and similarly for the other three vertices. Each line of our diagram thus yields a pair of Z-filtrations on the residue (over K or K ) of its vertices, which have opposed types in Z r where r is the rank of M , and the two pairs match along the ϕ-equivariant isomorphisms which are induced by the vertical maps. In particular, the Hodge Z-filtrations with equality if and only if for every γ ∈ R, the complex of K -vector spaces Proof We first claim that each M i is finite free over A. By descending induction on i, it is sufficient to establish that the following A-module has projective dimension 1: We will show that it is finite free over A(1) := A/Aξ . Since A(1) A/Aξ O K is a valuation ring, we just have to verify that X i is finitely generated and torsionfree over A(1). Since Q is finitely presented over A, it is finitely presented over A(n) := A/Aξ n , which is a coherent ring by [BMS16,3.24], thus Q i = Q[ξ i ] is finitely presented over A(n) and A for all i, and so is X i Q i /Q i−1 . On the other hand, Q[m ∞ ] = 0 by 2.2.1, thus also X i [m ∞ ] = 0, which means that X i is indeed torsion-free as an A(1)-module. We denote by x i the rank of X i over A(1).
Let S be any one of the valuations rings The triangular inequality of Lemma 1.9 then yields is the image of ξ in S and since also |ξ S | = q −1 in all three cases for the normalized absolute value on S, with exactly x i one's. Now observe that by definition of our various Hodge types, To establish the proposition, it is now sufficient to show that for S = B + dR , actually Being the completion of the Noetherian local ring A (ξ ) of A, S is flat over A, thus and this a priori real number actually belongs to Z. We call it the degree of M .

The functors of Fargues
Suppose from now on that K = C is algebraically closed. In this section, we will define and study the following commutative diagram of covariant ⊗-functors: In this diagram, the first two lines are equivalences of ⊗-categories, the top vertical arrows are faithful and the bottom ones fully faithful. The construction of E which is given below is a covariant version of the analytic construction of [Far15]. A slightly twisted version of it was sketched in Scholze's course [SW17] (for stukhas with one paw at m = Aξ). Our variant is meant to match the normalized construction of HT in [BMS16], where the paw was twisted from m to m = Aξ . Following [BMS16], we fix a compatible system of p-power roots of unity, ζ p r ∈ O × C for r 1, and set = (1, ζ p , ζ p 2 , . . . As suggested by the notations, ξ is a generator of ker(θ : A O C ). We have Moreover, θ(ϕ −1 (µ)) = ζ q − 1 = 0, and therefore ξ ϕ −1 (µ) and ξ µ. TOME 2 (2019)

The Fargues-Fontaine curve
Let X = X C ,E be the Fargues-Fontaine curve attached to (C , E) [FF18]. This is an integral noetherian regular 1-dimensional scheme over E which is a complete curve in the sense of [FF18,5.1.3]: the degree function on divisors factors through a degree function on the Picard group, deg : Pic(X) → N. We denote by η the generic point of X and by E(X) = O X,η the field of rational functions on X. In addition, there is a distinguished closed point ∞ ∈ |X| whose completed local ring O ∧ X,∞ is canonically isomorphic to the ring B + dR of Section 2.6.5, see also 3.4.12 below.

Vector bundles on the curve
Let Bun X be the E-linear ⊗-category of vector bundles E on X. Since X is a regular curve, it is a quasi-abelian category whose short exact sequences remain exact in the larger category of all sheaves on X, and the generic fiber E → E η yields an exact and faithful ⊗-functor which induces an isomorphism between the poset Sub(E) of strict subobjects of E in Bun X and the poset Sub(E η ) of E(X)-subspaces of E η .

Newton slope filtrations
The usual rank and degree functions rank : sk Bun X → N and deg : sk Bun X → Z are additive on short exact sequences in Bun X , and they are respectively constant and non-decreasing on mono-epis in Bun X . More precisely, if f : E 1 → E 2 is a monoepi, then rank(E 1 ) = rank(E 2 ) and deg(E 1 ) deg(E 2 ) with equality if and only if f is an isomorphism. These functions yield a Harder-Narasimhan filtration on Bun X , the Newton filtration F N with slopes µ = deg / rank in Q (introduced in [FF18, 5.5]). The filtration F N (E) on E ∈ Bun X is non-canonically split. More precisely for every µ ∈ Q, the full subcategory of semi-stable vector bundles of slope µ is abelian, equivalent to the category of right D µ -vector spaces, where D µ is the semi-simple division E-algebra whose invariant is the class of µ in Q/Z. We denote by O X (µ) its unique simple object [FF18,5.6.22]. Then for every vector bundle E on X, there is unique sequence µ 1 · · · µ s in Q for which there is a (non-unique) isomorphism E, and any such isomorphism maps i: [FF18,Chapter 8], particularly its main theorem [FF18,8.2.10]). We denote by t N (E) ∈ Q r the type of F N (E), where r = rank(E).
Proposition 3.1. -The Newton filtration is compatible with tensor products, duals, symmetric and exterior powers in Bun X . For any exact sequence in Bun X , with equality for s = 0 and s = r 2 . In particular, Proof. -The compatibility of F N with ⊗-products and duals comes from [FF18,5.6.23]. Since Bun X is an E-linear category, the compatibility of F N with symmetric and exterior powers follows from its additivity and compatibility with ⊗-products. For the remaining assertions, see [Cor18,Proposition 21] or [And09,4.4.4].

Modifications of vector bundles
We denote by Modif X the category of triples where E 1 and E 2 are vector bundles on X while f is an isomorphism This defines a quasi-abelian E-linear rigid ⊗-category with a Tate twist. The kernels and cokernels are induced by those of Bun X . The neutral object is the trivial modification O X = (O X , O X , Id), the tensor product and duals are given by Here E(1) := E ⊗ Zp Z p (1) with Z p (1) := lim ← − µ p n (C) and can : O X → O X (1) is the canonical morphism, dual to the embedding I(∞) → O X . There are also symmetric and exterior powers, given by the following formulae: for every k 0, The generic fiber E → E 1,η yields an exact faithful ⊗-functor which induces an isomorphism between the poset Sub(E) of strict subobjects of E in Modif X and the poset Sub(E 1,η ) of E(X)-subspaces of E 1,η . We say that a modification E = (E 1 , E 2 , f ) is effective if f extends to a (necessarily unique) morphism f : E 1 → E 2 , which is then a mono-epi in Bun X . For every E in Modif X , E{i} is effective for i 0.

Hodge and Newton filtrations
For E = (E 1 , E 2 , f ) as above, we denote by These are filtrations with opposed types t H,i (E) ∈ Z r , where The filtrations F N,i and F H,i are compatible with tensor products, duals, symmetric and exterior powers. In particular for every 0 k r, i (E)(k) viewing the right hand side terms as functions on [0, r]. Also, E is effective if and only if the slopes of F H,1 (resp. F H,2 ) are non-negative (resp. non-positive), in which case t H,1 (E) is the type t(Q) of the torsion O X -module Q = E 2 /f (E 1 ) supported at ∞, which means that if t H,1 = (n 1 · · · n r ) ∈ N r , then Proof. -Using a Tate twist, we may assume that E is effective. The left and right-hand side concave polygons then already have the same terminal point, since deg(E 2 ) = deg(E 1 ) + deg(Q) where Q = E 2 /f (E 1 ). By the formula for the exterior powers, it is then sufficient to establish that . Then E 2 is semi-stable of slope µ = t max N (E 2 ) and deg E 2 = deg E 1 + deg Q , thus t N (E 2 ) t N (E 1 ) + t(Q ) by concavity of the sum and equality of the terminal points. Considering the first (largest) slopes, we find that µ t max N (E 1 ) + t max (Q ). But E 1 ⊂ E 1 and Q ⊂ Q, thus t max N (E 1 ) t max N (E 1 ) and t max (Q ) t max (Q). This yields the desired inequality.

Admissible modifications
Let Modif ad X be the full subcategory of Modif X whose objects are the modifications E = (E 1 , E 2 , f ) such that E 1 is semi-stable of slope 0, i.e. t N,1 (E) = t N (E 1 ) = 0. This is a quasi-abelian E-linear rigid ⊗-category with Tate twists. The kernels, cokernels, duals, ⊗-products, Tate twist, symmetric and exterior powers are induced by those of Modif X . On Modif ad X , we set Proof. -This is the special case of Proposition 3.2 where t N,1 (E) = 0. The restriction of the generic fiber functor ( · ) 1,η : Modif X → Vect E(X) to the full subcategory Modif ad X of Mod X descends to an exact E-linear faithful ⊗-functor ω : Modif ad X → Vect E , ω(E) = Γ(X, E 1 ) inducing an isomorphism between the poset Sub ad (E) ⊂ Sub(E) of strict subobjects of E in Modif ad X and the poset Sub(ω(E)) ⊂ Sub(E 1,η ) of E-subspaces of ω(E).

The Fargues filtration
The rank and degree functions rank : sk Modif ad X → N and deg : sk Modif ad X → Z which are respectively defined by are additive on short exact sequences in Modif ad X , and they are respectively constant and non-decreasing on mono-epis in Modif ad X . More precisely if F = (F 1 , F 2 ) is a monoepi F : E → E , then F 1 : E 1 → E 1 is an isomorphism and F 2 : E 2 → E 2 is a mono-epi in Bun X , thus deg(E) = deg(E 2 ) deg(E 2 ) = deg(E ) with equality if and only if F 2 is an isomorphism in Bun X , which amounts to F = (F 1 , F 2 ) being an isomorphism in Modif ad X . These rank and degree functions thus induce a Harder-Narasimhan filtration on Modif ad X , the Fargues filtration F F with slopes µ = deg / rank in Q, and the full subcategory of Modif ad X of semi-stable objects of slope µ is abelian. We denote by t F (E) the type of F F (E).
if 0 s r 1 f 1 (r 1 ) + f 3 (s − r 1 ) if r 1 s r 2 with equality for s = 0 and s = r 2 . In particular, Proof. -The breaks of the concave polygon t F (E) have coordinates is a strict subobject of E 2 in Bun X , equal to E 2 for γ 0. Thus by definition of F N (E 2 ), we find that t F (E) lies below t N (E 2 ) = t N (E) and both polygons have the same terminal points, which proves the proposition.

3.1.8.
Let E = (E 1 , E 2 , α) be an admissible modification and set V = Γ(X, E 1 ), so that E 1,η = V E(X) and E 1 (∞) = V C . We view F H = F H (E) as an element of F(V C ), as an element of F(V E(X) ) and F * F := Γ(X, F F (E) 1 ) as an element of F(V ). For every F ∈ F(V E(X) ), define , we have for any i ∈ {1, 2} the following equality:

ANNALES HENRI LEBESGUE
Since E 1 is semi-stable of slope 0, E 1 , F 0 with equality if and only if each F γ j (E 1 ) is of degree 0. We thus obtain: for every F ∈ F(V E(X) ), Proposition 3.6. -With notations as above, the following conditions are equivalent: (1) F * N is the unique element F of F(V E(X) ) such that E 2 , G F, G for every G ∈ F(V E(X) ) with equality for G = F, and; (2) F * F is the unique element f of F(V ) such that E 2 , g f, g for every g ∈ F(V ) with equality for g = f . Thus F * F = F * N ⇔ F * N ∈ F(V ) and the proposition follows.
The kernel and cokernel of F are given by This defines a quasi-abelian rigid E-linear ⊗-category with tensor product neutral object (E, B + dR ) and duals, symmetric and exterior powers given by where the tensor product constructions are over E or B + dR .

3.2.3.
There is also an exact and fully faithful ⊗-functor from the category HT B dR E to the quasi-abelian ⊗-category denoted by This functor is plainly compatible with the rank and degree functions of both categories (for the appropriate normalization of the valuation on B dR ), and its essential image is stable under strict subobjects. It is therefore also compatible with the corresponding Harder-Narasimhan filtrations. Since the Harder-Narasimhan filtration on Norm B dR E is compatible with tensor products, duals, symmetric and exterior powers by [Cor18, Proposition 28], we obtain the following proposition: and Modif ad X are compatible with tensor products, duals, symmetric and exterior powers.

3.2.4.
Fix an admissible modification E of rank r and set HT(E) = (V, Ξ). Then where F γ (Q) is the image of F γ (E 2 ) in the torsion sheaf Q = E 2 /f (E 1 ) on X and Gr γ F (Q) = F γ (Q)/F >γ (Q). These are skyscraper sheaves supported at ∞, with Γ(X, Q) = Ξ/V + dR and Γ(X, where • is the gauge norm of V + dR ⊂ V dR and the right-hand side term is the Busemann scalar product, see [Cor,6.4.15]. This formula still holds true for a non-necessarily effective admissible modification E, since indeed for every i ∈ Z, Returning thus to the general case, we now obtain: Here t H = t(F H ) is the Hodge type of E and F C = loc(F dR ) is the R-filtration on in R r . The last pairing is the standard scalar product on R r ⊂ R r , and the two inequalities come from [Cor,4.2 & 5.5].
is the Newton type of E. Now we have already seen that , and we thus obtain the following inequalities: Proposition 3.9. -With notations as above, t N 2 t H , t N and Proof. -This now follows from Proposition 3.6.

3.2.5.
Let Any such morphism has a kernel and a cokernel, which are respectively given by (ker(f ), ker(f dR ) ∩ Ξ) and is the torsion submodule of T /f (T ). This defines a quasi-abelian rigid O E -linear ⊗-category with tensor product neutral object (O E , B + dR ) and duals, symmetric and exterior powers given by where the tensor product constructions are over O E or B + dR . There is also a Tate twist in 3.2.6.
The exact and faithful ⊗-functor induces a ⊗-equivalence of ⊗-categories This first square induces yet another commutative diagram of isomorphisms with notations as in 2.6.5. Restricting to lattices, we obtain the following commutative diagrams of isomorphisms (for the second diagram, note that µ ∈ (B + dR ) × ): Ξ It follows that our various Hodge Z-filtrations are related as follows: where ϕ : C → C is the residue of ϕ : B + dR → B + dR and the isomorphisms η M,C :

Compatibility with Tate objects
The Tate object of Mod ϕ A,f is given by Thus since µ is invertible in A(C), Since O E = A(C) ϕ=Id and t H (A{1}) = 1, it follows that

Fargues's theorem
The following theorem was conjectured by Fargues in [Far15]. The full faithfulness is established in [BMS16,4.29]. A proof of the essential surjectivity is sketched in Scholze's Berkeley lectures [SW17], where it is mostly attributed to Fargues. An expanded and referenced version of this sketch is given in Section 3.4 below.
Corollary 3.11. -The categories Mod ϕ A,f and Mod ϕ A,f ⊗ E are quasi-abelian. In particular, any morphism in these categories has a kernel and a cokernel. But we have no explicit and manageable formulas for them. Note also that we have two structures of exact category on Mod ϕ A,f and Mod ϕ A,f ⊗E: the canonical structure which any quasi-abelian category has, and the naive structure inherited from the abelian category Mod ϕ A . A three term complex which is naively exact is also canonically exact, but the converse is not true. We will investigate this in Section 3.5.

The analytic construction
3.4.1.
In a category C with duals and effective objects, let us say that an object X is antieffective if its dual X ∨ is effective. We denote by C and C the full subcategories of effective and anti-effective objects in C.

3.4.2.
We equip A with its (π, [ ])-topology. Following [SW17, 12.2], we give names to four special points of Spa(A) = Spa(A, A), labeled by their residue fields: y F , y C , y L and y C , corresponding respectively to the trivial valuation on the residue field F of A and to the fixed valuations on the A-algebras C , L and C. Then y F is the unique non-analytic point of Spa(A) and the complement Y = Spa(A) \ {y F } is equipped with a continuous surjective map κ : Y → [−∞, +∞] defined by where y is the maximal generalization of y, see [SW17,12.2]. We have The Frobenius ϕ of A induces an automorphism Spa(ϕ) of Spa(A) and Y , which we still denote by ϕ. It fixes y F , y C and y L , but not y C . We set y i = ϕ i (y C ) for every i ∈ Z, so that κ(y i ) = i since more generally κ(ϕ(y)) = ϕ(κ(y)) for every y ∈ Y , where the automorphism ϕ of [−∞, +∞] maps x to x + 1 (and fixes ±∞). Thus y 0 = y C while y −1 corresponds to A O C → C . For any interval I ⊂ [−∞, +∞], we denote by Y I the interior of the pre-image of I under κ. We set

3.4.3.
By [SW17, 13.1.1], Y is an honest (or sheafy) adic space. This means that the presheaf O Y of analytic functions on Y is a sheaf on Y . Thus there is a well-defined ⊗-category Bun Y I of vector bundles on Y I . A ϕ-equivariant bundle on Y I is a pair (E , ϕ E ) where E is a vector bundle on Y I and ϕ E : ϕ * E | Y ϕ −1 (I)∩I → E | Y ϕ −1 (I)∩I is an isomorphism. This defines a ⊗-category Bun ϕ Y I . By [Ked16a], the adic subspace Y • of Y is strongly Noetherian. Thus for any interval I ⊂ ]−∞, +∞[, there is also a well-behaved abelian category Coh Y I of coherent sheaves on Y I . A modification of vector bundles on Y I is a monomorphism f : E 1 → E 2 of vector bundles on Y I whose cokernel is a coherent sheaf supported at {y i : i ∈ Z} ∩ Y I . Similarly, there is a notion of ϕ-equivariant modification of ϕ-equivariant vector bundles on Y I .

3.4.4.
By [Ked16b, 3.9], the global section functor yields an equivalence of ⊗-categories In particular, every vector bundle E over Y is actually finite and free. Let Modif Y be the ⊗-category of pairs (E , ψ E ) where E is a vector bundle on Y and ψ E : E → ϕ * E is a modification supported at {y −1 }, i.e. ψ E is an isomorphism over Y \ {y −1 }. Then plainly Mod ϕ, are mutually inverse equivalences of ⊗-categories.
We claim that there are mutually inverse equivalences of ⊗-categories whose inverse Frobenius mappings are ϕ-equivariant vector bundles over respectively Y − and Y + , and is a ϕ-equivariant modification as desired. Conversely, starting from (E − , E + , f E ) on the right hand side, we define a vector bundle E on Y by gluing E − | Y [−∞,0[ and E + | Y ]−1,+∞] along the isomorphism induced by the restriction of f E to Y ]−1,0[ . Thus E | Y • is the subsheaf of E + | Y • made of those sections whose restriction to Y ]−∞,0[ belong to the image of f E . Since f E is a ϕ-equivariant modification, it follows that there is a commutative diagram . Therefore ψ E : E → ϕ * E is an isomorphism away from κ −1 (−1) ∩ {y i } = {y −1 }, i.e. ψ E is indeed a modification supported at y −1 .
One checks easily that these constructions yield mutually inverse ⊗-functors. 3.4.6.
Starting with Mod ϕ A[ 1 π ],f , we may analogously define ⊗-functors Mod ϕ, with the obvious definitions for the ⊗-categories Modif Y + and Modif ϕ, is the local ring of Y at y C ; this is the integral Robba ring, a Henselian discrete valuation ring with uniformizer π, residue field C and completion A(C) = W O E (C ) [FF18, 1.8.2]. The objects of the middle two categories are the finite free étale ϕ-modules (N, ϕ N ) over the indicated local rings, and the functor between them is the base change map (or π-adic completion) with respect to R int − → A(C). We have already encountered the last functor in 2.6.4: it maps (N, ϕ N In particular, every ϕ-bundle over Y − is actually finite free.

3.4.8.
There is also a commutative diagram of ⊗-categories [FF18, §11.4] in which all solid arrows are equivalences of ⊗-categories. In the first line, is the analog of the integral Robba ring R int − with y C replaced by y L , and is the adic version of the Fargues-Fontaine curve X, a strongly noetherian analytic space. There is a morphism of locally ringed space X → X which induces pull-back ⊗-functors ( · ) an : Coh X → Coh X and ( · ) an : Bun X → Bun X . The equivalence of ⊗-categories Bun ϕ Y • ↔ Bun X maps a vector bundle on X to its pull-back through the ϕ-invariant morphism π : Y • → X , and maps a ϕ-bundle E on Y • to the sheaf E /ϕ E of ϕ E -invariant sections of π * E . We denote by E → E (d) the Tate Moreover, B is a local domain with residue field L which is also a quotient of R int + . The Fargues-Fontaine curve X equals Proj(P ) where P : The ⊗-functor E : Bun ϕ B → Bun X maps a finite projective étale ϕ-module (N, ϕ N ) to the quasi-coherent sheaf on X associated with the graded P -module d 0 N ϕ N =π d . The ⊗-functor ( · ) alg : Bun X → Bun X maps an adic vector bundle E on X to the quasi-coherent sheaf on X associated with the graded P -module d 0 Γ(X , E (d)). In the second column of our diagram, the primes refer to the full ⊗-subcategories of finite free objects in the relevant ⊗-categories of ϕ-bundles. Thus plainly, are equivalence of ⊗-categories by the indicated references in [FF18,§11], and so are therefore also all of the above solid arrow functors. In particular, every ϕ-bundle on Y • is finite free and extends uniquely to a finite free ϕ-bundle on Y + .

3.4.9.
This is in sharp contrast with what happens at y C : not every ϕ-bundle on Y • extends to Y − , and those who do have many extensions. This is related to semistability as follows.
We then have three ⊗-functors given by lim This is a finite free étale ϕ-module over B + and (N, . It follows that we have equivalences of ⊗-categories In other words, a ϕ-bundle (E , ϕ E ) over Y • extends to a ϕ-bundle over Y − if and only if it is semi-stable of slope 0 and then, there is a functorial bijective correspondence between the set of all such extensions and the set of

3.4.10.
We shall now compute the equivalence of ⊗-categories Putting everything together, we obtain an equivalence of ⊗-categories E : Mod ϕ, A,f ⊗ E / / Modif ad, X . This is of course the restriction of the ⊗-functor E : Mod ϕ, but the first two components of the latter may not be equivalences.

Compatibility with Hodge filtrations
The morphisms of locally ringed space map y i ∈ |Y • | to respectively ∞ ∈ |X| and m i = Aϕ −i (ξ) ∈ |Spec(A)|. Moreover, they induce isomorphism between the corresponding completed local rings

Compatibility with Tate objects
The Tate object over A is anti-effective, The corresponding sequence · · · → E (i) → E (i + 1) → · · · is obtained from by tensoring with · ⊗ A (O Y ⊗ O E (1)). Thus by [BMS16,3.23],  Since the second square is cartesian, it is sufficient to establish that the outer rectangle is cartesian, for then so will be the first square, and its top row will thus be an equivalence of categories since so is the second row. We may again restrict our attention to anti-effective objects. The outer rectangle then factors as Mod ϕ, In this commutative diagram, the first square is cartesian since the two E 's are equivalences of ⊗-categories, the second square is obviously cartesian, and the third square is cartesian by Kedlaya's theory as explained in 3.4.9. So the outer rectangle is indeed cartesian. This finishes the proof of Theorem 3.10.

Final questions
Is it true that every ϕ-bundle over Y + is finite and free? Is there an integral version of the Fargues-Fontaine curve X corresponding to Y − /ϕ Z ? And is it true that E : Mod ϕ A[ 1 π ],f → Modif X is an equivalence of ⊗-categories?

3.5.1.
Recall that Theorem 3.10 implies that Mod ϕ A,f is a quasi-abelian category. In particular, any morphism f in Mod ϕ A,f has a kernel and a cokernel, whose underlying A-modules may however not be the kernel and cokernel of α as computed in the abelian category of all A-modules. Accordingly, we say that a three term complex in Mod ϕ A,f is canonically (resp. naively) exact if f = ker(g) and g = coker(f ) in Mod ϕ A,f (resp. in Mod A ). We want to investigate the difference between canonical and naive short exact sequences in Mod ϕ A,f .
Proposition 3.17. -The conditions of Proposition 3.14 are equivalent to: The exact sequence 0 → E 1,t → E 2,t → E 3,t → 0 is split.
Proof. -Using the compatibility of E : Mod ϕ A,f ⊗ E → Modif ad X with Tate twists, we may assume that our BKF-modules are anti-effective. Our claim then follows from the criterion (2b) of Proposition 3.15 since E i,t = ( · /ϕ) alg (E + i | Y • ) with the local notations, and ( · /ϕ) alg : Bun ϕ Y • → Bun X is an exact equivalence of categories.

Application
Let Modif ad, * X be the strictly full subcategory of Modif ad X whose objects are the admissible modifications (E 1 , E 2 , f E ) such that the Q-filtration on E 2 induced by the Fargues Q-filtration of (E 1 , E 2 , f E ) is split. Recall from 2.5.3 that Mod ϕ, * A,f is the strictly full subcategory of Mod ϕ A,f whose objects are the finite free BKF-modules of HN-type (Definition 2.9).
If both condition holds, then E maps the Fargues filtration F F on (M, ϕ M ) (from Proposition 2.10) to the Fargues filtration F F on (E 1 , E 2 , f E ) (defined in Section 3.1.7).
Proof. -The first claim follows from 3.4.14.
( (2) Suppose that (E 1 , E 2 , f E ) belongs to Modif ad, * X . We need to show that for all γ ∈ R, d f (s) where f = t F,∞ (M, ϕ M ) and (s, d) = (rank, deg) (F γ F (E 1 , E 2 , f E )). By assumption, Propositions 3.17 and 2.6, we may assume that the exact sequence  (1) and (2), we know that f = t F (E 1 , E 2 , f E ), thus also (r γ , d γ ) = (rank, deg)(F γ F (E 1 , E 2 , f E )). It then follows from Proposition 3.4 that E(F γ F M ) = F γ F E(F γ F M ). By functoriality of F F on Modif ad X , we find that