1.1.1 Prisms

For a $\mathbb {Z}_{(p)}$ -algebra A, a $\delta $ -structure is a map $\delta \colon A\to A$ with $\delta (0)=\delta (1)=0$ and

$$ \begin{align*} \delta(xy)=x^p\delta(y)+y^p\delta(x)+p\delta(x)\delta(y),\qquad \delta(x+y)=\delta(x)+\delta(y)+\tfrac{1}{p}(x^p+y^p-(x+y)^p). \end{align*} $$

Associated to $\delta $ is a Frobenius lift $\phi \colon A\to A$ given by $\phi (x)=x^p+p\delta (x)$ , which we call the Frobenius. If A is p-torsion-free then any Frobenius lift $\phi $ on A defines a $\delta $ structure by $\tfrac {1}{p}(\phi (x)-x^p)$ , establishing a bijection between the two types of structures, and we conflate the two notions. We call the pair $(A,\delta )$ a $\delta $ -ring. We often suppress $\delta $ from the notation, writing $\delta _A$ (or $\phi _A$ ) when we want to be clear. A morphism of $\delta $ -rings is a ring map that intertwines the $\delta $ -structures.

A prism is a pair $(A,I)$ where A is a $\delta $ -ring and $I\subseteq A$ is an invertible ideal with A derived $(p,I)$ -adically complete (see [Reference Bhatt, Morrow and ScholzeBMS18, §6.2]), and such that $I+\phi (I)$ contains p. Thus, I is finitely generated and $\operatorname {\mathrm {Spec}}(A)-V(I)$ is affine (see [SP, Tag 07ZT]), and we denote by the global sections of the structure sheaf. Let $j_{(A,I)}$ denote the inclusion of into $\operatorname {\mathrm {Spec}}(A)$ . For a prism $(A,I)$ , unless stated otherwise, we view A as being equipped with the $(p,I)$ -adic topology.

A prism $(A,I)$ is bounded if $A/I$ has bounded $p^\infty $ -torsion. The following two results will be used without comment in the sequel.

A morphism $(A,I)\to (B,J)$ is a morphism $A\to B$ of $\delta $ -rings mapping I into J. By the rigidity property of morphisms of prisms (see [Reference Bhatt and ScholzeBS22, Proposition 3.5]), if $(A,I)\to (B,J)$ is a morphism of prisms, then $I\otimes _A B$ maps isomorphically onto J and, in particular, $J=IB$ . A morphism $(A,I)\to (B,IB)$ is $(p,I)$ -completely (faithfully) flat (resp. étale, smooth) when $B\otimes ^L_A (A/(p,I))$ is concentrated in degree $0$ and $A/(p,I)\to B\otimes ^L_A (A/(p,I))$ is (faithfully) flat (resp. étale, smooth).

A prism $(A,I)$ is perfect if $\phi _A$ is an isomorphism, in which case it is bounded (see [Reference Bhatt and ScholzeBS22, Lemma 2.34]). For a perfectoid ring R, we have the perfect prism $({\mathrm {A}}_{\mathrm {inf}}(R),\ker (\theta _R))$ where ${\mathrm {A}}_{\mathrm {inf}}(R)=W(R^\flat )$ is Fontaine’s ring, which comes equipped with a natural Frobenius $\phi _R$ , and $\theta _R\colon {\mathrm {A}}_{\mathrm {inf}}(R)\to R$ is Fontaine’s map. We also have the perfect prism $({\mathrm {A}}_{\mathrm {inf}}(R),\ker (\widetilde {\theta }_R))$ , where , which is isomorphic to $({\mathrm {A}}_{\mathrm {inf}}(R),\ker (\theta _R))$ via $\phi _R$ (see Remark 1.10 to see why we introduce these two choices). We often fix a generator $\xi _R$ of $\ker (\theta _R)$ and set so that $\ker (\widetilde {\theta }_R)=(\tilde {\xi }_R)$ . When R is clear from context we shall omit the decoration of R at all places. When R is an $\mathcal {O}_C$ -algebra, we may take $\xi =\xi _0$ and $\tilde \xi =\tilde \xi _0$ which we often implicitly do.

1.1.2 The absolute prismatic site

Let $\mathfrak {X}$ be a p-adic formal scheme. Consider the category of triples $(A,I,s)$ where $(A,I)$ is a bounded prism, and $s\colon \operatorname {\mathrm {Spf}}(A/I)\to \mathfrak {X}$ is a morphism, and where morphisms are maps of prisms commuting with the maps to $\mathfrak {X}$ . We often omit s from the notation. The absolute prismatic site of $\mathfrak {X}$ (see [Reference Bhatt and ScholzeBS23, Definition 2.3]) is the opposite category of , endowed with the topology where $\{\alpha _i\colon (A,I)\to (B_i,J_i)\}$ in corresponds to a cover if $\{\operatorname {\mathrm {Spf}}(\alpha _i)\colon \operatorname {\mathrm {Spf}}(B_i)\to \operatorname {\mathrm {Spf}}(A)\}$ is a cover in $\operatorname {\mathrm {Spf}}(A)_{\mathrm {fl}}$ (see §A.4). That this is a site follows from the argument in [Reference Bhatt and ScholzeBS22, Corollary 3.12], which also shows that for a diagram in

$$ \begin{align*} (B,IB)\leftarrow (A,I)\to (C,IC), \end{align*} $$

with one of the maps adically faithfully flat, then its cofibered product is $(B\widehat {\otimes }_A C,I(B\widehat {\otimes }_A C))$ with the obvious $\delta $ -structure and map to $\mathfrak {X}$ . We often abuse notation when working in , writing objects and morphisms as in . We also shorten to , in which case we often write $s\colon \operatorname {\mathrm {Spf}}(A/I)\to \operatorname {\mathrm {Spf}}(R)$ as $s\colon R\to A/I$ .

By [Reference Bhatt and ScholzeBS22, Corollary 3.12], the presheaves and are sheaves. By the rigidity property of morphisms of prisms, is a quasi-coherent ideal sheaf of . The association is a sheaf of rings as is flat. Define

(see [Reference Bhatt and ScholzeBS23, Remark 2.4] and [Reference Bhatt and ScholzeBS15, Proposition 3.1.10] for the second equality). The morphisms $\phi _A$ collectively form a morphism of sheaves of rings . We often omit the $\mathfrak {X}$ from the above notation when it is clear from context, writing , , , , , and $\phi $ .

For a morphism $f\colon \mathfrak {X}\to \mathfrak {Y}$ , the association $((A,I),s)\mapsto ((A,I),f\circ s)$ is cocontinuous, so gives a morphism of topoi .

1.1.3 The quasi-syntomic site

As in [Reference Bhatt, Morrow and ScholzeBMS19], call a p-adically complete ring R quasi-syntomic if it has bounded $p^\infty $ -torsion and the cotangent complex $L_{R/\mathbb {Z}_p}$ has p-complete Tor-amplitude in $[-1,0]$ (see [Reference Bhatt, Morrow and ScholzeBMS19, Definition 4.1]), which we consider as having the p-adic topology. A map $R\to S$ of p-adically complete rings with bounded $p^\infty $ -torsion is called a quasi-syntomic morphism (resp. cover) if it is adically flat (resp. adically faithfully flat)Footnote ³ and $L_{S/R}$ has p-complete Tor-amplitude in $[-1,0]$ . By [Reference Bhatt, Morrow and ScholzeBMS19, Proposition 4.19] a perfectoid ring R is quasi-syntomic.

One extends these definitions to (maps of) p-adic formal schemes by working affine locally. For a quasi-syntomic p-adic formal scheme $\mathfrak {X}$ the big (resp. small) quasi-syntomic site of $\mathfrak {X}$ , denoted $\mathfrak {X}_{\mathrm {QSYN}}$ (resp. $\mathfrak {X}_{\mathrm {qsyn}}$ ), has objects maps (resp. quasi-syntomic maps) $\operatorname {\mathrm {Spf}}(R)\to \mathfrak {X}$ for R a p-adically complete ring with bounded $p^\infty $ -torsion, morphisms given by $\mathfrak {X}$ -morphisms, and covers given by quasi-syntomic covers (see [Reference Bhatt, Morrow and ScholzeBMS19, Lemma 4.17]).

The functor

is cocontinuous (see [Reference Anschütz and Le BrasALB23, Corollary 3.24]), and therefore gives rise to a morphism of topoi . The inclusion $\mathfrak {X}_{\mathrm {qsyn}}\to \mathfrak {X}_{\mathrm {QSYN}}$ , while continuous, may not induce a morphism of sites (see [Reference Anschütz and Le BrasALB23, §4.1]), but we may still consider the functor

Following [Reference Anschütz and Le BrasALB23, Definition 4.1], we use $\mathcal {O}^{\mathrm {pris}}_{\mathfrak {X}}$ to denote and ${\mathcal {I}}_{\mathfrak {X}}^{\mathrm {pris}}$ for . Define

There is a morphism of sheaves of rings . When no confusion will arise, we omit $\mathfrak {X}$ from notation, writing $\mathcal {O}^{\mathrm {pris}}$ , $\mathcal {I}^{\mathrm {pris}}$ , , and $\phi $ . There are also obvious analogues of these objects using $u_\ast $ in place of $v_\ast $ , which we denote by $\mathcal {O}^{\mathrm {PRIS}}$ , etc.

A quasi-syntomic ring R is called quasi-regular semi-perfectoid (see [Reference Bhatt, Morrow and ScholzeBMS19, Definition 4.20 and Remark 4.22]), abbreviated qrsp, if there exists a surjection $S\to R$ with S a perfectoid ring.

Denote by $\mathfrak {X}_{\mathrm {qrsp}}$ the full subcategory of $\mathfrak {X}_{\mathrm {qsyn}}$ consisting of qrsp objects, with the induced topology. By [Reference Bhatt, Morrow and ScholzeBMS19, Lemma 4.28 and Proposition 4.31], $\mathfrak {X}_{\mathrm {qrsp}}$ is a basis for $\mathfrak {X}_{\mathrm {qsyn}}$ . By a qrsp cover $\{\operatorname {\mathrm {Spf}}(R_i)\to \mathfrak {X}\}$ , we mean a quasi-syntomic cover where each $R_i$ is qrsp.

1.1.4 Initial prisms

When R is a qrsp ring, the category has an initial object , necessarily unique up to unique isomorphism (see [Reference Bhatt and ScholzeBS22, Proposition 7.2]).

Example 1.8. Let R be a qrsp $\mathbb {F}_p$ -algebra. If and J denotes the kernel of the composition $W(R^\flat )\to R^\flat \to R$ , set to be the p-completed divided power envelope of $(W(R^\flat ),J)$ , which constitutes the universal pro-(PD thickening) of R over W (see [Reference FontaineFon94, Théorème 2.2.1]). Let $\phi _R\colon {\mathrm {A}}_{\mathrm {crys}}(R)\to {\mathrm {A}}_{\mathrm {crys}}(R)$ denote the morphism induced from $F_{R}$ by this universality. Then, by [Reference Bhatt, Morrow and ScholzeBMS19, Theorem 8.14], ${\mathrm {A}}_{\mathrm {crys}}(R)$ is p-torsion-free and $\phi _R$ is a Frobenius lift on ${\mathrm {A}}_{\mathrm {crys}}(R)$ and so $({\mathrm {A}}_{\mathrm {crys}}(R),(p))$ is a prism.

Under the natural Frobenius-equivariant morphism $W(R^\flat )\to {\mathrm {A}}_{\mathrm {crys}}(R)$ , the ideal $\phi _R(J)$ maps to $(p)$ . Indeed, as $\phi _R(J)$ contains p it suffices to show that if x is in J, then p divides $\phi _R(x)$ in ${\mathrm {A}}_{\mathrm {crys}}(R)$ . But, observe that $x^p= p(p-1)!\tfrac {x^p}{p!}$ and so $\phi _R(x)=x^p=0 \mod p{\mathrm {A}}_{\mathrm {crys}}(R)$ . We then obtain a morphism $\widetilde {\mathrm {nat}}.\colon R\to {\mathrm {A}}_{\mathrm {crys}}(R)/p$ obtained as the composition

So, $({\mathrm {A}}_{\mathrm {crys}}(R),(p),\widetilde {\mathrm {nat}}.)$ constitutes an element of , and is initial by [Reference Anschütz and Le BrasALB23, Lemma 3.27] and [Reference Bhatt and ScholzeBS22, Proposition 7.10].

If $\operatorname {\mathrm {Spf}}(R)\to \mathfrak {X}$ is an object of $\mathfrak {X}_{\mathrm {qrsp}}$ then $u^{-1}(h_{\operatorname {\mathrm {Spf}}(R)})$ is equal to . Using this and the cocontinuity of u (see [Reference Anschütz and Le BrasALB23, Corollary 3.24]), one deduces the following (cf. Lemma A.8).

1.1.5 Small and base $\mathcal {O}_K$ -algebras

We now discuss the rings of main interest in this article.

First definitions

Let R be a p-adically complete $\mathcal {O}_K$ -algebra with $\operatorname {\mathrm {Spec}}(R)$ connected. Call R a base $\mathcal {O}_K$ -algebra if $R=R_0\otimes _W\mathcal {O}_K$ where $R_0$ is a $J_{R_0}$ -adically complete ring for which the pair $(R_0,J_{R_0})=(A_n,I_n)$ , is obtained from the following iterative procedure. For some $d\geqslant 0$ , let $A_0$ be , and set $I_0=(p)$ . For each $i=0,\ldots ,n-1$ iteratively form the pair $(A_{i+1},I_{i+1})$ by one of the following operations:

• ○ $A_{i+1}$ is the p-adic completion of an étale $A_i$ -algebra,Footnote ⁴ and $I_{i+1}=I_iA_{i+1}$ ,

• ○ $A_{i+1}$ is the p-adic completion of a localization $A_i\to (A_i)_{\mathfrak {p}}$ at a prime $\mathfrak {p}$ containing p, and $I_{i+1}=I_iA_{i+1}$ ,

• ○ $A_{i+1}$ is the I-adic completion of $A_i$ with respect to an ideal $I\subseteq A_i$ containing p, and $I_{i+1}=(I_i,I)A_{i+1}$ .

While the discussion of base $\mathcal {O}_K$ -algebras implicitly entails other topologies, we always think of a base $\mathcal {O}_K$ -algebra as being equipped with the p-adic topology.

A map $t\colon T_d\to R_0$ (where we implicitly have $R=R_0\otimes _W \mathcal {O}_K$ ) of the form constructed above is called a presentation. In the following we use terminology from [Reference KimKim15, §2.2]. Moreover, for a map of rings $R\to S$ and an ideal $J\subseteq S$ , we say that $R\to S$ is J-formally étale if the following solid diagram

can be uniquely completed along the dotted arrow assuming that A is a ring and $I\subseteq A$ is a square-zero ideal such that the image J in $A/I$ is nilpotent, and similarly for J-formally smooth.

Proposition 1.12. A base $\mathcal {O}_K$ -algebra R is excellent and regular, and $R/\pi $ has a finite p-basis. For a presentation $t\colon T_d\to R_0$ , the k-algebra $R_0/J_{R_0}$ is finite type, and t is $J_{R_0}$ -formally étale.

Proof. Fix a presentation $t\colon T_d\to R_0$ . We first prove that R is excellent. As $R_0\to R$ is finite type, it suffices to prove that $R_0$ is excellent (see [SP, Tag 07QU]). We prove that $A_i$ is excellent by induction on i. As $T_d$ is the p-adic completion of $W[t_1^{\pm 1},\ldots ,t_d^{\pm 1}]$ , which is excellent by loc. cit., we know that $T_d$ is excellent by [Reference Kurano and ShimomotoKS21, Main Theorem 2]. If $A_i$ is excellent, then $A_{i+1}$ is excellent regardless of which of the three constructions is applied to $A_i$ by combining [SP, Tag 07QU] and [Reference Kurano and ShimomotoKS21, Main Theorem 2]. To prove that t is $J_{R_0}$ -formally étale, it suffices by induction to prove that $A_i\to A_{i+1}$ is $I_{i+1}$ -formally étale for each $i=0,\ldots ,n-1$ , but this is clear. Thus, $T_d\otimes _W \mathcal {O}_K\to R$ is $J_{R_0}$ -formally étale, and thus it follows from [Reference Majadas and RodicioMR10, Theorem 6.4.2] that $T_d\otimes _W \mathcal {O}_K\to R$ is regular and thus R is regular.Footnote ⁵ The final claims concerning $R/\pi =R_0/p$ and $R_0/J_{R_0}$ may be checked iteratively, the latter being obvious. The former being preserved by our three operations requires [Reference de JongdJ95a, Lemma 1.1.3], and the observation that if $\{x_\alpha \}$ is a p-basis for an $\mathbb {F}_p$ -algebra A, and $A\to B$ is étale then $\{x_\alpha \}$ is a p-basis for B as $F_A\otimes 1$ is identified with $F_B$ via the isomorphism .

We call a decomposition $R=R_0\otimes _W \mathcal {O}_K$ and a $J_{R_0}$ -formally étale map $t\colon T_d\to R_0$ , where $J_{R_0}$ is any ideal coming from a presentation, a formal framing. For a formal framing t, the ring $R_0$ carries a unique Frobenius lift $\phi _{t}$ such that $\phi _t\circ t=t \circ \phi _0$ , where $\phi _0$ is the Frobenius lift on $T_d$ acting as usual on W and with $\phi _0(t_i)=t_i^p$ .

Various morphisms of prisms

Let $R=R_0\otimes _W\mathcal {O}_K$ be a base $\mathcal {O}_K$ -algebra and choose a formal framing $t\colon T_d\to R_0$ . Denote by $t^\flat $ the choice of $p^{\text {th}}$ -power roots $t_i^\flat $ of $t_i$ in $\check {T}_d^\flat $ .

Define $R_0\to {\mathrm {A}}_{\mathrm {inf}}(\check {R})$ , using the $J_{R_0}$ -formal étaleness of t, as the unique extension of the map $T_d\to {\mathrm {A}}_{\mathrm {inf}}(\check {R})$ sending $t_i$ to $[t_i^\flat ]$ and inducing the natural map $R_0\to \check {R}$ after composition with $\widetilde {\theta }$ . We further define $\alpha _{\inf }=\alpha _{\inf ,t^{\flat }}\colon \mathfrak {S}_R\to {\mathrm {A}}_{\mathrm {inf}}(\check {R})$ to be the unique extension of this map such that $\alpha _{\inf ,t^\flat }(u)=[\pi ^\flat ]$ . We then have the following diagram of prisms, where each inclusion arrow is the obvious inclusion, and all other not previously defined arrows are determined uniquely by commutativity.

(1.1.2)

Note that for all of these triples, save $(R_0^{(\phi _t)},(p),q)$ and $(R_0^{(\phi _t)},(p),F_{R_0/p}\circ q)$ , the structure morphism is unambiguous given the first two entries, so we often omit it. If we write $(R_0^{(\phi _t)},(p))$ then the reader should assume we mean $(R_0^{(\phi _t)},(p),q)$ .

While every morphism in Diagram (1.1.2) is a morphism of prisms, the arrow labeled $(\ast )$ is the only map which may not be a morphism in . In fact, this happens precisely when $\mathcal {O}_K=W$ .

Lemma 1.14. For an integer $a\geqslant 0$ , let $s_{a+1}$ , denote the following composition:

$$ \begin{align*} R\xrightarrow{\mathrm{nat}.}\mathfrak{S}_R/(E)\xrightarrow{\bar\phi_t} \mathfrak{S}_R/(\phi_t(E))\xrightarrow{\overline{i}}S_R/p\xrightarrow{F_{S_R/p}^a} S_R/p. \end{align*} $$

Then $(R_0,(p),F^{a+1}_{R_0/p}\circ q)\hookrightarrow (S_R,(p),s_{a+1})$ is a morphism in if and only if $p^{a}\geqslant e$ .

Proof. It suffices to observe that the following diagram commutes if and only if $p^a\geqslant e$ :

Note that $R=R_0[\pi ]$ and that the diagram always commutes on $R_0$ . Then chasing where $\pi $ is sent, one sees this commutativity holds if and only if $u^{p^{a+1}}$ is divisible by p in $S_R$ . Using Equation (1.1.1), one easily sees that this happens if and only if $\tfrac {p^{a+1}}{e}\geqslant p$ or, equivalently, that $p^a\geqslant e$ .

1.2.1 Prismatic F-crystals

Define the category of prismatic crystals in vector bundles (resp. perfect complexes) over T as follows:

Concretely, a prismatic crystal in vector bundles (resp. perfect complexes) is a collection of finite projective A-modules $M_{(A,I)}$ (resp. perfect complexes $K^{\bullet }_{(A,I)}$ ), indexed by objects $(A,I)$ of , together with (quasi-)isomorphisms (resp. ) for any morphism $(A,I)\to (B,J)$ in (the crystal property), satisfying the obvious compatibility conditions. The category of prismatic crystals in vector bundles carries the structure of an exact $\mathbb {Z}_p$ -linear $\otimes $ -category where exactness and tensor products are defined term-by-term.

Proposition 1.17 (cf. [Reference Bhatt and ScholzeBS23, Proposition 2.7])

The global sections functor

is a bi-exact $\mathbb {Z}_p$ -linear $\otimes $ -equivalence. Moreover, the derived global sections functor

is an equivalence of $\infty $ -categories.

As in [Reference Bhatt and ScholzeBS23, Definition 4.1], define the category of prismatic F-crystals over T to have objects $(\mathcal {E},\varphi _{\mathcal {E}})$ where $\mathcal {E}$ is an object of and

is an isomorphism in , called the Frobenius, and morphisms are morphisms in commuting with the Frobenii. Likewise, define the category of prismatic F-crystals in perfect complexes to be the category of pairs $({\mathcal {E}}^{\bullet },\varphi _{{\mathcal {E}}^{\bullet }})$ where ${\mathcal {E}}^{\bullet }$ is an object of together with a Frobenius isomorphism in

and with morphisms being those in commuting with Frobenii. By Proposition 1.17, when $\mathbf {Vect}^\varphi (A,I)$ and $\mathbf {D}^\varphi _{\mathrm {perf}}(A,I)$ are given the obvious meanings, then

Observe that inherits from the structure of an exact $\mathbb {Z}_p$ -linear $\otimes $ -category. We say $(\mathcal {E},\varphi _{\mathcal {E}})$ is effective if $\varphi _{\mathcal {E}}$ is induced from a morphism $\varphi _{\mathcal {E},0}\colon \phi ^\ast \mathcal {E}\to \mathcal {E}$ , which is automatically injective, and the minimal r such kills $\mathrm {coker}(\varphi _{\mathcal {E},0})$ is the height of $(\mathcal {E},\varphi _{\mathcal {E}})$ .

Remark 1.19. Suppose that $(A,I,s)$ is an object of with the property that $\phi _A(I)$ is an invertible ideal (e.g., if $\phi _A$ is flat, see [SP, Tag 02OO], or $I=(p)$ ). In this case $(A,\phi _A(I),s\circ \operatorname {\mathrm {Spf}}(\overline {\phi }_A))$ is an object of , where $\overline {\phi }_A\colon A/I\to A/\phi _A(I)$ is the natural map. Indeed, if $p=i+\phi _A(i')$ , with i and $i'$ in I, then evidently $p=\phi _A(i)+\phi _A(\phi _A(i'))$ so that $p\in \phi _A(I)+\phi _A(\phi _A(I))$ , and A is $(p,\phi _A(I))$ -adically complete as $(p,I)^p\subseteq (p,\phi _A(I))\subseteq (p,I)$ . Moreover, we observe that $\phi _A\colon (A,I,s)\to (A,\phi _A(I),s\circ \operatorname {\mathrm {Spf}}(\overline {\phi }_A))$ is a morphism in so that, by the crystal property, we have an identification of A-modules

Thus, we, in particular, see that via this identification there is an isomorphism

which provides $\phi ^\ast \mathcal {E}(A,I,s)$ with a Frobenius-like structure.

1.2.2 Analytic prismatic F-crystals

We now discuss the notion of analytic prismatic F-crystals as defined in [Reference Guo and ReineckeGR24], which are necessary to model every crystalline $\mathbb {Z}_p$ -local system.

Basic definitions

Following [Reference Guo and ReineckeGR24, Definition 3.1], define the category of analytic prismatic crystals over T as follows (where we recall that ):

We denote an object of as $\mathcal {V}=(\mathcal {V}_{(A,I)})$ . By the argument given in [Reference Guo and ReineckeGR24, Proposition 3.7], the following restriction is fully faithful:

Endow with the structure of a $\mathbb {Z}_p$ -linear $\otimes $ -category defined term-by-term in the two-limit. Then the above restriction is a $\mathbb {Z}_p$ -linear $\otimes $ -functor. Denote the object $(\mathcal {O}_{U(A,I)})$ by . As in [Reference Guo and ReineckeGR24, Definition 3.5], we then define the category of analytic prismatic F-crystals, denoted typically by $(\mathcal {V},\varphi _{\mathcal {V}})$ , in the same way and endowed with the analogous structure of a $\mathbb {Z}_p$ -linear $\otimes $ -category.

We say that a sequence of analytic prismatic crystals

$$ \begin{align*} 0\to \mathcal{V}^1\to \mathcal{V}^2\to \mathcal{V}^3\to 0 \end{align*} $$

is exact if the sequence of vector bundles

$$ \begin{align*} 0\to \mathcal{V}^1_{(A,I)}\to \mathcal{V}^2_{(A,I)}\to \mathcal{V}^3_{(A,I)}\to 0 \end{align*} $$

on $\operatorname {\mathrm {Spec}}(A)-V(p,I)$ is exact for all $(A,I)$ in . We define a sequence of objects of to be exact if the underlying sequence of analytic prismatic crystals is. This endows and with the structures of exact $\mathbb {Z}_p$ -linear $\otimes $ -categories.

Criteria for exactness

Our next goal is to establish the following basic result.

Proposition 1.20. A sequence

$$ \begin{align*} 0\to \mathcal{V}^1\to \mathcal{V}^2\to \mathcal{V}^3\to 0 \end{align*} $$

in is exact if and only if there exists a cover $\{(A_i,I_i)\}$ of $\ast $ such that

$$ \begin{align*} 0\to \mathcal{V}^1_{(A_i,I_i)}\to \mathcal{V}^2_{(A_i,I_i)}\to \mathcal{V}^3_{(A_i,I_i)}\to 0 \end{align*} $$

is exact for all i.

Remark 1.21. Proposition 1.20 does not easily follow using fpqc descent. The latter condition implies (see Lemma A.1) that for every $(A,I)$ there exists a cover $\{(B_j,IB_j)\}$ of $(A,I)$ such that sequence over $(A,I)$ is exact when pulled back to each $U(B_j,IB_j)$ . If $S=\{U(B_j,IB_j)\to U(A,I)\}$ were an fpqc cover we would be done. But, a priori all we know is that $\{\operatorname {\mathrm {Spf}}(B_j)\to \operatorname {\mathrm {Spf}}(A)\}$ is an adically flat cover. With Noetherian hypotheses this is sufficient (cf. [SP, Tag 0912]), but in our generality it is not even clear that S is jointly surjective (although see Proposition 1.25 below).

Using Lemma A.1, and the fact that an exact sequence of vector bundles is universally exact (see [SP, Tag 058H]), the desired result is a special case of the following.

Proposition 1.22. Let $f\colon A\to B$ be a map of rings, and suppose that the finitely generated ideal $J\subseteq A$ is contained in the Jacobson radical of A. If $f\colon A/J^n\to B/J^nB$ is faithfully flat for all n then, a sequence of vector bundles

$$ \begin{align*} P\colon \quad 0\to \mathcal{M}_1\to\mathcal{M}_2\to\mathcal{M}_3\to 0, \end{align*} $$

on $\operatorname {\mathrm {Spec}}(A)-V(J)$ is exact if and only if the sequence

$$ \begin{align*} f^\ast P\colon \quad 0\to f^\ast\mathcal{M}_1\to f^\ast\mathcal{M}_2\to f^\ast\mathcal{M}_3\to 0, \end{align*} $$

of vector bundles on $\operatorname {\mathrm {Spec}}(B)-V(JB)$ is exact.

Proposition 1.22 follows from the following two lemmas as $\operatorname {\mathrm {Spec}}(B)-V(JB)$ is quasi-compact (as J is finitely generated) so that any point contains a closed point in its closure.

Lemma 1.23. Let $(Y,\mathcal {O}_Y)$ be a locally ringed space and S a subset of Y such that every point of Y admits an element of S as a specialization. Then, a sequence

$$ \begin{align*} Q\colon \quad 0\to V_1\to V_2\to V_3\to 0, \end{align*} $$

of $\mathcal {O}_Y$ -modules, where each $V_i$ is a vector bundle on $(Y,\mathcal {O}_Y)$ , is exact if and only if for all y in S the induced sequence

$$ \begin{align*} 0\to (V_1)_{k(y)}\to (V_2)_{k(y)}\to (V_3)_{k(y)}\to 0, \end{align*} $$

of vector spaces over the residue field $k(y)$ is exact. In particular, if $f\colon (Y',\mathcal {O}_Y)\to (Y,\mathcal {O}_Y)$ is a map of locally ringed spaces containing S in its image, then Q is exact if and only if

$$ \begin{align*} f^\ast Q\colon \quad 0\to f^\ast V_1\to f^\ast V_2\to f^\ast V_3\to 0, \end{align*} $$

is exact.

Proof. The second claim follows easily from the first, and only the if condition of the first statement requires proof. To prove this, it suffices to show that for each y in S, the sequence of projective $\mathcal {O}_{Y,y}$ -modules

$$ \begin{align*} Q_y\colon \quad 0\to (V_1)_y\to (V_2)_y\to (V_3)_y\to 0, \end{align*} $$

is exact. Indeed, if $y'$ is an arbitrary point of Y and y in S is a specialization, then the sequence $Q_{y'}$ is obtained by base changing the sequence $Q_y$ along the flat map $\mathcal {O}_{Y,y}\to \mathcal {O}_{Y,y'}$ .

To show that $Q_y$ is exact, let $n_i$ , for $i=1,2,3$ , be the rank of the finite free $\mathcal {O}_{Y,y}$ -module $(V_i)_y$ . That $(V_2)_y\to (V_3)_y$ is surjective follows from Nakayama’s lemma. We then see that the sequence

(1.2.1) $$ \begin{align} 0\to \ker((V_2)_y\to (V_3)_y)\to (V_2)_y\to (V_3)_y\to 0 \end{align} $$

is exact. As $V_3$ is a vector bundle, this sequence splits. So, $\ker ((V_2)_y\to (V_3)_y)$ is locally free of rank equal to $n_2-n_3$ . As rank can be computed over $k(y)$ , we have that $n_1=n_2-n_3$ . Thus, $(V_1)_y\to \ker ((V_2)_y\to (V_3)_y)$ is a map of finite free $\mathcal {O}_{Y,y}$ -modules of the same rank. Thus, it is an isomorphism if and only if it is surjective (see [Reference OrzechOrz71] for a generalization of this fact). But, by Nakayama’s lemma, it suffices to check this claim after base change to $k(y)$ . But, as (1.2.1) is exact, and $(V_3)_y$ flat, this is equivalent to checking that $(V_1)_{k(y)}\to \ker ((V_2)_{k(y)}\to (V_3)_{k(y)})$ is surjective, which is true by assumption.

Lemma 1.24. Let $f\colon A\to B$ be a map of rings, and suppose that the finitely generated ideal $J\subseteq A$ is contained in the Jacobson radical. If $f\colon A/J^n\to B/J^nB$ is faithfully flat for all n, then the map $\operatorname {\mathrm {Spec}}(B)-V(JB)\to \operatorname {\mathrm {Spec}}(A)-V(J)$ is surjective on closed points.

Proof. Let $\mathfrak {p}$ be a prime of A constituting a closed point of $\operatorname {\mathrm {Spec}}(A)-V(J)$ . We show that $\mathfrak {p}$ is in the image of f. Suppose first that $\mathfrak {p}=(0)$ . In this case we see that it suffices to show that $\operatorname {\mathrm {Spec}}(B)-V(JB)$ is nonempty. If it were empty, then $JB$ is contained in the nilradical of B and as J is finitely generated, this implies that $J^nB=0$ for some n. This implies that $J^nB/J^{n+1}B$ is zero, and as $A/J^{n+1}\to B/J^{n+1}B$ is faithfully flat, this implies that $J^n/J^{n+1}$ is zero. Since J is contained in the Jacobson radical of A, we deduce from Nakayama’s lemma (see [SP, Tag 00DV]) that $J^n$ is zero, but this implies that $\operatorname {\mathrm {Spec}}(A)-V(J)$ is empty, which is a contradiction.

In general, let , , and . Then, $\overline {J}$ is in the Jacobson radical of $\overline {A}$ , and $\overline {A}/\overline {J}^n\to \overline {B}/\overline {J}^n\overline {B}$ is equal to the base change of $A/J^n\to B/J^n B$ along $A\to \overline {A}$ , and so faithfully flat. From the argument in the previous paragraph there exists some $\overline {\mathfrak {q}}$ in $\operatorname {\mathrm {Spec}}(\overline {B})-V(\overline {J}\overline {B})$ . Let $\mathfrak {q}$ be a prime of B lying over $\overline {\mathfrak {q}}$ . Observe that $\mathfrak {q}$ belongs to $\operatorname {\mathrm {Spec}}(B)-V(JB)$ , and by construction $f(\mathfrak {q})$ lies in $V(\mathfrak {p})\cap (\operatorname {\mathrm {Spec}}(A)-V(J))$ . But, as $\mathfrak {p}$ is closed in $\operatorname {\mathrm {Spec}}(A)-V(J)$ , one checks that $V(\mathfrak {p})\cap (\operatorname {\mathrm {Spec}}(A)-V(J))=\{\mathfrak {p}\}$ from where the claim follows.

Finally, we include the following beautiful observation of Ofer Gabber showing that the restriction to closed points is not necessary when the rings are complete.

Proposition 1.25 (Gabber)

Let $J\subseteq A$ be a finitely generated ideal, and $f\colon A\to B$ a ring map. Suppose that A (resp. B) is complete with respect to J (resp. $JB$ ), and that $A/J^n\to B/J^nB$ is faithfully flat for all n. Then, the map $\operatorname {\mathrm {Spec}}(B)\to \operatorname {\mathrm {Spec}}(A)$ is surjective.

Proof. For an ideal I of A, let us denote by $I^{\mathrm {ec}}$ the ideal $f^{-1}(IB)$ , and similarly for the ring maps $f_n\colon A/J^n\to B/J^nB$ . Observe that by assumption that each $f_n$ is faithfully flat, we have that $I^{\mathrm {ec}}=I$ for any ideal $I\subseteq A/J^n$ . Thus, if I is an ideal of A closed in the J-adic topology, then by passing to the limit we see that $I^{\mathrm {ec}}=I$ .

Claim 1: For a finitely generated ideal $I\subseteq A$ , we have that $\overline {I}^2\subseteq I$ , where $\overline {I}$ is the closure of I.

Proof. More generally we show that for finitely generated ideals $I_1$ and $I_2$ of A, the product of their closures is contained in $I_1+I_2$ . If x is in the closure of $I_1$ we can write $x=\sum _n x_n$ with $x_n$ in $I_1\cap J^n$ . Similarly if y is in the closure of $I_2$ then $y=\sum _n y_n$ , with $y_n$ in $I_2\cap J^n$ . Thus,

(1.2.2) $$ \begin{align} xy = \sum_{n\leqslant m}x_n y_m+ \sum_{n>m}x_n y_m. \end{align} $$

Let $f_k$ be a finite system of generators of $I_1$ and write $x_n=\sum _k x_{nk} f_k$ . Then the first term on the right-hand side of (1.2.2) is $\sum _k(\sum _m(\sum _{n\leqslant m}x_{nk}) y_m)f_k$ , which is in $I_1$ , and similarly the second term on the right-hand side of (1.2.2) is in $I_2$ .

Claim 2: For every ideal I of A, one has $(I^{\mathrm {ec}})^2\subseteq I$ .

Proof. One reduces to the case when I is finitely generated, in which case we observe that

$$ \begin{align*} (I^{\mathrm{ec}})^2\subseteq (\overline{I}^{\mathrm{ec}})^2=\overline{I}^2\subseteq I, \end{align*} $$

the second equality by our initial observations, and the second containment by Claim 1.

If $\mathfrak {p}\subseteq A$ is prime, then from Claim 2 we have $(\mathfrak {p}^{\mathrm {ec}})^2\subseteq \mathfrak {p}$ . As $\mathfrak {p}$ is radical this implies that $\mathfrak {p}^{\mathrm {ec}}\subseteq \mathfrak {p}$ and thus $\mathfrak {p}^{\mathrm {ec}}=\mathfrak {p}$ , and so $\mathfrak {p}$ is in the image of $\operatorname {\mathrm {Spec}}(B)\to \operatorname {\mathrm {Spec}}(A)$ . Indeed, $B_{\mathfrak {p}}/\mathfrak {p}B_{\mathfrak {p}}$ is nonzero as $A_{\mathfrak {p}}/\mathfrak {p}=A_{\mathfrak {p}}/f^{-1}(\mathfrak {p}B)A_{\mathfrak {p}}$ is nonzero and embeds into this ring via f.

A criterion to be a prismatic F-crystal

We now observe a simple criterion for when an analytic prismatic (F-)crystal comes from a prismatic (F-)crystal.

Proposition 1.26. Let $\mathfrak {X}$ be a base formal $\mathcal {O}_K$ -scheme and let $\{\operatorname {\mathrm {Spf}}(R_i)\}$ be an open cover of $\mathfrak {X}$ where $R_i$ is a (formally framed) base $\mathcal {O}_K$ -algebra. Then, the essential image of

consists of those $\mathcal {W}$ and those $(\mathcal {V},\varphi _{\mathcal {V}})$ , such that $(j_{(\mathfrak {S}_{R_i},(E))})_\ast \mathcal {W}_{(\mathfrak {S}_{R_i},(E))}$ and $(j_{(\mathfrak {S}_{R_i},(E))})_\ast \mathcal {V}_{(\mathfrak {S}_{R_i},(E))}$ are vector bundles on $\mathfrak {S}_{R_i}$ for all i, respectively.

Proof. It suffices to prove the first claim. Moreover, we are clearly reduced to the case when $\mathfrak {X}=\operatorname {\mathrm {Spf}}(R)$ . For $i=1,2,3$ , let $\mathfrak {S}_R^{(i)}$ be the i-fold self-product of $\mathfrak {S}_R$ in the topos (see [Reference Du, Liu, Moon and ShimizuDLMS24, Example 3.4]), and let $p^{(i)}_k$ for $k=1,\ldots , i$ denote the projection maps, which are flat by [Reference Du, Liu, Moon and ShimizuDLMS24, Lemma 3.5]. Let $j^{(i)}$ , for $i=1,2,3$ denote the inclusion of $U(\mathfrak {S}_R^{(i)},(E))$ into $\operatorname {\mathrm {Spec}}(\mathfrak {S}_R^{(i)})$ . Observe that we have the $2$ -commutative diagram of categories.

The top row is obviously exact, and the bottom row is exact by [Reference MathewMat22] (cf. [Reference Bhatt and ScholzeBS23, Theorem 2.2]). Observe that, by inspection, $(p,(E))$ has height $2$ in $\mathfrak {S}_R$ , and thus by the flatness of $p_k^{(i)}$ , the same holds for $(p,(E))$ in $\mathfrak {S}^{(i)}$ for $i=1,2,3$ . Thus, we observe that the maps $(j^{(i)})^\ast $ are fully faithful by Proposition A.23. Suppose now that $j^{(1)}_\ast \mathcal {W}_{(\mathfrak {S}_R,(E))}$ is a vector bundle. Let us observe that

$$ \begin{align*} (j^{(i)})^\ast (p_k^{(i)})^\ast j^{(1)}_\ast \mathcal{W}_{(\mathfrak{S}_R,(E))}\simeq (p_k^{(i)})^\ast \mathcal{W}_{(\mathfrak{S}_R,(E))}, \end{align*} $$

for each $i=1,2,3$ and $k=1,\ldots ,i$ , again by Proposition A.23. Thus, by the full faithfulness of the maps $(j^{(i)})^\ast $ , we may pull back the descent data on $\mathcal {W}_{(\mathfrak {S}_R,(E))}$ relative to the covering $\{p_k^{(2)}\colon U(\mathfrak {S}_R^{(2)},(E))\to U(\mathfrak {S}_R,(E))\}_{k=1,2}$ to descent data on $j^{(1)}_\ast \mathcal {W}_{(\mathfrak {S}_R,(E))}$ relative to the covering $\{p_k^{(2)}\colon \operatorname {\mathrm {Spec}}(\mathfrak {S}_R^{(2)})\to \operatorname {\mathrm {Spec}}(\mathfrak {S}_R)\}_{k=1,2}$ . This gives an object $\mathcal {F}$ of whose image in is isomorphic to $\mathcal {W}$ .

We have shown that anything satisfying the condition on the pushforward in the proposition statement is in the essential image of . Conversely, if $\mathcal {W}$ is in the essential image, then this pushforward condition holds again by applying Proposition A.23.

1.3.1 Prismatic F-crystals

We begin by shortening some notation from §A.5:

Combining Proposition 1.17 and Theorem A.19, we see that is reconstructible in , and every object of is locally trivial.

Set to be the morphism of group sheaves associating to every $(A,I)$ the map $\mathcal {G}(A)\to \mathcal {G}(A)$ obtained from $\phi _A\colon A\to A$ . For an object $\mathcal {A}$ of , denote by $\phi ^\ast \mathcal {A}$ the -torsor $\phi _\ast \mathcal {A}$ , in the notation of §A.1. Denote by the group sheaf and let be the obvious monomorphism. Finally, set to be $\iota _\ast \mathcal {A}$ .

Definition 1.28. The category of prismatic $\mathcal {G}$ -torsors with F-structure over T has objects $(\mathcal {A},\varphi _{\mathcal {A}})$ where $\mathcal {A}$ is an object of and $\varphi $ is a Frobenius isomorphism

and morphisms are morphisms in commuting with the Frobenii.

Objects of are pairs $(\omega ,\varphi _\omega )$ with $\omega $ an object of and an isomorphism in . For an object $\mathcal {A}$ in there is a natural identification $\phi ^\ast \omega _{\mathcal {A}}\simeq \omega _{\phi ^\ast \mathcal {A}}$ . So, for an object $(\mathcal {A},\varphi _{\mathcal {A}})$ of there is a natural Frobenius $\varphi _{\omega _{\mathcal {A}}}$ , and $(\omega _{\mathcal {A}},\varphi _{\omega _{\mathcal {A}}})$ is an object of .

On the other hand, define to be the groupoid consisting of pairs where $(\mathcal {E},\varphi _{\mathcal {E}})$ is an object of , the pair is an object of , and constitutes a set of tensors on $(\mathcal {E},\varphi _{\mathcal {E}})$ or, equivalently, each tensor in is Frobenius invariant: fixed under the composition $\mathcal {E}\to \phi ^\ast \mathcal {E}\xrightarrow {\varphi _{\mathcal {E}}}\mathcal {E}$ , where the first map sends x to $x\otimes 1$ .

Finally, observe that there is a natural identification

for an object $\mathcal {E}$ of . Moreover, there is an identification between and . So, if is an object of , then the -torsor has a natural Frobenius also denoted $\varphi _{\mathcal {E}}$ .

Our main Tannakian result concerning prismatic F-crystals is the following, where for a prism $(A,I)$ the category $\mathbf {Tors}^\varphi _{\mathcal {G}}(A,I)$ has the obvious meaning.

Proposition 1.29. The natural functor

is an equivalence. Moreover, we have a commuting triangle of equivalences

Results from Appendix A (notably Proposition A.18 applied to both and ) and Proposition 1.17 reduces the proof of this statement to the claim that the natural functor

is an equivalence.

To prove this, we introduce an ancillary site. Denote by the subcategory of with the same objects as and with those morphisms $(A,I)\to (B,J)$ with $\operatorname {\mathrm {Spf}}(B)\to \operatorname {\mathrm {Spf}}(A)$ adically étale, with the induced topology. By Proposition 1.4 (and [SP, Tag 00X5]), the functor induces a morphism of sites, and there are equivalences of sites .

Proposition 1.30. The following restriction functors are equivalences of categories

Proof. Fix an object $(A,I)$ of and set to be the full subcategory of consisting of covers $(A,I)\to (B,J)$ , with the induced topology. We use this notation in a similar way for other sites. The naturally defined functor is continuous and has exact pullback by [SP, Tag 00X6] and [Reference Du, Liu, Moon and ShimizuDLMS24, Lemma 3.3], and we have the equality . The same properties hold mutatis mutandis for . From the following commutative diagram

(where the vertical functors are the natural inclusions), we obtain the diagram

Note that $\mu _4^\ast $ and $\mu _2^\ast $ are evidently equivalences, $\nu _2^\ast $ and $\nu _3^\ast $ are equivalences by Corollary A.11, and $\mu _3^\ast $ is an equivalence by above discussion. By a diagram chase we see that $\mu _1^\ast $ is essentially surjective and so it is an equivalence by Corollary A.6. Thus, $\nu _1^\ast $ is an equivalence. We deduce that for any object of , its restriction to $(A,I)$ can be trivialized on an étale cover, and thus the restriction is an equivalence by Corollary A.6.

To prove that is an equivalence we exhibit a quasi-inverse. For an object $(\mathcal {A}_{(A,I)})$ of the 2-limit category define the object $\mathcal {A}$ of to be the one sending $(A,I)$ to $\mathcal {A}_{(A,I)}(A)$ . This is a presheaf carrying an action of and for which the action on $(A,I)$ -points is simply transitive whenever nonempty. Thus, we are done as it is simple to see that this presheaf is locally isomorphic to .

Finally, we mention the relationship between the above discussion and analytic prismatic F-crystals. By Theorem A.19, and the ideas applied in the proof of Proposition 1.29, one has

(1.3.1)

and the restriction map

is fully faithful by [Reference Guo and ReineckeGR24, Proposition 3.7], where $\mathbf {Tors}^\varphi _{\mathcal {G}}(U(A,I))$ has the obvious meaning. For this reason, we occasionally identify with the category which, by definition, is the right-most term in (1.3.1).

1.3.2 Quasi-syntomic torsors with F-structure

For the sake of completeness, we compare the material from §1.3.1 to the analogous theory in the quasi-syntomic setting.

Abbreviate $\mathcal {G}_{\mathcal {O}^{\mathrm {pris}}}$ to $\mathcal {G}^{\mathrm {pris}}$ and to . Write $\mathbf {Tors}_{\mathcal {G}}(\mathfrak {X}_{\mathrm {qsyn}})$ instead of $\mathbf {Tors}_{\mathcal {G}^{\mathrm {pris}}}(\mathfrak {X}_{\mathrm {qsyn}})$ . We make similar notational conventions concerning $(\mathfrak {X}_{\mathrm {QSYN}},\mathcal {O}^{\mathrm {PRIS}})$ .

Proposition 1.31. The following are well-defined equivalences of categories functorial in $\mathcal {G}$ :

The proof of the following is obtained mutatis mutandis from the proof of Proposition 1.31.

Proposition 1.32 (cf. [Reference Anschütz and Le BrasALB23, Proposition 4.4] and [Reference Bhatt and ScholzeBS23, Proposition 2.13 and Proposition 2.14])

The following are well-defined rank-preserving bi-exact $\mathbb {Z}_p$ -linear $\otimes $ -equivalences:

Set $\phi =v_\ast (\phi )$ on . For an object $\mathcal {B}$ of $\mathbf {Tors}_{\mathcal {G}}(\mathfrak {X}_{\mathrm {qsyn}})$ , define as the pushforward along , and . There are identifications $v_\ast \phi ^\ast \mathcal {A}=\phi ^\ast v_\ast \mathcal {A}$ and . Define a quasi-syntomic $\mathcal {G}$ -torsor with F-structure to be a pair $(\mathcal {B},\varphi )$ where $\mathcal {B}$ is an object of $\mathbf {Tors}_{\mathcal {G}}(\mathfrak {X}_{\mathrm {qsyn}})$ and $\varphi $ a Frobenius isomorphism

A morphism of quasi-syntomic $\mathcal {G}$ -torsors with F-structure is a morphism of $\mathcal {G}^{\mathrm {pris}}$ -torsors commuting with the Frobenii. Denote the category of such objects by $\mathbf {Tors}^\varphi _{\mathcal {G}}(\mathfrak {X}_{\mathrm {qsyn}})$ . One can similarly define the categories $\mathbf {Vect}^\varphi (\mathfrak {X}_{\mathrm {qsyn}})$ and $\mathcal {G}\text {-}\mathbf {Vect}^\varphi (\mathfrak {X}_{\mathrm {qsyn}})$ , as well as their analogues for $\mathfrak {X}_{\mathrm {QSYN}}$ or $\mathfrak {X}_{\mathrm {qrsp}}$ .

Using Proposition 1.29, Proposition 1.31, and Proposition 1.32 one may deduce the following.

Proposition 1.33. There is a commuting diagram of well-defined equivalences

which is functorial in $\mathcal {G}$ . Similar assertions hold with $\mathfrak {X}_{\mathrm {qsyn}}$ replaced by $\mathfrak {X}_{\mathrm {QSYN}}$ or $\mathfrak {X}_{\mathrm {qrsp}}$ .

2.1.1 $\mathcal {G}(\mathbb {Z}_p)$ -local systems on a scheme

For a locally topologically Noetherian scheme S (in the sense in [Reference Bhatt and ScholzeBS15, Definition 6.6.9]), recall from [Reference Bhatt and ScholzeBS15, Definition 4.1.1] that a morphism of schemes $f\colon S'\to S$ is weakly étale if f and $\Delta _f$ are flat. Moreover, we can associate to S the proétale site $S_{\mathrm {pro\acute {e}t}}$ which is the full subcategory of $S_{\mathrm {fpqc}}$ consisting of weakly étale morphisms $S'\to S$ , with the induced topology.

We will be interested in the ringed site $(S_{\mathrm {pro\acute {e}t}},\underline {\mathbb {Z}_p}_S)$ , where

where the second equality follows from [Reference Bhatt and ScholzeBS15, Proposition 3.2.3, Proposition 3.1.10, and Proposition 4.2.8]. We often use the notation . More generally, for a topological space T, denote by $\underline {T}_S$ (or just $\underline {T}$ when S is clear from context) the sheaf (see [Reference Bhatt and ScholzeBS15, Lemma 4.2.12])

$$ \begin{align*} \underline{T}_S\colon S_{\mathrm{pro\acute{e}t}}\to \mathbf{Set},\qquad U\mapsto \operatorname{\mathrm{Hom}}_{\text{cont.}}(U,T). \end{align*} $$

If T is totally disconnected, then $\operatorname {\mathrm {Hom}}_{\text {cont.}}(U,T)$ is equal to $\text {Hom}_{\text {cont.}}(\pi _0(U),T)$ , and so if $T=\varprojlim T_i$ with each $T_i$ finite then $\underline {T}=\varprojlim \underline {T_i}$ where each $\underline {T_i}$ is the constant sheaf.

Proof. We handle only the case where n is finite as the case for $n=\infty $ follows by passing to the limit. We provide for U in $S_{\mathrm {pro\acute {e}t}}$ an isomorphism bi-functorial in U and $\mathcal {G}$ and compatible in n. Let $\{D_i\}$ be the set of discrete quotient spaces of U. Then, for any discrete space X there is an identification between $\operatorname {\mathrm {Hom}}_{\text {cont.}}(U,X)$ and $\varinjlim \operatorname {\mathrm {Hom}}(D_i,X)$ , bi-functorial in U and X. As $\mathcal {G}$ is locally of finite presentation over $\mathbb {Z}_p$

$$ \begin{align*} \begin{aligned}\mathcal{G}_{\underline{\mathbb{Z}/p^n}}(U) &=\mathcal{G}(\operatorname{\mathrm{Hom}}_{\text{cont.}}(U,\mathbb{Z}/p^n))\\ &=\varinjlim \mathcal{G}(\operatorname{\mathrm{Hom}}(D_i,\mathbb{Z}/p^n))\\ &= \varinjlim \text{Hom}_{\mathbf{Alg}_{\mathbb{Z}_p}}(A_{\mathcal{G}},\text{Hom}(D_i,\mathbb{Z}/p^n)),\end{aligned} \end{align*} $$

(cf. [SP, Tag 01ZC]), where . On the other hand we see that

$$ \begin{align*} \begin{aligned} \underline{\mathcal{G}(\mathbb{Z}_p/p^n\mathbb{Z})}(U) &= \text{Hom}_{\text{cont.}}(U,\mathcal{G}(\mathbb{Z}/p^n))\\ &= \varinjlim \text{Hom}(D_i,\mathcal{G}(\mathbb{Z}/p^n))\\ &= \varinjlim \operatorname{\mathrm{Hom}}(D_i,\operatorname{\mathrm{Hom}}_{\mathbf{Alg}_{\mathbb{Z}_p}}(A_{\mathcal{G}},\mathbb{Z}/p^n)). \end{aligned} \end{align*} $$

Thus, we are done via the natural isomorphism of groups

$$ \begin{align*} \operatorname{\mathrm{Hom}}(D_i,\operatorname{\mathrm{Hom}}_{\mathbf{Alg}_{\mathbb{Z}_p}}(A_{\mathcal{G}},\mathbb{Z}/p^n))\to\text{Hom}_{\mathbf{Alg}_{\mathbb{Z}_p}}(A_{\mathcal{G}},\text{Hom}(D_i,\mathbb{Z}/p^n)) \end{align*} $$

bi-functorial in $D_i$ and $\mathcal {G}$ , given by currying.

Denote by $\mathbf {Loc}_{\mathbb {Z}/p^n}(S)$ the category $\mathbf {Vect}(S_{\mathrm {pro\acute {e}t}},\underline {\mathbb {Z}/p^n})$ of $\mathbb {Z}/p^n$ -local systems. By [Reference Bhatt and ScholzeBS15, Corollary 5.1.5 and Proposition 6.8.4], this is equivalent to $\mathbf {Loc}_{\mathbb {Z}/p^n}(S_{\mathrm {\acute {e}t}})$ (see [SGA5, Exposé VI]) as an exact $\mathbb {Z}_p$ -linear $\otimes $ -category. We refer to objects of $\mathcal {G}\text {-}\mathbf {Loc}_{\mathbb {Z}/p^n}(S)$ as $\mathcal {G}(\mathbb {Z}/p^n)$ -local systems.

It will also be convenient to consider the category $\mathbf {Loc}_{\mathbb {Q}_p}(S)$ of $\mathbb {Q}_p$ -local systems on S. This has the same objects as $\mathbf {Loc}_{\mathbb {Z}_p}(S)$ , denoted by $\mathbb {L}$ of when clarity is necessary, but with

(2.1.1)

where $\mathcal {H}om(\mathbb {L},\mathbb {L}')$ is the internal Hom in $\mathbf {Loc}_{\mathbb {Z}_p}(S)$ . The category $\mathbf {Loc}_{\mathbb {Q}_p}(S)$ admits a fully faithful embedding into $\mathbf {Vect}(S_{\mathrm {pro\acute {e}t}},\underline {\mathbb {Q}_p})$ , and so inherits an exact $\mathbb {Q}_p$ -linear $\otimes $ -structure.

Proof. Assume first that S is locally Noetherian. For the first statement, we deal only with the case $n=\infty $ . Let U in $S_{\mathrm {pro\acute {e}t}}$ be arbitrary. One may identify the natural map $\mathcal {G}_{\underline {\mathbb {Z}_p}}(U)\to \underline {\operatorname {\mathrm {Aut}}}(\omega _{\mathrm {triv}})(U)$ with the map from $\operatorname {\mathrm {Hom}}_{\text {cont.}}(\pi _0(U),\mathcal {G}(\mathbb {Z}_p))$ to the system $(g_\Lambda )$ of elements of $\operatorname {\mathrm {Hom}}_{\text {cont.}}(\pi _0(U),\operatorname {\mathrm {GL}}(\Lambda ))$ for $\Lambda $ in ${\mathbf {Rep}}_{\mathbb {Z}_p}(\mathcal {G})$ , which are compatible in $\Lambda $ . The projection $(g_\Lambda )\mapsto g_{\Lambda _0}$ from $\underline {\operatorname {\mathrm {Aut}}}(\omega _{\mathrm {triv}})(U)$ to $\operatorname {\mathrm {Hom}}_{\text {cont.}}(\pi _0(U),\operatorname {\mathrm {GL}}(\Lambda _0))$ is injective as every object $\Lambda $ of ${\mathbf {Rep}}_{\mathbb {Z}_p}(\mathcal {G})$ is a subquotient of $\Lambda _0^{\otimes }$ (see [Reference dos SantosdS09, Proposition 12]). As the diagram

commutes, and a and b are injective, to show that a is an isomorphism it suffices to show that $\mathrm {im}(b)\subseteq \mathrm {im}(c)$ . But, by functoriality $\mathrm {im}(b)$ fixes and thus lies in $\mathrm {im}(c)$ .

Let $\omega $ be an object of $\mathcal {G}\text {-}\mathbf {Loc}_{\mathbb {Z}_p}(S)$ . For $n\in \mathbb {N}\cup \{\infty \}$ consider the sheaf $\underline {\operatorname {\mathrm {Isom}}}(\omega _{\mathrm {triv}},\omega _n)$ , associating to a locally Noetherian S-scheme $f\colon T\to S$ the set of isomorphisms $\omega _{\mathrm {triv}}\to \omega _{n,T}$ , where is considered as an object of $\mathcal {G}\text {-}\mathbf {Loc}_{\mathbb {Z}/p^n}(T)$ . To prove the second claim it suffices to show that $\underline {\operatorname {\mathrm {Isom}}}(\omega _{\mathrm {triv}},\omega _\infty )$ is representable by a weakly étale cover of S. Moreover, as $\underline {\operatorname {\mathrm {Isom}}}(\omega _{\mathrm {triv}},\omega _\infty )$ is the limit of $\underline {\operatorname {\mathrm {Isom}}}(\omega _{\mathrm {triv}},\omega _n)$ (cf. [Reference Bhatt and ScholzeBS15, Proposition 6.8.4]) it further suffices to show that $\underline {\operatorname {\mathrm {Isom}}}(\omega _{\mathrm {triv}},\omega _n)$ is represented by a finite étale cover of S for every n in $\mathbb {N}$ . To see this, observe that the natural map

$$ \begin{align*} \underline{\operatorname{\mathrm{Isom}}}(\omega_{\mathrm{triv}},\omega_n)\to \underline{\operatorname{\mathrm{Isom}}}(\Lambda_0\otimes_{\mathbb{Z}_p} \underline{\mathbb{Z}/p^n},\omega_n(\Lambda_0)) \end{align*} $$

is a closed embedding, cut out by the intersection of the following conditions (with terminology from [Reference dos SantosdS09, Definition 10 and Proposition 12]) on an f in $\underline {\operatorname {\mathrm {Isom}}}(\Lambda _0\otimes _{\mathbb {Z}_p} \underline {\mathbb {Z}/p^n},\omega _n(\Lambda _0))$ : for every tuple $(a,b,W,U,q)$ , where $a,b$ are multi-indices, W is a special subrepresentation of $(\Lambda _0)^a_b$ , and $q\colon W\to U$ is a surjection of representations, f and $f^{-1}$ preserve W and $\ker q$ . Thus, $\underline {\operatorname {\mathrm {Isom}}}(\omega _{\mathrm {triv}},\omega _n)\to S$ is finite. Moreover, by the topological invariance of the pro-étale topos (see [Reference Bhatt and ScholzeBS15, Lemma 5.4.2]), pullback along $i\colon \operatorname {\mathrm {Spec}}(A/I)\to \operatorname {\mathrm {Spec}}(A)$ , for a square-zero ideal I of a Noetherian ring A, defines an equivalence

$$ \begin{align*} i^{-1}\colon \mathbf{Loc}_{\mathbb{Z}/p^n}(\operatorname{\mathrm{Spec}}(A))\to \mathbf{Loc}_{\mathbb{Z}/p^n}(\operatorname{\mathrm{Spec}}(A/I)) \end{align*} $$

for $n\in \mathbb {N}\cup \{\infty \}$ . From this we deduce that $\underline {\operatorname {\mathrm {Isom}}}(\omega _{\mathrm {triv}},\omega _n)\to S$ is formally étale in the sense of [SP, Tag 02HG], and thus finite étale by [SP, Tag 02HM].

To show that $\underline {\operatorname {\mathrm {Isom}}}(\omega _{\mathrm {triv}},\omega _n)\to S$ is surjective, it suffices to show that the pullback to each geometric point of S is a trivial torsor, and so we may assume that S is the spectrum of an algebraically closed field. In this case, there is a bi-exact $\mathbb {Z}_p$ -linear $\otimes $ -equivalence between $\mathbf {Loc}_{\mathbb {Z}/p^n}(S)$ and $\mathbf {Vect}(\mathbb {Z}/p^n)$ . But, by Theorem A.19, if $\nu _n$ belongs to $\mathcal {G}\text {-}\mathbf {Vect}(\mathbb {Z}/p^n)$ then $\underline {\operatorname {\mathrm {Isom}}}(\omega _{\mathrm {triv}},\nu _n)$ is a $\mathcal {G}$ -torsor. But, $H^1(\operatorname {\mathrm {Spec}}(\mathbb {Z}/p^n),\mathcal {G})$ is trivial (e.g. by [Reference MilneMil17, Corollary 17.98] and [Reference Bouthier and ČesnavičiusBČ22, Theorem 2.1.6]), and so the claim follows.

To remove the locally Noetherian hypothesis, we may proceed as follows. We may assume that n is in $\mathbb {N}$ and S is connected. Consider W in ${\mathbf {Rep}}_{\mathbb {Z}_p}(\mathcal {G})$ with a saturated injection $W \hookrightarrow A_{\mathcal {G}}^*=\operatorname {\mathrm {Hom}}_{\mathbb {Z}_p}(\mathcal {O}_{\mathcal {G}}(\mathcal {G}),\mathbb {Z}_p)$ whose image in $A_{\mathcal {G}}^*/p^n$ contains $\mathcal {G}(\mathbb {Z}/p^n)$ . We may replace S by a finite étale connected cover that trivializes $\omega _n(W)$ . We claim then that $\omega _n(V)$ is trivial for all V. By the locally Noetherian case, for each s in S, we have an isomorphism

$$ \begin{align*} \psi \colon \underline{\operatorname{\mathrm{Isom}}}(\omega_{\mathrm{triv}},\omega_n)_s \times^{\mathcal{G}(\mathbb{Z}/p^n)} W(\mathbb{Z}/p^n) \simeq \omega_n(W)_s. \end{align*} $$

Take a trivial $\mathcal {G}(\mathbb {Z}/p^n)$ -torsor $\mathcal {T}$ over S and an isomorphism $\mathcal {T} \times ^{\mathcal {G}(\mathbb {Z}/p^n)} W(\mathbb {Z}/p^n) \simeq \omega _n(W)$ extending $\psi $ . For an object V of ${\mathbf {Rep}}_{\mathbb {Z}_p}(\mathcal {G})$ and v in $V(\mathbb {Z}_p)$ , let $f_v \colon A_{\mathcal {G}}^* \to V$ be the morphism induced by the map $\mathcal {G} \to V$ sending g to $gv$ . Let $\mathrm {ev}_1$ in $A_{\mathcal {G}}^*/p^n$ be the unit section. Then

$$ \begin{align*} \mathcal{T} \xrightarrow{\operatorname{\mathrm{id}} \times \mathrm{ev}_1} \mathcal{T} \times^{\mathcal{G}(\mathbb{Z}/p^n)} W(\mathbb{Z}/p^n) \simeq \omega_n(W) \xrightarrow{\omega_n(f_v|_W)} \omega_n(V) \end{align*} $$

gives a map $\mathcal {T} \times ^{\mathcal {G}(\mathbb {Z}/p^n)} V(\mathbb {Z}/p^n) \to \omega _n(V)$ . One checks this is an isomorphism over s, and so is an isomorphism as the source and target are finite étale over S.

Given Lemma 2.1, we write $\mathbf {Tors}_{\mathcal {G}(\mathbb {Z}/p^n)}(S_{\mathrm {pro\acute {e}t}})$ instead of $\mathbf {Tors}_{\mathcal {G}_{\underline {\mathbb {Z}/p^n}}}(S_{\mathrm {pro\acute {e}t}})$ , and call objects of this category $\mathcal {G}(\mathbb {Z}/p^n)$ -torsors. By Proposition 2.3 and Proposition A.18 we know that there is a natural equivalence of categories between $\mathcal {G}\text {-}\mathbf {Loc}_{\mathbb {Z}/p^n}(S)$ and $\mathbf {Tors}_{\mathcal {G}(\mathbb {Z}/p^n)}(S_{\mathrm {pro\acute {e}t}})$ . Explicitly, this functor associates to a $\mathcal {G}(\mathbb {Z}/p^n)$ -torsor $\mathcal {A}$ the $\mathcal {G}(\mathbb {Z}/p^n)$ -local system $\omega _{\mathcal {A}}$ given by .

By a $G(\mathbb {Q}_p)$ -local system on S, we mean an exact $\mathbb {Q}_p$ -linear $\otimes $ -functor ${\mathbf {Rep}}_{\mathbb {Q}_p}(G)\to \mathbf {Loc}_{\mathbb {Q}_p}(S)$ , the category of which we denote by $G\text {-}\mathbf {Loc}_{\mathbb {Q}_p}(S)$ . The following trivial observation allows us to define a natural functor . Let us denote by the category which has the same objects as ${\mathbf {Rep}}_{\mathbb {Z}_p}(\mathcal {G})$ , denoted by either $\Lambda $ or for clarity, but where we set to be .

Lemma 2.4. The functor

is an equivalence of categories.

Finally, it is useful to have a more concrete description of torsors for a pro-finite group H on $S_{\mathrm {pro\acute {e}t}}$ . We set the following notation.

Consider a projective system of schemes $\{X_n\}$ , each equipped with the structure of a principal homogeneous space over S for $H_n$ (see [SP, Tag 049A]). The quotient sheaf $X_n/K_n$ , where , is representable by [SGA3-1, Exposé V, §7, Théorème 7.1], and $X_n/K_n\to S$ is naturally a principal homogeneous space for $H_{n-1}$ . We say $(X_n)$ is an H-covering if each $X_n\to X_{n-1}$ is $K_n$ -equivariant (when the target is given the trivial action) and the induced map $X_n/K_n\to X_{n-1}$ is an $H_{n-1}$ -equivariant isomorphism. A morphism of H-coverings $(X_n)\to (Y_n)$ is a compatible family of $H_n$ -equivariant morphisms $X_n\to Y_n$ .

An S-scheme X equipped with an action of H is a principal homogeneous space for H if $h_X$ with the induced action of $\underline {H}$ is an $\underline {H}$ -torsor. Morphisms of principal homogeneous spaces are H-equivariant morphisms of S-schemes. If $(X_n)$ is an H-covering then is a principal homogeneous space for H. Conversely, if X is a principal homogeneous space for H, the system $(X_n)$ with (a scheme by Proposition A.10) is an H-covering with $X_\infty = X$ .

Applying [Reference Bhatt and ScholzeBS15, Lemma 7.3.9] to $\mathcal {T}\times ^{\underline {H}}\underline {H}_n$ for every n, every object $\mathcal {T}$ of $\mathbf {Tors}_{\underline {H}}(S_{\mathrm {pro\acute {e}t}})$ is representable. We summarize the above as follows.

Proposition 2.6. For locally topologically Noetherian S, the following are equivalences:

$$ \begin{align*} \left\{\begin{matrix}H\mathrm{-coverings}\\ \mathrm{of}\ S\end{matrix}\right\}\xrightarrow{(X_n)\mapsto X_\infty} \left\{\begin{matrix}\mathrm{Principal\ homogenous}\\ \mathrm{spaces\ for}\ H\ \mathrm{on}\ S\end{matrix}\right\} \xrightarrow{X\mapsto h_X}\mathbf{Tors}_{\underline{H}}(S_{\mathrm{pro\acute{e}t}}). \end{align*} $$

2.1.2 $\mathcal {G}(\mathbb {Z}_p)$ -local systems on an adic space

Let X be a locally Noetherian adic space over $\mathbb {Q}_p$ (cf. [Reference ScholzeSch13, p. 17]), and let $X_{\mathrm {pro\acute {e}t}}$ be the pro-étale site as in [Reference Bhatt, Morrow and ScholzeBMS18, §5.1]. As in [Reference ScholzeSch13], we call an object of $X_{\mathrm {pro\acute {e}t}}$ affinoid perfectoid if it can be represented as $(\operatorname {\mathrm {Spa}}(R_i,R_i^+))$ with finite étale surjective transition maps, and the Huber pair $(R,R^+)=((\varinjlim R_i)^\wedge ,(\varinjlim R_i^+)^\wedge )$ (endowed with the unique topology such that each $R_i\to R$ is adic) is perfectoid. In this case, $\operatorname {\mathrm {Spa}}(R,R^+)\sim \varprojlim \operatorname {\mathrm {Spa}}(R_i,R_i^+)$ (see [Reference Scholze and WeinsteinSW13, §2.4]). The affinoid perfectoid objects of $X_{\mathrm {pro\acute {e}t}}$ form a basis of $X_{\mathrm {pro\acute {e}t}}$ (cf. [Reference ScholzeSch13, Proposition 4.8]). The site $X_{\mathrm {pro\acute {e}t}}$ is not subcanonical (see [Reference Achinger, Lara and YoucisALY21, Example 4.1.7]), but $h_Y$ and $h_Y^\#$ have the same value on affinoid perfectoid objects (see [Reference Achinger, Lara and YoucisALY21, Proposition 4.1.8]). So, we abusively conflate the two.

We will again be interested in the ringed site $(X_{\mathrm {pro\acute {e}t}},\underline {\mathbb {Z}_p}_X)$ , where

$$ \begin{align*} \underline{\mathbb{Z}_p}_X=R\varprojlim \underline{\mathbb{Z}/p^n}_X=\varprojlim \underline{\mathbb{Z}/p^n}_X\in \mathbf{Shv}(X_{\mathrm{pro\acute{e}t}}), \end{align*} $$

where $\underline {\mathbb {Z}/p^n}_X$ is the constant sheaf on $X_{\mathrm {pro\acute {e}t}}$ associated to $\mathbb {Z}/p^n$ , and the second equality follows now from [Reference ScholzeSch13, Proposition 8.2]. Again, more generally, for a topological space T, we denote by $\underline {T}_X$ (or just $\underline {T}$ when X is clear from context) the sheafFootnote ⁷