1 Introduction
Let p be a prime number. The goal of the present paper is to establish classification theorems for pdivisible groups over quasisyntomic rings. This class of rings is a nonNoetherian generalisation of the class of pcomplete locally complete intersection rings and contains also big rings, such as perfectoid rings. Our main theorem is as follows.
Theorem. Let R be a quasisyntomic ring. There is a natural functor from the category of pdivisible groups over R to the category $\mathrm {DM}^{\mathrm {adm}}(R)$ of admissible prismatic Dieudonné crystals over R, which is an antiequivalence.
A more precise version of this statement and a detailed explanation will be given later in this Introduction. For now, let us just say that the category $\mathrm {DM}^{\mathrm {adm}}(R)$ is formed by objects of semilinear algebraic nature. The problem of classifying pdivisible groups and finite locally free group schemes by semilinear algebraic structures has a long history, going back to the work of Dieudonné on formal groups over characteristic p perfect fields. In characteristic p, as envisionned by Grothendieck, and later developed by Messing ([Reference Messing44]), MazurMessing ([Reference Mazur and Messing43]) and BerthelotBreenMessing ([Reference Berthelot, Breen and Messing6], [Reference Berthelot and Messing7]), the formalism of crystalline cohomology provides a natural way to attach such invariants to pdivisible groups. This theory goes by the name of crystalline Dieudonné theory and leads to classification theorems for pdivisible groups over a characteristic p base in a wide variety of situations, which we will not try to survey but for which we refer the reader, for instance, to [Reference Lau37]. In mixed characteristic, the existing results have been more limited. Fontaine ([Reference Fontaine23]) obtained complete results when the base is the ring of integers of a finite totally ramified extension K of the ring of Witt vectors $W(k)$ of a perfect field k of characteristic p, with ramification index $e<p1$ . This ramification hypothesis was later removed by Breuil ([Reference Breuil16]) for $p>2$ , who also conjectured an alternative reformulation of his classification in [Reference Breuil15], simpler and likely to hold even for $p=2$ , which was proved by Kisin ([Reference Kisin30]), for odd p, and extended by Kim ([Reference Kim29]), Lau ([Reference Lau35]) and Liu ([Reference Liu38]) to all p. Zink, and then Lau, gave a classification of formal pdivisible groups over very general bases using his theory of displays ([Reference Zink54]). More recently, pdivisible groups have been classified over perfectoid rings ([Reference Lau36], [Reference Scholze and Weinstein51, Appendix to Lecture XVII]). The main interest of our approach is that it gives a uniform and geometric construction of the classifying functor on quasisyntomic rings. This is made possible by the recent spectacular work of BhattScholze on prisms and prismatic cohomology ([Reference Bhatt8], [Reference Bhatt and Scholze13]). So far, such a cohomological construction of the functor had been available only in characteristic p, using the crystalline theory. This led, in practice, to some restrictions, when trying to study pdivisible groups in mixed characteristic by reduction to characteristic p, of which BreuilKisin theory is a prototypical example: there, no direct definition of the functor was available when $p=2$ ! Replacing the crystalline formalism by the prismatic formalism, we give a definition of the classifying functor very close in spirit to the one used by BerthelotBreenMessing ([Reference Berthelot, Breen and Messing6]) and which now makes sense without the limitation to characteristic p. Over a quasisyntomic ring R, our functor takes values in the category of admissible prismatic Dieudonné crystals over R. As the name suggests, prismatic Dieudonné crystals are prismatic analogues of the classical notion of a Dieudonné crystal on the crystalline site.
Before stating precisely the main results of this paper and explaining the techniques involved, let us note that several natural questions are not addressed in this paper.

1. It would be interesting to go beyond quasisyntomic rings. By analogy with the characteristic p story, one would expect that the prismatic theory should also shed light on more general rings. In the general case, admissible prismatic Dieudonné crystals will not be the right objects to work with. One should instead define analogues of the divided Dieudonné crystals introduced recently by Lau [Reference Lau37] in characteristic p.

2. Even for quasisyntomic rings, our classification is explicit for the socalled quasiregular semiperfectoid rings or for complete regular local rings with perfect residue field of characteristic p (cf. Section 5.2), as will be explained below, but quite abstract in general. Classical Dieudonné crystals can be described as modules over the pcompletion of the divided power envelope of a smooth presentation, together with a Frobenius and a connection satisfying various conditions. Is there an analogous concrete description of (admissible) prismatic Dieudonné crystals?

3. Finally, it would also be interesting and useful to study deformation theory (in the spirit of GrothendieckMessing theory) for the prismatic Dieudonné functor.
We now discuss in more detail the content of this paper.
1.1 Quasisyntomic rings
Let us first define the class of rings over which we study pdivisible groups.
Definition 1.1 (cf. Definition 3.15).
A ring R is quasisyntomic if R is pcomplete with bounded $p^{\infty }$ torsion and if the cotangent complex $L_{R/\mathbb {Z}_p}$ has pcomplete Toramplitude in $[1,0]$ Footnote ^{1} . The category of all quasisyntomic rings is denoted by $\mathrm {QSyn}$ .
Similarly, a map $R \to R'$ of pcomplete rings with bounded $p^{\infty }$ torsion is a quasisyntomic morphism if $R'$ is pcompletely flat over R and $L_{R'/R} \in D(R')$ has pcomplete Toramplitude in $[1,0]$ .
Remark 1.2. This definition is due to BhattMorrowScholze [Reference Bhatt, Morrow and Scholze12] and extends (in the pcomplete world) the usual notion of locally complete intersection (l.c.i.) rings and syntomic morphisms (flat and l.c.i.) to the nonNoetherian, non finitetype setting. The interest of this definition, apart from being more general, is that it more clearly shows why this category of rings is relevant: the key property of (quasi)syntomic rings is that they have a wellbehaved (pcompleted) cotangent complex. The work of Avramov shows that the cotangent complex is very badly behaved for all other rings, at least in the Noetherian setting: it is left unbounded (cf. [Reference Avramov2]).
Example 1.3. Any pcomplete l.c.i. Noetherian ring is in $\mathrm {QSyn}$ . But there are also big rings in $\mathrm {QSyn}$ : for example, any (integral) perfectoid ring is in $\mathrm {QSyn}$ (cf. Example 3.17). As a consequence of this, the pcompletion of a smooth algebra over a perfectoid ring is also quasisyntomic, as well as any bounded $p^{\infty }$ torsion pcomplete ring which can be presented as the quotient of an integral perfectoid ring by a finite regular sequence. For example, the rings
are quasisyntomic.
The category of quasisyntomic rings is endowed with a natural topology: the Grothendieck topology for which covers are given by quasisyntomic covers, that is, morphisms $R \to R'$ of pcomplete rings which are quasisyntomic and pcompletely faithfully flat.
An important property of the quasisyntomic topology is that quasiregular semiperfectoid rings form a basis of the topology (cf. Proposition 3.21).
Definition 1.4 (cf. Definition 3.19).
A ring R is quasiregular semiperfectoid if $R \in \mathrm {QSyn}$ and there exists a perfectoid ring S mapping surjectively to R.
As an example, any perfectoid ring, or any pcomplete bounded $p^{\infty }$ torsion quotient of a perfectoid ring by a finite regular sequence, is quasiregular semiperfectoid.
1.2 Prisms and prismatic cohomology (after BhattScholze)
Our main tool for studying pdivisible groups over quasisyntomic rings is the recent prismatic theory of BhattScholze [Reference Bhatt8], [Reference Bhatt and Scholze13]. This theory relies on the seemingly simple notions of $\delta $ rings and prisms. In what follows, all the rings considered are assumed to be $\mathbb {Z}_{(p)}$ algebras.
A $\delta $ ring is a commutative ring A, together with a map of sets $\delta : A\to A$ , with $\delta (0)=0$ , $\delta (1)=0$ and satisfying the following identities:
for all $x, y \in A$ . For any $\delta $ ring $(A,\delta )$ , denote by $\varphi $ the map defined by
The identities satisfied by $\delta $ are made to make $\varphi $ a ring endomorphism lifting Frobenius modulo p. Conversely, a ptorsion free ring equipped with a lift of Frobenius gives rise to a $\delta $ ring. A pair $(A,I)$ formed by a $\delta $ ring A and an ideal $I \subset A$ is a prism if I defines a Cartier divisor on $\mathrm {Spec}(A)$ , if A is (derived) $(p,I)$ complete and if I is proZariski locally generatedFootnote ^{2} by a distinguished element, that is, an element d, such that $\delta (d)$ is a unit.
Example 1.5.

1. For any pcomplete ptorsion free $\delta $ ring A, the pair $(A,(p)$ ) is a prism.

2. Say that a prism is perfect if the Frobenius $\varphi $ on the underlying $\delta $ ring is an isomorphism. Then the category of perfect prisms is equivalent to the category of (integral) perfectoid rings: in one direction, one maps a perfectoid ring R to the pair $(A_{\mathrm {inf}}(R):=W(R^{\flat }), \mathrm {ker}(\theta ))$ (here, $\theta : A_{\mathrm {inf}}(R) \to R$ is Fontaine’s theta map); in the other direction, one maps $(A,I)$ to $A/I$ . Therefore, one sees that, in the words of the authors of [Reference Bhatt and Scholze13], prisms are some kind of ‘deperfection’ of perfectoid rings.
The crucial definition for us is the following. We stick to the affine case for simplicity, but it admits an immediate extension to padic formal schemes.
Definition 1.6. Let R be a pcomplete ring. The (absolute) prismatic site of R is the opposite of the category of boundedFootnote ^{3} prisms $(A,I)$ together with a map $R \to A/I$ , endowed with the Grothendieck topology for which covers are morphisms of prisms $(A, I) \to (B,J)$ , such that the underlying ring map $A\to B$ is $(p,I)$ completely faithfully flat.
Bhatt and Scholze prove that the functor (respectively, ) on the prismatic site valued in $(p,I)$ complete $\delta $ rings (respectively, in pcomplete Ralgebras), sending to A (respectively, $A/I$ ), is a sheaf. The sheaf (respectively, ) is called the prismatic structure sheaf (respectively, the reduced prismatic structure sheaf).
From this, one easily deduces that the presheaves (respectively, ) sending $(A,I)$ to I (respectively, $\mathcal {N}^{\geq 1} A:=\varphi ^{1}(I)$ ) are also sheaves on .
Let R be a pcomplete ring. One proves the existence of a morphism of topoi:
Set:
The sheaf $\mathcal {O}^{\mathrm {pris}}$ is endowed with a Frobenius lift $\varphi $ . Moreover, if R is quasisyntomic, the quotient sheaf $\mathcal {O}^{\mathrm {pris}} / \mathcal {N}^{\geq 1} \mathcal {O}^{\mathrm {pris}}$ is naturally isomorphic to the structure sheaf $\mathcal {O}$ of $(R)_{\mathrm {qsyn}}$ .
1.3 Admissible prismatic Dieudonné crystals and modules
We are now in position to define the category of objects classifying pdivisible groups.
Definition 1.7. Let R be a quasisyntomic ring. A prismatic Dieudonné crystal over R is a finite locally free $\mathcal {O}^{\mathrm {pris}}$ module $\mathcal {M}$ together with $\varphi $ linear morphism
whose linearisation $\varphi ^{\ast } \mathcal {M}\to \mathcal {M}$ has its cokernel is killed by $\mathcal {I}^{\mathrm {pris}}$ . It is said to be admissible if the image of the composition
is a finite locally free $\mathcal {O}$ module $\mathcal {F}_{\mathcal {M}}$ , such that the map $(\mathcal {O}^{\mathrm {pris}}/\mathcal {I}^{\mathrm {pris}}) \otimes _{\mathcal {O}} \mathcal {F}_{\mathcal {M}} \to \mathcal {M}/\mathcal {I}^{\mathrm {pris}}\mathcal {M}$ induced by $\varphi _{\mathcal {M}}$ is a monomorphism.
Definition 1.8. Let R be a quasisyntomic ring. We denote by $\mathrm {DM}(R)$ the category of prismatic Dieudonné crystals over R (with morphisms the $\mathcal {O}^{\mathrm {pris}}$ linear morphisms commuting with the Frobenius), and by $\mathrm {DM}^{\mathrm {adm}}(R)$ its full subcategory of admissible prismatic Dieudonné crystals.
Remark 1.9. In a former version of the paper, we used the notion of filtered prismatic Dieudonné crystal. A filtered prismatic Dieudonné crystal over a quasisyntomic ring R is a collection $(\mathcal {M}, \mathrm {Fil} \mathcal {M}, \varphi _{\mathcal {M}})$ consisting of a finite locally free $\mathcal {O}^{\mathrm {pris}}$ module $\mathcal {M}$ , a $\mathcal {O}^{\mathrm {pris}}$ submodule $\mathrm {Fil} \mathcal {M}$ and a $\varphi $ linear map $\varphi _{\mathcal {M}}: \mathcal {M} \to \mathcal {M}$ , satisfying the following conditions:

1. $\varphi _{\mathcal {M}}(\mathrm {Fil} \mathcal {M}) \subset \mathcal {I}^{\mathrm {pris}}.\mathcal {M}$ .

2. $\mathcal {N}^{\geq 1}\mathcal {O}^{\mathrm {pris}}. \mathcal {M} \subset \mathrm {Fil} \mathcal {M}$ and $\mathcal {M}/\mathrm {Fil} \mathcal {M}$ is a finite locally free $\mathcal {O}$ module.

3. $\varphi _{\mathcal {M}}(\mathrm {Fil} \mathcal {M})$ generates $\mathcal {I}^{\mathrm {pris}}.\mathcal {M}$ as an $\mathcal {O}^{\mathrm {pris}}$ module.
However, as was pointed out to us by the referee, the category of filtered prismatic Dieudonné crystals embeds fully faithfully in the category of prismatic Dieudonné crystals, with essential image given by the admissible objects (this essentially follows from Proposition 4.29 below). Since admissible prismatic Dieudonné crystals are easier to work with than filtered prismatic Dieudonné crystals, we decided to work only with the first; hence, the results stayed the same, but their formulation changed slightly.
For quasiregular semiperfectoid rings, these abstract objects have a concrete incarnation. Let R be a quasiregular semiperfectoid ring. The prismatic site admits a final object .
Example 1.10.

1. If R is a perfectoid ring, .

2. If R is quasiregular semiperfectoid and $pR=0$ , .
Definition 1.11. A prismatic Dieudonné module over R is a finite locally free module M together with a $\varphi $ linear morphism
whose linearisation $\varphi ^{\ast } M \to M$ has its cokernel is killed by I. It is said to be admissible if the composition
is a finite locally free module $F_M$ , such that the map induced by $\varphi _{M}$ is a monomorphism.
Proposition 1.12 (Proposition 4.13).
Let R be a quasiregular semiperfectoid ring. The functor of global sections induces an equivalence between the category of (admissible) prismatic Dieudonné crystals over R and the category of (admissible) prismatic Dieudonné modules over R.
1.4 Statements of the main results
In all this paragraph, R is a quasisyntomic ring.
Theorem 1.13 (Theorem 4.71).
Let G be a pdivisible group over R. The pair
where the $\mathcal {E}xt$ is an Extgroup of abelian sheaves on $(R)_{\mathrm {qsyn}}$ and is the Frobenius induced by the Frobenius of $\mathcal {O}^{\mathrm {pris}}$ , is an admissible prismatic Dieudonné crystal over R, often denoted simply by .
Remark 1.14. The rank of the finite locally free $\mathcal {O}^{\mathrm {pris}}$ module is the height of G, and the quotient is naturally isomorphic to $\mathrm {Lie}(\check {G})$ , where $\check {G}$ is the Cartier dual of G.
Remark 1.15. When $pR=0$ , the crystalline comparison theorem for prismatic cohomology allows us to prove that this construction coincides with the functor usually considered in crystalline Dieudonné theory, relying on BerthelotBreenMessing’s constructions ([Reference Berthelot, Breen and Messing6]).
Theorem 1.16 (Theorem 4.74).
The prismatic Dieudonné functor
induces an antiequivalence between the category $\mathrm {BT}(R)$ of pdivisible groups over R and the category $\mathrm {DM}^{\mathrm {adm}}(R)$ of admissible prismatic Dieudonné crystals over R.
Remark 1.17. Theorems 1.13 and 1.16 immediately extend to pdivisible groups over a quasisyntomic formal scheme.
Remark 1.18. It is easy to write down a formula for a functor attaching to an admissible prismatic Dieudonné crystal an abelian sheaf on $(R)_{\mathrm {qsyn}}$ , which will be a quasiinverse of the prismatic Dieudonné functor: see Remark 4.91. But such a formula does not look very useful.
Remark 1.19. As a corollary of the theorem and the comparison with the crystalline functor, one obtains that the (contravariant) Dieudonné functor from crystalline Dieudonné theory is an antiequivalence for quasisyntomic rings in characteristic p. For excellent l.c.i. rings, fully faithfulness was proved by de JongMessing; the antiequivalence was proved by Lau for Ffinite l.c.i. rings (which are, in particular, excellent rings).
Remark 1.20. It is not difficult to prove that if R is perfectoid, admissible prismatic Dieudonné crystals (or modules) over R are equivalent to minuscule BreuilKisinFargues modules for R, in the sense of [Reference Bhatt, Morrow and Scholze11]. Therefore, Theorem 1.16 contains, as a special case, the results of Lau and ScholzeWeinstein. But the proof of the theorem actually requires this special caseFootnote ^{4} as an input.
Remark 1.21. In general, the prismatic Dieudonné functor (without the admissibility condition) is not essentially surjective, but we prove it is an antiequivalence for complete regular (Noetherian) local rings in Proposition 5.10, that is, in this case, the admissibility condition is automatic.
Moreover, we explain in Section 5.2 how to recover BreuilKisin’s classification (as extended by Kim, Lau and Liu to all p) of pdivisible groups over $\mathcal {O}_K$ , where K is a discretely valued extension of $\mathbb {Q}_p$ with perfect residue field, from Theorem 1.16.
Remark 1.22. Section 5.3 shows how to extract from the admissible prismatic Dieudonné functor a functor from $\mathrm {BT}(R)$ to the category of displays of Zink over R. Even though the actual argument is slightly involved for technical reasons, the main result there ultimately comes from the following fact: if R is a quasiregular semiperfectoid ring, the natural morphism gives rise by adjunction to a morphism of $\delta $ rings , mapping to the image of Verschiebung on Witt vectors. Zink’s classification by displays works on very general bases but is restricted (by design) to formal pdivisible groups or to odd p; by contrast, our classification is limited to quasisyntomic rings, but do not make these restrictions.
Remark 1.23. As in Kisin’s article [Reference Kisin30], it should be possible to deduce from Theorem 1.16 a classification result for finite locally free group schemes. We only write this down over a perfectoid ring, in which case, it was already known for $p>2$ by the work of Lau, [Reference Lau36]. This result is used in the recent work of $\breve{\mathrm{C}}$ esnavi $\breve{\mathrm{c}}$ ius and Scholze [Reference Cesnavic̆ius and Scholze18].
1.5 Overview of the proof and plan of the paper
Sections 2 and 3 contain some useful basic results concerning prisms and prismatic cohomology, with special emphasis on the case of quasisyntomic rings. Most of them are extracted from [Reference Bhatt, Morrow and Scholze12] and [Reference Bhatt and Scholze13], but some are not contained in loc. cit. (for instance, the definition of the qlogarithm, Section 2.2, or the Künneth formula, Section 3.5), or only briefly discussed there (for instance, the description of truncated HodgeTate cohomology, Section 3.2).
Section 4 is the heart of this paper. We first introduce the category $\mathrm {DM}^{\mathrm {adm}}(R)$ of admissible prismatic Dieudonné crystals over a quasisyntomic ring R and discuss some of its abstract properties (Section 4.1). We then introduce a candidate functor from pdivisible groups over R to $\mathrm {DM}^{\mathrm {adm}}(R)$ (Section 4.2). That it, indeed, takes values in the category $\mathrm {DM}^{\mathrm {adm}}(R)$ is the content of Theorem 1.13, which we do not prove immediately. We first relate this functor to other existing functors, for characteristic p rings or perfectoid rings (Section 4.3). The next three sections are devoted to the proof of Theorem 1.13. This proof follows a road similar to the one of [Reference Berthelot, Breen and Messing6, Chapters 2, 3]. The basic idea is to reduce many statements to the case of pdivisible groups attached to abelian schemes, using a theorem of Raynaud ensuring that a finite locally free group scheme on R can always be realised as the kernel of an isogeny between two abelian schemes over R, Zariskilocally on R. For abelian schemes, via the general device, explained in [Reference Berthelot, Breen and Messing6, Chapter 2] and recalled in Section 4.4, for computing Extgroups in low degrees in a topos, one needs a good understanding of the prismatic cohomology. It relies on the degeneration of the conjugate spectral sequence abutting to reduced prismatic cohomology, in the same way as the description of the crystalline cohomology of abelian schemes is based on the degeneration of the Hodgede Rham spectral sequence. We prove it in Section 4.5 by appealing to the group structure on the abelian scheme. Alternatively, one could use an identification of some truncation of the reduced prismatic complex with some cotangent complex, in the spirit of DeligneIllusie (or, more recently, [Reference Bhatt, Morrow and Scholze11]), proved in Section 3.2. To prove Theorem 1.16, stated as Theorem 4.74 below, one first observes that the functors
on $\mathrm {QSyn}$ are both stacks for the quasisyntomic topology (for $\mathrm {BT}$ , this is done in the Appendix). Therefore, to prove that the functor is an antiequivalence, it is enough to prove it for R quasiregular semiperfectoid, since these rings form a basis of the topology, in which case, one can simply consider the more concrete functor taking values in admissible prismatic Dieudonné modules over R, defined by taking global sections of . Therefore, one sees that, even if one is ultimately interested only by Noetherian rings, the structure of the argument forces to consider large quasisyntomic ringsFootnote ^{5} . Assume from now on that R is quasiregular semiperfectoid. The proof of fully faithfulness is ultimately reduced to the identification of the syntomic sheaf $\mathbb {Z}_p(1)$ (as defined using prismatic cohomology) to the padic Tate module of $\mathbb {G}_m$ , a result of BhattMorrowScholze recently reproved without Ktheory by BhattLurie ([Reference Bhatt and Lurie10, Theorem 7.5.6]). (A former version of this paper attempted to prove fully faithfulness using the strategy of [Reference Scholze and Weinstein50], following an idea of de JongMessing: one first proves it for morphisms from $\mathbb {Q}_p/\mathbb {Z}_p$ to $\mu _{p^{\infty }}$ and then reduces to this special case. This reduction step works fine in many cases of interest — such as characteristic p or ptorsion free quasiregular semiperfectoid rings — but we encountered several technical difficulties while trying to push it to the general case.) Once fully faithfulness is acquired, the proof of essential surjectivity is by reduction to the perfectoid case. One can actually even reduce to the case of perfectoid valuation rings with algebraically closed fraction field. In this case, the result is known, and due — depending whether one is in characteristic p or in mixed characteristic — to Berthelot and ScholzeWeinstein.
Finally, Section 5 gathers several complements to the main theorems, already mentioned above: the classification of finite locally free group schemes of ppower order over a perfectoid ring, BreuilKisin’s classification of pdivisible groups over the ring of integers of a finite extension of $\mathbb {Q}_p$ , the relation with the theory of displays and the description of the Tate module of the generic fibre of a pdivisible group from its prismatic Dieudonné crystal.
1.6 Notations and conventions
In all the text, we fix a prime number p.

• All finite locally free group schemes will be assumed to be commutative.

• If R is a ring, we denote by $\mathrm {BT}(R)$ the category of pdivisible groups over R.

• If A is a ring, $I \subset A$ an ideal and $K \in D(A)$ an object of the derived category of Amodules, K is said to be derived Icomplete if for every $f \in I$ , the derived limit of the inverse system
$$\begin{align*}\dots K \overset{f} \to K \overset{f} \to K \end{align*}$$vanishes. Equivalently, when $I=(f_1,\dots ,f_r)$ is finitely generated, K is derived Icomplete if the natural map$$\begin{align*}K \to R\lim(K \otimes_A^{\mathbb{L}} K_n^{\bullet}) \end{align*}$$is an isomorphism in $D(A)$ , where for each $n\geq 1$ , $K_n^{\bullet }$ denotes the Koszul complex $K_{\bullet }(A;f_1^n,\dots ,f_r^n)$ (one has $H^0(K_n^{\bullet })=A/(f_1^n,\dots ,f_r^n)$ , but beware that, in general, $K_n^{\bullet }$ may also have cohomology in negative degrees, unless $(f_1,\dots ,f_r)$ forms a regular sequence). An Amodule M is said to be derived Icomplete if $K=M[0] \in D(A)$ is derived Icomplete. The following properties are useful in practice:
1. A complex $K \in D(A)$ is derived Icomplete if and only if for each integer i, $H^i(K)$ is derived Icomplete (this implies, in particular, that the category of derived Icomplete Amodules form a weak Serre subcategory of the category of Amodules).

2. If $I=(f_1,\dots ,f_r)$ is finitely generated, the inclusion of the full subcategory of derived Icomplete complexes in $D(A)$ admits a left adjoint, sending $K \in D(A)$ to its derived Icompletion
$$\begin{align*}\widehat{K} = R\lim(K \otimes_A^{\mathbb{L}} K_n^{\bullet}). \end{align*}$$ 
3. (Derived Nakayama) If I is finitely generated, a derived Icomplete complex $K \in D(A)$ (respectively, a derived Icomplete Amodule M) is zero if and only if $K\otimes _A^{\mathbb {L}} A/I=0$ (respectively, $M/IM=0$ ).

4. If I is finitely generated, an Amodule M is (classically) Iadically complete if and only if it is derived Icomplete and Iadically separated.

5. $I=(f)$ is principal and M is an Amodule with bounded $f^{\infty }$ torsion (i.e. such that $M[f^{\infty }]=M[f^N]$ for some N), the derived Icompletion of M (as a complex) is discrete and coincides with its (classical) Iadic completion.
A useful reference for derived completions is [Reference Project52, Tag 091N].


• Let A be a ring, I a finitely generated ideal. A complex $K\in D(A)$ is Icompletely flat (respectively, Icompletely faithfully flat) if $K \otimes _A^{\mathbb {L}} A/I$ is concentrated in degree $0$ and flat (respectively, faithfully flat), cf. [Reference Bhatt, Morrow and Scholze12, Definition 4.1]. If an Amodule M is flat, its derived completion $\widehat {M}$ is Icompletely flat. Assume that I is principal, generated by $f \in A$ (in the sequel, f will often be p). Let $A\to B$ be a map of derived fcomplete rings. If A has bounded $f^{\infty }$ torsion and $A\to B$ is fcompletely flat, then B has bounded $f^{\infty }$ torsion. Conversely, if B has bounded $f^{\infty }$ torsion and $A\to B$ is fcompletely faithfully flat, A has bounded $f^{\infty }$ torsion. Moreover, if A and B both have bounded $f^{\infty }$ torsion, then $A\to B$ is fcompletely (faithfully) flat if and only if $A/f^n \to B/f^n$ is (faithfully) flat for all $n\geq 1$ (see [Reference Bhatt, Morrow and Scholze12, Corollary 4.8]).

• A derived Icomplete Aalgebra R is Icompletely étale (respectively, Icompletely smooth) if $R \otimes _A^{\mathbb {L}} A/I$ is concentrated in degree $0$ and étale (respectively, smooth).
2 Generalities on prisms
In this section, we review the theory of prisms and collect some additional results. In particular, we present the definition of the qlogarithm (cf. Section 2.2).
2.1 Prisms and perfectoid rings
We list here some basic definitions and results from [Reference Bhatt and Scholze13], of which we will make constant use in the paper. Let us first recall the definition of a $\delta $ ring A. In the following, all rings will be assumed to be $\mathbb {Z}_{(p)}$ algebras.
Definition 2.1. A $\delta $ ring is a pair $(A,\delta )$ with A a commutative ring and $\delta \colon A\to A$ a map (of sets), such that for $x,y\in A$ , the following equalities hold:
A morphism of $\delta $ rings $f\colon (A,\delta )\to (A^{\prime },\delta ^{\prime })$ is a morphism $f\colon A\to A^{\prime }$ of rings, such that $f\circ \delta =\delta ^{\prime }\circ f$ .
By design, the morphism
for a $\delta $ ring $(A,\delta )$ is a ring homomorphism lifting the Frobenius on $A/p$ . Using $\varphi $ , the second property of $\delta $ can be rephrased as
which looks close to that of a derivation. If A is ptorsion free, then any Frobenius lift $\psi \colon A\to A$ defines a $\delta $ structure on A by setting
Thus, in the ptorsion free case, a $\delta $ ring is the same as a ring with a Frobenius lift.
Remark 2.2. The category of $\delta $ rings has all limits and colimits and these are calculated on the underlying ringsFootnote ^{6} (cf. [Reference Bhatt and Scholze13, Section 1]). In particular, there exist free $\delta $ rings (by the adjoint functor theorem). Concretely, if A is a $\delta $ ring and X is a set, then the free $\delta $ ring $A\{X\}$ on X is a polynomial ring over A with variables $\delta ^n(x)$ for $n\geq 0$ and $x\in X$ (cf. [Reference Bhatt and Scholze13, Lemma 2.11]). Moreover, the Frobenius on $\mathbb {Z}_{(p)}\{X\}$ is faithfully flat (cf. [Reference Bhatt and Scholze13, Lemma 2.11]).
Definition 2.3. Let $(A,\delta )$ be a $\delta $ ring.

1. An element $x\in A$ is called of rank $1$ if $\delta (x)=0$ .

2. An element $d\in A$ is called distinguished if $\delta (d)\in A^{\times }$ is a unit.
In particular, $\varphi (x)=x^p$ if $x\in A$ is of rank $1$ .
Here is a useful lemma showing how to find rank $1$ elements in a padically separated $\delta $ ring.
Lemma 2.4. Let A be a $\delta $ ring, and let $x\in A$ . Then $\delta (x^{p^n})\in p^nA$ for all n. In particular, if A is padically separated and $y\in A$ admits a $p^n$ th root for all $n\geq 0$ , then $\delta (y)=0$ , that is, y has rank $1$ .
Proof. Cf. [Reference Bhatt and Scholze13, Lemma 2.31].
We can now state the definition of a prism (cf. [Reference Bhatt and Scholze13, Definition 3.2]). Recall that a $\delta $ pair $(A,I)$ is simply a $\delta $ ring A together with an ideal $I\subseteq A$ .
Definition 2.5. A $\delta $ pair $(A,I)$ is a prism if $I\subseteq A$ is an invertible ideal, such that A is derived $(p,I)$ complete, and $p\in I+\varphi (I)A$ . A prism $(A,I)$ is called bounded if $A/I$ has bounded $p^{\infty }$ torsion.
Remark 2.6. Some comments about these definitions are in order:

1. By [Reference Bhatt and Scholze13, Lemma 3.1], the condition $p\in I+\varphi (I)A$ is equivalent to the fact that I is proZariski locally on $\mathrm {Spec}(A)$ generated by a distinguished element. Thus, it is usually not much harm to assume that $I=(d)$ is actually principalFootnote ^{7} .

2. If $(A,I)\to (B,J)$ is a morphism of prisms, i.e., $A\to B$ is a morphism of $\delta $ rings carrying I to J, then [Reference Bhatt and Scholze13, Lemma 3.5] implies that $J=IB$ .

3. An important example of a prism is provided by
$$ \begin{align*} (A,I)=(\mathbb{Z}_p[[q1]],([p]_q)), \end{align*} $$where$$ \begin{align*} [p]_q:=\frac{q^p1}{q1} \end{align*} $$is the qanalog of p. Many other interesting examples will appear below. 
4. The prism $(A,I)$ being bounded implies that A is classically $(p,I)$ adically complete (cf. [Reference Bhatt8, Exercise 3.4]), and thus, in particular, padically separated.
Lemma 2.7. Let $(A,I)$ be a prism, and let $d\in I$ be distinguished. If $(p,d)$ is a regular sequence in A, then for all $r,s\geq 0$ , $r\neq s$ , the sequences
are regular.
Proof. Note that for the second case, one can always assume $\min (r,s)=0$ , up to replacing d by $\varphi ^{\min (r,s)}(d)$ . Then the statement is proven in [Reference Anschütz and Le Bras1, Lemma 3.3] and [Reference Anschütz and Le Bras1, Lemma 3.6].
Previous work in padic Hodge theory used, in one form or another, the theory of perfectoid spaces. From the prismatic perspective, this is explained as follows. We recall that a $\delta $ ring A (or prism $(A,I)$ ) is called perfect if the Frobenius $\varphi \colon A\to A$ is an isomorphism. If A is perfect, then necessarily $A\cong W(R)$ for some perfect $\mathbb {F}_p$ algebra R (cf. [Reference Bhatt and Scholze13, Corollary 2.30]).
Proposition 2.8. The functor
is an equivalence of categories with inverse $R\mapsto (A_{\mathrm {inf}}(R),\mathrm {ker}(\tilde {\theta }))$ , where $A_{\mathrm {inf}}(R):=W(R^{\flat })$ and $\tilde {\theta }=\theta \circ \varphi ^{1}$ , $\theta $ being Fontaine’s theta map.
Proof. Cf. [Reference Bhatt and Scholze13, Theorem 3.9].
Remark 2.9.

(1) Of course, one could use $\theta $ instead of $\tilde {\theta }$ . We make this (slightly strange) choice for coherence with later choices.

(2) The theorem implies, in particular, that for every perfect prism $(A,I)$ , the ideal I is principal.
As a corollary, we get the following easy case of almost purity.
Corollary 2.10. Let R be a perfectoid ring, and let $R\to R^{\prime }$ be pcompletely étale. Then $R^{\prime }$ is perfectoid. Moreover, if $J\subseteq R$ is an ideal, then the pcompletion $R^{\prime }$ of the henselisation of R at J is perfectoid.
Proof. We can lift $R^{\prime }$ to a $(p,\ker (\theta ))$ completely étale $A_{\mathrm {inf}}(R)$ algebra B. By [Reference Bhatt and Scholze13, Lemma 2.18], the $\delta $ structure on $A_{\mathrm {inf}}(R)$ extends uniquely to B. Reducing modulo p, we see that B is a perfect $\delta $ ring as it is $(p,\ker (\theta ))$ completely étale over $A_{\mathrm {inf}}(R)$ . Using Proposition 2.8, $R^{\prime }\cong B/\ker (\theta )B$ is therefore perfectoid. The statement on henselisations follows from this as henselisations are colimits along étale maps (cf. the proof of [Reference Project52, Tag 0A02]). (Note that since R has bounded $p^{\infty }$ torsion, the pcompletion of an étale Ralgebra is pcompletely étale.)
Moreover, perfectoid rings enjoy the following fundamental property.
Proposition 2.11. Let $(A,I)$ be a perfect prism. Then for every prism $(B,J)$ , the map
is a bijection.
Proof. Cf. [Reference Bhatt and Scholze13, Lemma 4.7].
2.2 The qlogarithm
Each prism is endowed with its Nygaard filtration (cf. [Reference Bhatt8, Definition 11.2]).
Definition 2.12. Let $(A,I)$ be a prism. Then we set
for $i\geq 0$ . The filtration $\mathcal {N}^{\geq \bullet }A$ is called the Nygaard filtration of $(A,I)$ .
This filtration (or rather the first piece of this filtration) will play an important role in the rest of this text. It already shows up when proving the existence of the qlogarithm
for a prism $(A,I)$ over $(\mathbb {Z}_p[[q1]],([p]_q))$ from Remark 2.6, as we now explain. Here,
is the functor sending a ring R to $T_p(R^{\times })= \varprojlim \limits_n\mu _{p^n}(R)$ and
is the Teichmüller lift sending a ppower compatible system
to the limit
where $\tilde {x}_n\in A$ is a lift of $x_n\in A/I$ . By definition,
is the subset of the inverse limit consisting of sequences that start with a $1$ . Moreover, on $\varprojlim \limits_{x\mapsto x^p} A/I$ , one can take pth roots
In [Reference Anschütz and Le Bras1, Lemma 4.10], there is the following lemma on the qlogarithm. For $n\in \mathbb {Z}$ , we recall that the qnumber $[n]_q$ is defined as
Lemma 2.13. Let $(B,J)$ be a prism over $(\mathbb {Z}_p[[q1]],([p]_q))$ . Then for every element $x\in 1+\mathcal {N}^{\geq 1} B$ of rank $1$ , that is, $\delta (x)=0$ , the series
is welldefined and converges in B. Moreover, $\log _q(x)\in \mathcal {N}^{\geq 1}B$ and, in
one has the relation $\log _q(x)=\frac {q1}{\log (q)}\log (x)$ , where $\mathrm {log}(x):=\sum \limits _{n=1}^{\infty } (1)^{n1}\frac {{(x1)^n}}{n}$ .
The defining properties of the qlogarithm are that $\log _q(1)=0$ and that its qderivative is $\frac {d_qx}{x}$ (cf. [Reference Anschütz and Le Bras1, Lemma 4.6]).
One derives easily the existence of the ‘divided qlogarithm’.
Lemma 2.14. Let $(B,J)$ be a bounded prism over $(\mathbb {Z}_p[[q1]],([p]_q))$ , and let $x\in \mathbb {Z}_p(1)(B/J)$ . Then $[x^{1/p}]_{\tilde {\theta }}\in B$ is of rank $1$ and lies in $1+\mathcal {N}^{\geq 1}B$ . Thus
exists in B.
Proof. By Lemma 2.4 (which applies to B as B is bounded and, thus, classically $(p,[p]_q)$ complete, by [Reference Bhatt and Scholze13, Lemma 3.7 (1)]), the element $[x^{1/p}]_{\tilde {\theta }}$ is of rank $1$ as it admits arbitrary $p^n$ roots. Moreover, $[x^{1/p}]_{\tilde {\theta }}\in 1+\mathcal {N}^{\geq 1}B$ as $\varphi ([x^{1/p}]_{\tilde {\theta }})=[x]_{\tilde {\theta }}\equiv 1$ modulo J. By Lemma 2.13, we can therefore conclude.
3 Generalities on prismatic cohomology
3.1 Prismatic site and prismatic cohomology
In this paragraph, we shortly recall, mostly for the convenience of the reader and to fix notations, some fundamental definitions and results, without proofs, from [Reference Bhatt and Scholze13]. Fix a bounded prism $(A,I)$ . Let R be a pcomplete $A/I$ algebra.
Definition 3.1. The prismatic site of R relative to A, denoted , is the category whose objects are given by bounded prisms $(B,IB)$ over $(A,I)$ together with an $A/I$ algebra map $R \to B/IB$ , with the obvious morphisms, endowed with the Grothendieck topology for which covers are given by $(p,I)$ completely faithfully flat morphisms of prisms over $(A,I)$ .
Remark 3.2. In this remark, we deal with the settheoretic issues arising from Definition 3.1. For example, as it stands, there does not exist a sheafification functor for presheaves on . We will implicitly fix a cutoff cardinal $\kappa $ like in [Reference Scholze49, Lemma 4.1] and assume that all objects appearing in Definition 3.1 (or Definition 3.4) have cardinality $<\kappa $ . The results of this paper will not change under enlarging $\kappa $ . For example, the category of prismatic Dieudonné crystals on will be independent of the choice of $\kappa $ . Also, the prismatic cohomology does not change (because it can be calculated via a $\breve{\mathrm{C}}$ echAlexander complex), and, thus, the prismatic Dieudonné crystals will be independent of $\kappa $ (by Section 4.4).
This affine definition admits an immediate extension to padic formal schemes over $\mathrm {Spf}(A/I)$ , cf [Reference Bhatt and Scholze13].
Proposition 3.3 ([Reference Bhatt and Scholze13], Corollary 3.12).
The functor (respectively, ) on the prismatic site valued in $(p,I)$ complete $\delta A$ algebras (respectively, in pcomplete Ralgebras), sending to B (respectively, $B/IB$ ), is a sheaf. The sheaf (respectively, ) is called the prismatic structure sheaf (respectively, the reduced prismatic structure sheaf).
These constructions have absolute variants, where one does not fix a base prism. Let R be a pcomplete ring.
Definition 3.4. The (absolute) prismatic site of R, denoted , is the category whose objects are given by bounded prisms $(B,J)$ together with a ring map $R \to B/J$ , with the obvious morphisms, endowed with the Grothendieck topology for which covers are given by morphisms of prism $(B,J) \to (C,JC)$ which are $(p,I)$ completely faithfully flat.
Exactly as before, one defines functors and , which are sheaves on .
We turn to the definition of (derived) prismatic cohomology. Fix a bounded prism $(A,I)$ . The prismatic cohomology of R over A is defined in two steps. One starts with the case where R is pcompletely smooth over $A/I$ .
Definition 3.5. Let R be a pcomplete pcompletely smooth $A/I$ algebra. The prismatic complex of R over A is defined to be the cohomology of the sheaf on the prismatic site:
This is a $(p,I)$ complete commutative algebra object in $D(A)$ endowed with a semilinear map , induced by the Frobenius of .
Similarly, one defines the reduced prismatic complex or HodgeTate complex:
This is a pcomplete commutative algebra object in $D(R)$ .
A fundamental property of prismatic cohomology is the HodgeTate comparison theorem, which relates the HodgeTate complex to differential forms. For this, first recall that for any $A/I$ module M and integer n, the nthBreuilKisin twist of M is defined as
The Bockstein maps
for each $i \geq 0$ , make a graded commutative $A/I$ differential graded algebraFootnote ^{8} , which comes with a map .
Theorem 3.6 ([Reference Bhatt and Scholze13], Theorem 4.10).
The map $\eta $ extends to a map
which is an isomorphism.
While proving Theorem 3.6, Bhatt and Scholze also relate prismatic and crystalline cohomology when the ring R is an $\mathbb {F}_p$ algebra. The precise statement is the following. Assume that $I=(p)$ , that is that $(A,I)$ is a crystalline prism. Let $J \subset A$ be a PDideal with $p\in J$ . Let R be a smooth $A/J$ algebra and
where the map $A/J \to A/p$ is the map induced by Frobenius and the fact that J is a PDideal.
Theorem 3.7 ([Reference Bhatt and Scholze13], Theorem 5.2).
Under the previous assumptions, there is a canonical isomorphism of $E_{\infty }A$ algebras
compatible with Frobenius.
Remark 3.8.

1. If $J=(p)$ , $R^{(1)}$ is just the Frobenius twist of R.

2. The proof of Theorem 3.7 goes through for a syntomic $A/J$ algebra R. The important point is that in the proof in [Reference Bhatt and Scholze13, Theorem 5.2], in each simplicial degree, the kernel of the morphism $B^{\bullet }\to \tilde {R}$ is the inductive limit of ideals of the form $(p,x_1,\ldots , x_r)$ , with $(x_1,\ldots , x_r)$ being pcompletely regular relative to A, which allows to apply [Reference Bhatt and Scholze13, Proposition 3.13]. The statement extends by descent from the quasiregular semiperfect case to all quasisyntomic rings over $\mathbb {F}_p$ (cf. Lemma 3.27).
Definition 3.5 of course makes sense without the hypothesis that R is pcompletely smooth over $A/I$ . But it would not give wellbehaved objects; for instance, the HodgeTate comparison would not hold in generalFootnote ^{9} . The formalism of nonabelian derived functors allows to extend the definition of the prismatic and HodgeTate complexes to all pcomplete $A/I$ algebras in a manner compatible with the HodgeTate comparison theorem.
Definition 3.9. The derived prismatic cohomology functor (respectively, the derived HodgeTate cohomology functor ) is the left Kan extension (cf. [Reference Bhatt, Morrow and Scholze12, Construction 2.1]) of the functor (respectively, ) from pcompletely smooth $A/I$ algebras to $(p,I)$ complete commutative algebra objects in (the $\infty $ category) $D(A)$ (respectively, pcomplete commutative algebra objects in $D(R)$ ), to the category of pcomplete $A/I$ algebras.
For short, we will just write (respectively, ) for (respectively, ) in the following.
Left Kan extension of the Postnikov (or canonical filtration) filtration leads to an extension of HodgeTate comparison to derived prismatic cohomology.
Proposition 3.10. For any pcomplete $A/I$ algebra R, the derived HodgeTate complex comes equipped with a functorial increasing multiplicative exhaustive filtration $\mathrm {Fil}_*^{\mathrm { conj}}$ in the category of pcomplete objects in $D(R)$ and canonical identifications
Finally, let us indicate how these affine statements globalise.
Proposition 3.11. Let X be a padic formal scheme over $\mathrm {Spf}(A/I)$ , which is locally the formal spectrum of a pcomplete ring with bounded $p^{\infty }$ torsion. There exists a functorially defined $(p,I)$ complete commutative algebra object , equipped with a $\varphi _A$ linear map , and having the following properties:

• For any affine open $U=\mathrm {Spf}(R)$ in X, there is a natural isomorphism of $(p,I)$ complete commutative algebra objects in $D(A)$ between and , compatible with Frobenius.

• Set . Then is naturally an object of $D(X)$ , which comes with a functorial increasing multiplicative exhaustive filtration $\mathrm {Fil}_*^{\mathrm {conj}}$ in the category of pcomplete objects in $D(X)$ and canonical identifications
3.2 Truncated HodgeTate cohomology and the cotangent complex
Let $(A,I)$ be a bounded prism, and let X be a padic $A/I$ formal scheme. The following result also appears in [Reference Bhatt and Scholze13, Proposition 4.14]Footnote ^{10} . We give a similar argument (suggested to us by Bhatt), with more details than in loc. cit. Since this result is not strictly necessary for the rest of the paper, the reader can safely skip this subsection.
Proposition 3.12. There is a canonical isomorphism:
where the righthand side is the first piece of the increasing filtration on introduced in Proposition 3.11.
Proof. We can assume that $X=\mathrm {Spf}(R)$ is affine. Write $\bar {A}=A/I$ . We want to prove that there is a canonical isomorphism
First, let us note that by the transitivity triangle for $A\to {\bar {A}} \to R$ , the cotangent complex $L_{R/A}\{1\}[1]^{\wedge _p}$ sits inside a triangle
and the outer terms are isomorphic to and
To construct the isomorphism $\alpha _R$ , it suffices to restrict to ${\bar {A}}\to R p$ completely smooth first, and then Kan extend. Thus, assume from now on that R is pcompletely smooth over ${\bar {A}}$ .
Let
, that is, $(B,J)$ is a prism over $(A,I)$ with a morphism $\iota \colon R\to B/J$ . Pulling back the extension of Aalgebras
along $\iota \colon R\to B/J$ defines an extension of R by $J/J^2\cong B/J\{1\}$ , and as such, is thus classified by a morphism
Passing to the (homotopy) limit over all
then defines (after shifting and twisting) the morphism
Concretely, if $R={\bar {A}}\langle x\rangle $ , then
On the summand $R\otimes _{{\bar {A}}}I/I^2[1]$ , the morphism $\alpha _R^{\prime }$ is simply the base extension of $I/I^2\to J/J^2$ as follows by considering the case ${\bar {A}}=R$ . On the summand $Rdx$ , the morphism $\alpha _R^{\prime }$ is (canonically) represented by the $J/J^2$ torsor of preimages of $\iota (x)$ in $B/J^2$ and factors as $R\xrightarrow {\iota }B/J\to B/J\{1\}[1]$ with the second morphism the connecting morphism for $0\to B/J\{1\}\to B/J^2\to B/J\to 0$ . Thus, after passing to the limit, we get a diagram
and on $H^0$ , the horizontal morphism induces the Bockstein differential
Thus, the image of $dx\in H^0(L_{R/A}^{\wedge _p})$ under $\alpha _R$ is $\beta (\iota (x))$ . Therefore, we see that on $H^0$ , the morphism $\alpha _R$ induces the identity under the identifications
and
(the second is the HodgeTate comparison). Moreover, the morphism
is the canonical one obtained by tensoring
with $I/I^2$ . By functoriality (and as $\Omega ^1_{R/A}$ is generated by $dr$ for $r\in R$ ), we can conclude that for every pcompletely smooth algebra R over A
induces the canonical morphism, and thus, that $\alpha _R$ is an isomorphism in general.
Recall the following proposition, which is a general consequence of the theory of the cotangent complex.
Proposition 3.13. Let S be a ring, $I\subseteq S$ an invertible ideal and X a flat $\overline {S}:=S/I$ scheme. Then the class $\gamma \in \mathrm {Ext}^2_{\mathcal {O}_X}(L_{X/\mathrm {Spec}(\overline {S})},I/I^2\otimes _{\overline {S}}\mathcal {O}_X)$ defined by $L_{X/\mathrm {Spec}(S)}$ is $\pm $ the obstruction class for lifting X to a flat $S/I^2$ scheme.
Proof. See [Reference Illusie24, Chapter III.2.1.2.3], respectively, [Reference Illusie24, Chapter III.2.1.3.3].
As before, let $(A,I)$ be a bounded prism.
Corollary 3.14. Let X be a pcompletely flat padic formal scheme over $A/I$ . The complex splits in $D(X)$ (i.e. is isomorphic in $D(X)$ to a complex with zero differentials) if and only if X admits a lifting to a pcompletely flat formal scheme over $A/I^2$ .
Proof. Indeed, splits if and only if the class in
defined by vanishes. Proposition 3.12 shows that this class is the same as the class defined by the pcompleted cotangent complex $L_{X/\mathrm {Spf}(A)}^{\wedge _p}\{1\}$ . Lifting X to a pcompletely flat formal scheme over $A/I^2$ is the same as lifting $X\otimes _{A/I}A/(I,p^n)$ to a flat scheme over $A/(I^2,p^n)$ for all $n\geq 1$ (here, we use that $(A,I)$ is bounded in order to know that $A/I$ is classically pcomplete). One concludes by applying Proposition 3.13, together with the fact that the pcompletion of the cotangent complex does not affect the (derived) reduction modulo $p^n$ .
3.3 Quasisyntomic rings
We shortly recall some key definitions from [Reference Bhatt, Morrow and Scholze12, Chapter 4].
Definition 3.15. A ring R is quasisyntomic if R is pcomplete with bounded $p^{\infty }$ torsion and if the cotangent complex $L_{R/\mathbb {Z}_p}$ has pcomplete Toramplitude in $[1,0]$ Footnote ^{11} . The category of all quasisyntomic rings is denoted by $\mathrm {QSyn}$ .
Similarly, a map $R \to R'$ of pcomplete rings with bounded $p^{\infty }$ torsion is a quasisyntomic morphism (respectively, a quasisyntomic cover) if $R'$ is pcompletely flat (respectively, pcompletely faithfully flat) over R and $L_{R'/R} \in D(R')$ has pcomplete Toramplitude in $[1,0]$ .
For a quasisyntomic ring R, the pcompleted cotangent complex $(L_{R/\mathbb {Z}_p})^{\wedge }_p$ will thus be in $D^{[1,0]}$ (cf. [Reference Bhatt, Morrow and Scholze12, Lemma 4.6]).
Remark 3.16. This definition extends (in the pcomplete world) the usual notion of locally complete intersection ring and syntomic morphism (flat and local complete intersection) to the nonNoetherian, non finitetype setting, as shown by the next example.
Example 3.17.

1. Any pcomplete l.c.i. Noetherian ring is in $\mathrm {QSyn}$ (cf. [Reference Avramov2, Theorem 1.2]).

2. There are also big rings in $\mathrm {QSyn}$ . For example, any (integral) perfectoid ring (i.e. a ring R which is pcomplete, such that $\pi ^p=pu$ for some $\pi \in R$ and $u \in R^{\times }$ , Frobenius is surjective on $R/p$ and $\ker (\theta )$ is principal) is in $\mathrm {QSyn}$ (cf. [Reference Bhatt, Morrow and Scholze12, Proposition 4.18]). We give a short explanation: if R is such a ring, the transitivity triangle for
$$\begin{align*}\mathbb{Z}_p \to A_{\mathrm{inf}}(R) \to R \end{align*}$$and the fact that $A_{\mathrm {inf}}(R)$ is relatively perfect over $\mathbb {Z}_p$ modulo p imply that after applying $ \otimes _R^{\mathbb {L}} R/p$ , $L_{R/\mathbb {Z}_p}$ and $L_{R/A_{\mathrm {inf}}(R)}$ identify. But$$\begin{align*}L_{R/A_{\mathrm{inf}}(R)} = \ker(\theta)/\ker(\theta)^2 [1]= R[1], \end{align*}$$as $\ker (\theta )$ is generated by a nonzero divisorFootnote ^{12} . 
3. As a consequence of (ii), the pcompletion of a smooth algebra over a perfectoid ring is also quasisyntomic, as well as any pcomplete bounded $p^{\infty }$ torsion ring which can be presented as the quotient of an integral perfectoid ring by a finite regular sequence.
The (opposite of the) category $\mathrm {QSyn}$ is endowed with the structure of a site.
Definition 3.18. Let $\mathrm {QSyn}_{\mathrm {qsyn}}^{\mathrm {op}}$ be the site whose underlying category is the opposite category of the category $\mathrm {QSyn}$ and endowed with the Grothendieck topology generated by quasisyntomic covers.
If $R \in \mathrm {QSyn}$ , we will denote by $(R)_{\mathrm {QSYN}}$ (respectively, $(R)_{\mathrm {qsyn}}$ ) the big (respectively, the small) quasisyntomic site of R, given by all pcomplete with bounded $p^{\infty }$ torsion Ralgebras (respectively, by all quasisyntomic Ralgebras, i.e. all pcomplete with bounded $p^{\infty }$ torsion Ralgebras S, such that the structure map $R \to S$ is quasisyntomic) endowed with the quasisyntomic topology.
The authors of [Reference Bhatt, Morrow and Scholze12] isolated an interesting class of quasisyntomic rings.
Definition 3.19. A ring R is quasiregular semiperfectoid if $R \in \mathrm {QSyn}$ and there exists a perfectoid ring S mapping surjectively to R.
Example 3.20. Any perfectoid ring, or any pcomplete bounded $p^{\infty }$ torsion quotient of a perfectoid ring by a finite regular sequence, is quasiregular semiperfectoid.
The interest in quasiregular semiperfectoid rings comes from the fact that they form a basis of the site $\mathrm {QSyn}_{\mathrm {qsyn}}^{\mathrm {op}}$ .
Proposition 3.21. Let R be quasisyntomic ring. There exists a quasisyntomic cover $R \to R'$ , with $R'$ quasiregular semiperfectoid. Moreover, all terms of the $\breve{C}$ ech nerve $R^{'\bullet }$ are quasiregular semiperfectoid.
Proof. See [Reference Bhatt, Morrow and Scholze12, Lemma 4.27] and [Reference Bhatt, Morrow and Scholze12, Lemma 4.29].
Finally, recall the following result, which is [Reference Bhatt and Scholze13, Proposition 7.11].
Proposition 3.22. Let $(A,I)$ be a bounded prism and R be a quasisyntomic $A/I$ algebra. There exists a prism , such that the map $R \to B/IB$ is pcompletely faithfully flat. In particular, if $A/I \to R$ is a quasisyntomic cover, then $(A,I) \to (B,IB)$ is a faithfully flat map of prisms.
Proof. Since the proof is short, we recall it. Choose a surjection
for some index set J. Set
Then $R \to S$ is a quasisyntomic cover, and by assumption, $A/I \to R$ is quasisyntomic: hence, the map $A/I \to S$ is quasisyntomic. Moreover the pcompletion of $\Omega _{S/(A/I)}^1$ is zero. We deduce that the map $A/I \to S$ is such that $(L_{S/(A/I)})^{\wedge _p}$ has pcomplete Toramplitude in degree $[1,1]$ . Therefore, by the HodgeTate comparison, the derived prismatic cohomology is concentrated in degree $0$ and the map is pcompletely faithfully flat. One can thus just take .
As observed in [Reference Bhatt and Scholze13], a corollary of Proposition 3.22 is André’s lemma.
Theorem 3.23 (André’s lemma).
Let R be perfectoid ring. Then there exists a pcompletely faithfully flat map $R\to S$ of perfectoid rings, such that S is absolutely integrally closed, that is, every monic polynomial with coefficients in S has a solution.
Proof. This is [Reference Bhatt and Scholze13, Theorem 7.12]. Since the proof is also short, we recall it. Write $R=A/I$ , for a perfect prism $(A,I)$ (Proposition 2.8). The pcomplete Ralgebra $\tilde {R}$ obtained by adding roots of all possible monic polynomials over R is a quasisyntomic cover, so by Proposition 3.22, we can find a prism $(B,J)$ over $(A,I)$ with a pcompletely faithfully flat morphism $\tilde {R} \to R_1:=B/J$ . Moreover, we can (and do) assume that $(B,J)$ is a perfect prism. Indeed, as $(A,I)$ is perfect, the underlying Aalgebra of the perfectionFootnote ^{13} of $(B,J)$ is the $(p,I)$ completion of a filtered colimit of $(p,I)$ completely faithfully flat Aalgebras, hence is $(p,I)$ completely faithfully flat. Transfinitely iterating the construction $R\mapsto R_1$ produces the desired ring S.
Let us recall that a functor $u\colon \mathcal {C}\to \mathcal {D}$ between sites is cocontinuous (cf. [Reference Project52, Tag 00XI]) if for every object $C\in \mathcal {C}$ and any covering $\{V_j\to u(C)\}_{j\in J}$ of $u(C)$ in $\mathcal {D}$ there exists a covering $\{ C_i\to C\}_{i\in I}$ of C in $\mathcal {C}$ , such that the family $\{ u(C_i)\to u(C)\}_{i\in I}$ refines the covering $\{V_j\to u(C)\}_{j\in J}$ . For a cocontinuous functor $u\colon \mathcal {C}\to \mathcal {D}$ , the functor
(here, $()^{\sharp }$ denotes sheafification) is leftexact (even exact) with right adjoint
Thus, a cocontinuous functor $u\colon \mathcal {C}\to \mathcal {D}$ induces a morphism of topoi
Note that in the definition of a cocontinuous functor, the morphisms $u(C_j)\to u(C)$ are not required to form a covering of C.
Corollary 3.24. Let R be a pcomplete ring. The functor , sending $(A,I)$ to
is cocontinuous. Consequently, it defines a morphism of topoi, still denoted by u:
Proof. Immediate from the definition (cf. [Reference Project52, Tag 00XJ]) and the previous proposition.
This yields the following important corollary.
Corollary 3.25. Let R be a pcomplete ring. Let
be a short exact sequence of abelian sheaves on $(R)_{\mathrm {QSYN}}$ . Then the sequence
is an exact sequence on . This applies, for example, when $G_1, G_2, G_3$ are finite locally free group schemes over R.
Proof. The first assertion is just saying that $u^{1}$ is exact, as u is a cocontinuous functor ([Reference Project52, Tag 00XL]). The second assertion follows, as any finite locally free group scheme is syntomic (cf. [Reference Breuil16, Proposition 2.2.2]).
3.4 Prismatic cohomology of quasiregular semiperfectoid rings
In this short subsection, we collect a few facts about prismatic cohomology of quasiregular semiperfectoid rings for later reference.
For the moment, fix a bounded base prism $(A,I)$ and let R be pcomplete $A/I$ algebra. There are several cohomologies attached to R:

1. The derived prismatic cohomology

2. The cohomology

3. Finally (and only for technical purposes),
Assume from now on that $(A,I)$ is a perfect prism and that $A/I\to R$ is a surjection with R quasiregular semiperfectoid. The prism admits then a more concrete (but, in general, rather untractable) description. Let K be the kernel of $A\to R$ . Then
is the prismatic envelope of the $\delta $ pair $(A,K)$ from [Reference Bhatt8, Lemma V.5.1] as follows from the universal property of the latter. In particular, the site has a final objectFootnote ^{14} .
Proposition 3.26. Let as above $(A,I)$ be a perfect prism and R quasiregular semiperfectoid with a surjection $A/I\twoheadrightarrow R$ . Then the canonical morphisms induce isomorphisms
as $\delta $ rings.
Proof. This is [Reference Bhatt and Scholze13, Proposition 7.10] (the second isomorphism, i.e. the fact that is bounded, follows from the last assertion of loc. cit.).
If $pR=0$ , that is, R is quasiregular semiperfect, there is, moreover, the universal pcomplete PDthickening
of R (cf. [Reference Scholze and Weinstein50, Proposition 4.1.3]). The ring $A_{\mathrm {crys}}(R)$ is ptorsion free by [Reference Bhatt, Morrow and Scholze12, Theorem 8.14].
Lemma 3.27. Let $(A,I)$ , R be as above, and assume that $pR=0$ . Then there is a canonical $\varphi $ equivariant isomorphism
Proof. As $A_{\mathrm {crys}}(R)$ is ptorsion free (cf. [Reference Bhatt, Morrow and Scholze12, Theorem 8.14]) and carries a canonical Frobenius lift, there we get a natural morphism
Conversely, the kernel of the natural morphism (cf. Theorem 3.29, which does not depend on this lemma)
has divided powers (as one checks similarly to [Reference Bhatt, Morrow and Scholze12, Proposition 8.12], using that the proof of Theorem 3.7 goes through in the syntomic case, cf. Remark 3.8). This provides a canonical morphism
in the other direction. Similarly, to [Reference Bhatt, Morrow and Scholze12, Theorem 8.14], one checks that both are inverse to each other.
Remark 3.28. Both rings and $A_{\mathrm {crys}}(R)$ are naturally $W(R^{\flat })$ algebras, but the isomorphism of Lemma 3.27 restricts to the Frobenius on $W(R^{\flat })$ . Concretely, if $R=R^{\flat }/x$ for some nonzero divisor $x\in R^{\flat }$ , then
and (cf. [Reference Bhatt and Scholze13, Corollary 2.37])
The prismatic cohomology of a quasiregular semiperfectoid ring R comes equipped with its Nygaard filtration, [Reference Bhatt and Scholze13, Section 12], an $\mathbb {N}$ indexed decreasing multiplicative filtration defined for $i\geq 0$ by
d denoting a generator of the ideal I. The graded pieces of the Nygaard filtration can be described as follows.
Theorem 3.29. Let R be a quasiregular semiperfectoid ring. Then
for $i\geq 0$ . In particular, .
Here, denotes the conjugate filtration on with graded pieces given by , for any choice of perfectoid ring S mapping to R (cf. Proposition 3.10).
Proof. See [Reference Bhatt and Scholze13, Theorem 12.2].
3.5 The Künneth formula in prismatic cohomology
The HodgeTate comparison implies a Künneth formula. Here is the precise statement. Note that for a bounded prism $(A,I)$ , the functor is naturally defined on all derived pcomplete simplicial $A/I$ algebras.
Proposition 3.30. Let $(A,I)$ be a bounded prism. Then the functor
from derived pcomplete simplicial rings over $A/I$ to derived $(p,I)$ complete $E_{\infty }$ algebras over A preserves tensor products, that is, for all morphism $R_1\leftarrow R_3 \to R_2$ the canonical morphism
is an equivalence.
Proof. Using [Reference Bhatt, Morrow and Scholze12, Construction 2.1] (respectively, [Reference Lurie40, Proposition 5.5.8.15]) the functor , which is the left Kan extension from pcompletely smooth algebras to all derived pcomplete simplicial $A/I$ algebras, commutes with colimits if it preserves finite coproducts. Clearly, , that is preserves the final object. Moreover, for $R,S p$ completely smooth over $A/I$ , the canonical morphism
is an isomorphisms because this Icompleteness may be checked for where it follows from the HodgeTate comparison.
Gluing the isomorphism in Proposition 3.30, we can derive, using as well the projection formula and flat base change for quasicoherent cohomology, the following statement.
Corollary 3.31. If X and Y are quasicompact quasiseparated pcompletely smooth padic formal schemes over $\mathrm {Spf}(A/I)$ ), then
4 Prismatic Dieudonné theory for pdivisible groups
This chapter is the heart of this paper. We construct the prismatic Dieudonné functor over any quasisyntomic ring and prove that it gives an antiequivalence between pdivisible groups over R and admissible prismatic Dieudonné crystals over R. The strategy to do this is to use quasisyntomic descent to reduce to the case where R is quasiregular semiperfectoid, in which case, the (admissible) prismatic Dieudonné crystals over R can be replaced by simpler objects, the (admissible) prismatic Dieudonné modules.
4.1 Abstract prismatic Dieudonné crystals and modules
Let R be a pcomplete ring. We defined in Corollary 3.24 a morphism of topoi:
We let
be the functor defined by $\epsilon _{\ast } \mathcal {F}(R') = \mathcal {F}(R')$ for $\mathcal {F} \in \mathrm {Shv}((R)_{\mathrm {QSYN}})$ and $R' \in (R)_{\mathrm {qsyn}}$ . It has a left adjoint $\epsilon ^{\natural }: \mathrm {Shv}((R)_{\mathrm { qsyn}}) \to \mathrm {Shv}((R)_{\mathrm {QSYN}})$ . We warn the reader that the restriction functor from the big to the small quasisyntomic site does not induce a morphism of sitesFootnote ^{15} , that is this left adjoint need not preserve finite limits (which explains why we denoted it $\epsilon ^{\natural }$ instead of $\epsilon ^{1}$ ).
We let
and
We still have the formula $Rv_{\ast }\cong R\varepsilon _{\ast }\circ Ru_{\ast }$ as $\varepsilon _{\ast }$ is exact.
Definition 4.1. Let R be a pcomplete ring. We define:
where denotes the canonical invertible ideal sheaf sending a prism to J. The sheaf $\mathcal {O}^{\mathrm {pris}}$ is endowed with a Frobenius lift $\varphi $ .
Although these sheaves are defined in general, we will only use them over quasisyntomic rings.
Proposition 4.2. Let R be a quasisyntomic ring. The quotient sheaf
is isomorphic to the structure sheaf $\mathcal {O}$ of $(R)_{\mathrm {qsyn}}$ .
Proof. It is enough to produce such an isomorphism functorially on a basis of $(R)_{\mathrm {qsyn}}$ . By Proposition 3.21, we can thus assume that R is quasiregular semiperfectoid. In this case, we conclude by Theorem 3.29.
Definition 4.3. Let R be a pcomplete ring. A prismatic crystal over R is an module $\mathcal {M}$ on the prismatic site of R, such that for all morphisms $(B,J)\to (B^{\prime },J^{\prime })$ in the canonical morphism
is an isomorphism.
Note that a prismatic crystal in finitely generated projective modules (respectively, in finitely generated projective modules) is the same thing as a finite locally free module (respectively, a finite locally free module). In what follows, we will essentially consider only this kind of prismatic crystal.
Proposition 4.4. Let R be a quasisyntomic ring. The functors $v_*$ and induce equivalences between the category of finite locally free modules and the category of finite locally free $\mathcal {O}^{\mathrm {pris}}$ modules.
Proof. Because , it is clear that for all finite locally free $\mathcal {O}^{\mathrm {pris}}$ modules $\mathcal {M}$ , the canonical morphism
is an isomorphism as this can be checked locally on $(R)_{\mathrm {qsyn}}$ . Conversely, let $\mathcal {N}$ be a finite locally free module. We have to show that the counit
is an isomorphism. For any morphism $R\to R^{\prime }$ with $R^{\prime }$ quasisyntomic, there are equivalences
of slice topoi, where $h_{R^{\prime }}(B,J):=\mathrm {Hom}_R(R^{\prime },B/J)$ . By passing to a quasisyntomic cover $R\to R^{\prime }$ , we can therefore assume that R is quasiregular semiperfectoid, in particular, that the site has a final object given by . By $(p,I)$ completely faithfully flat descent of finitely generated projective modules over $(p,I)$ complete rings of bounded $(p,I)$ torsion (cf. Proposition A.3), the category of finite locally free modules on is equivalent to finitely generated projective modulesFootnote ^{16} . As the morphism (the ‘ $\theta $ ’map) is henselian along its kernel, cf. Lemma 4.28, finite locally free modules split on the pullback of an open cover of $\mathrm {Spf}(R)$ . Thus, after passing to a quasisyntomic cover of $\mathrm {Spf}(R)$ , we may assume that $\mathcal {N}$ is finite free. Then the isomorphism
is clear.
Definition 4.5. Let R be a quasisyntomic ring. A prismatic Dieudonné crystal over R is a finite locally free $\mathcal {O}^{\mathrm {pris}}$ module $\mathcal {M}$ together with $\varphi $ linear morphism
whose linearisation $\varphi ^{\ast } \mathcal {M}\to \mathcal {M}$ has its cokernel killed by $\mathcal {I}^{\mathrm {pris}}$ . We call a prismatic Dieudonné crystal $(\mathcal {M},\varphi _{\mathcal {M}})$ admissible if the image of the composition
is a finite locally free $\mathcal {O}$ module $\mathcal {F}_{\mathcal {M}}$ , such that the map $(\mathcal {O}^{\mathrm {pris}}/\mathcal {I}^{\mathrm {pris}}) \otimes _{\mathcal {O}} \mathcal {F}_{\mathcal {M}} \to \mathcal {M}/\mathcal {I}^{\mathrm {pris}}\mathcal {M}$ induced by $\varphi _{\mathcal {M}}$ is a monomorphism.
Here, $\mathcal {M}/\mathcal {I}^{\mathrm {pris}}\cdot \mathcal {M}$ is an $\mathcal {O}\cong \mathcal {O}^{\mathrm {pris}}/\mathcal {N}^{\geq 1}\mathcal {O}^{\mathrm {pris}}$ module, cf. Proposition 4.2, via the composition $\mathcal {O}^{\mathrm {pris}}\xrightarrow {\varphi }\mathcal {O}^{\mathrm {pris}}\to \mathcal {O}^{\mathrm {pris}}/\mathcal {I}^{\mathrm {pris}}\mathcal {O}$ .
Remark 4.6. For a prismatic Dieudonné crystal $(\mathcal {M},\varphi _{\mathcal {M}})$ , the linearisation $\varphi ^{\ast } \mathcal {M} \to \mathcal {M}$ of the morphism $\varphi _{\mathcal {M}}\colon \mathcal {M} \to \mathcal {M}$ is an isomorphism after inverting a local generator $\tilde {\xi }$ of $\mathcal {I}^{\mathrm {pris}}$ and, in particular, is injective, since $\varphi ^{\ast } \mathcal {M}$ is $\tilde {\xi }$ torsion free.
Remark 4.7. Let $(\mathcal {M},\varphi _{\mathcal {M}})$ be a prismatic Dieudonné crystal. Write $\mathrm {Fil} \mathcal {M}=\varphi _{\mathcal {M}}^{1}(\mathcal {I}^{\mathrm {pris}}.\mathcal {M})$ . Consider the diagram (defining $Q,K$ )
As $\mathcal {I}^{\mathrm {pris}}.K=0$ (by definition of a prismatic Dieudonné crystal), the map $\alpha $ is zero. The snake lemma implies, therefore, that there exists a short exact sequence
(where, as in Definition 4.5, we wrote $\mathcal {F}_{\mathcal {M}}= \mathcal {M}/\mathrm {Fil} \mathcal {M}$ ). Hence, we see that the injectivity of $\beta $ (condition required in the definition of admissibility) is equivalent to the condition that $Q=0$ .
Definition 4.8. Let R be a quasisyntomic ring. We denote by $\mathrm {DM}(R)$ the category of prismatic Dieudonné crystals over R (with $\mathcal {O}^{\mathrm {pris}}$ linear morphisms commuting with Frobenius) and by $\mathrm {DM}^{\mathrm {adm}}(R)$ the full subcategory of admissible objects.
Proposition 4.9. The fibred category of (usual or admissible) prismatic Dieudonné crystals over the category $\mathrm {QSyn}$ of quasisyntomic rings endowed with the quasisyntomic topology is a stack.
Proof. This follows from the definition, because by general properties of topoi, modules under $\mathcal {O}^{\mathrm {pris}}$ and $\mathcal {O}$ form a stack for the quasisyntomic topology on $(R)_{\mathrm {qsyn}}$ .
For quasiregular semiperfectoid rings, these abstract objects have a more concrete incarnation, which we explain now. Let R be a quasiregular semiperfectoid ring, and let be the prism associated with R. Note that I is necessarily principal as there exists a perfectoid ring mapping to R. Recall (Theorem 3.29) that
is an isomorphism.
Definition 4.10. A prismatic Dieudonné module over R is a finite locally free module M together with a $\varphi $ linear morphism
whose linearisation $\varphi ^{\ast } M \to M$ has its cokernel killed by I. As in 4.5, we call a prismatic Dieudonné module $(M,\varphi _M)$ over R admissible if the image of the composition
is a finite locally free module $F_M$ , such that the map induced by $\varphi _{M}$ is a monomorphism.
Remark 4.11. For a prismatic Dieudonné module $(M,\varphi _M)$ , the linearisation $\varphi ^{\ast } M \to M$ of the morphism $\varphi _M\colon M\to M$ is an isomorphism after inverting a generator $\tilde {\xi }$ of I and, in particular, is injective, since $\varphi ^{\ast }M$ is $\tilde {\xi }$ torsion free. In 4.25, we will prove that these properties imply that the cokernel of $\varphi ^{\ast } M\to M$ is a finite projective module.
If R is perfectoid, one has
A prismatic Dieudonné module is the same thing as a minuscule BreuilKisinFargues module ([Reference Bhatt, Morrow and Scholze11]) over $A_{\mathrm {inf}}(R)$ with respect to $\tilde {\xi }$ . In fact, the situation for perfectoid rings is simple, as shown by the following proposition.
Proposition 4.12. Let R be a perfectoid ring. Any prismatic Dieudonné module over R is admissible.
We postpone the proof, it will be given below after Proposition 4.29.
Proposition 4.13. Let R be a quasiregular semiperfectoid ring. The functor
of evaluation on the initial prism induces an equivalence between the category of (usual or admissible) prismatic Dieudonné crystals over R and the category of (usual or admissible) prismatic Dieudonné modules over R, with quasiinverse
Proof. Let us call $G_R$ , respectively, $F_R$ , the first, respectively, the second, functor displayed in the statement of the proposition. Using Proposition 4.4 and the equivalence between finite locally free modules and finite locally free modules, one immediately gets that $F_R$ is an equivalence between the category of prismatic Dieudonné crystals over R and the category of prismatic Dieudonné modules over R, with quasiinverse given by $G_R$ . Hence, we only need to check that the admissibility conditions on both sides agree.
Let $(M,\varphi _M)$ be an admissible Dieudonné module over R.
Lemma 4.14. Let $R \to R^{\prime }$ be a quasisyntomic morphism, with $R^{\prime }$ being also quasiregular semiperfectoid. Let be the base change of $(M,\varphi _M)$ . Then
The lemma follows from Proposition 4.29 (and Remark 4.21), which will be proved below; let us take it for granted and finish the proof. For any quasiregular semiperfectoid ring $R^{\prime }$ quasisyntomic over R, note that, using the notations from the lemma,
The lemma tells us that, in particular
This being true for any quasiregular semiperfectoid ring $R^{\prime }$ quasisyntomic over R, we deduce that we have a short exact sequence of sheaves on $(R)_{\mathrm {qsyn}}$
By admissibility of $(M,\varphi _M)$ , the rightmost term is a finite locally free $\mathcal {O}$ module, and thus, $F_R(M)$ is admissible.
Conversely, let $(\mathcal {M},\varphi _{\mathcal {M}})$ be an admissible Dieudonné crystal. Consider the exact sequence of sheaves
and apply to it the functor $\Gamma (R,)$ . We get an exact sequence
Since