Galois representations, $(\varphi, \Gamma)$-modules and prismatic F-crystals

We prove that both local Galois representations and $(\varphi,\Gamma)$-modules can be recovered from prismatic F-crystals, from which we obtain a new proof of the equivalence of Galois representations and $(\varphi,\Gamma)$-modules.


Introduction
The theory of (φ, Γ)-modules was developed by Fontaine ( [1]) to study local Galois representations.It plays an important role in the study of families of Galois representations and p-adic Langlands correspondence.
The main result of the theory is that (φ, Γ)-modules are equivalent to Galois representations.The rough idea is to encode the difficult deeply ramified part of Galois theory into complicated rings.In other words, representations of complicated groups with simple coefficients are traded with representations of simple groups but with complicated coefficients.
The key part of the theory is then the construction of these complicated coefficient rings.It is based on the theory of fields of norms, which is a machine to switch between characteristic 0 and characteristic p worlds.There is another, probably more well-known, theory that serves the same purpose, namely perfectoid fields.Indeed, theory of fields of norms can be viewed as a deperfection of perfectoid fields.The coefficient rings appearing in (φ, Γ)-modules are certain infinitesimal lifting of fields of norms along the p-direction, in technical terms, they are Cohen rings of fields of norms.
The insight of this work is to put these rings in a world in which they naturally live, namely the framework of prisms as developed by Bhatt and Scholze ([2]).More precisely, these mysterious rings have natural integral structures which can be viewed as prisms, and the rings themselves are viewed as a structure sheaf on the prismatic site.Then (φ, Γ)-modules are vector bundles (with extra structures) on the prismatic site.This perspective is useful since prismatic sites are very rich.In particular, we can encode infinitesimal lifting of perfectoid fields into the prismatic world as well, which links with Galois representations.We can deduce the classical equivalence of Galois representations and (φ, Γ)-modules from this perspective, namely they are both equivalent to a third, arguably more fundamental object, the prismatic F-crystals.In summary, we have Theorem 1.1.Let k be a perfect field of characteristic p, and K be a finite extension of W (k) [ 1  p ]. Then (φ, Γ)-modules over A K are equivalent to prismatic F-crystals in O ∆ [ 1 Moreover, continuous finite free Z p -representations of the absolute Galois group G K are also equivalent to prismatic F-crystals in O ∆ [ 1 Corollary 1.2.The category of continuous finite free Z p -representations of the absolute Galois group G K is equivalent to the category of (φ, Γ)-modules over A K .
See the main text for explanations of the notation.The equivalence of Galois representations and prismatic F-crystals in O ∆ [ 1 I ∆ ] ∧ p -modules is also contained in the work of Bhatt and Scholze [3].The proof for (φ, Γ)-modules follows the same line as the proof for Galois representations.
Acknowledgments.I would like to thank Peter Scholze for helpful discussions and encouragement.I would like to thank Heng Du, Koji Shimiz, Yu Min and Liang Xiao for discussions on the initial draft.I would like to thank Tong Liu for pointing out a mistake in the early version of the paper.I would also like to thank the anonymous referee for many suggestions any corrections.I am grateful to Max Planck Institute for Mathematics in Bonn for its hospitality and financial support.
Convention.We follow the notation of [2].Fixing a prime p, a δ-ring is a Z (p) -algebra R equipped with a map δ : R → R such that δ(0) = δ(1) = 0 satisfying δ(x + y) = δ(x) + δ(y) + x p + y p − (x + y) p p δ(xy) = x p δ(y) + y p δ(x) + pδ(x)δ(y) for any x, y ∈ R. We write ϕ(x) := x p + pδ(x) which is a ring homomorphism lifting the Frobenius.An element Following the notation of Scholze, for an integral perfectoid ring R, we denote by R ♭ := lim ϕ R/p the tilt of R. We can also identify R ♭ with lim x→x p R as a multiplicative monoid, then there exists a natural monoid map R ♭ → R given by As a standard notation in p-adic Hodge theory, we denote A inf (R) := W (R ♭ ).There exists a canonical surjection of rings Moreover, we know that Ker(θ) is principal and is generated by any distinguished element in the kernel, see [2] lemma 3.8, lemma 2.33 and lemma 2.24 for example.In particular, if R contains a compatible system of p-power roots of unit ζ p n , then generates Ker(θ).We will use the theory of diamonds as developed in [4] and [5] in a rudimentary way.Recall that a diamond is a sheaf on the pro-étale site of characteristic p perfectoid spaces which can be written as the quotient of a representable sheaf by a pro-étale equivalence relation.There is a functor from analytic adic spaces over Spa(Z p ) into diamonds, which sends where X ⋄ is the sheaf whose value on a characteristic p perfectoid space S is the set of untilts S ♯ of S together with a map S ♯ → X of adic spaces.When X = Spa(R, R + ) is affinoid, we sometimes denote Spd(R, R + ) := Spa(R, R + ) ⋄ .When A = R ⊗ S R and M is an R-module, we will write when A is viewed as an R-algebra with respect to the first factor, i.e. the R-algebra structure is when A is equipped with the R-algebra structure with respect to the second factor.The convention applies also to other situations when A has two structure maps, such as A = R ⊗S R.

Prisms
We recall the basic theory of prisms as developed in [2], and introduce the primary examples that will be relevant to us.Definition 2.2.Let X be a p-adic formal scheme, then the (absolute) prismatic site X ∆ of X has objects bounded prisms (A, I) together with a map of formal schemes Spf (A/I) → X.The arrows are morphisms of prisms preserving the structure map to X.An arrow (A, is affine, we simplify the notation by writing R ∆ := X ∆ .
Example 2.3.Let k be a perfect field of characteristic p, then is a prism in W (k) ∆ , where [p] q := q p −1 q−1 , and the δ-structure is given by the usual δ-structure on W (k) and δ(q) = 0.It is clearly (p, q − 1)-complete, which is equivalent to being (p, [p] q )-complete as [p] q ≡ p mod(q − 1) and [p] q ≡ (q − 1) p−1 mod p.Moreover, from [p] q ≡ p mod(q − 1) we have Example 2.4.Let C be the completion of algebraic closure of W (k), and O C its ring of integers.Then (W (O ♭ C ), Ker(θ)) is a prism, where θ : We choose a compatible system {ζ p n } of p-power roots in C, and let then it is well-known that 1 + [ϵ ] generates Ker(θ), and we have a map of δ-rings and the condition p ∈ (Ker(θ), ϕ(Ker(θ))) follows from the same condition for the prism We made use of an embedding p ] in the previous example.This is not standard, and from now on we view It is the ϕ-twist of the previous embedding.
We now recall the theory of fields of norms, see [6] for details.Let K be a finite totally ramified extension of W (k) [ 1  p ] contained in a fixed completed algebraically closure C of W (k)[ 1 p ], then we can associate the cyclotomic tower Then E + K is a complete discrete valuation ring of characteristic p, which by construction contains We observe that . By the henselian property of W (k)((q − 1)), the extension ], then we have that y satisfies the minimal polynomial equation y n + a 1 p x n−1 +• • •+ an p n = 0 with coefficients in the fraction field of the complete discrete valuation ring W (k)((q−1)), but y ∈ A K is integral over W (k)((q−1)) as it is an extension of complete DVRs, which tells us that a Next we note that by definition

Proof. Being a subring of A
Proposition 2.12.([2] lemma 3.8) Let (A, I) be a prism, and then (A perf , IA perf ) is a perfect prism.Moreover, it is the universal perfect prism over (A, I).
Perfect prisms are canonically equivalent to integral perfectoid rings as defined in [8].We recall the definition of integral perfectoid rings first.Definition 2.13.A ring R is integral perfectoid if it is π-adically complete for some π ∈ R such that π p divides p, the Frobenius on R/p is surjective, and the canonical map θ : The desired equivalence with perfect prisms is the following theorem.
There is another notion of perfectoid rings used in the theory of perfectoid spaces.We recall the definition and compare it with integral perfectoid rings.Recall that a complete Tate ring is a complete Huber ring that contains a topological nilpotent unit.In more concrete terms, it is a complete topological ring R which contains an open subring R + whose topology is π-adic for some element π ∈ R + , and For any Huber ring R, we denote by R • the subring of power bounded elements.Definition 2.15.A perfectoid Tate ring is a uniform complete Tate ring R, i.e.R • is bounded, such that there exists a topological nilpotent unit π ∈ R • such that π p divides p, and Frobenius is surjective on R • /π p .
We have the following comparison between the two notions of perfectoid rings.Recall that a ring of integral elements of a Huber ring R is an open and integrally closed subring R + of R • .Proposition 2.16.([8] lemma 3.20) Let R be a complete Tate ring, and R + ⊂ R be a ring of integral elements.Then R is a perfectoid Tate ring if and only if R + is bounded in R and integral perfectoid.
The proposition characterizes integral perfectoid subrings of a perfectoid Tate ring.We can also build a perfectoid Tate ring from a integral perfectoid ring as in the following proposition.Proposition 2.17.Let R be an integral perfectoid ring and π ∈ R be an element such that R is π-adically complete and π p divides p, then R/Ann(π) is integral perfectoid, and R[ 1 π ] is a perfectoid Tate ring with ring of definition R/Ann(π).
Similarly, for R integral perfectoid, then R/Ann(p) is integral perfectoid and R[ 1 p ] is a perfectoid Tate ring with ring of definition R/Ann R (p).
Proof.By [8] lemma 3.9, there are units u, v of R such that both πu and pv has compatible systems of p-power roots in R. Then by [9] 16.3.69,R/Ann R (πu), resp.R/Ann R (pv), is integral perfectoid without π-, resp.p-, torsion (the definition of perfectoid in [9] is the same as being integral perfectoid, as [8] , is a perfectoid Tate ring with ring of definition R/Ann R (πu), resp.R/Ann R (πu), by [8] lemma 3.21.□ Remark 2.18.We have not excluded the zero ring in the proposition.For example, any perfect ring of characteristic p is integral perfectoid with π = 0, then the rings produced in the proposition are all zero.This tells us that in some sense the integral perfectoid rings are more general than being perfectoid Tate.For example, finite fields are integral perfectoid, but can not be nonzero (ring of integers of ) perfectoid Tate in any way.
We can compute the perfection of the prism (A + K , (ϕ n ([p] q ))).Lemma 2.19.We have Proof.Let I = (ϕ n ([p] q )).By [2] corollary 2.31, we have where we use lemma 10.96.1 (1) of Stacks project, and commutation of colimit with tensoring with Z/p, in the third equality.From the theory of fields of norms, we know that where the completion is with respect to the natural valuation.We know that ϕ n ([p] q ) ≡ (q − 1) , hence the completion is the same as completion with respect to I. Then we have as desired.□ Corollary 2.20.The automorphism group of Proof.Let γ be an automorphism of (A + K , (ϕ n ([p] q ))), then γ by definition is a δ-ring morphism, and is continuous with respect to (p, ϕ n ([p] q ))-topology, hence γ extends to an automorphism of (A + K , (ϕ n ([p] q ))) perf .By theorem 3.10 of [2], the automorphism group of (A + K , (ϕ n ([p] q ))) perf as an abstract prism is the same as the automorphism group of the corresponding integral perfectoid ring, which is But we observe that Γ already acts on (A + K , (ϕ n ([p] q ))).The action on A + K is clear from its construction, and we need to check that it preserves the ideal (ϕ n ([p] q )). First, . This follows from the definition q = [ϵ], and α is the value at γ of the cyclotomic character.Then and we claim that q p n −1 q αp n −1 × q αp n+1 −1 is a unit, which is what we want to prove.This follows from the computation

Now the corollary follows from the fact that
□ Moreover, we have the following proposition.
Proposition 2.21.There exists n ∈ N such that In other words, A + K cannot see the difference between K(ζ p k ) and K, and this is taken care by the choice of ϕ n ([p] q ).Proof.Let I = ([p] q ), we have by lemma 2.19 , where the last isomorphism follows from and ϕ being an automorphism and Ker(θ) = (ϕ −1 ([p] q )), for θ : On the other hand, we also have and we claim that this implies that Observe that . We look at the short exact sequence of (colim and Stacks project lemma 10.96.1, we have M ∧ p = 0.The identification after completion also implies that colim have the same fraction field, in other words, M is ptorsion.Further, we observe that colim This implies that M is a finitely generated (colim ϕ A + K /ϕ i (I))-module, which, together with being p-torsion, tells us that M is killed by p k for some k, so M is p-adically complete.Then we have M = M ∧ p = 0, proving the claim.Now as , lifts to an element z 1 ∈ A + K /ϕ n (I) for some n > 0, and the relation f (z) = 0 in the colimit forces that ϕ m (f (z 1 )) = 0 in the ring A + K /ϕ n+m (I) for some m, which implies that O K → A + K /ϕ n+m (I).Note that we have used the fact that , where the first isomorphism follows from the perfectness of W (k). □ Remark 2.23.We have used freely in the above proof the fact that there is no algebraic extension of nonarchimedean fields in their completion.More precisely, let L be an algebraic extension of W (k)[ 1 p ], then there is no nontrivial algebraic extension in its completion L ∧ .Equivalently, completion induces an equivalence between algebraic extensions M/L of L and algebraic extension M ∧ /L ∧ of L ∧ .This follows immediately from the fact that C Gal( L/M ) = M ∧ , where C is a completed algebraic closure of L.
We extract the following lemma from the proof, which will be useful later.

Lemma 2.24. The map O
is flat, so if x ∈ O K \ 0 satisfies xy = 0 for some y ∈ A + K /ϕ n ([p] q ), then y defines a zero-divisor of x in the colimit, which has to be 0 by the flatness of O K over O K(ζ p ∞ ) .Thus ϕ k (y) = 0 for some k, which implies that y = 0 by the faithfully flatness of ϕ established in lemma 2.10.□

Prismatic F-crystals
We recall the definition of prismatic F-crystals and make explicit an example that is relevant for us.Recall that we have a natural structure sheaf of δ-rings O ∆ on X ∆ , together with an ideal sheaf I ∆ .] ∧ p , the p-adic completion of the structure sheaf O ∆ with (locally) a generator of I ∆ inverted.A prismatic F -crystal on X in R-modules is a finite locally free R-module M over X ∆ such that for any arrow (A, I) → (B, IB) in X ∆ , together with an isomorphism A concrete way to work with prismatic F-crystals in O ∆ [ 1 ] ∧ p -modules is to choose a cover (A, I) of the final object * of the topos X ∆ , suppose that (A, I) × * (A, I) is representable, and p -modules satisfying cocycle conditions.Indeed, it is obvious that we can obtain such an object from a prismatic F -crystal.Conversely, if we are given such data, we can build a prismatic F -crystal M as follows.Given a prism (C, K) ∈ X ∆ , since (A, I) covers the final object, there exists a cover (C ′ , K ′ ) of (C, K), which also lies over (A, I).Then we can define where (C ′′ , K ′′ ) := (C ′ , K ′ ) × (C,K) (C ′ , K ′ ), which lives over (B, J), and the two arrows comes from the base change descent data Now by the next proposition, the descent data is effective on finite projective modules, so M(C, K) is a finite projective module over Proposition 3.2.Let (A, I) → (B, IB) be a cover in the category of bounded prisms, and (B • , IB • ) be the corresponding Čech nerve, then we have an equivalence of categories where Vect(R) denotes the category of finite projective modules over the ring R.
. Thus it suffices to prove the equivalence We know from the proof of [2] corollary 3.12 that B 2 is the derived (p, I)-completion of B ⊗ L A B, which is proved in loc.cit. to be discrete and classically (p, I)-complete.Then lemma 3.
, where the third equality follows from lemma 3.3.Then we have Proof.For any R-module M , we denote M its derived I-completion by viewing M as a complex concentrated in zero degree.Recall that there is a natural map M → H 0 ( M ) which is initial among maps M → N with N a derived I-complete R-module, see [10] definition 2.27 for example.By the spectral sequence in [7] tag 0BKE, we have where N ′ is classical I-complete.This is exactly the universal property of the classical I-completion of H 0 (K), whence H 0 ( K) ∧ I = H 0 (K) ∧ I .
We now prove the claim.Let H 0 (K) → N ′ where N ′ be as given, since classical I-completeness implies derived I-completeness ( [7] tag 091T), we have a unique factorization Let k be a perfect field of characteristic p, K be a finite totally ramified extension of W (k)[ 1 p ], and X = Spf(O K ).Let n ∈ N be chosen as in proposition 2.21 such that )) ∈ X ∆ , then it is a cover of the final object by the following lemma 3.5.Moreover, we have by lemma 3.6 where The ring C is difficult to understand explicitly, we will see below how we can bypass this difficulty by passing to perfections.Lemma 3.5.Let K be a finite totally ramified extension of W (k)[ 1 p ], X = Spf (O K ), and n ∈ N be chosen as in proposition 2.21 such that is a quasisyntomic cover of A/I.By [2] proposition 7.11, we can find a prism (C, J) that covers (A, I) such that there is a morphism to M as étale φ-modules over R n .Iterate the φ-module structure F : φ * M ∼ = M , we obtain an R-module isomorphism we need to check that this is an étale φ-module isomorphism, i.e.G • (φ n ) * F = F • φ * G, but this is clear.This proves essential surjectivity.
For fully faithfullness, we need to show that for (M, F M ), (N, The arrow comes from an arrow in ÉM /Rn for some n by standard finiteness argument, and we are reduced to showing that any arrow has a unique descent to R 0 .We show that it descends uniquely to R n−1 , which proves the claim by iteration.Now we can assume n = 1, the previous paragraph shows that are isomorphisms as étale φ-modules over R (being denoted by G in the previous paragraph).Let Next we show that p-adic completion of the perfection does not lose information of étale φmodules over p-complete rings.
Proof.By p-completeness of (colim φ R) ∧ p , we have where we use the commutativity of colimit with tensoring with Z/p n in the third equality, proposition 4.2 in the fourth equality and p-completeness in the last one.□ We now specialize to the case of prisms and closely related rings.Let (A, I) be a bounded prism, we want to study étale φ-modules over A[ 1 I ] ∧ p .There are two natural ways to form a perfection of the ring.The first is take the perfection directly and then p-complete it, namely while the second is to take the perfection of the prism (A, I) first, then inverting I and p-complete, i.e. ((colim It is the second one that will ultimately help us, and we want to understand étale φ-modules over it.Note that we have already understand étale φ-modules over the first ring, namely lemma 4.3 tells us that étale φ-modules over (colim Proof.We compute (colim Being a p-complete perfect δ-ring, we know from [2] corollary 2.31 that where we use again the commutation of colimit with tensoring with Z/p.A is (p, I)-complete as (A, I) is a bounded prism, so A/p is I-adically complete.By [2] lemma 3.6, ϕ(I)A is principal, so ϕ(I) ≡ I p mod p is principal which is generated by a non-zero divisor by assumption.It follows that A/p[ 1 I ] is a Tate ring with ring of definition A/p.
On the other hand, by [2] corollary 2.31 again.
We know that étale φ-modules over W (R) are equivalent to lisse sheaves on R for a perfect ring R by [11] proposition 3.2.7,hence it is enough to compare the finite étale sites of (colim Proof.Let R u be the uniformization of R as defined in [11] definition 2.8.13, then by [11] proposition 2.8.16, the finite étale site of R u is equivalent to that of R under base change.Moreover, by [11]  .By [11] lemma 2.6.2,there exists an affionid system R i (see [11] definition 2.6.1 for where we use that T red perf = T perf for any ring T of characteristic p, which can be checked directly.□ Combining all the equivalences we have established, we have the following theorem.
Theorem 4.6.With assumptions as in theorem 4.4, we have an equivalence of categories for a uniformizer ϖ = (a 0 , a 1 , • • • ) ∈ (B/I) ♭ which we can choose so that a 0 divides p in B/I.Thus where the second isomorphism follows directly from the description F ♭ = lim x→x p F , and the same descrition for C 0 (Γ, F ) ♭ .The last isomorphism follows from the concrete description of the Witt vector, namely W (F ♭ ) = (F ♭ ) N .This clearly commutes with taking continuous functions.Moreover, the ring structure is defined by polynomial equations that is independent of the input ring, whence the last isomorphism.The description of the two structure maps is straightforward by chasing through the various canonical isomorphisms.□ Remark 5.4.The action of Γ can be directly detected as follows.For a prismatic F-crystal M over O K , and γ ∈ Γ.The action of γ on M ((A + K , (ϕ n ([p] q )))) is induced by the base change isomorphism (the crystal structure) With the help of theorem 4.6, this proves that a prismatic F-crystal is the same as an étale φmodule over W (C ♭ ) together with a G-action which is semilinear with respect to the action of G on W (C ♭ ).Now as C ♭ is algebraically closed, it is well-known that the category of étale φ-modules over W (C ♭ ) is equivalent to the category of finite free Z p -modules via taking F -invariants, see [11] proposition 3.2.7 for example.This shows that the prismatic F-crystals are equivalent to finite free Z p -representations of G. □ T → (T ⊗ Zp W (C ♭ )) H=1 deperfection −→ (•) where M is a (φ, Γ)-module over A K , T is a finite free Z p -representation of G, and The action of G is diagonal on both T ⊗ Zp W (C ♭ ) and M ⊗ A K W (C ♭ ), where G acts on M through the canonical quotient G → Γ.The φ-structure on T ⊗ Zp W (C ♭ ) is defined by ϕ on the second factor.Moreover, the deperfection functor is the equivalence from the category of (φ, Γ)-modules over W (K(ζ p ∞ ) ♭ ) to the category of (φ, Γ)-modules over A K , as induced from theorem 4.6.

2. 1 .
Definitions and Examples.Definition 2.1.A prism is a pair (A, I), where A is a δ-ring, and I is an ideal of A such that A is derived (p, I)-complete, I defines a Cartier divisor on Spec(A), and p ∈ I + ϕ(I)A.The category of prisms has objects the prisms, and the arrows are δ-ring maps preserving the given ideals.A prism (A, I) is called bounded if A/I[p ∞ ] = A/I[p n ] for some n.

Definition 3 . 1 .
Let X be a p-adic formal scheme, and R be O ∆ [ 1 I ∆

3 shows that B 2 =
(B ⊗ A B) ∧ (p,I) , i.e. it is the classical (p, I)-completion of B ⊗ A B, and similarly for B • .Indeed, by the discreteness and classical (p, I)-completeness of B ⊗ L A B, the derived (p, I)-completion of B ⊗ L A B, we have

ILemma 3 . 3 .
and similarly for B • /p n .Since A → B is assumed to be (p, I)-completely faithfully flat, A/p n → B/p n is I-completely faithfully flat, and (2) follows from[10] theorem 7.8.□Let R be a ring and I be a finitely generated ideal of R. Let K ∈ D ≤0 (R) be a complex of R-modules with zero cohomology in positive degrees, and K its derived I-completion, then we have a canonical identificationH 0 ( K) ∧ I ∼ = H 0 (K) ∧I , in other words, the classical I-completion of H 0 ( K) is the same as the classical I-completion of H 0 (K).
proving the existence of the descent.It is unique since any descent h of f satisfies the relation φ * h = f by definition, and h satisfies the relation h

Lemma 4 . 3 .
Let R be a p-adically complete ring equipped with a ring morphism φ : R −→ R, then base change induces an equivalence of categories

ϕ A[ 1 Theorem 4 . 4 .
I ] ∧ p ) ∧ p is the same as étale φ-modules over A[ 1 I ] the following theorem characterizing étale φ-modules over ((colim ϕ A) ∧ (p,I) [ 1 I ]) ∧ p .Let (A, I) be a bounded prism such that ϕ(I) mod p is generated by a non-zero divisor in A/p, then we have an equivalence of categories ÉM /(colim ϕ

1 ILemma 4 . 5 .
] is the completed perfection of the Tate ring A/p[ 1 I ], and by the following lemma the finite étale site of (colimϕ A/p) ∧ I [ 1 I ] is the same as that of A/p[ 1 I ].But perfection also does not change finite étale site (see[11] theorem 3.1.15(a)),hence finite étale site of colim ϕ (A/p[ 1 I ]) is also identified with A/p[ 1 I ].This proves that the finite étale sites of (colim ϕ A/p) ∧ I [ 1 I ] and colim ϕ (A/p[ 1 I ]) are equivalent via base change (as all intermediate equivalences here are through base change).□Let R be a Banach ring of characteristic p (in the sense of[11] definition 2.2.1), then the finite étale site of (colim ϕ R) ∧ is equivalent to that of R via base change.
theorem 3.1.15(b), the finite étale sites of (colim ϕ R u ) ∧ and R u are equivalent, so we have the comparison between finite étale sites of R and (colim ϕ R u ) ∧ .We claim that (colim ϕ

∼
e. the site of perfect prisms over Spf (O K ), and the equivalence is induced by the obvious restriction functor.Indeed, prismaticF -crystals in O ∆ [ 1 I ∆ ] ∧ p -modules over Spf(O K ) (resp.Spf(O K ) perf ∆ ) are equivalent to φ-modules over A + K [ 1 I ] ∧ p together with descent data over C[ 1 I ] ∧ p (resp.φ-modules over (A + K ) perf [ 1 I ] ∧ p together with descent data over C perf [ 1 I ] ∧ p ),and the proof shows that the latter objects are equivalent.With exactly the same idea, we can recover Galois representations from prismatic F-crystals in O ∆ [ 1 I ∆ ] ∧ p -modules.As it is along the same reasoning as above, we only sketch the argument.Theorem 5.6.The category of prismatic F-crystals in O ∆ [ 1 I ∆ ] ∧ p -modules over Spf (O K ) is equivalent to the category of finite free continuous Z p -representations of G := Gal(K/K).Proof.Let C := K and O C be the ring of integers of it.By remark 5.5, it enough to work in Spf(O K ) perf ∆ .We can evaluate a prismatic F-crystal at (A inf (O C ), Ker(θ)).Let (B, J) be the product (A inf (O C ), Ker(θ)) × (A inf (O C ), Ker(θ))in Spf(O K ) perf ∆ , and we need to compute B/J[ 1 p ] is a perfectoid Tate K-algebra and can be interpreted asSpa(B/J[ 1 p ], B/J[ 1 p ] + ) ⋄ = Spd(C, O C ) × Spd(K,O K ) Spd(C, O C ).We know that Spd(C, O C ) × Spd(K,O K ) Spd(C, O C ) ∼ = Spd(C, O C ) × G = Spd(C 0 (G, C), C 0 (G, O C )= C 0 (G, W (C ♭ )).