Prismatic Dieudonné Theory

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:

$$ \begin{align*}\begin{matrix} \delta(0)=\delta(1)=0 \\ \delta(xy)=x^p\delta(y)+y^p\delta(x)+p\delta(x)\delta(y)\\ \delta(x+y)=\delta(x)+\delta(y)+\frac{x^p+y^p-(x+y)^p}{p}. \end{matrix} \end{align*} $$

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

$$ \begin{align*}\varphi\colon A\to A,\ x\mapsto x^p+p\delta(x) \end{align*} $$

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

$$ \begin{align*}\delta(xy)=\varphi(x)\delta(y)+y^p\delta(x)=x^p\delta(y)+\varphi(y)\delta(x), \end{align*} $$

which looks close to that of a derivation. If A is p-torsion free, then any Frobenius lift $\psi \colon A\to A$ defines a $\delta $ -structure on A by setting

$$ \begin{align*}\delta(x):=\frac{\psi(x)-x^p}{p}. \end{align*} $$

Thus, in the p-torsion free case, a $\delta $ -ring is the same as a ring with a Frobenius lift.

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 p-adically separated $\delta $ -ring.

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 $I\subseteq A$ .

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

$$ \begin{align*} (p,\varphi^r(d)),(\varphi^r(d),\varphi^s(d)) \end{align*} $$

are regular.

Previous work in p-adic 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

$$ \begin{align*}\{\textrm{perfect prisms} ~ (A,I) \}\to \{ \textrm{(integral) perfectoid rings} ~ R \},\ (A,I)\mapsto A/I \end{align*} $$

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.

As a corollary, we get the following easy case of almost purity.

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

$$ \begin{align*} \mathrm{Hom}((A,I),(B,J))\to \mathrm{Hom}(A/I,B/J) \end{align*} $$

is a bijection.

2.2 The q-logarithm

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

$$ \begin{align*}\mathcal{N}^{\geq i}A:=\varphi^{-1}(I^i) \end{align*} $$

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 q-logarithm

$$ \begin{align*}\log_q\colon \mathbb{Z}_p(1)(B/J)\to B,\ x\mapsto \log_q([x^{1/p}]_{\tilde{\theta}}) \end{align*} $$

for a prism $(A,I)$ over $(\mathbb {Z}_p[[q-1]],([p]_q))$ from Remark 2.6, as we now explain. Here,

$$ \begin{align*} \mathbb{Z}_p(1):=T_p(\mu_{p^{\infty}}) \end{align*} $$

is the functor sending a ring R to $T_p(R^{\times })= \varprojlim \limits_n\mu _{p^n}(R)$ and

$$ \begin{align*} [-]_{\tilde{\theta}}\colon \varprojlim\limits_{x\mapsto x^p} A/I \to A \end{align*} $$

is the Teichmüller lift sending a p-power compatible system

$$ \begin{align*} x:=(x_0,x_1,\ldots)\in \varprojlim\limits_{x\mapsto x^p} A/I \end{align*} $$

to the limit

$$ \begin{align*} [x]_{\tilde{\theta}}:=\varinjlim\limits_{n\to \infty} \tilde{x}_n^{p^n}, \end{align*} $$

where $\tilde {x}_n\in A$ is a lift of $x_n\in A/I$ . By definition,

$$ \begin{align*} \mathbb{Z}_p(1)(A/I)\subseteq \varprojlim\limits_{x\mapsto x^p} A/I \end{align*} $$

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 p-th roots

$$ \begin{align*} (-)^{1/p}\colon \varprojlim\limits_{x\mapsto x^p} A/I\to \varprojlim\limits_{x\mapsto x^p} A/I,\ (x_0,x_1,\ldots)\mapsto (x_1,x_2,\ldots). \end{align*} $$

In [Reference Anschütz and Le Bras1, Lemma 4.10], there is the following lemma on the q-logarithm. For $n\in \mathbb {Z}$ , we recall that the q-number $[n]_q$ is defined as

$$ \begin{align*} [n]_q:=\frac{q^n-1}{q-1}\in \mathbb{Z}_p[[q-1]]. \end{align*} $$

Lemma 2.13. Let $(B,J)$ be a prism over $(\mathbb {Z}_p[[q-1]],([p]_q))$ . Then for every element $x\in 1+\mathcal {N}^{\geq 1} B$ of rank $1$ , that is, $\delta (x)=0$ , the series

$$ \begin{align*}\mathrm{log}_q(x)=\sum\limits_{n=1}^{\infty} (-1)^{n-1}q^{-n(n-1)/2}\frac{(x-1)(x-q)\cdots (x-q^{n-1})}{[n]_q} \end{align*} $$

is well-defined and converges in B. Moreover, $\log _q(x)\in \mathcal {N}^{\geq 1}B$ and, in

$$ \begin{align*}B[1/p][[x-1]]^{\wedge (q-1)}, \end{align*} $$

one has the relation $\log _q(x)=\frac {q-1}{\log (q)}\log (x)$ , where $\mathrm {log}(x):=\sum \limits _{n=1}^{\infty } (-1)^{n-1}\frac {{(x-1)^n}}{n}$ .

The defining properties of the q-logarithm are that $\log _q(1)=0$ and that its q-derivative is $\frac {d_qx}{x}$ (cf. [Reference Anschütz and Le Bras1, Lemma 4.6]).

One derives easily the existence of the ‘divided q-logarithm’.

Lemma 2.14. Let $(B,J)$ be a bounded prism over $(\mathbb {Z}_p[[q-1]],([p]_q))$ , 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

$$ \begin{align*}\mathrm{log}_q([x^{1/p}]_{\tilde{\theta}})=\sum\limits_{n=1}^{\infty} (-1)^{n-1}q^{-n(n-1)/2}\frac{([x^{1/p}]_{\tilde{\theta}}-1)\ldots ([x^{1/p}]_{\tilde{\theta}}-q^{n-1})}{[n]_q} \end{align*} $$

exists in B.

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 p-complete $A/I$ -algebra.

This affine definition admits an immediate extension to p-adic formal schemes over $\mathrm {Spf}(A/I)$ , cf [Reference Bhatt and Scholze13].

These constructions have absolute variants, where one does not fix a base prism. Let R be a p-complete ring.

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 p-completely smooth over $A/I$ .

Definition 3.5. Let R be a p-complete p-completely 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 Hodge-Tate complex:

This is a p-complete commutative algebra object in $D(R)$ .

A fundamental property of prismatic cohomology is the Hodge-Tate comparison theorem, which relates the Hodge-Tate complex to differential forms. For this, first recall that for any $A/I$ -module M and integer n, the nth-Breuil-Kisin twist of M is defined as

$$\begin{align*}M\{n\}:= M \otimes_{A/I} (I/I^2)^{\otimes n}. \end{align*}$$

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 PD-ideal with $p\in J$ . Let R be a smooth $A/J$ -algebra and

$$\begin{align*}R^{(1)} = R \otimes_{A/J} A/p, \end{align*}$$

where the map $A/J \to A/p$ is the map induced by Frobenius and the fact that J is a PD-ideal.

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.

Definition 3.5 of course makes sense without the hypothesis that R is p-completely smooth over $A/I$ . But it would not give well-behaved objects; for instance, the Hodge-Tate comparison would not hold in generalFootnote ⁹ . The formalism of nonabelian derived functors allows to extend the definition of the prismatic and Hodge-Tate complexes to all p-complete $A/I$ -algebras in a manner compatible with the Hodge-Tate comparison theorem.

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 Hodge-Tate comparison to derived prismatic cohomology.

Proposition 3.10. For any p-complete $A/I$ -algebra R, the derived Hodge-Tate complex comes equipped with a functorial increasing multiplicative exhaustive filtration $\mathrm {Fil}_*^{\mathrm { conj}}$ in the category of p-complete objects in $D(R)$ and canonical identifications

Finally, let us indicate how these affine statements globalise.

3.2 Truncated Hodge-Tate cohomology and the cotangent complex

Let $(A,I)$ be a bounded prism, and let X be a p-adic $A/I$ -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 right-hand 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

$$ \begin{align*} R\cong R\otimes_{{\bar{A}}}L_{{\bar{A}}/A}\{-1\}[-1]^{\wedge_p}\to L_{R/A}\{-1\}[-1]^{\wedge_p}\to L_{R/{\bar{A}}}\{-1\}[-1]^{\wedge_p}, \end{align*} $$

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 p-completely 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 A-algebras

$$ \begin{align*} 0\to J/J^2\to B/J^2\to B/J\to 0 \end{align*} $$

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

$$ \begin{align*} \alpha_R^{\prime}\colon L_{R/A}^{\wedge_p}\to B/J\{1\}[1]. \end{align*} $$

Passing to the (homotopy) limit over all

then defines (after shifting and twisting) the morphism

Concretely, if $R={\bar {A}}\langle x\rangle $ , then

$$ \begin{align*} L_{R/A}^{\wedge_p}\cong R\otimes_{{\bar{A}}} I/I^2[1]\oplus Rdx. \end{align*} $$

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

$$ \begin{align*}(\Omega^{1}_{R/{\bar{A}}})^{\wedge_p}\cong H^0(L_{R/A}^{\wedge_p}) \end{align*} $$

and

(the second is the Hodge-Tate 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 p-completely 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.

As before, let $(A,I)$ be a bounded prism.

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 p-completed cotangent complex $L_{X/\mathrm {Spf}(A)}^{\wedge _p}\{-1\}$ . Lifting X to a p-completely 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 p-complete). One concludes by applying Proposition 3.13, together with the fact that the p-completion 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].

For a quasisyntomic ring R, the p-completed 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]).

The (opposite of the) category $\mathrm {QSyn}$ is endowed with the structure of a site.

The authors of [Reference Bhatt, Morrow and Scholze12] isolated an interesting class of quasisyntomic rings.

The interest in quasiregular semiperfectoid rings comes from the fact that they form a basis of the site $\mathrm {QSyn}_{\mathrm {qsyn}}^{\mathrm {op}}$ .

Finally, recall the following result, which is [Reference Bhatt and Scholze13, Proposition 7.11].

Proof. Since the proof is short, we recall it. Choose a surjection

$$ \begin{align*} A/I\langle x_j, j\in J \rangle \to R, \end{align*} $$

for some index set J. Set

$$\begin{align*}S = A/I\langle x_j^{1/p^{\infty}} \rangle \hat{\otimes}_{A/I\langle x_j, j\in J \rangle}^{\mathbb{L}} R. \end{align*}$$

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 p-completion 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 p-complete Tor-amplitude in degree $[-1,-1]$ . Therefore, by the Hodge-Tate comparison, the derived prismatic cohomology is concentrated in degree $0$ and the map is p-completely faithfully flat. One can thus just take .

As observed in [Reference Bhatt and Scholze13], a corollary of Proposition 3.22 is André’s lemma.

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

$$ \begin{align*}u^{-1} \colon \mathrm{Shv}(\mathcal{D})\to \mathrm{Shv}(\mathcal{C}),\ \mathcal{F}\to (\mathcal{F}\circ u)^{\sharp} \end{align*} $$

(here, $()^{\sharp }$ denotes sheafification) is left-exact (even exact) with right adjoint

$$ \begin{align*}\mathcal{G}\in \mathrm{Shv}(\mathcal{C})\mapsto (D\mapsto \varprojlim\limits_{\{ u(C)\to D\}^{\mathrm{op}}} \mathcal{G}(C)). \end{align*} $$

Thus, a cocontinuous functor $u\colon \mathcal {C}\to \mathcal {D}$ induces a morphism of topoi

$$ \begin{align*}u\colon \mathrm{Shv}(\mathcal{C})\to \mathrm{Shv}(\mathcal{D}). \end{align*} $$

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 p-complete ring. The functor , sending $(A,I)$ to

$$ \begin{align*} R \to A/I, \end{align*} $$

is cocontinuous. Consequently, it defines a morphism of topoi, still denoted by u:

This yields the following important corollary.

Corollary 3.25. Let R be a p-complete ring. Let

$$ \begin{align*} 0\to G_1\to G_2\to G_3\to 0 \end{align*} $$

be a short exact sequence of abelian sheaves on $(R)_{\mathrm {QSYN}}$ . Then the sequence

$$ \begin{align*}0\to u^{-1}(G_1)\to u^{-1}(G_2)\to u^{-1}(G_3)\to 0 \end{align*} $$

is an exact sequence on . This applies, for example, when $G_1, G_2, G_3$ are finite locally free group schemes over R.

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 p-complete $A/I$ -algebra. There are several cohomologies attached to R:

1. 1. The derived prismatic cohomology

   
   of R over $(A,I)$ defined in Definition 3.9 via left Kan extension of prismatic cohomology.

2. 2. The cohomology

   
   of the prismatic site of (with its p-completely faithfully flat topology).

3. 3. Finally (and only for technical purposes),

   
   the prismatic cohomology of R with respect to the site of not necessarily bounded prisms $(B,J)$ over $(A,I)$ together with a morphism $R\to B/J$ of $A/I$ -algebras. We equip with the chaotic topology.

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.

If $pR=0$ , that is, R is quasiregular semiperfect, there is, moreover, the universal p-complete PD-thickening

$$ \begin{align*}A_{\mathrm{crys}}(R) \end{align*} $$

of R (cf. [Reference Scholze and Weinstein50, Proposition 4.1.3]). The ring $A_{\mathrm {crys}}(R)$ is p-torsion 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 p-torsion 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).

3.5 The Künneth formula in prismatic cohomology

The Hodge-Tate 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 p-complete simplicial $A/I$ -algebras.

Proposition 3.30. Let $(A,I)$ be a bounded prism. Then the functor

from derived p-complete 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 p-completely smooth algebras to all derived p-complete 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 I-completeness may be checked for where it follows from the Hodge-Tate 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 p-completely smooth p-adic formal schemes over $\mathrm {Spf}(A/I)$ ), then

4.1 Abstract prismatic Dieudonné crystals and modules

Let R be a p-complete ring. We defined in Corollary 3.24 a morphism of topoi:

We let

$$ \begin{align*}\epsilon_{\ast}: \mathrm{Shv}((R)_{\mathrm{QSYN}}) \to \mathrm{Shv}((R)_{\mathrm{qsyn}}) \end{align*} $$

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 p-complete 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

$$ \begin{align*} \mathcal{O}^{\mathrm{pris}} / \mathcal{N}^{\geq 1} \mathcal{O}^{\mathrm{pris}} \end{align*} $$

is isomorphic to the structure sheaf $\mathcal {O}$ of $(R)_{\mathrm {qsyn}}$ .

Definition 4.3. Let R be a p-complete 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

$$ \begin{align*}\mathcal{M}(B,J)\otimes_{B}B^{\prime}{\to} \mathcal{M}(B^{\prime},J^{\prime}) \end{align*} $$

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.

Proof. Because , it is clear that for all finite locally free $\mathcal {O}^{\mathrm {pris}}$ -modules $\mathcal {M}$ , the canonical morphism

$$ \begin{align*} \mathcal{M}\to v_{\ast}(v^{\ast}(\mathcal{M})) \end{align*} $$

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

$$ \begin{align*} v^{\ast}v_{\ast}(\mathcal{N})\to \mathcal{N} \end{align*} $$

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

$$ \begin{align*} v^{\ast} v_{\ast}(\mathcal{N})\cong \mathcal{N} \end{align*} $$

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

$$ \begin{align*} \varphi_{\mathcal{M}} \colon \mathcal{M}\to \mathcal{M} \end{align*} $$

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

$$\begin{align*}\mathcal{M}\xrightarrow{\varphi_{\mathcal{M}}} \mathcal{M}\to \mathcal{M}/\mathcal{I}^{\mathrm{pris}}\cdot \mathcal{M} \end{align*}$$

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 4.7. 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

$$\begin{align*}0\to Q \to \varphi^{\ast} \mathcal{M}/{\varphi^{\ast} \mathrm{Fil} \mathcal{M}}\cong \mathcal{O}^{\mathrm{pris}}/\mathcal{I}^{\mathrm{pris}}\otimes_{\mathcal{O}} \mathcal{F}_{\mathcal{M}} \xrightarrow{\beta} \mathcal{M}/\mathcal{I}^{\mathrm{pris}}\mathcal{M} \to K \to 0 \end{align*}$$

(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 $Q=0$ .

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

$$ \begin{align*}\varphi_M \colon M\to M, \end{align*} $$

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

$$\begin{align*}M\xrightarrow{\varphi_M}M\to M/I\cdot M \end{align*}$$

is a finite locally free -module $F_M$ , such that the map induced by $\varphi _{M}$ is a monomorphism.

If R is perfectoid, one has

A prismatic Dieudonné module is the same thing as a minuscule Breuil-Kisin-Fargues 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.

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 quasi-inverse

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 quasi-inverse 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 $(R)_{\mathrm {qsyn}}$

$$ \begin{align*}0 \to \varphi_{F_R(M)}^{-1}(\mathcal{I}^{\mathrm{pris}}.F_R(M)) \to F_R(M) \to \mathcal{O} \otimes_R M/\varphi_M^{-1}(I.M) \to 0. \end{align*} $$

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

$$ \begin{align*}0 \to \varphi_{\mathcal{M}}^{-1}(\mathcal{I}^{\mathrm{pris}}.\mathcal{M}) \to \mathcal{M} \to \mathcal{M}/\varphi_{\mathcal{M}}^{-1}(\mathcal{I}^{\mathrm{pris}}.\mathcal{M}) \to 0, \end{align*} $$

and apply to it the functor $\Gamma (R,-)$ . We get an exact sequence

$$ \begin{align*}0 \to \Gamma(R, \varphi_{\mathcal{M}}^{-1}(\mathcal{I}^{\mathrm{pris}}.\mathcal{M})) = \varphi_{G_R(\mathcal{M})}^{-1}(I.G_R(\mathcal{M})) \to G_R(\mathcal{M}) \to \Gamma(R,\mathcal{M}/\varphi_{\mathcal{M}}^{-1}(\mathcal{I}^{\mathrm{pris}}.\mathcal{M})). \end{align*} $$

Since