# Generalized Motives through Witt Vectors

Xin Tong

###### Abstract

We generalize condensed motives, through Witt vectors and the associated
moduli stacks of generalized line bundles.

###### Contents

  1. 1 $p$-adic Motives
    1. 1.1 Generalized Prismatization
    2. 1.2 Perfectoid Picture
  2. 2 $z$-adic Motives
    1. 2.1 Generalized Prismatization
    2. 2.2 Perfectoid Picture
  3. 3 Applications
    1. 3.1 Rigid Analytic Spaces
    2. 3.2 Generalized Langlands Program

######  Reference Summarization 1.

We follow [G] on the corresponding motives. We start from the foundation in
[BSI], [BLI], [DI], [SchI], [ALBRCS], [TI], [TII], [TIII], [TVI]. For
condensed mathematics consideration see [CS1], [CS2], [CS3]. This paper is
also closely following and inspired by [TIV] and [TV]. We apply the
generalization then to the level of consideration closely after [Ta], [FI],
[KLI], [KLII], [SchII], [SchIII], [SchIV], as well as closely after [L], [FS],
[DHKM], [VL], [LL], [EGH], [DIII], [DII], [Z], [GL]. Witt vectors are very
significant in this paper since all the corresponding prismatizations are
defined to be the Cartier moduli stacks parametrizing those divisors living in
the Witt vector spaces, in either perfect setting (see [SchIII], [SchII],
[KLI], [KLII] as well for more inspiration and development) and the imperfect
setting. Prismatization will mean taking the spectrum (up to the Cartier
primes) of the Witt vectors, while perfect prismatization relates to stacks of
untilts such as in the work of Scholze and Kedlaya-Liu in [SchIII], [SchII],
[KLI], [KLII]. By Witt vectors we will mean the most generalized version such
as in [KLI], [KLII], [FS], which allow us to consider the $z$-adic setting,
for instance the Fontaine-Wintenberger parametrization of the Cartier divisors
is considered extensively in the $z$-adic setting in [KLI] and [KLII].

This paper generalizes motives. Motives are universalization of cohomologies
after [G], therefore generalization of universal cohomology theory is a
completely mysterious and sophisticated consideration. Extending the original
idea on the motives can be very tricky. What we considered here is the so
called mixed-parity filtred generalization of the usual motives after [BSB].
Even in the situation where we do not consider Galois representations, this
can be for instance realized by using generalized Langlands program.
Representation of Galois groups can be regarded as certain sheaves over more
generalized Hodge theoretic spaces and even fiber categories. For any $p$-adic
formal scheme we generalize the corresponding prismatization from [BSI], [BLI]
and [DI]. We also have the $z$-adic version as well the corrsponding $v$-stack
version, where the idea is to consider certain generalized paramatrization
stacks of Cartier sheaves, extensively by using the Witt vectors attached to
the field $T$. We then apply this construction to rigid analytic space $R$ by
using the $v$-stack consideration, which is completely compatible with [TI],
[TII], [TIII], [SchI], [ALBRCS]. This immediately allows us to generalize the
$z$-adic cohomology theory after [KLI], [KLII], [SchII], [SchIII], [SchIV].
Finally we consider the generalized Langlands program after [TIV], [TV],
[TVI], [L], [FS], [VL], [DII], [DIII]. See the final chapter.

##  1 $p$-adic Motives

###  1.1 Generalized Prismatization

We start from the foundation in [BSI], [BLI], [DI], [SchI], [ALBRCS], [TI],
[TII], [TIII], [TVI]. We start from the corresponding formal scheme situation.
We will start from a finite extension $T$ of $\mathbb{Q}_{p}$. We consider a
general $F$ which is a $p$-adic formal scheme over $\mathcal{O}_{T}$. Attached
to $F$ we have the corresponding primatic stack:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},F}$ (1)

which is defined actually over rings where $p$ is nilpotent. We use the
notation:

$\displaystyle\mathrm{C}_{p\mathrm{nil}}$ (2)

to be this category. Therefore the stack

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},F}$ (3)

is then defined over this category as the family of Cartier divisors (closed
immersingly) living in the corresponding generalized Witt vector of any ring
$H$ in the under category:

$\displaystyle?\rightarrow\mathrm{WVector}_{\mathcal{O}_{T}}(H).$ (4)

This stack is actually formal stack. One then have the corresponding
quasicoherent sheaves over this stack as the corresponding $\infty$-category
of all the quasicoherent modules over the structure sheaf:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},F}}$
(5)

which can be presented as the inverse limit of certain $\infty$-category of
all the quasicoherent modules over a family of prisms in coherent way:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},F}}=\projlim_{i}\mathrm{Quasicoherent}_{\mathrm{Spec}U_{i}}$
(6)

where $U_{i}$ is the desired family of such sort of prisms111In fact one can
care more about the quasicoherent sheaves instead of the actual stacks. This
point of view is quite well-known in the field of algebraic geometry where
even in certain situation the underlying stack can be reconstructed from the
corresponding category of quasicoherent sheaves.. This presentation can be
directly use to construct the corresponding de Rham stacks for instance:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},F}:=\injlim_{i}\mathrm{Spec}U_{i}[1/p]_{I_{U_{i}}}$
(7)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},F}}=\projlim_{i}\mathrm{Quasicoherent}_{\mathrm{Spec}U_{i}[1/p]_{I_{U_{i}}}}.$
(8)

One then have the analytification version of this stack:

$\displaystyle\mathrm{\Gamma}^{\sharp}_{\mathrm{prismatization},\mathrm{deRham},F}:=\injlim_{i}\mathrm{Spec}^{\sharp}U_{i}[1/p]_{I_{U_{i}}}$
(9)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},F}}=\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/p]_{I_{U_{i}}}}.$
(10)

We then have the corresponding cristalline version as well. This presentation
can be directly use to construct the corresponding cristalline stacks for
instance:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},F}:=\injlim_{i}\mathrm{Spec}U_{i}[1/p]_{I_{U_{i}}}$
(11)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},F}}=\projlim_{i}\mathrm{Quasicoherent}_{\mathrm{Spec}U_{i}[1/p]_{I_{U_{i}}}}.$
(12)

One then have the analytification version of this stack:

$\displaystyle\mathrm{\Gamma}^{\sharp}_{\mathrm{prismatization},\mathrm{cristalline},F}:=\injlim_{i}\mathrm{Spec}^{\sharp}U_{i}[1/p]_{I_{U_{i}}}$
(13)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},F}}=\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/p]_{I_{U_{i}}}}.$
(14)

######  Definition 1.

We now generalize the corresponding foundation above to [BSB] in the following
sense. First starting with the stack:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},F}$ (15)

we have the corresponding $p$-adic primitive element $h$ from [BSB] which is
usual the corresponding logrithmic with respect to the imperfect Robba ring
(in variable $X$, then it will be the log of $1+X$) attached to the field $T$.
We then add $h^{1/2}$ to the following map in the definition of the
primatization:

$\displaystyle?\rightarrow\mathrm{WVector}_{\mathcal{O}_{T}}(H)[h^{1/2}].$
(16)

Then we have the corresponding stack which we will denote it by:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},F,2}.$ (17)

Then the de Rham and cristalline situations are generalized accordingly. This
stack is actually formal stack. One then has the corresponding quasicoherent
sheaves over this stack as the corresponding $\infty$-category of all the
quasicoherent modules over the structure sheaf:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},F,2}}$
(18)

which can be presented as the inverse limit of certain $\infty$-category of
all the quasicoherent modules over a family of prisms in coherent way:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},F,2}}=\projlim_{i}\mathrm{Quasicoherent}_{\mathrm{Spec}U_{i}[h^{1/2}]}$
(19)

where $U_{i}$ is the desired family of such sort of prisms. This presentation
can be directly use to construct the corresponding de Rham stacks for
instance:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},F,2}:=\injlim_{i}\mathrm{Spec}U_{i}[1/p]_{I_{U_{i}}}[h^{1/2}]$
(20)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},F,2}}=\projlim_{i}\mathrm{Quasicoherent}_{\mathrm{Spec}U_{i}[1/p]_{I_{U_{i}}}[h^{1/2}]}.$
(21)

One then have the analytification version of this stack:

$\displaystyle\mathrm{\Gamma}^{\sharp}_{\mathrm{prismatization},\mathrm{deRham},F,2}:=\injlim_{i}\mathrm{Spec}^{\sharp}U_{i}[1/p]_{I_{U_{i}}}[h^{1/2}]$
(22)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},F,2}}=\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/p]_{I_{U_{i}}}[h^{1/2}]}.$
(23)

We then have the corresponding cristalline version as well. This presentation
can be directly use to construct the corresponding cristalline stacks for
instance:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},F,2}:=\injlim_{i}\mathrm{Spec}U_{i}[1/p][h^{1/2}]$
(24)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},F,2}}=\projlim_{i}\mathrm{Quasicoherent}_{\mathrm{Spec}U_{i}[1/p][h^{1/2}]}.$
(25)

One then have the analytification version of this stack:

$\displaystyle\mathrm{\Gamma}^{\sharp}_{\mathrm{prismatization},\mathrm{cristalline},F,2}:=\injlim_{i}\mathrm{Spec}^{\sharp}U_{i}[1/p][h^{1/2}]$
(26)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},F,2}}=\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/p][h^{1/2}]}.$
(27)

######  Theorem 1.

When $F$ is $\mathcal{O}_{T}$ we have a generalized condensed prismatization,
which is well-defined.

######  Remark 1.

We remark that in order for us to relate this generalization to Galois
representations, one has to use Fargues-Fontaine curves. The reason behind
this is on homotopicalization. Recall that Fargues-Fontaine curves are
encoding the Galois actions. Therefore over $\mathcal{C}_{T}$ we have the
corresponding fundamental groups of FF curves as the corresponding Galois
groups of $T$. Therefore in order to have action from two fold covering groups
of the Galois groups of $T$ and even in certain relative fashion, one has to
use the generalized FF curves by extending the action of the action from
Galois groups of $T$ to the extension of these Galois groups, then we have a
functor from the corresponding generalized prismatization to the corresponding
$G_{T,2}$-equivariant sheaves over generalized FF curves. This parallels to
$z$-adic situation as well.

###  1.2 Perfectoid Picture

We then consider the contact with the work of Kedlaya-Liu and Scholze, where
untilts of perfectoids are parametrized by certain similar stacks, i.e. the
Fargues-Fontaine stacks, from [KLI], [KLII], [SchII], [SchIII], [SchIV]. We
now consider any small $v$-stack $S$ over $\mathrm{Spd}\mathcal{O}_{T}$. For
any local chart ${F}$ in the $v$-topology over this stack $S$, we can get the
corresponding prismatization directly in a transparent way, namely the
corresponding prismatization is just the corresponding space of the Witt
vector attached to $F$. However this is a prismatic method. Attached to $F$ we
have the corresponding primatic stack:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},F}$ (28)

which is defined actually over rings where $p$ is nilpotent. We use the
notation:

$\displaystyle\mathrm{C}_{p\mathrm{nil}}$ (29)

to be this category. Therefore the stack

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},F}$ (30)

is then defined over this category as the family of Cartier divisors (closed
immersingly) living in the corresponding generalized Witt vector of any ring
$H$ in the under category:

$\displaystyle?\rightarrow\mathrm{WVector}_{\mathcal{O}_{T}}(H).$ (31)

This stack is actually formal stack. One then have the corresponding
quasicoherent sheaves over this stack as the corresponding $\infty$-category
of all the quasicoherent modules over the structure sheaf:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},F}}$
(32)

which can be presented as the inverse limit of certain $\infty$-category of
all the quasicoherent modules over a family of prisms in coherent way:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},F}}=\projlim_{i}\mathrm{Quasicoherent}_{\mathrm{Spec}U_{i}}$
(33)

where $U_{i}$ is the desired family of such sort of prisms. This presentation
can be directly use to construct the corresponding de Rham stacks for
instance:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},F}:=\injlim_{i}\mathrm{Spec}U_{i}[1/p]_{I_{U_{i}}}$
(34)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},F}}=\projlim_{i}\mathrm{Quasicoherent}_{\mathrm{Spec}U_{i}[1/p]_{I_{U_{i}}}}.$
(35)

One then have the analytification version of this stack:

$\displaystyle\mathrm{\Gamma}^{\sharp}_{\mathrm{prismatization},\mathrm{deRham},F}:=\injlim_{i}\mathrm{Spec}^{\sharp}U_{i}[1/p]_{I_{U_{i}}}$
(36)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},F}}=\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/p]_{I_{U_{i}}}}.$
(37)

We then have the corresponding cristalline version as well. This presentation
can be directly use to construct the corresponding cristalline stacks for
instance:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},F}:=\injlim_{i}\mathrm{Spec}U_{i}[1/p]_{I_{U_{i}}}$
(38)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},F}}=\projlim_{i}\mathrm{Quasicoherent}_{\mathrm{Spec}U_{i}[1/p]_{I_{U_{i}}}}.$
(39)

One then have the analytification version of this stack:

$\displaystyle\mathrm{\Gamma}^{\sharp}_{\mathrm{prismatization},\mathrm{cristalline},F}:=\injlim_{i}\mathrm{Spec}^{\sharp}U_{i}[1/p]_{I_{U_{i}}}$
(40)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},F}}=\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/p]_{I_{U_{i}}}}.$
(41)

######  Definition 2.

We now generlize the corresponding foundation above to [BSB] in the following
sense. First starting with the stack:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},F}$ (42)

we have the corresponding $p$-adic primitive element $h$ from [BSB] which is
usual the corresponding logrithmic with respect to the imperfect Robba ring
(in variable $X$, then it will be the log of $1+X$) attached to the field $T$.
We then add $h^{1/2}$ to the following map in the definition of the
primatization:

$\displaystyle?\rightarrow\mathrm{WVector}_{\mathcal{O}_{T}}(H)[h^{1/2}].$
(43)

Then we have the corresponding stack which we will denote it by:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},F,2}.$ (44)

Then the de Rham and cristalline situations are generalized accordingly. This
stack is actually formal stack. One then have the corresponding quasicoherent
sheaves over this stack as the corresponding $\infty$-category of all the
quasicoherent modules over the structure sheaf:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},F,2}}$
(45)

which can be presented as the inverse limit of certain $\infty$-category of
all the quasicoherent modules over a family of prisms in coherent way:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},F,2}}=\projlim_{i}\mathrm{Quasicoherent}_{\mathrm{Spec}U_{i}[h^{1/2}]}$
(46)

where $U_{i}$ is the desired family of such sort of prisms. This presentation
can be directly use to construct the corresponding de Rham stacks for
instance:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},F,2}:=\injlim_{i}\mathrm{Spec}U_{i}[1/p]_{I_{U_{i}}}[h^{1/2}]$
(47)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},F,2}}=\projlim_{i}\mathrm{Quasicoherent}_{\mathrm{Spec}U_{i}[1/p]_{I_{U_{i}}}[h^{1/2}]}.$
(48)

One then have the analytification version of this stack:

$\displaystyle\mathrm{\Gamma}^{\sharp}_{\mathrm{prismatization},\mathrm{deRham},F,2}:=\injlim_{i}\mathrm{Spec}^{\sharp}U_{i}[1/p]_{I_{U_{i}}}[h^{1/2}]$
(49)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},F,2}}=\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/p]_{I_{U_{i}}}[h^{1/2}]}.$
(50)

We then have the corresponding cristalline version as well. This presentation
can be directly use to construct the corresponding cristalline stacks for
instance:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},F,2}:=\injlim_{i}\mathrm{Spec}U_{i}[1/p][h^{1/2}]$
(51)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},F,2}}=\projlim_{i}\mathrm{Quasicoherent}_{\mathrm{Spec}U_{i}[1/p][h^{1/2}]}.$
(52)

One then have the analytification version of this stack:

$\displaystyle\mathrm{\Gamma}^{\sharp}_{\mathrm{prismatization},\mathrm{cristalline},F,2}:=\injlim_{i}\mathrm{Spec}^{\sharp}U_{i}[1/p][h^{1/2}]$
(53)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},F,2}}=\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/p][h^{1/2}]}.$
(54)

Then we vary $F$ in the $v$-site for $S$ we then have the corresponding the
condensed prismatization of the $v$-stack $S$ in the mixed-parity
generalization fashion.

##  2 $z$-adic Motives

###  2.1 Generalized Prismatization

Since there is $z$-adic version of Fargues-Fontaine stacks, there should be at
least222Over real number $\mathbb{R}$ or complex number $\mathbb{C}$, one can
also consider the prismatization by using the moduli stacks of line bundles in
the Witt vectors, at least by deformation to:
$\displaystyle\mathbb{R}((u)),\mathbb{C}((u)),$ (55) namely the
nonarchimedeanizations. two versions of prismatizations over
$T/\mathbb{Q}_{p}$ or $T/\mathbb{F}_{p}((z))$. We start from the foundation in
[BSI], [BLI], [DI], [SchI], [ALBRCS], [TI], [TII], [TIII], [TVI]. We start
from the corresponding formal scheme situation. We will start from a finite
extension $T$ of $\mathbb{F}_{p}((z))$. We consider a general $F$ which is a
$z$-adic formal scheme over $\mathcal{O}_{T}$. Attached to $F$ we have the
corresponding primatic stack:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},F}$ (56)

which is defined actually over rings where $z$ is nilpotent333Yes as in [KLI],
[KLII], [FS], [SchII] the underlying category of rings can be chosen to be the
same..

######  Remark 2.

Here the definition goes completely the parallel as in [BSI], [BLI], [DI],
[SchI], [ALBRCS], [TI], [TII], [TIII], [TVI] where we just use the
corresponding $\mathrm{WVector}_{\mathcal{O}_{T}}$ to define the family of
Cartier sheaves.

We use the notation:

$\displaystyle\mathrm{C}_{z\mathrm{nil}}$ (57)

to be this category. Therefore the stack

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},F}$ (58)

is then defined over this category as the family of Cartier divisors (closed
immersingly) living in the corresponding generalized Witt vector of any ring
$H$ in the under category:

$\displaystyle?\rightarrow\mathrm{WVector}_{\mathcal{O}_{T}}(H).$ (59)

This stack is actually formal stack. One then have the corresponding
quasicoherent sheaves over this stack as the corresponding $\infty$-category
of all the quasicoherent modules over the structure sheaf:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},F}}$
(60)

which can be presented as the inverse limit of certain $\infty$-category of
all the quasicoherent modules over a family of prisms in coherent way:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},F}}=\projlim_{i}\mathrm{Quasicoherent}_{\mathrm{Spec}U_{i}}$
(61)

where $U_{i}$ is the desired family of such sort of prisms444Here we require
the prisms to take the form of a fixed map
$?\rightarrow\mathrm{WVector}_{\mathcal{O}_{T}}(H)$. And as in [BSI] the
completeness is also required.. This presentation can be directly use to
construct the corresponding de Rham stacks for instance:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},F}:=\injlim_{i}\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}$
(62)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},F}}=\projlim_{i}\mathrm{Quasicoherent}_{\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}}.$
(63)

One then have the analytification version of this stack:

$\displaystyle\mathrm{\Gamma}^{\sharp}_{\mathrm{prismatization},\mathrm{deRham},F}:=\injlim_{i}\mathrm{Spec}^{\sharp}U_{i}[1/\eta]_{I_{U_{i}}}$
(64)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},F}}=\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}}.$
(65)

We then have the corresponding cristalline version as well. This presentation
can be directly use to construct the corresponding cristalline stacks for
instance:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},F}:=\injlim_{i}\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}$
(66)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},F}}=\projlim_{i}\mathrm{Quasicoherent}_{\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}}.$
(67)

One then have the analytification version of this stack:

$\displaystyle\mathrm{\Gamma}^{\sharp}_{\mathrm{prismatization},\mathrm{cristalline},F}:=\injlim_{i}\mathrm{Spec}^{\sharp}U_{i}[1/\eta]_{I_{U_{i}}}$
(68)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},F}}=\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}}.$
(69)

######  Definition 3.

We now generlize the corresponding foundation above to [BSB] in the following
sense. First starting with the stack:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},F}$ (70)

we have the corresponding $p$-adic primitive element $h$ from [BSB] which is
usual the corresponding logrithmic with respect to the imperfect Robba ring
(in variable $X$, then it will be the log of $1+X$) attached to the field $T$.
We then add $h^{1/2}$ to the following map in the definition of the
primatization:

$\displaystyle?\rightarrow\mathrm{WVector}_{\mathcal{O}_{T}}(H)[h^{1/2}].$
(71)

Then we have the corresponding stack which we will denote it by:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},F,2}.$ (72)

Then the de Rham and cristalline situations are generalized accordingly. This
stack is actually formal stack. One then have the corresponding quasicoherent
sheaves over this stack as the corresponding $\infty$-category of all the
quasicoherent modules over the structure sheaf:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},F,2}}$
(73)

which can be presented as the inverse limit of certain $\infty$-category of
all the quasicoherent modules over a family of prisms in coherent way:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},F,2}}=\projlim_{i}\mathrm{Quasicoherent}_{\mathrm{Spec}U_{i}[h^{1/2}]}$
(74)

where $U_{i}$ is the desired family of such sort of prisms. This presentation
can be directly use to construct the corresponding de Rham stacks for
instance:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},F,2}:=\injlim_{i}\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}[h^{1/2}]$
(75)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},F,2}}=\projlim_{i}\mathrm{Quasicoherent}_{\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}[h^{1/2}]}.$
(76)

One then have the analytification version of this stack:

$\displaystyle\mathrm{\Gamma}^{\sharp}_{\mathrm{prismatization},\mathrm{deRham},F,2}:=\injlim_{i}\mathrm{Spec}^{\sharp}U_{i}[1/p]_{I_{U_{i}}}[h^{1/2}]$
(77)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},F,2}}=\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}[h^{1/2}]}.$
(78)

We then have the corresponding cristalline version as well. This presentation
can be directly use to construct the corresponding cristalline stacks for
instance:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},F,2}:=\injlim_{i}\mathrm{Spec}U_{i}[1/\eta][h^{1/2}]$
(79)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},F,2}}=\projlim_{i}\mathrm{Quasicoherent}_{\mathrm{Spec}U_{i}[1/\eta][h^{1/2}]}.$
(80)

One then have the analytification version of this stack:

$\displaystyle\mathrm{\Gamma}^{\sharp}_{\mathrm{prismatization},\mathrm{cristalline},F,2}:=\injlim_{i}\mathrm{Spec}^{\sharp}U_{i}[1/\eta][h^{1/2}]$
(81)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},F,2}}=\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/\eta][h^{1/2}]}.$
(82)

######  Theorem 2.

When $F$ is $\mathcal{O}_{T}$ we have a generalized condensed prismatization,
which is well-defined.

###  2.2 Perfectoid Picture

We then consider the contact with the work of Kedlaya-Liu and Scholze, where
untilts of perfectoids are parametrized by certain similar stacks, i.e. the
Fargues-Fontaine stacks, from [KLI], [KLII], [SchII], [SchIII], [SchIV]. We
now consider any small $v$-stack $S$ over $\mathrm{Spd}\mathcal{O}_{T}$. For
any local chart ${F}$ in the $v$-topology over this stack $S$, we can get the
corresponding prismatization directly in a transparent way, namely the
corresponding prismatization is just the corresponding space of the Witt
vector attached to $F$. However this is a prismatic method. Attached to $F$ we
have the corresponding primatic stack:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},F}$ (83)

which is defined actually over rings where $z$ is nilpotent.

######  Remark 3.

Here the definition goes completely the parallel as in [BSI], [BLI], [DI],
[SchI], [ALBRCS], [TI], [TII], [TIII], [TVI] where we just use the
corresponding $\mathrm{WVector}_{\mathcal{O}_{T}}$ to define the family of
Cartier sheaves.

We use the notation:

$\displaystyle\mathrm{C}_{z\mathrm{nil}}$ (84)

to be this category. Therefore the stack

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},F}$ (85)

is then defined over this category as the family of Cartier divisors (closed
immersingly) living in the corresponding generalized Witt vector of any ring
$H$ in the under category:

$\displaystyle?\rightarrow\mathrm{WVector}_{\mathcal{O}_{T}}(H).$ (86)

This stack is actually formal stack. One then have the corresponding
quasicoherent sheaves over this stack as the corresponding $\infty$-category
of all the quasicoherent modules over the structure sheaf:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},F}}$
(87)

which can be presented as the inverse limit of certain $\infty$-category of
all the quasicoherent modules over a family of prisms in coherent way:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},F}}=\projlim_{i}\mathrm{Quasicoherent}_{\mathrm{Spec}U_{i}}$
(88)

where $U_{i}$ is the desired family of such sort of prisms.555Here we require
the prisms to take the form of a fixed map
$?\rightarrow\mathrm{WVector}_{\mathcal{O}_{T}}(H)$. And as in [BSI] the
completeness is also required. Completeness is defined to be the same as in
[BSI] where we just replace $p$ by $z$. This presentation can be directly use
to construct the corresponding de Rham stacks for instance:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},F}:=\injlim_{i}\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}$
(89)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},F}}=\projlim_{i}\mathrm{Quasicoherent}_{\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}}.$
(90)

One then have the analytification version of this stack:

$\displaystyle\mathrm{\Gamma}^{\sharp}_{\mathrm{prismatization},\mathrm{deRham},F}:=\injlim_{i}\mathrm{Spec}^{\sharp}U_{i}[1/\eta]_{I_{U_{i}}}$
(91)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},F}}=\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}}.$
(92)

We then have the corresponding cristalline version as well. This presentation
can be directly use to construct the corresponding cristalline stacks for
instance:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},F}:=\injlim_{i}\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}$
(93)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},F}}=\projlim_{i}\mathrm{Quasicoherent}_{\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}}.$
(94)

One then have the analytification version of this stack:

$\displaystyle\mathrm{\Gamma}^{\sharp}_{\mathrm{prismatization},\mathrm{cristalline},F}:=\injlim_{i}\mathrm{Spec}^{\sharp}U_{i}[1/\eta]_{I_{U_{i}}}$
(95)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},F}}=\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}}.$
(96)

######  Definition 4.

We now generlize the corresponding foundation above to [BSB] in the following
sense. First starting with the stack:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},F}$ (97)

we have the corresponding $p$-adic primitive element $h$ from [BSB] which is
usual the corresponding logrithmic with respect to the imperfect Robba ring
(in variable $X$, then it will be the log of $1+X$) attached to the field $T$.
We then add $h^{1/2}$ to the following map in the definition of the
primatization:

$\displaystyle?\rightarrow\mathrm{WVector}_{\mathcal{O}_{T}}(H)[h^{1/2}].$
(98)

Then we have the corresponding stack which we will denote it by:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},F,2}.$ (99)

Then the de Rham and cristalline situations are generalized accordingly. This
stack is actually formal stack. One then have the corresponding quasicoherent
sheaves over this stack as the corresponding $\infty$-category of all the
quasicoherent modules over the structure sheaf:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},F,2}}$
(100)

which can be presented as the inverse limit of certain $\infty$-category of
all the quasicoherent modules over a family of prisms in coherent way:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},F,2}}=\projlim_{i}\mathrm{Quasicoherent}_{\mathrm{Spec}U_{i}[h^{1/2}]}$
(101)

where $U_{i}$ is the desired family of such sort of prisms. 666Here we require
the prisms to take the form of a fixed map
$?\rightarrow\mathrm{WVector}_{\mathcal{O}_{T}}(H)$. And as in [BSI] the
completeness is also required. Completeness is defined to be the same as in
[BSI] where we just replace $p$ by $z$. This presentation can be directly use
to construct the corresponding de Rham stacks for instance:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},F,2}:=\injlim_{i}\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}[h^{1/2}]$
(102)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},F,2}}=\projlim_{i}\mathrm{Quasicoherent}_{\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}[h^{1/2}]}.$
(103)

One then have the analytification version of this stack:

$\displaystyle\mathrm{\Gamma}^{\sharp}_{\mathrm{prismatization},\mathrm{deRham},F,2}:=\injlim_{i}\mathrm{Spec}^{\sharp}U_{i}[1/\eta]_{I_{U_{i}}}[h^{1/2}]$
(104)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},F,2}}=\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}[h^{1/2}]}.$
(105)

We then have the corresponding cristalline version as well. This presentation
can be directly use to construct the corresponding cristalline stacks for
instance:

$\displaystyle\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},F,2}:=\injlim_{i}\mathrm{Spec}U_{i}[1/\eta][h^{1/2}]$
(106)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},F,2}}=\projlim_{i}\mathrm{Quasicoherent}_{\mathrm{Spec}U_{i}[1/\eta][h^{1/2}]}.$
(107)

One then have the analytification version of this stack:

$\displaystyle\mathrm{\Gamma}^{\sharp}_{\mathrm{prismatization},\mathrm{cristalline},F,2}:=\injlim_{i}\mathrm{Spec}^{\sharp}U_{i}[1/\eta][h^{1/2}]$
(108)

with the corresponding quasicoherent sheaves category:

$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},F,2}}=\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/\eta][h^{1/2}]}.$
(109)

Then we vary $F$ in the $v$-site for $S$ we then have the corresponding the
condensed prismatization of the $v$-stack $S$ in the mixed-parity
generalization fashion.

##  3 Applications

###  3.1 Rigid Analytic Spaces

######  Definition 5.

We now construct the condensed motives for rigid analytic spave $R$ over $T$
through the perfectoid consideration above from $v$-stacks. One can certainly
go along [SchI] and [ALBRCS] to glue along the formal model locally to reach
the whole motives for the spaces. In the perfectoid setting we regard $R$ in
genral as a corresponding $v$-stacks over $\mathrm{Spd}T$. Then we have the
following $\infty$-categories in the generalization we considered here:

$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},R,2}}=\lim_{F}\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[h^{1/2}]},$
(110)
$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},R,2}}=\lim_{F}\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}[h^{1/2}]},$
(111)
$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},R,2}}=\lim_{F}\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/\eta][h^{1/2}]}.$
(112)

Here $\eta$ is either $p$ or $z$. Here in the $z$-adic setting the picture
will be completely $z$-adic following [SchIV], [KLI], [KLII]. The $v$-site
relative consideration in [SchIV], [KLI], [KLII] enriced over the pro-étale
consideration actually relates to the above picture by taking the projective
limit onto the corresponding perfectoid prisms in the limit sequence. This
also generalizes [TIV] significantly.

In [TIV] we discuss only the $p$-adic situation with generalization from
[BSB]. Go along those consideration in [TIV] we have a completely parallel
picture in $z$-adic setting with the $z$-adic $h$, where $h$ is defined
parallel to the $p$-adic setting.

######  Definition 6.

We now construct the condensed motives for rigid analytic spave $R$ over $T$
through the perfectoid consideration above from $v$-stacks. One can certainly
go along [SchI] and [ALBRCS] to glue along the formal model locally to reach
the whole motives for the spaces. In the perfectoid setting we regard $R$ in
genral as a corresponding $v$-stacks over $\mathrm{Spd}T$. Then we have the
following $\infty$-categories in the generalization we considered here:

$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},R,2}}=\lim_{F}\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[h^{1/2}]},$
(113)
$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},R,2}}=\lim_{F}\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}[h^{1/2}]},$
(114)
$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},R,2}}=\lim_{F}\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/\eta][h^{1/2}]}.$
(115)

Here $\eta$ is either $p$ or $z$. Here in the $z$-adic setting the picture
will be completely $z$-adic following [SchIV], [KLI], [KLII]. The $v$-site
relative consideration in [SchIV], [KLI], [KLII] enriced over the pro-étale
consideration actually relates to the above picture by taking the projective
limit onto the corresponding perfectoid prisms in the limit sequence. This
also generalizes [TIV] significantly. We then have the following Fontaine
style functors:

$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},R,2}}=\lim_{F}\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[h^{1/2}]}$
(116)
$\displaystyle\rightarrow\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},T,2}},$
(117)
$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},R,2}}=\lim_{F}\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}[h^{1/2}]}$
(118)
$\displaystyle\rightarrow\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},T,2}},$
(119)
$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},R,2}}=\lim_{F}\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/\eta][h^{1/2}]}$
(120)
$\displaystyle\rightarrow\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},T,2}},$
(121)

through the push forward along the structure morphism for $R$ to
$\mathrm{Spd}T$. For any bundle $B$ (we allow infinite rank bundle here beyond
the vector bundle situation, and we do not require the existence of Frobenius
structure while one can definitely add the shtukas structures) over the Robba
sheaf $\Pi_{\mathrm{perf},\blacksquare,v,2}$ (adding the element $h^{1/2}$ to
the usual Robba sheaf in the perfect setting) from [KLI], [KLII], we can
define the corresponding functor of the generalized de Rham functor as:

$\displaystyle(\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},R,2}}=\lim_{F}\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}[h^{1/2}]}$
(122)
$\displaystyle\rightarrow\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},T,2}})_{\sharp}(B\otimes\mathcal{O}_{\mathrm{\Gamma}^{\sharp}_{\mathrm{prismatization},\mathrm{deRham},F,2}}).$
(123)

We call this sheaf de Rham in the mixed parity situation if an isomorphism
retains after the base change back over to
$\mathcal{O}_{\mathrm{\Gamma}^{\sharp}_{\mathrm{prismatization},\mathrm{deRham},F,2}}$.
One then defines the corresponding cristalline sheaves over the Robba rings
$\Pi_{\mathrm{perf},\blacksquare,v,2}$ in the same fashion.

We remind the readers that this is not just $z$-adic generalization from
[TIV], it is a prismatization generalization, where we will see the motivic
structures richer than the context of [TIV] based on the pro-étale cohomology.

######  Theorem 3.

We have a well-defined functor from solid quasicoherent sheaves over mixed-
parity pre-Fargues-Fontaine stack attached to $R$ (taking the analytic stack
in the sense of [CS3] of $\Pi_{\mathrm{perf},\blacksquare,v,2}$ without taking
the Frobenius quotients) to solid quasicoherent sheaves over the de Rham stack
we considered in the mixed-parity setting. We have a well-defined functor from
solid quasicoherent sheaves over mixed-parity pre-Fargues-Fontaine stack
attached to $R$ (taking the analytic stack in the sense of [CS3] of
$\Pi_{\mathrm{perf},\blacksquare,v,2}$ without taking the Frobenius quotients)
to solid quasicoherent sheaves over the cristalline stack we considered in the
mixed-parity setting.

######  Theorem 4.

We have a well-defined functor from solid quasicoherent sheaves over mixed-
parity Fargues-Fontaine stack attached to $R$ (taking the analytic stack in
the sense of [CS3] of $\Pi_{\mathrm{perf},\blacksquare,v,2}$) to solid
quasicoherent sheaves over the de Rham stack we considered in the mixed-parity
setting. We have a well-defined functor from solid quasicoherent sheaves over
mixed-parity Fargues-Fontaine stack attached to $R$ (taking the analytic stack
in the sense of [CS3] of $\Pi_{\mathrm{perf},\blacksquare,v,2}$) to solid
quasicoherent sheaves over the cristalline stack we considered in the mixed-
parity setting.

###  3.2 Generalized Langlands Program

In this section we study the applications of the generalization we discussed
above on the prismatization. Here we following [FS] to make certain $p$-adic
cohomologicalization to the picture in the $\ell$-adic setting in the [FS].
This will also in some sense generalize the following papers: [TIV], [TV],
[TVI] in the sense that we will also consider mixed-parity generalization of
the motives we are considering while the underlying $v$-stacks are also
generalized from [FS] by directly considering the Fargues-Fontaine stacks
constructed using Robba rings where we allow the square root of $h$ in the
previos discussion in the previous sections. Recall that for such FF stack:

$\displaystyle\mathrm{\Gamma}_{\mathrm{FF},h^{1/2}}(.)$ (124)

we have the associated $G$-bundle stack in the moduli sense:

$\displaystyle\mathrm{\Gamma}_{\mathrm{Bun},h^{1/2}}(.)^{G}$ (125)

where $G/T$ over $T$ have two different form in different characteristic. In
our current consideration we can then apply the consideration above to achive
the corresponding motives over this stack by using basis like the $F$ in the
previous sections.

######  Theorem 5.

Over the stack:

$\displaystyle\mathrm{\Gamma}_{\mathrm{Bun},h^{1/2}}(.)^{G}$ (126)

we have the quasicoherent motives over the stackification by using the two
different forms of the corresponding prismatizations. In both situations we
have the corresponding generalized condensation of the corresponding
prismatization for $v$-stacks:

$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{\Gamma}_{\mathrm{Bun},h^{1/2}}(.)^{G},2}}=\lim_{F}\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[h^{1/2}]},$
(127)
$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},\mathrm{\Gamma}_{\mathrm{Bun},h^{1/2}}(.)^{G},2}}=\lim_{F}\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}[h^{1/2}]},$
(128)
$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},\mathrm{\Gamma}_{\mathrm{Bun},h^{1/2}}(.)^{G},2}}=\lim_{F}\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/\eta][h^{1/2}]}.$
(129)

Here $\eta$ is either $p$ or $z$. In the $p$-adic situation, suppose we
consider the following two $\infty$-categories:

$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},\mathrm{\Gamma}_{\mathrm{Bun},h^{1/2}}(.)^{G},2}}=\lim_{F}\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}[h^{1/2}]},$
(130)
$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},\mathrm{\Gamma}_{\mathrm{Bun},h^{1/2}}(.)^{G},2}}=\lim_{F}\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/\eta][h^{1/2}]}.$
(131)

And we assume that we tensor with the large coefficient field
$\overline{\mathbb{Q}}_{p}$:

$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},\mathrm{\Gamma}_{\mathrm{Bun},h^{1/2}}(.)^{G},2,\overline{\mathbb{Q}}_{p}}}=\lim_{F}\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}[h^{1/2}]}\otimes^{\sharp}\overline{\mathbb{Q}}_{p},$
(132)
$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},\mathrm{\Gamma}_{\mathrm{Bun},h^{1/2}}(.)^{G},2,\overline{\mathbb{Q}}_{p}}}=\lim_{F}\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/\eta][h^{1/2}]}\otimes^{\sharp}\overline{\mathbb{Q}}_{p}.$
(133)

Then we can realize a generalized Langlands parametrization with
$\overline{\mathbb{Q}}_{p}$ coefficient with operation over the categories
from the Weil groups in the mixed-parity setting after [FS], [TIV], [TV],
[TVI].

###### Proof.

See the proof in [FS, Chapter VIII, IX], [VL], [TIV], [TV], [TVI]. Here the
key consideration comes from reaching a corresponding correspondence in
between the $p$-adic coefficients and the $\ell$-adic coefficients through the
algebraic isomorphism:

$\displaystyle\overline{\mathbb{Q}}_{p}\overset{\sim}{\longrightarrow}\overline{\mathbb{Q}}_{\ell}.$
(134)

From this isomorphism we start from the corresponding representation of
$p$-adic points ($\overline{\mathbb{Q}}_{p}$-values) of the full Langlands
dual group (with the action from the Weil group), then we can end up with the
corresponding representation of the corresponding representation of
$\ell$-adic points ($\overline{\mathbb{Q}}_{\ell}$-values) of the full
Langlands dual group (with the action from the Weil group), which produces a
motivic complex through the Hecke operator, which then produces a motivic
complex with $\overline{\mathbb{Q}}_{p}$ coefficients. This will then be sent
to the categories in our generalized setting by taking the condensed product
with the prismatization directly. ∎

######  Definition 7.

We use the following to define the solid geometric generalized Banach space
representation of $G(T)$:

$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{\Gamma}_{\mathrm{Bun},h^{1/2}}(.)^{G},2}}=\lim_{F}\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[h^{1/2}]},$
(135)
$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{deRham},\mathrm{\Gamma}_{\mathrm{Bun},h^{1/2}}(.)^{G},2}}=\lim_{F}\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/\eta]_{I_{U_{i}}}[h^{1/2}]},$
(136)
$\displaystyle\mathrm{Quasicoherent}^{\sharp}_{\mathrm{\Gamma}_{\mathrm{prismatization},\mathrm{cristalline},\mathrm{\Gamma}_{\mathrm{Bun},h^{1/2}}(.)^{G},2}}=\lim_{F}\projlim_{i}\mathrm{Quasicoherent}^{\sharp}_{\mathrm{Spec}U_{i}[1/\eta][h^{1/2}]}.$
(137)

######  Remark 4.

In some situation, the coverings of the Galois groups after [BSB] will be
trivial such as splitting as a pure product with the group scheme $\mu_{n}$,
but this does not matter since on the other hand the Hodge structure is
already generalized by adding the roots of the $p$-adic or $z$-adic ’2$\pi$i’.

### Acknowledgements

Being far beyond the original (with further originality from previous work of
Breuil and Schneider) perspective from Professor Sorensen, we express our
thankfulness to Professor Sorensen for sharing understanding on the Breuil-
Schneider conjecture and the philosophy. We also express our thankfulness to
Professor Kedlaya for sharing understanding on the philosophy of Witt
vectorial approach to the $p$-adic motives. This paper is profoundly
influenced by previous philosophicalization from Professor Kedlaya on Witt
vectors and the applications to motives.

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