#  $f$-cohomology and motives over number rings

Jakob Scholbach 111Universität Münster, Mathematisches Institut, Einsteinstr.
62, D-48149 Münster, Germany<EMAIL_ADDRESS>
###### Abstract

This paper is concerned with an interpretation of $f$-cohomology, a
modification of motivic cohomology of motives over number fields, in terms of
motives over number rings. Under standard assumptions on mixed motives over
finite fields, number fields and number rings, we show that the two extant
definitions of $f$-cohomology of mixed motives $M_{\eta}$ over $F$—one via
ramification conditions on $\ell$-adic realizations, another one via the
$K$-theory of proper regular models—both agree with motivic cohomology of
$\eta_{!*}M_{\eta}[1]$. Here $\eta_{!*}$ is constructed by a limiting process
in terms of intermediate extension functors $j_{!*}$ defined in analogy to
perverse sheaves.

The aim of this paper is to give an interpretation of $f$-cohomology in terms
of motives over number rings. The notion of $f$-cohomology goes back to
Beilinson who used it to formulate a conjecture about special $L$-values
[Beĭ84, Beĭ86]. The most classical example is what is now called
$\mathrm{H}^{1}_{f}(F,\mathbf{1}(1))$, $f$-cohomology of $\mathbf{1}(1)$, the
motive of a number field $F$, twisted by one. This group is
${\mathcal{O}_{F}}^{\times}{\otimes}_{\mathbb{Z}}\mathbb{Q}$, as opposed to
the full motivic cohomology
$\mathrm{H}^{1}(F,\mathbf{1}(1))=F^{\times}{\otimes}\mathbb{Q}$. Together with
the Dirichlet regulator, it explains the residue of the Dedekind zeta function
$\zeta_{F}(s)$ at $s=1$. In general, for any mixed motive $M_{\eta}$ over $F$,
there are two independent yet conjecturally equivalent ways to define
$\mathrm{H}^{1}_{f}(F,M_{\eta})\subset\mathrm{H}^{1}(F,M_{\eta})$. We propose
a third way to obtain these groups, by interpreting $f$-cohomology as motivic
cohomology of suitable motives over ${\mathcal{O}_{F}}$. This idea is due to
Huber.

Let us briefly recall the two definitions of $\mathrm{H}^{1}_{f}(M_{\eta})$.
The first is due to Beilinson [Beĭ87a, Remark 4.0.1.b], Bloch and Kato [BK90,
Conj. 5.3.] and Fontaine [Fon92, FPR94]. It is given by picking elements in
motivic cohomology acted on by the local Galois groups in a prescribed way
(Definition 6.1, Definition 6.4, Definition 6.6). The second definition of
$\mathrm{H}^{1}_{f}(M_{\eta})$, due to Beilinson [Beĭ86, Section 8], applies
to $M_{\eta}=\mathrm{h}^{i-1}(X_{\eta})(n)$, with $X_{\eta}$ smooth and
projective over $F$, $i-2n<0$. It is given by the image of $K$-theory of a
regular proper model $X$ of $X_{\eta}$ (Definition 6.11). Such a model may not
exist, but there is a unique meaningful extension of this definition to all
Chow motives over $F$ due to Scholl [Sch00].

Our main results (Theorems 6.9, 6.12, 6.14) show that both definitions of
$\mathrm{H}^{1}_{f}(M_{\eta})$ agree with
$\mathrm{H}^{0}(\eta_{!*}\mathrm{h}^{i-1}(X_{\eta},n)[1])$. Here $\eta_{!*}$
is a functor that attaches to any suitable mixed motive over $F$ one over
${\mathcal{O}_{F}}$. It is defined by a limiting process using the
intermediate extension $j_{!*}$ familiar from perverse sheaves [BBD82] along
open immersions $j:U\rightarrow{\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$.
Even to formulate such a definition, one has to rely on profound conjectures,
namely the existence of mixed motives over (open subschemes of)
${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$. The proof of the two main
theorems also requires us to assume a number of properties related to weights
of motives and, for the first comparison result, Beilinson’s conjecture about
the agreement of rational and numerical equivalence on smooth projective
varieties over finite fields up to torsion (Conjecture 6.7).

A motivation for the results of this work lies in an application to special
$L$-values conjecures [Sch10]. Very briefly, Beilinson’s conjecture concerning
special $L$-values for mixed motives $M_{\eta}$ over $\mathbb{Q}$ has
$f$-cohomology as motivic input. $L$-functions of such motives can be
generalized to motives over $\mathbb{Z}$ such that the classical $L$-function
of $M_{\eta}$ agrees with the $L$-function (over $\mathbb{Z}$) of
$\eta_{!*}M_{\eta}[1]$. Thereby the $L$-function and the motivic data in
Beilinson’s conjecture belong to the same motive over $\mathbb{Z}$, thus
giving content to a general conjecture about special $L$-values for motives
over $\mathbb{Z}$. In this light it is noteworthy that
$\mathrm{H}^{0}(\eta_{!*}\mathrm{h}^{2n-1}(X_{\eta},n)[1])$ identifies with
the group that occurs in the part of Beilinson’s conjecture that describes
special values at the central point.

The contents of the paper are as follows: Section 1 is the basis of the
remainder; it lists a number of axioms on triangulated categories of motives.
Such categories $\mathbf{DM}_{\mathrm{gm}}(S)$ have been constructed by
Voevodsky [Voe00] and Hanamura [Han95] (over fields) and Levine [Lev98] (over
bases $S$ over a field). The various approaches are known to be
(anti-)equivalent, at least for rational coefficients [Lev98, Section VI.2.5],
[Bon09, Section 4]. Over more general bases $S$, a candidate for the category
$\mathbf{DM}(S)$ has been constructed by Ivorra [Ivo07]. Cisinski and Déglise
are developing a robust theory of such motives [CD10]. We sum up the
properties of this construction by specifying a number of axioms concerning
triangulated categories of motives that will be used in the sequel. They are
concerned with the “core” behavior of $\mathbf{DM}(S)$, that is:
functoriality, compacity, the monoidal structure and the relation to algebraic
$K$-theory, as well as localization, purity, base-change and resolution of
singularities. We work with motives with rational coefficients only, since
this is sufficient for all our purposes. We use a contravariant notation for
motives, that is to say the functor that maps any scheme $X$ to its motive
$\operatorname{M}(X)$ shall be contravariant. This is in line with most pre-
Voevodsky papers.

Section 2 is a very brief reminder on realizations. The existence of various
realizations, due to Huber and Ivorra [Hub95, Hub00, Ivo07], is pinning down
the intuition that motives should be universal among (reasonable) cohomology
theories.

Section 4 spells out a number of conjectural properties (also called axioms in
the sequel) of $\mathbf{DM}_{\mathrm{gm}}(S)$, where $S$ is either a finite
field ${\mathbb{F}_{\mathfrak{p}}}$, a number field $F$ or a number ring
${\mathcal{O}_{F}}$. The first group of these properties centers around the
existence of a category of mixed motives $\mathbf{MM}(S)$, which is to be the
heart of the so-called motivic $t$-structure. The link between mixed motives
over ${\mathcal{O}_{F}}$ and ${\mathbb{F}_{\mathfrak{p}}}$ or $F$ is
axiomatized by mimicking the exactness properties familiar from perverse
sheaves (Axiom 4.2). On the triangulated subcategory
$\mathbf{DATM}({\mathcal{O}_{F}})\subset\mathbf{DM}_{\mathrm{gm}}({\mathcal{O}_{F}})$
of Artin-Tate motives, a $t$-structure satisfying these properties is
constructed in a separate paper [Scha]. A key requirement on mixed motives is
that the realization functors on motives should be exact (Axiom 4.8). For the
$\ell$-adic realization over ${\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}[1/\ell]$, this requires a notion of perverse sheaves
over that base, which is provided in Section 3. Another important conjectural
facet of mixed motives are weights. Weights are an additional structure
encountered in both Hodge structures and $\ell$-adic cohomology of algebraic
varieties over finite fields, both due to Deligne [Del74, Del80]. They are
important in that morphisms between Hodge structures or $\ell$-adic cohomology
groups are known to be strictly compatible with weights, moreover, they are
respected to a certain extent by smooth maps and proper maps. It is commonly
assumed that this should be the case for mixed motives, too.

The remaining two sections assume the validity of the axiomatic framework set
up so far. The first key notion in Section 5 is the intermediate extension
$j_{!*}M$ of a mixed motive $M$ along some open embedding $j$ inside
${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$. This is done in strict
parallelity to the case of perverse sheaves, due to Beilinson, Bernstein and
Deligne [BBD82]. Quite generally, much of this paper is built on the idea that
the abstract properties of mixed perverse sheaves (should) give a good model
for mixed motives over number rings. Next we develop a notion of smooth
motives, which is an analog of lisse étale sheaves. This is needed to extend
the intermediate extension to an extension functor $\eta_{!*}$ that extends
motives over $F$ to ones over ${\mathcal{O}_{F}}$. Finally, we apply the axiom
on the exactness of $\ell$-adic realization to show that intermediate
extensions commute with the realization functors. This will be a stepstone in
a separate work on $L$-functions of motives [Sch10].

Section 6 gives the comparison theorems on $f$-cohomology mentioned above. The
two definitions of $f$-cohomology being quite different, the proofs of the
comparison statements are different, too: the first is essentially based on
the Hochschild-Serre spectral sequence. The crystalline case of that
definition of $f$-cohomology is disregarded throughout. The second proof is a
purely formal, if occasionally intricate bookkeeping of cohomological degrees
and weights.

The problem of finding a motivic interpretation of terms such as
$\mathrm{H}^{1}_{f}(M_{\eta})$ underlying the formulation of Beilinson’s
conjecture has been studied by Scholl [Sch91, Sch00, Schb], who develops an
abelian category $\mathbf{MM}(F/{\mathcal{O}_{F}})$ of mixed motives over
${\mathcal{O}_{F}}$ by taking mixed motives over $F$, and imposing additional
non-ramification conditions. Conjecturally, the group
$\operatorname{Ext}^{i}_{\mathbf{MM}(F/{\mathcal{O}_{F}})}(\mathbf{1},\mathrm{h}^{i}(X_{\eta},n))$
for $X_{\eta}/F$ smooth and projective, $i=0$, $1$, agrees with what amounts
to $\mathrm{H}^{i-1}(\eta_{!*}\mathrm{h}^{2n-1}(X_{\eta},n)[1])$.

No originality is claimed for Sections 1, 2, and 4, except perhaps for the
formulation of the relation of mixed motives over ${\mathcal{O}_{F}}$ and $F$
and the residue fields ${\mathbb{F}_{\mathfrak{p}}}$, which however is a
natural and immediate translation of the theory of perverse sheaves. I would
like to thank Denis-Charles Cisinski and Frédéric Déglise for communicating to
me their work on $\mathbf{DM}_{\mathrm{gm}}(S)$ over general bases [CD10].
Finally, I gratefully acknowledge Annette Huber’s advice in writing my thesis,
of which this paper is a part.

##  1 Geometric motives

Throughout this paper, $F$ is a number field, ${\mathcal{O}_{F}}$ its ring of
integers, $\mathfrak{p}$ stands for a place of $F$. For finite places, the
residue field is denoted ${\mathbb{F}_{\mathfrak{p}}}$. By scheme we mean a
Noetherian separated scheme. Actually, it suffices to think of schemes of
finite type over one of the rings just mentioned. In this section $S$ denotes
a fixed base scheme.

This section is setting up a number of axioms describing a triangulated
category $\mathbf{DM}_{\mathrm{gm}}(S)$ of geometric motives over $S$. They
will be used throughout this work. As pointed out in the introduction, the
material of this section is due to Cisinski and Deglise [CD10], who build such
a category of motives using Ayoub’s base change formalism [Ayo07].

######  Axiom 1.1.

(Motivic complexes and functoriality)

  * •

There is a triangulated $\mathbb{Q}$-linear category $\mathbf{DM}(S)$. It is
called category of _motivic complexes_ over $S$ (with rational coefficients).
It has all limits and colimits.

  * •

(Tensor structure) The category $\mathbf{DM}(S)$ is a triangulated symmetric
monoidal category (see e.g. [Lev98, Part 2, II.2.1.3]). Tensor products
commute with direct sums. The unit of the tensor structure is denoted
$\mathbf{1}_{S}$ or $\mathbf{1}$. Also, there are internal
$\mathrm{Hom}$-objects in $\mathbf{DM}$, denoted $\underline{\mathrm{Hom}}$.
The _dual_ $M^{\vee}$ of an object $M\in\mathbf{DM}(S)$ is defined by
$M^{\vee}:=\underline{\mathrm{Hom}}(M,\mathbf{1})$.

  * •

For any map $f:X\rightarrow Y$ of schemes, there are pairs of adjoint functors

$f^{*}:\mathbf{DM}(Y)\leftrightarrows\mathbf{DM}(X):f_{*}$

such that $f^{*}\mathbf{1}_{Y}=\mathbf{1}_{X}$ and, if $f$ is quasi-
projective,

$f_{!}:\mathbf{DM}(X)\leftrightarrows\mathbf{DM}(Y):f^{!}.$

The existence of $f_{!}$ and $f^{!}$ is restricted to quasi-projective maps
since the abstract construction of these functors in Ayoub’s work [Ayo07,
Section 1.6.5], on which Cisinski’s and Déglise’s construction of motives over
general bases [CD10] relies, has a similar restriction.

Recall that an object $X$ in a triangulated category $\mathcal{T}$ closed
under arbitrary direct sums is _compact_ if $\mathrm{Hom}(X,-)$ commutes with
direct sums. The subcategory of $\mathcal{T}$ of compact objects is
triangulated and closed under direct summands (a.k.a. a _thick subcategory_)
[Nee01, Lemma 4.2.4]. The category $\mathcal{T}$ is called _compactly
generated_ if the smallest triangulated subcategory closed under arbitrary
sums containing the compact objects is the whole category $\mathcal{T}$.

######  Axiom 1.2.

(Compact objects) The motive $\mathbf{1}\in\mathbf{DM}(S)$ is compact. The
functors $f^{*}$ and $f^{!}$, whenever defined, and ${\otimes}$ and
$\underline{\mathrm{Hom}}$ preserve compact objects. The same is true for
$f_{*}$ and $f_{!}$ if $f$ is of finite type. The canonical map
$M\rightarrow(M^{\vee})^{\vee}$ is an isomorphism for any compact object $M$.

######  Definition 1.3.

The subcategory of compact objects of $\mathbf{DM}(S)$ is denoted
$\mathbf{DM}_{\mathrm{gm}}(S)$ and called the category of _geometric motives_
over $S$.

For any map $f:X\rightarrow S$ of finite type, the object
$\operatorname{M}_{S}(X):=\operatorname{M}(X):=f_{*}f^{*}\mathbf{1}\in\mathbf{DM}_{\mathrm{gm}}(S)$
is called the _motive_ of $X$ over $S$. By adjunction, $\operatorname{M}$ is a
contravariant functor from schemes of finite type over $S$ to
$\mathbf{DM}_{\mathrm{gm}}(S)$. For any quasi-projective $f:X\rightarrow S$,
the _motive with compact support_ of $X$, $\operatorname{M}_{\mathrm{c}}(X)$,
is defined as $f_{!}f^{*}\mathbf{1}\in\mathbf{DM}_{\mathrm{gm}}(S)$.

The smallest thick subcategory of $\mathbf{DM}(S)$ containing the image of
$\operatorname{M}$ is denoted ${\mathbf{DM}^{\mathrm{eff}}_{\mathrm{gm}}}(S)$
and called the category of _effective geometric motives_. The closure of that
subcategory under all direct sums is called the category of _effective
motives_ , ${\mathbf{DM}_{\mathrm{eff}}}(S)$.

######  Axiom 1.4.

(Tensor product vs. fiber product) The functor $\operatorname{M}$ is an
additive tensor functor, i.e., maps disjoint unions of schemes over $S$ to
direct sums and fiber products of schemes over $S$ to tensor products in
$\mathbf{DM}_{\mathrm{gm}}(S)$.

######  Axiom 1.5.

(Compact generation) The categories $\mathbf{DM}(S)$ and
${\mathbf{DM}_{\mathrm{eff}}}(S)$ are compactly generated.

There is a decomposition
$\operatorname{M}(\mathbb{P}^{1}_{S})=\mathbf{1}\oplus\mathbf{1}(-1)[-2]$,
where

$\mathbf{1}(-1):=\ker(\operatorname{M}(\mathbb{P}^{1}_{S})\rightarrow\operatorname{M}(S)\rightarrow\operatorname{M}(\mathbb{P}^{1}_{S}))[2],$

where the first map is induced by the projection onto the base, the second map
stems from the rational point $0\in\mathbb{P}^{1}_{S}$. The object
$\mathbf{1}(-1)$ is called _Tate object_ or Tate motive. The category
$\mathbf{DM}(S)$, being closed under countable direct sums is pseudo-abelian
[Lev98, Lemma II.2.2.4.8.1], i.e., it contains kernels of projectors. It
follows from the preceding axioms that
$\mathbf{1}(-1)\in{\mathbf{DM}^{\mathrm{eff}}_{\mathrm{gm}}}(S)$.

######  Axiom 1.6.

(Cancellation and Effectivity) In $\mathbf{DM}_{\mathrm{gm}}(S)$ (and thus in
$\mathbf{DM}(S)$), the Tate object $\mathbf{1}(-1)$ has a tensor-inverse
denoted $\mathbf{1}(1)$. For any $M\in\mathbf{DM}(S)$, $n\in\mathbb{Z}$, set
$M(n):=M{\otimes}\mathbf{1}(1)^{{\otimes}n}$. Then there is a canonical
isomorphism called _cancellation_ isomorphism ($n\in\mathbb{Z}$,
$M,N\in\mathbf{DM}(S)$):

$\mathrm{Hom}_{\mathbf{DM}(S)}(M,N)\cong\mathrm{Hom}_{\mathbf{DM}(S)}(M(n),N(n)).$

The smallest tensor subcategory of $\mathbf{DM}_{\mathrm{gm}}(S)$ that
contains ${\mathbf{DM}^{\mathrm{eff}}_{\mathrm{gm}}}(S)$ and $\mathbf{1}(1)$
is $\mathbf{DM}_{\mathrm{gm}}(S)$. In other words,
$\mathbf{DM}_{\mathrm{gm}}(S)$ is obtained from
${\mathbf{DM}^{\mathrm{eff}}_{\mathrm{gm}}}(S)$ by tensor-inverting
$\mathbf{1}(-1)$.

######  Definition 1.7.

Let $M$ be any geometric motive over $S$. We write
$\mathrm{H}^{i}(M):=\mathrm{H}^{i}(S,M):=\mathrm{Hom}_{\mathbf{DM}(S)}(\mathbf{1},M[i])$.
For $M=\operatorname{M}(X)(n)$ for any $X$ over $S$ we also write
$\mathrm{H}^{i}(X,n):=\mathrm{H}^{i}(\operatorname{M}(X)(n))=\mathrm{Hom}_{\mathbf{DM}_{\mathrm{gm}}(S)}(\mathbf{1},\operatorname{M}(X)(n)[i])$.
This is called _motivic cohomology_ of $M$ and $X$, respectively.

######  Axiom 1.8.

(Motivic cohomology vs. K-theory) For any regular scheme $X$, there is an
isomorphism $\mathrm{H}^{i}(X,n)\cong K_{2n-i}(X)^{(n)}_{\mathbb{Q}}$, where
the right hand term denotes the Adams eigenspace of algebraic $K$-theory
tensored with $\mathbb{Q}$ [Qui73].

This is a key property of motives, since algebraic $K$-theory is a universal
cohomology theory in the sense that Chern characters map from algebraic
$K$-theory to any other (reasonable) cohomology theory of algebraic varieties
[Gil80]. For $S$ a perfect field, this axiom is given by [Voe00, Prop. 4.2.9]
and its non-effective analogue. See also [Lev98, Theorem I.III.3.6.12.].

Recall Grothendieck’s category of pure motives $\mathbf{M}_{\sim}(K)$ with
respect to an adequate equivalence relation $\sim$, see e.g. [And04, Section
4]. For rational equivalence they are also called Chow motives, since, for any
smooth projective variety $X$ over a field $K$,

$\mathrm{Hom}_{\mathbf{M}_{\mathrm{rat}}(K)}(\mathbf{1}(-n),h(X))=\mathrm{CH}^{n}(X),$
(1)

where $h(X)$ denotes the Chow motive of $X$ and the right hand term is the
_Chow group_ of cycles of codimension $n$ in $X$. This way, the above axiom
models the fact [Voe00, 2.1.4] that Chow motives are a full subcategory of
$\mathbf{DM}_{\mathrm{gm}}(K)$. Under the embedding
$\mathbf{M}_{\mathrm{rat}}(K)\subset\mathbf{DM}_{\mathrm{gm}}(K)$, $h(X,n)$
maps to $\operatorname{M}(X)(n)[2n]$.

######  Remark 1.9.

We do not need to assume _expressis verbis_ homotopy invariance (i.e.,
$\mathbf{1}\stackrel{{\scriptstyle\cong}}{{\rightarrow}}pr_{*}pr^{*}\mathbf{1}\in\mathbf{DM}_{\mathrm{gm}}(S)$
for $pr:S{\times}\mathbb{A}^{1}\rightarrow S$) nor the projective bundle
formula [Voe00, Prop. 3.5.1]. (Note, however, that $K^{\prime}$-theory does
have such properties.)

######  Axiom 1.10.

(Localization) Let $i:Z\rightarrow S$ be any closed immersion and
$j:V\rightarrow S$ the open complement. The adjointness maps give rise to the
following distinguished triangles in $\mathbf{DM}(S)$:

$j_{!}j^{*}\rightarrow\mathrm{id}\rightarrow i_{*}i^{*},$
$i_{*}i^{!}\rightarrow\mathrm{id}\rightarrow j_{*}j^{*}.$

(In particular, $f_{*}f^{*}\cong\mathrm{id}$, where
$f:X_{\mathrm{red}}\rightarrow X$ denotes the canonical map of the reduced
subscheme structure.) In addition, one has $j^{*}j_{*}=\mathrm{id}$ and
$i^{*}i_{*}=\mathrm{id}$, equivalently $j^{*}i_{*}=i^{*}j_{!}=0$.

It should be emphasized that the localization axiom requires rational
coefficients.

######  Axiom 1.11.

(Purity and base change)

  * •

For any quasi-projective map $f$, there is a functorial transformation of
functors $f_{!}\rightarrow f_{*}$. It is an isomorphism if $f$ is projective.

  * •

(_Relative purity_): If $f$ is quasi-projective and smooth of constant
relative dimension $d$, there is a functorial (in $f$) isomorphism $f^{!}\cong
f^{*}(d)[2d]$.

  * •

(_Absolute purity_): If $i:Z\rightarrow U$ is a closed immersion of
codimension $c$ of two regular schemes $Z$ and $U$, there is a natural
isomorphism $i^{!}\mathbf{1}\cong\mathbf{1}(-c)[-2c]$.

  * •

(_Base change_): For any two quasi-projective maps $f$ and $g$ let
$f^{\prime}$ and $g^{\prime}$ denote the pullback maps:

$\textstyle{X^{\prime}{\times}_{X}Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$$\scriptstyle{f^{\prime}}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{X}$
(2)

Then there is canonical isomorphism of functors

$f^{*}g_{!}\stackrel{{\scriptstyle\cong}}{{{\longrightarrow}}}g^{\prime}_{!}f^{\prime*}.$

This axiom is proven by Cisinski & Déglise using Ayoub’s general base change
formalism. See in particular [Ayo07, 1.4.11, 12] for the construction of the
base change map. See also [Lev98, Theorem I.I.2.4.9] for a similar statement
in Levine’s category of motives.

######  Definition 1.12.

Let $f:S\rightarrow{\mathrm{Spec}\text{ }}{\mathbb{Z}}$ be the structural map.
Assume $f$ is quasi-projective. Then
$D(M):=\underline{\mathrm{Hom}}(M,f^{!}\mathbf{1}(1)[2])$ is called _Verdier
dual_ of $M$.

By the preceding axioms, $D$ induces a contravariant endofunctor of
$\mathbf{DM}_{\mathrm{gm}}(S)$. The shift and twist in the definition is
motivated as follows: given some complex analytic space $X$, the Verdier dual
of a sheaf $\mathcal{F}$ on $X$ is defined by

$D(\mathcal{F}):=\underline{{\mathrm{R}}{\mathrm{Hom}}}_{\mathbf{D}(\mathbf{Shv}\left(X\right))}(\mathcal{F},f^{!}\mathbb{Z}),$

where $f$ denotes the projection to a point, see e.g. [Ive86, Ch. VI]. When
$X$ is smooth of dimension $d$, one has
$f^{!}\mathbb{Z}=f^{*}\mathbb{Z}(d)[2d]=\mathbb{Z}(d)[2d]$. A similar fact
holds for $\ell$-adic sheaves (see e.g. [KW01, Section II.7-8]). The above
definition mimics this situation insofar as ${\mathrm{Spec}\text{
}}{\mathbb{Z}}$ is seen as an analogue of a smooth affine curve.

Let us give a number of consequences of the preceding axioms, in particular
purity, base change and localization: in (2), suppose that $f$ is smooth and
$X^{\prime}\subset X$ is a codimension one closed immersion between regular
schemes. Then there is a canonical isomorphism

$i^{!}\operatorname{M}_{S}(X)=\operatorname{M}_{Z}(X{\times}_{S}Z)(-1)[-2].$
(3)

Let $Z\subset X$ be a closed immersion of quasiprojective schemes over $S$.
Then there is a distinguished triangle of motives with compact support

$\operatorname{M}_{\mathrm{c}}(Z)\rightarrow\operatorname{M}_{\mathrm{c}}(X)\rightarrow\operatorname{M}_{\mathrm{c}}(X\backslash
Z).$

Let $S$ be a regular scheme of equidimension $d$ that is affine or projective
over ${\mathrm{Spec}\text{ }}{\mathbb{Z}}$. Let
$f:S\rightarrow{\mathrm{Spec}\text{ }}{\mathbb{Z}}$ be the structural map.
Then $f^{!}\mathbf{1}=\mathbf{1}(d-1)[2d-2]$, as one sees by applying relative
and absolute purity to the maps $p$ and $i$, where $f=p\circ i$ is a
factorization into a affine or projective projection map and a closed
immersion. In particular, the Verdier duality functor on any open subscheme
$S$ of ${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$ is given by
$D_{\mathbf{DM}_{\mathrm{gm}}(S)}(?)=\underline{\mathrm{Hom}}(?,\mathbf{1}(1)[2])$
while on $\mathbf{DM}_{\mathrm{gm}}({\mathbb{F}_{\mathfrak{p}}})$ it is given
by $\underline{\mathrm{Hom}}(?,\mathbf{1})=?^{\vee}$.

######  Axiom 1.13.

(Verdier dual) The Verdier dual functor $D$ exchanges “$!$” and “$*$”
throughout, e.g., there are natural isomorphisms $D(f^{!}M)\cong f^{*}D(M)$
for any quasi-projective map $f:X\rightarrow Y$ and $M\in\mathbf{DM}(Y)$ and
similarly with $f_{!}$ and $f_{*}$.

######  Lemma 1.14.

Let $S$ be such that $f^{!}\mathbf{1}=f^{*}\mathbf{1}(d)[2d]$ for some integer
$d$, where $f:S\rightarrow{\mathrm{Spec}\text{ }}{\mathbb{Z}}$ is the
structural map. For example, $S$ might be regular and affine or projective
over $\mathbb{Z}$ (see above), or smooth over ${\mathrm{Spec}\text{
}}{\mathbb{Z}}$ (purity). Then, for any compact object
$M\in\mathbf{DM}_{\mathrm{gm}}(S)$, the canonical map $M\rightarrow D(D(M))$
is an isomorphism. This will be referred to as _reflexivity_ of Verdier
duality.

######  Proof: .

By Axiom 1.5, it suffices to check it for $M=\pi_{*}\pi^{*}\mathbf{1}$, where
$\pi:X\rightarrow S$ is some map of finite type. In this case it follows for
adjointness reasons and the assumption. ∎

######  Axiom 1.15.

(Resolution of singularities) Let $K$ be a field. As a triangulated additive
tensor category (i.e., closed under triangles, arbitrary direct sums and
tensor product), $\mathbf{DM}(K)$ is generated by $\mathbf{1}(-1)$ and all
$\operatorname{M}(X)$, where $X/K$ is a smooth projective variety.

When $S={\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$, the generators of
$\mathbf{DM}(S)$ are $\mathbf{1}(-1)$,
${i_{\mathfrak{p}}}_{*}\operatorname{M}(X_{\mathfrak{p}})$, and
$\operatorname{M}(X)$, instead, where $X_{\mathfrak{p}}$ is any projective and
smooth variety over ${\mathbb{F}_{\mathfrak{p}}}$, $i_{\mathfrak{p}}$ denotes
the immersion of any closed point ${\mathbb{F}_{\mathfrak{p}}}$ of $S$, and
$X$ is any regular, flat projective scheme over ${\mathcal{O}_{F}}$.

Consequently, the subcategories of compact objects
$\mathbf{DM}_{\mathrm{gm}}(-)$ are generated as a thick tensor subcategory by
the mentioned objects. In Voevodsky’s theory of motives over a field of
characteristic zero, this is [Voe00, Section 4.1]. This uses Hironaka’s
resolution of singularities. Over a field of positive characteristic and
number rings, one has to use de Jong’s resolution result, see [HK06, Lemma
B.4].

We also need a limit property of the generic point. Let $S$ be an open
subscheme of ${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$, let
$\eta:{\mathrm{Spec}\text{ }}{F}\rightarrow S$ be the generic point.

######  Axiom 1.16.

(Generic point) Let $M$ be any geometric motive over $S$. The natural maps
$j_{*}j^{*}M\rightarrow\eta_{*}\eta^{*}M$ give rise to an isomorphism
$\varinjlim j_{*}j^{*}M=\eta_{*}\eta^{*}M$, where the colimit is over all open
subschemes $j:S^{\prime}\rightarrow S$. It induces a distinguished triangle in
$\mathbf{DM}(S)$

$\oplus_{\mathfrak{p}\in
S}{i_{\mathfrak{p}}}_{*}i_{\mathfrak{p}}^{!}M\rightarrow\mathrm{id}\rightarrow\eta_{*}\eta^{*}M,$
(4)

where the sum runs over all closed points $\mathfrak{p}\in S$ and
$i_{\mathfrak{p}}$ is the closed immersion.

##  2 Realizations

One of the main interests in motives lies in the fact that they are explaining
(or are supposed to explain) common phenomena in various cohomology theories.
These cohomology functors are commonly referred to as _realization_ functors.
They typically have the form
$\mathbf{DM}_{\mathrm{gm}}(S)\rightarrow\mathbf{D}^{\mathrm{b}}(\mathcal{C})$,
where $\mathcal{C}$ is an abelian category whose objects are amenable with the
methods of (linear) algebra, such as finite-dimensional vector spaces or
finite-dimensional continuous group representations or constructible sheaves.

For example, let $\ell$ be a prime and let $S$ be either a field of
characteristic different from $\ell$ or a scheme of finite type over
${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}[1/\ell]$. The $\ell$-adic
cohomology maps any scheme $X$ over $S$ to

${\mathrm{R}}{\Gamma}_{\ell}(X):=\mathrm{R}\pi_{*}\pi^{*}{\mathbb{Q}_{{\ell}}}\in\mathbf{D}^{\mathrm{b}}_{c}\left(S,{\mathbb{Q}_{{\ell}}}\right),$

where $\pi:X\rightarrow S$ is the structural map and the right hand category
denotes the “derived” category of constructible
${\mathbb{Q}_{{\ell}}}$-sheaves on $S$ (committing the standard abuse of
notation, see e.g. [KW01, II.6., II.7.]). This functor factors over the
_$\ell$ -adic realization functor_ ([Hub00, p. 772], [Ivo07])
${\mathrm{R}}{\Gamma}_{\ell}:\mathbf{DM}_{\mathrm{gm}}(S)\rightarrow\mathbf{D}^{\mathrm{b}}_{c}(S,{\mathbb{Q}_{{\ell}}})$.
When $S$ is of finite type over ${\mathbb{F}_{\mathfrak{p}}}$, the realization
functor actually maps to
$\mathbf{D}^{\mathrm{b}}_{c,m}(S,{\overline{{\mathbb{Q}_{{\ell}}}}})$, the
full subcategory of complexes $C$ in
$\mathbf{D}^{\mathrm{b}}_{c}(S,{\overline{{\mathbb{Q}_{{\ell}}}}})$ such that
all $\mathrm{H}^{n}(C)$ are mixed sheaves [Del80, 1.2].

Further realization functors include Betti, de Rham and Hodge realization. See
e.g. [Hub00, 2.3.5]. The following axiom says (in particular) that the
$\ell$-adic realization of $\operatorname{M}(X)$ does give the $\ell$-adic
cohomology groups.

######  Axiom 2.1.

(Functoriality and realizations) The $\ell$-adic realization functor commutes
with the six Grothendieck functors $f_{*}$, $f_{!}$, $f^{!}$, $f^{*}$,
${\otimes}$ and $\underline{\mathrm{Hom}}$ (where applicable). For example,
for any map $f:S^{\prime}\rightarrow S$ and any geometric motive $M$ over
$S^{\prime}$:

$(f_{*}M)_{\ell}=f_{*}(M_{\ell}).$

##  3 Interlude: Perverse sheaves over number rings

This section is devoted to a modest extension of $\ell$-adic perverse sheaves
[BBD82] to the situation where the base $S$ is an open subscheme of
${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}[1/\ell]$. It is needed to
formulate Axiom 4.8 for the $\ell$-adic realization of motives over number
rings. In a nutshell, the theory of perverse sheaves on varieties over
${\mathbb{F}_{q}}$ stakes on relative purity, that is
$f^{!}{\mathbb{Z}_{{\ell}}}=f^{*}{\mathbb{Z}_{{\ell}}}(n)[2n]$ for a smooth
map $f$ of relative dimension $n$. The analogous identity for a closed
immersion $i:{\mathrm{Spec}\text{ }}{{\mathbb{F}_{\mathfrak{p}}}}\rightarrow
S$ reads

$i^{!}{\mathbb{Z}_{{\ell}}}=i^{*}{\mathbb{Z}_{{\ell}}}(-1)[-2].$ (5)

It is a reformulation of well-known cohomological properties of the inertia
group: $\mathrm{H}^{1}(I_{\mathfrak{p}},V)=(V(-1))_{I_{\mathfrak{p}}}$ for any
$\ell$-adic module with continuous $I_{\mathfrak{p}}$-action
($\mathfrak{p}\nmid\ell$). All higher group cohomologies of $I_{\mathfrak{p}}$
vanish.

Let $\mathbf{D}^{\mathrm{b}}(S,{\mathbb{Z}_{{\ell}}})$ be the bounded
“derived” category of ${\mathbb{Z}_{{\ell}}}$-sheaves on $S$ as constructed by
Ekedahl [Eke90]. All following constructions can be done for
${\mathbb{Q}_{{\ell}}}$ instead of ${\mathbb{Z}_{{\ell}}}$, as well. We keep
writing $j_{*}$ for the total derived functor, commonly denoted
$\mathrm{R}j_{*}$ etc. However, $\mathrm{R}^{n}j_{*}$ etc. keep their original
meaning.

As in loc. cit., see especially [2.2.10, 2.1.2, 2.1.3, 1.4.10]222In the
sequel, any reference in brackets refers to [BBD82]., one first defines a
notion of stratification, and secondly obtains a $t$-structure on the
subcategory $\mathbf{D}^{\mathrm{b}}_{(\Sigma,L)}(S,{\mathbb{Z}_{{\ell}}})$
that are constructible with respect to a given stratification
$\Sigma=\\{\Sigma_{i}\\}$ and a set $L$ of irreducible lisse sheaves on the
strata. Thirdly, one takes the “limit” over the stratifications. The union of
all $\mathbf{D}^{\mathrm{b}}_{(\Sigma,L)}(S,{\mathbb{Z}_{{\ell}}})$ is the
“derived” category $\mathbf{D}^{\mathrm{b}}_{c}(S,{\mathbb{Z}_{{\ell}}})$ of
constructible sheaves. In order to extend the $t$-structure on the
subcategories to one on
$\mathbf{D}^{\mathrm{b}}_{c}(S,{\mathbb{Z}_{{\ell}}})$, one has to check that
the inclusion
$\mathbf{D}^{\mathrm{b}}_{(\Sigma^{\prime},L^{\prime})}(S,{\mathbb{Z}_{{\ell}}})\rightarrow\mathbf{D}^{\mathrm{b}}_{(\Sigma,L)}(S,{\mathbb{Z}_{{\ell}}})$
is $t$-exact for any refinement of stratifications. Here we employ a different
argument. The proof of [2.1.14, 2.2.11] relies on relative purity for
$\ell$-adic sheaves [SGA73, Exp. XVI, 3.7]. As in the proof of [2.1.14] we
have to check the following: let $\Sigma_{i}\stackrel{{\scriptstyle
a}}{{\rightarrow}}\Sigma^{\prime}_{i}\stackrel{{\scriptstyle
b}}{{{\longrightarrow}}}S$ be the inclusions of some strata and let
$C\in\mathbf{D}^{\mathrm{b},\geq
0}_{(\Sigma^{\prime},L^{\prime})}(S,{\mathbb{Z}_{{\ell}}})$. Then
$C\in\mathbf{D}^{\mathrm{b},\geq 0}_{(\Sigma,L)}(S,{\mathbb{Z}_{{\ell}}})$. We
can assume $\dim\Sigma_{i}=0$, $\dim\Sigma^{\prime}_{i}=1$, since all other
cases are clear. Thus, $b$ is an open immersion. We may also assume for
notational simplicity that $\Sigma_{i}={\mathrm{Spec}\text{
}}{{\mathbb{F}_{\mathfrak{p}}}}$. Let $j$ be the complementary open immersion
to $a$. By definition,
$\mathrm{H}^{n}b^{!}C=b^{!}\mathrm{H}^{n}C=b^{*}\mathrm{H}^{n}C$ is locally
constant and vanishes for $n<-1$. In the parlance of Galois modules this means
that, viewed as a $\pi_{1}(\Sigma^{\prime}_{i})$-representation, the action of
the inertia group $I_{\mathfrak{p}}\subset\pi_{1}(\Sigma^{\prime}_{i})$ on
that sheaf is trivial. Thus

$a^{!}\mathrm{H}^{n}b^{*}C=a^{*}(\mathrm{R}^{1}j_{*}j^{*}\mathrm{H}^{n}b^{*}C)[-2]=\mathrm{H}^{1}(I_{\mathfrak{p}},\mathrm{H}^{n}b^{*}C)[-2]=a^{*}\mathrm{H}^{n}b^{*}C(-1)[-2].$

(We have used $\mathfrak{p}\nmid\ell$ at this point.) The spectral sequence

$\mathrm{H}^{p-2}a^{*}\mathrm{H}^{q}b^{!}C(-1)=\mathrm{H}^{p}a^{!}\mathrm{H}^{q}b^{!}C\Rightarrow\mathrm{H}^{n}a^{!}b^{!}C$

is such that the left hand term vanishes for $p\neq 2$ since $a^{*}$ is exact
w.r.t. the standard $t$-structure. It also vanishes for $q<-1$ by the above.
Hence the right hand term vanishes for $n=p+q<1$. A fortiori it vanishes for
$n<-\dim{\mathbb{F}_{\mathfrak{p}}}=0$.

Objects in the heart of this $t$-structure on
$\mathbf{D}^{\mathrm{b}}_{c}(S,{\mathbb{Z}_{{\ell}}})$ are called _perverse
sheaves_ on $S$. For example ${\mathbb{Z}_{{\ell}}}[1]$ and
$i_{*}{\mathbb{Z}_{{\ell}}}$ for any immersion $i$ of a closed point are
perverse sheaves on $S$. The _Verdier dual_ of any
$\mathcal{C}\in\mathbf{D}^{\mathrm{b}}_{c}(S,{\mathbb{Z}_{{\ell}}})$ is
defined by $D(C):=\underline{\mathrm{Hom}}(C,{\mathbb{Z}_{{\ell}}}(1)[2])$. As
above, we have dropped “$\mathrm{R}$” from the notation, so that this
$\underline{\mathrm{Hom}}$ means what is usually denoted
$\underline{{\mathrm{R}}{\mathrm{Hom}}}$.

######  Lemma 3.1.

Let $j:S^{\prime}\rightarrow S$ be an open immersion and $i:Z\rightarrow S$ a
closed immersion. Let $\eta:{\mathrm{Spec}\text{ }}{F}\rightarrow S$ be the
generic point. Then $j_{*}$, $j_{!}$, $i_{*}$, $\eta^{*}[-1]$, $j^{*}$ and $D$
are $t$-exact, while $i^{*}$ ($i^{!}$) is of cohomological amplitude $[-1,0]$
($[0,1]$), in particular right-exact (left- exact, respectively). Finally, the
$t$-structure on $\mathbf{D}^{\mathrm{b}}_{c}(S,{\mathbb{Z}_{{\ell}}})$ is
non-degenerate.

######  Proof: .

The only non-formal statement is the exactness of $j_{*}$. The corresponding
precursor result [4.1.10] is a reformulation of [SGA72, Th. 3.1., Exp. XIV],
which says for any affine map $j:X\rightarrow Y$ over schemes over a field
$K$, and any (honest) sheaf $\mathcal{F}$ which is torsion (prime to
$\operatorname{char}K$)

$d(\mathrm{R}^{q}j_{*}\mathcal{F}))\leq d(\mathcal{F})-q$

where
$d(\mathcal{G}):=\sup\\{\dim{\overline{\\{x\\}}},\mathcal{G}_{{\overline{x}}}\neq
0\\}$ for any sheaf $\mathcal{G}$. In our situation, we are given a locally
constant sheaf $\mathcal{F}$ on $S^{\prime}$ whose torsion is prime to all
characteristics of $S$. The conclusion of the theorem also holds for $j$, as
follows from the cohomological dimension of $I_{\mathfrak{p}}$, which is one.
∎

Let $\mathcal{F}$ be any perverse sheaf on $S^{\prime}$. Following [1.4.22],
let the _intermediate extension_ $j_{!*}\mathcal{F}$ be the image of the map
$j_{!}\mathcal{F}\rightarrow j_{*}\mathcal{F}$ of perverse sheaves on $S$. As
in [2.1.11] one sees that it can be calculated in terms of the good truncation
with respect to the standard $t$-structure:
$j_{!*}\mathcal{F}=\tau_{\leq-1}^{can}j_{*}\mathcal{F}.$ If
$\mathcal{F}=\mathcal{G}[1]$, where $\mathcal{G}$ is a lisse (honest) sheaf on
$S^{\prime}$, this gives $(\mathrm{R}^{0}j_{*}\mathcal{G})[1]$.

##  4 Mixed motives

Throughout this section, let $S={\mathrm{Spec}\text{ }}{F}$ or
${\mathrm{Spec}\text{ }}{{\mathbb{F}_{\mathfrak{p}}}}$ or an open subscheme of
${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$.

This section formulates a number of axioms concerning weights and the motivic
$t$-structure on triangulated categories of motives over $S$. In contrast to
the axioms listed in Section 1, the axioms mentioned in this section are wide
open, so it might be more appropriate to call them conjectures instead.

######  Axiom 4.1.

(Motivic $t$-structure and cohomological dimension) The category of geometric
motives $\mathbf{DM}_{\mathrm{gm}}(S)$ has a non-degenerate $t$-structure
[BBD82, Def. 1.3.1] called _motivic $t$-structure_. Its heart is denoted
$\mathbf{MM}(S)$. Objects of $\mathbf{MM}(S)$ are called _mixed motives_ over
$S$.

For any $M\in\mathbf{DM}_{\mathrm{gm}}(S)$, there are $a$, $b\in\mathbb{Z}$
such that $\tau_{\leq a}M=\tau_{\geq b}M=0$. Here and in the sequel,
$\tau_{\leq-}$ and $\tau_{\geq-}$ denote the truncation functors with respect
to the motivic $t$-structure.

The _cohomological dimension_ of
$\mathbf{DM}_{\mathrm{gm}}({\mathbb{F}_{\mathfrak{p}}})$ and
$\mathbf{DM}_{\mathrm{gm}}(F)$ is 0 and 1, respectively, in the sense that

$\mathrm{Hom}_{\mathbf{DM}({\mathbb{F}_{\mathfrak{p}}})}(M,N[i])=0$

for all mixed motives $M,N$ over ${\mathbb{F}_{\mathfrak{p}}}$ and $i>0$ and
similarly for mixed motives over $F$ and $i>1$. (For $i<0$ the term vanishes
by the $t$-structure axioms.)

The $t$-structures are such that over $S=\mathrm{Spec}\text{ }F$ or
${\mathrm{Spec}\text{ }}{{\mathbb{F}_{\mathfrak{p}}}}$,
$\mathbf{1}\in\mathbf{MM}(S)$, while for an open subscheme
$S\subset{\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$,
$\mathbf{1}[1]\in\mathbf{MM}(S)$.

The existence of the motivic $t$-structure on $\mathbf{DM}_{\mathrm{gm}}(K)$
satisfying the axioms listed in this section is part of the general motivic
conjectural framework, see e.g. [Beĭ87a, App. A], [And04, Ch. 21]. The idea of
building a triangulated category of motives and descending to mixed motives by
means of a $t$-structure is due to Deligne. The existence of a motivic
$t$-structure on $\mathbf{DM}_{\mathrm{gm}}(K)$ is only known in low
dimensions: the subcategory of Artin motives, i.e., motives of zero-
dimensional varieties, carries such a $t$-structure [Voe00, Section 3.4.]. By
loc. cit., [Org04], the subcategory of $\mathbf{DM}_{\mathrm{gm}}(K)$
generated by motives of smooth varieties of dimension $\leq 1$ is equivalent
to the bounded derived category of 1-motives [Del74, Section 10] up to
isogeny. Finally, if $K$ is a field satisfying the Beilinson-Soulé vanishing
conjecture, such as a finite field or a number field, the category of Artin-
Tate motives over $K$ enjoys a motivic $t$-structure [Lev93, Wil]. The results
on Artin-Tate motives are generalized to bases $S$ which are open subschemes
of ${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$ in [Scha].

The conjecture about the cohomological dimension is due to Beilinson. A
(fairly weak) evidence for this conjecture is the cohomological dimension of
Tate motives over $F$ and ${\mathbb{F}_{\mathfrak{p}}}$, which is one and
zero, respectively. This follows from vanishing properties of $K$-theory of
these fields.

The normalization in the last item is merely a matter of bookkeeping, but is
motivated by similar shifts in perverse sheaves (Section 3). The existence of
a motivic $t$-structure is not expected to hold for motives with integral
coefficients.

We do not (need to) assume that the canonical functor
$\mathbf{D}^{\mathrm{b}}(\mathbf{MM}(S))\rightarrow\mathbf{DM}_{\mathrm{gm}}(S)$
is an equivalence of categories or, equivalently [Beĭ87b, Lemma 1.4.],
$\operatorname{Ext}^{i}_{\mathbf{MM}(S)}(A,B)=\mathrm{Hom}_{\mathbf{DM}_{\mathrm{gm}}(S)}(A,B[i])$
for all mixed motives $A$ and $B$.

######  Axiom 4.2.

(Exactness properties) Let $S\subset{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}$ be an open subscheme, let $i:{\mathrm{Spec}\text{
}}{{\mathbb{F}_{\mathfrak{p}}}}\rightarrow{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}$ be a closed point, $j:U\rightarrow S$ an open immersion
and $\eta:\mathrm{Spec}\text{ }F\rightarrow S$ the generic point.

Then $j^{*}=j^{!}$, $\eta^{*}[-1]$, $i_{*}$, $j_{*}$ and $j_{!}$ are exact
with respect to the motivic $t$-structures on the involved categories of
geometric motives. Further, $i^{*}$ is right-exact, more precisely it maps
objects in cohomological degree $0$ to degrees $[-1,0]$. Dually, $i^{!}$ has
cohomological amplitude $[0,1]$. Verdier duality $D$ is “anti-exact”, i.e.,
maps objects in positive degrees to ones in negative degrees and vice versa.

The axiom is motivated by the same exactness properties in the situation of
perverse sheaves over ${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}[1/\ell]$
(Section 3). The corresponding exactness properties of the above functors on
Artin-Tate motives, where the motivic $t$-structure is available, is
established in [Scha].

######  Definition 4.3.

The cohomology functor with respect to the motivic $t$-structure on
$\mathbf{DM}_{\mathrm{gm}}(S)$ is denoted ${{{}^{\mathrm{p}}}\mathrm{H}}^{*}$.
For any scheme $X/S$, we write

$\mathrm{h}^{i}(X,n):={{{}^{\mathrm{p}}}\mathrm{H}}^{i}\operatorname{M}_{S}(X)(n).$

######  Axiom 4.4.

(Hard Lefschetz) Let $X_{\eta}/F$ be smooth and projective of constant
dimension $d_{\eta}$. Let $i\leq d_{\eta}$ and $a$ any integer. Then, taking
the $(d_{\eta}-i)$-fold cup product with the cycle class of a hyperplane
section with respect to an embedding of $X_{\eta}$ into some projective space
over $F$ yields an isomorphism (“hard Lefschetz isomorphism”)

$\mathrm{h}^{i}(X_{\eta},a)\stackrel{{\scriptstyle\cong}}{{{\longrightarrow}}}\mathrm{h}^{2d_{\eta}-i}(X_{\eta},d_{\eta}-i+a).$

######  Axiom 4.5.

(Decomposition of smooth projective varieties) Let $X/S$ be smooth and
projective. In $\mathbf{DM}_{\mathrm{gm}}(S)$, there is a non-canonical
isomorphism

$\phi_{X}:\operatorname{M}(X)\cong\bigoplus_{n}\mathrm{h}^{n}(X)[-n].$

For open subschemes $S\subset{\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$, this
isomorphism is compatible with pullbacks along all closed points
$i:{\mathrm{Spec}\text{ }}{{\mathbb{F}_{\mathfrak{p}}}}\rightarrow S$ in the
following sense: let $X_{\mathfrak{p}}$ be the fiber of $X$ over
${\mathbb{F}_{\mathfrak{p}}}$, and let $\psi$ be the isomorphism making the
following diagram commutative. Its left hand isomorphism is an instance of
base change.

$\textstyle{i^{*}\operatorname{M}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}\phi_{X}}$$\scriptstyle{\cong}$$\textstyle{\oplus_{n}i^{*}\mathrm{h}^{n}(X)[-n]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{\operatorname{M}(X_{\mathfrak{p}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{X_{\mathfrak{p}}}}$$\textstyle{\oplus_{m}\mathrm{h}^{m}(X_{\mathfrak{p}})[-m]}$

Then $\psi$ respects the direct summands, i.e., induces isomorphisms

$i^{*}\mathrm{h}^{n}(X)[-n]\cong\mathrm{h}^{n-1}(X_{\mathfrak{p}})[-n+1].$

######  Remark 4.6.

Axiom 4.4 is part of the classical conjectural framework of motives (see e.g.
[And04, Section 5.2, Prop. 5.2.5.1]). It implies the decomposition of Axiom
4.5 [Del94]. The latter is used in Lemma 5.11, which in turn is crucial in
Section 6. The index shift in Axiom 4.5 is due to the normalization in Axiom
4.1: for $S={\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$ and a closed point $i$
as above, take for example $X=S$,
$\operatorname{M}(S)=\mathbf{1}=\mathrm{h}^{1}(S)[-1]$ (sic) and
$i^{*}\operatorname{M}(S)=\mathbf{1}_{{\mathbb{F}_{\mathfrak{p}}}}=\mathrm{h}^{0}(\mathrm{Spec}\text{
}{\mathbb{F}_{\mathfrak{p}}})$.

Consider any smooth projective variety $X_{\eta}$ over $F$. Using that the
cohomological dimension of $\mathbf{DM}_{\mathrm{gm}}(F)$ is one, the term
$\mathrm{H}^{2n}(X_{\eta},n)$, that is identified with
$\mathrm{CH}^{n}(X_{\eta})_{\mathbb{Q}}$ by Axiom 1.8, appears in an exact
sequence of the following form

$0\rightarrow\mathrm{H}^{1}(\mathrm{h}^{2n-1}(X_{\eta},n))\rightarrow\mathrm{H}^{2n}(X_{\eta},n)\rightarrow\mathrm{H}^{0}(\mathrm{h}^{2n}(X_{\eta},n))\rightarrow
0.$

Consider the cycle class map from the $m$-th Chow group to $\ell$-adic
cohomology of $X_{\eta}$,
$\mathrm{CH}^{m}(X_{\eta})_{\mathbb{Q}}\rightarrow\mathrm{H}^{2m}(X_{\eta},{\mathbb{Q}_{{\ell}}}(m))$
[Mil80, VI.9]. The groups
$\mathrm{CH}^{m}(X_{\eta})_{\mathbb{Q},\mathrm{hom}}$ and
$\mathrm{CH}^{m}(X_{\eta})_{\mathbb{Q}}/\mathrm{hom}$ are by definition the
kernel and the image (seen as a quotient of the Chow group) of this map.
According to standard conjectures, homological equivalence should not depend
on the choice of the cohomology theory, since they are all supposed to agree
with numerical equivalence, see e.g. [And04, Chapter 5].

######  Axiom 4.7.

(Morphisms in $\mathbf{MM}(F)$ and Chow groups) For such $X_{\eta}$, the
sequence above identifies with

$0\rightarrow\mathrm{CH}^{n}(X_{\eta})_{\mathbb{Q},\mathrm{hom}}\rightarrow\mathrm{CH}^{n}(X_{\eta})_{\mathbb{Q}}\rightarrow\mathrm{CH}^{n}(X_{\eta})_{\mathbb{Q}}/\mathrm{hom}\rightarrow
0.$

Let either $S$ be a field and let $C$ stand for the $\ell$-adic realization
(in case $\text{char }S\neq\ell$), Betti, de Rham or absolute Hodge
realization or let $S\subset{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}[1/\ell]$ be an open subscheme and let $C$ be the
$\ell$-adicrealization. We write
${\mathrm{R}}{\Gamma}_{C}:\mathbf{DM}_{\mathrm{gm}}(S)\rightarrow\mathbf{D}^{\mathrm{b}}(\mathcal{C})$
for the realization functor, where $\mathbf{D}^{\mathrm{b}}(\mathcal{C})$ is
understood as a placeholder of the target category of $C$. (We abuse the
notation insofar as that target category is not a derived category in the
strict sense when $C$ is the $\ell$-adic realization.) For all realizations
over a field, this category is endowed with the usual $t$-structure, e.g. on
$\mathbf{D}^{\mathrm{b}}_{c}(K,{\mathbb{Q}_{{\ell}}})$ for $\ell$-adic
realization. When $S$ is an open subscheme of ${\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}[1/\ell]$ and $C$ is the $\ell$-adic realization, we take
the perverse $t$-structure on
$\mathbf{D}^{\mathrm{b}}_{c}(S,{\mathbb{Q}_{{\ell}}})$ defined in Section 3.
Using this, we have the following axiom:

######  Axiom 4.8.

(Exactness of realization functors) Realization functors
${\mathrm{R}}{\Gamma}_{C}$ are exact with respect to the motivic $t$-structure
on $\mathbf{DM}_{\mathrm{gm}}(S)$. Equivalently, as the $t$-structure on
$\mathbf{D}^{\mathrm{b}}(\mathcal{C})$ is non-degenerate,
${\mathrm{R}}{\Gamma}_{C}({{{}^{\mathrm{p}}}\mathrm{H}}^{0}M)={{{}^{\mathrm{p}}}\mathrm{H}}^{0}{\mathrm{R}}{\Gamma}_{C}(M)$
for any geometric motive $M$ over $S$. On the left,
${{{}^{\mathrm{p}}}\mathrm{H}}^{0}$ denotes the cohomology functor belonging
to the motivic $t$-structure on $\mathbf{DM}_{\mathrm{gm}}(S)$, while on the
right hand side it is the one belonging to the $t$-structure on
$\mathbf{D}^{\mathrm{b}}(\mathcal{C})$.

This axiom is, if fairly loosely, motivated by a similar fact in the theory of
mixed Hodge modules: let $X$ be any complex algebraic variety. Then, under the
faithful “forgetful functor” from the derived category of mixed Hodge modules
to the derived category of constructible sheaves with rational coefficients

$\mathbf{D}^{\mathrm{b}}(\mathbf{MHM}(X))\rightarrow\mathbf{D}^{\mathrm{b}}_{c}(X,\mathbb{Q})$

the category $\mathbf{MHM}(X)$ corresponds to perverse sheaves on $X$.

Recall that in an abelian category $\mathcal{C}$, a morphism
$f:(X,W^{*})\rightarrow(Y,W^{*})$ between filtered objects is called _strict_
if $f(W^{n}X)=f(X)\cap W^{n}Y$ for all $n$.

######  Axiom 4.9.

(Weights) Any mixed motive $M$ over $S$ has a functorial finite exhaustive
separated filtration $W_{*}M$ called _weight filtration_ , i.e., a sequence of
subobjects in the abelian category $\mathbf{MM}(S)$

$0=W_{a}M\subset W_{a+1}M\subset\dots\subset W_{b}M=M.$

Any morphism between mixed motives is strict with respect to the weight
filtration.

Given two motives of pure weight, their tensor product is pure; its weight is
the sum of the two individual weights. In particular, the unit motive
$\mathbf{1}$ is pure of weight zero.

Let
${\mathrm{R}}{\Gamma}_{C}:\mathbf{DM}_{\mathrm{gm}}(S)\rightarrow\mathbf{D}^{\mathrm{b}}(\mathcal{C})$
be any realization functor that has a notion of weights (such as the
$\ell$-adic realization when $S={\mathrm{Spec}\text{
}}{{\mathbb{F}_{\mathfrak{p}}}}$ or the Hodge realization when
$S=\mathrm{Spec}\text{ }\mathbb{Q}$). Then

$\operatorname{gr}_{n}^{W}{\mathrm{R}}{\Gamma}_{C}(M)={\mathrm{R}}{\Gamma}_{C}(\operatorname{gr}_{n}^{W}M)$

for any mixed motive $M$ over $S$.

######  Definition 4.10.

For any $M\in\mathbf{MM}(S)$, we write $\operatorname{wt}(M)$ for the (finite)
set of integers $n$ such that $\operatorname{gr}^{W}_{n}M\neq 0$. For
$M\in\mathbf{DM}_{\mathrm{gm}}(S)$, define
$\operatorname{wt}(M):=\cup_{i\in\mathbb{Z}}\operatorname{wt}({{{}^{\mathrm{p}}}\mathrm{H}}^{i}(M))-i$.

######  Axiom 4.11.

(Preservation of weights) Let $f:X\rightarrow S$ be a quasi-projective map.
Then the functors $f_{!}f^{*}$ preserve negativity of weights, i.e., given a
geometric motive $M$ over $S$ with weights $\leq 0$, $f_{!}f^{*}M$ also has
weights $\leq 0$. Dually, $f_{*}f^{!}$ preserves positive weights.

In the particular case $S\subset{\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$
(open), let $j:U\rightarrow S$ and $\eta:{\mathrm{Spec}\text{ }}{F}\rightarrow
S$ be an open immersion into $S$ and the generic point of $S$, respectively.
Let $i:{\mathrm{Spec}\text{ }}{{\mathbb{F}_{\mathfrak{p}}}}\rightarrow S$ be a
closed point. Then, $i^{*}$ and $j_{!}$ preserve negativity of weights and
dually for $i^{!}$ and $j_{*}$. Finally, $j^{*}$ and $\eta^{*}$ both preserve
both positivity and negativity of weights.

The preceding weight axioms are motivated by the very same properties of
$\ell$-adic perverse sheaves on schemes over $\mathbb{C}$ or finite fields
[BBD82, 5.1.14], number fields [Hub97] as well as Hodge structures [Del74, Th.
8.2.4] and Hodge modules (see [PS08, Chapter 14.1] for a synopsis). In these
settings, actually $f_{!}$ and $f^{*}$ preserve negative weights, but we do
not need weights for motives over more general bases than the ones above. The
weight formalism we require is stronger than the one provided by the
differential-graded interpretation of $\mathbf{DM}_{\mathrm{gm}}$ over a field
[Bon09] or [BV08, 6.7.4].

######  Example 4.12.

For any projective (smooth) scheme $X$ of finite type over $S$, the weights of
$\mathrm{h}^{i}(X)(n)$ are $\leq i-2n$ ($\geq i-2n$, respectively).

######  Axiom 4.13.

(Mixed vs. pure motives) For any field $K$, the subcategory of pure objects in
$\mathbf{MM}(K)$ identifies with $\mathbf{M}_{\mathrm{num}}(K)$, the category
of numerical pure motives over $K$.

As a consequence of the weight filtration, every mixed motive is obtained in
finitely many steps by taking extensions of motives in
$\mathbf{M}_{\mathrm{num}}(K)$.

##  5 Motives over number rings

In the following sections we assume the axioms of Sections 1, 2, and 4. Unless
explicitly mentioned otherwise, let $S$ be an open subscheme of
${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$, let $i:{\mathrm{Spec}\text{
}}{{\mathbb{F}_{\mathfrak{p}}}}\rightarrow{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}$ be a closed point, $j:S^{\prime}\rightarrow S$ an open
subscheme and $\eta:\mathrm{Spec}\text{ }F\rightarrow S$ the generic point.

This section derives a number of basic results about motives over $S$ from the
axioms spelled out above. We define and study the _intermediate extension_
$j_{!*}:\mathbf{MM}(S^{\prime})\rightarrow\mathbf{MM}(S)$ in analogy to
perverse sheaves (Definition 5.3). An “explicit” set of generators of
$\mathbf{DM}_{\mathrm{gm}}(S)$ (Proposition 5.7) is obtained using $j_{!*}$.
We introduce a notion of _smooth motives_ (Definition 5.8), which should be
thought of as analogs of lisse sheaves. Using this notion, we extend the
intermediate extension to a functor $\eta_{!*}$ spreading out motives over $F$
with a certain smoothness property to motives over $S$, cf. Definition 5.14.
This functor will be the main technical tool in dealing with $f$-cohomology in
Section 6. In Lemma 5.16 we express the $\ell$-adic realization of motives of
the form $j_{!*}M$ in sheaf-theoretic language.

###  5.1 Cohomological dimension

The following is an immediate consequence of Axiom 4.2:

######  Lemma 5.1.

For any scheme $X$ over $S$ we have
$\eta^{*}[-1]\mathrm{h}^{i}(X,n)=\mathrm{h}^{i-1}(X{\times}_{S}F,n)$.

The following lemma parallels (and is a consequence of) Axiom 4.1.

######  Lemma 5.2.

The cohomological dimension of $\mathbf{DM}_{\mathrm{gm}}(S)$ is two, that is
to say, for any two mixed motives $M,N$ over $S$,

$\mathrm{Hom}_{\mathbf{DM}_{\mathrm{gm}}(S)}(M,N[i])=0$

for all $i>2$. In particular $\mathrm{H}^{i}(M)$ vanishes for $|i|>1$.

######  Proof: .

Apply $\mathrm{Hom}(M,-)$ to the localization triangle
$\oplus_{\mathfrak{p}\in
S}{i_{\mathfrak{p}}}_{*}i_{\mathfrak{p}}^{!}N\rightarrow
N\rightarrow\eta_{*}\eta^{*}N$ of Axiom 1.16, where $i_{\mathfrak{p}}$ are the
immersions of the closed points of $S$. The terms adjacent to
$\mathrm{Hom}(M,N[i])$ are
$\mathrm{Hom}(M,\oplus_{\mathfrak{p}}{i_{\mathfrak{p}}}_{*}i_{\mathfrak{p}}^{!}N[i])=\oplus_{\mathfrak{p}}\mathrm{Hom}(i_{\mathfrak{p}}^{*}M,i_{\mathfrak{p}}^{!}N[i])$
(as $M$ is compact) and
$\mathrm{Hom}(M,\eta_{*}\eta^{*}N[i])=\mathrm{Hom}(\eta^{*}M,\eta^{*}N[i])$.
The latter term vanishes for $i>1$ since $\eta^{*}[-1]$ is exact and the
cohomological dimension of $\mathbf{DM}_{\mathrm{gm}}(F)$ is one.

To deal with the former term, we have to take into account that
$i_{\mathfrak{p}}^{!}$ and $i_{\mathfrak{p}}^{*}$ are not $t$-exact, but of
cohomological amplitude $[0,1]$ and $[-1,0]$, respectively. By decomposing
$i_{\mathfrak{p}}^{!}N$ into its ${{{}^{\mathrm{p}}}\mathrm{H}}^{1}$\- and
${{{}^{\mathrm{p}}}\mathrm{H}}^{0}$-part and similarly with
$i_{\mathfrak{p}}^{*}M$ and using that the cohomological dimension of
$\mathbf{DM}_{\mathrm{gm}}({\mathbb{F}_{\mathfrak{p}}})$ is zero, the term
vanishes for $i>2$. Using general $t$-structure properties, the second claim
is a particular case of the first one. ∎

###  5.2 Intermediate extension

######  Definition 5.3.

(Motivic analog of [BBD82, Def. 1.4.22]) The _intermediate extension_ $j_{!*}$
of some mixed motive $M$ over $S^{\prime}$ is defined as

$j_{!*}M:=\mathrm{im}(j_{!}M\rightarrow j_{*}M).$

The image is taken in the abelian category $\mathbf{MM}(S)$, using the
exactness of $j_{!}$ and $j_{*}$, Axiom 4.2.

######  Remark 5.4.

Let $i:Z\rightarrow S$ be the complement of $j$. The localization triangles
(Axiom 1.10) and cohomological amplitude of $i^{*}$ (Axiom 4.2) yield short
exact sequences in $\mathbf{MM}(S)$

$0\rightarrow i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{-1}i^{*}j_{*}M\rightarrow
j_{!}M\rightarrow j_{!*}M\rightarrow 0,$ (6) $0\rightarrow j_{!*}M\rightarrow
j_{*}M\rightarrow i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{*}j_{*}M\rightarrow
0.$ (7)

These triangles are the same as for perverse sheaves in the situation that the
analog of Axiom 4.2, [BBD82, 4.1.10], is applicable.

######  Lemma 5.5.

  * •

Given any mixed motive $M$ over $S^{\prime}$, $j_{!*}M$ is, up to a unique
isomorphism, the unique mixed extension of $M$ (i.e., an object $X$ in
$\mathbf{MM}(S)$ such that $j^{*}X=M$) not having nonzero subobjects or
quotients of the form $i_{*}N$, where $i:Z\rightarrow S$ is the closed
complement of $j$ and $N$ is a mixed motive on $Z$.

  * •

For any two composable open immersions $j_{1}$ and $j_{2}$ we have
${j_{1}}_{!*}\circ{j_{2}}_{!*}=({j_{1}}\circ j_{2})_{!*}$.

  * •

$j_{!*}$ commutes with duals, i.e., $D(j_{!*}-)\cong j_{!*}D(-)$.

######  Proof: .

The proofs of the same facts for perverse sheaves [BBD82, Cor. 1.4.25,
2.1.7.1] carry over to this setting. The first statement easily implies the
last one. ∎

######  Lemma 5.6.

(compare [BBD82, Cor. 5.3.2]) The intermediate extension $j_{!*}$ preserves
weights of pure objects: given a pure motive $M\in\mathbf{MM}(S^{\prime})$,
$j_{!*}M$ is pure of the same weight.

######  Proof: .

This is a consequence of the triangles (6), (7) and the strictness of the
weight filtration. ∎

The following proposition makes precise the intuition that any motive $M$ over
$S$ should be reconstructed by its generic fiber (over $F$) and a finite
number of special fibers (over various ${\mathbb{F}_{\mathfrak{p}}}$).

######  Proposition 5.7.

As a thick subcategory of $\mathbf{DM}(S)$, $\mathbf{DM}_{\mathrm{gm}}(S)$ is
generated by motives of the form

  * •

$i_{*}\operatorname{M}(X_{\mathfrak{p}})(m)$, where
$X_{\mathfrak{p}}/{\mathbb{F}_{\mathfrak{p}}}$ is smooth projective,
$m\in\mathbb{Z}$ and $i:{\mathrm{Spec}\text{
}}{{\mathbb{F}_{\mathfrak{p}}}}\rightarrow S$ is any closed point and

  * •

$j_{!*}j^{*}\mathrm{h}^{k}(X,m)$, where $X$ is regular, flat and projective
over $S$ with smooth generic fiber, and $j:S^{\prime}\rightarrow S$ is such
that $X{\times}_{S}S^{\prime}$ is smooth over $S^{\prime}$ and $k$ and $m$ are
arbitrary.

######  Proof: .

Let $\mathcal{D}\subset\mathbf{DM}_{\mathrm{gm}}(S)$ be the thick category
generated by the objects in the statement. By resolution of singularities over
$S$ (Axiom 1.15), $\mathbf{DM}_{\mathrm{gm}}(S)$ is the thick subcategory of
$\mathbf{DM}(S)$ generated by objects
$i_{*}\operatorname{M}(X_{\mathfrak{p}})(m)$ and $\operatorname{M}(X)(m)$,
where $X_{\mathfrak{p}}$ and $X$ are as in the statement and $m\in\mathbb{Z}$.

It is therefore sufficient to see $M:=\operatorname{M}(X)\in\mathcal{D}$. Let
$j:S^{\prime}\rightarrow S$ be such that $X_{S^{\prime}}$ is smooth over
$S^{\prime}$. By 1.10 it is enough to show $j_{*}j^{*}M\in\mathcal{D}$, since
motives over finite fields are already covered. Applying the truncations with
respect to the motivic $t$-structure to $j_{*}j^{*}M$ and exactness of
$j_{*}$, $j^{*}$ (Axiom 4.2) shows that we may deal with
$j_{*}j^{*}\mathrm{h}^{k}(X,m)$ for all $k$ instead of $j_{*}j^{*}M$. (Only
finitely many $k$ yield a nonzero term by Axiom 4.1.) By Remark 5.4, there is
a short exact sequence of mixed motives

$0\rightarrow j_{!*}j^{*}\mathrm{h}^{k}(X,m)\rightarrow
j_{*}j^{*}\mathrm{h}^{k}(X,m)\rightarrow
i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{*}j_{*}j^{*}\mathrm{h}^{k}(X,m)\rightarrow
0.$

Here $i$ is the complement of $j$. The left and right hand motives are in
$\mathcal{D}$, hence so is the middle one. ∎

###  5.3 Smooth motives

The notion of smooth motives (a neologism leaning on lisse sheaves) is a
technical stepstone for the definition of the generic intermediate extension
$\eta_{!*}$, cf. Definition 5.14. Roughly speaking, smoothness for mixed
motives $M$ means that $i^{*}M$ and $i^{!}M$ do not intermingle in the sense
that their cohomological degrees are disjoint.

######  Definition 5.8.

Let $M$ be a geometric motive over $S$. It is called _smooth_ if for any
closed point $i:{\mathrm{Spec}\text{
}}{{\mathbb{F}_{\mathfrak{p}}}}\rightarrow S$ there is an isomorphism

$i^{!}M\cong i^{*}M(-1)[-2].$

$M$ is called _generically smooth_ if there is an open (non-empty) immersion
$j:S^{\prime}\rightarrow S$ such that $j^{*}M$ is smooth.

Let $X/S$ be a scheme with smooth generic fiber $X_{\eta}$. Then
$\operatorname{M}_{S}(X)$ is a generically smooth motive.

The isomorphism in Definition 5.8 is not required to be canonical in any
sense. Therefore, the subcategory of smooth motives is _not_ triangulated in
$\mathbf{DM}_{\mathrm{gm}}(S)$.

######  Lemma 5.9.

Let $M$ be a smooth mixed motive over $S$. Let $i:Z\rightarrow S$ be proper
closed subscheme, let $j:S^{\prime}\rightarrow S$ be its complement. Then
$i^{!}M=({{{}^{\mathrm{p}}}\mathrm{H}}^{1}i^{!}M)[-1]$ and dually
$i^{*}M=({{{}^{\mathrm{p}}}\mathrm{H}}^{-1}i^{*}M)[1]$.

######  Proof: .

By assumption $i^{!}M\cong i^{*}M(-1)[-2]$. By Axiom 4.2, the left hand side
of that isomorphism is concentrated in degrees $[0,1]$. The right hand side is
in degrees $[1,2]$. This shows
$i^{!}M={{{}^{\mathrm{p}}}\mathrm{H}}^{1}(i^{!}M)[-1]$ by Axiom 4.1 and
similarly for $i^{*}M$. ∎

The following is the key relation of smooth motives and the intermediate
extension. Note the similarity with Lemma 5.15.

######  Lemma 5.10.

Let $M$ be a smooth mixed motive over $S$. Then $M$ is canonically isomorphic
to $j_{!*}j^{*}M$.

######  Proof: .

Let $i:Z\rightarrow S$ be the complement of $j$. Given any $i_{*}N\subset M$
with $N\in\mathbf{MM}(Z)$, we apply the left-exact functor $i^{!}$ and see
$N\subset{{{}^{\mathrm{p}}}\mathrm{H}}^{0}(i^{!}M)\stackrel{{\scriptstyle\ref{lemm_basic}}}{{=}}0$.
Quotients of $M$ of the form $i_{*}N$ are treated dually. We now invoke Lemma
5.5. ∎

######  Lemma 5.11.

Let $X$ be any smooth projective scheme over $S$. Set
$M:=\operatorname{M}(X)$. Then all
$\mathrm{h}^{n}X={{{}^{\mathrm{p}}}\mathrm{H}}^{n}M$ are smooth.

######  Proof: .

Let $f_{m,n}$ be the $(m,n)$-component of the bottom isomorphism making the
following commutative:

$\textstyle{i^{!}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong,\text{
see
(\ref{eqn_purityExample})}}$$\scriptstyle{\cong,\ref{axio_hardLefschetz}}$$\textstyle{i^{*}M(-1)[-2]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong,\ref{axio_hardLefschetz}}$$\textstyle{\oplus_{m}A_{m}:=\oplus
i^{!}({{{}^{\mathrm{p}}}\mathrm{H}}^{m}M)[-m]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\oplus_{n}B_{n}:=\oplus
i^{*}({{{}^{\mathrm{p}}}\mathrm{H}}^{n}M)(-1)[-n-2].}$

We claim $f_{m,n}=0$ for all $m\neq n$, from which the lemma follows. By Axiom
4.5 we have $B_{n}\cong\mathrm{h}^{n-1}(X_{\mathfrak{p}})[-n-1](-1)$. Using
this and the reflexivity of the Verdier dual functor, we obtain an isomorphism
$A_{m}\cong({{{}^{\mathrm{p}}}\mathrm{H}}^{m+1}i^{!}M)[-1-m]$. Hence $B_{n}$
is concentrated in cohomological degree $n+1$, while $A_{m}$ is in degree
$n+2$. (The a priori bounds of Axiom 4.2 would be $[m,m+1]$ and $[n+1,n+2]$,
respectively.) As the cohomological dimension of motives over
${\mathbb{F}_{\mathfrak{p}}}$ is zero (Axiom 4.1), the only way for
$f_{m,n}\neq 0$ is $m=n$. ∎

###  5.4 Generic intermediate extension

######  Lemma 5.12.

(Spreading out morphisms) Given two geometric motives $M$ and $M^{\prime}$
over $S$ together with a map
$\phi_{\eta}:\eta^{*}M\rightarrow\eta^{*}M^{\prime}$, there is an open
subscheme $j:S^{\prime}\subset S$ and a map
$\phi_{S^{\prime}}:j^{*}M\rightarrow j^{*}M^{\prime}$ which extends
$\phi_{\eta}$. Any two such extensions agree when restricted to a possibly
smaller open subscheme. In particular, if $\phi_{\eta}$ is an isomorphism,
then $\phi_{S^{\prime}}$ is an isomorphism for sufficiently small
$S^{\prime}$.

######  Proof: .

The adjunction map $M\rightarrow\eta_{*}\eta^{*}M$ and $\eta_{*}\phi_{\eta}$
give a map $M\rightarrow\eta_{*}\eta^{*}M^{\prime}$, hence by (4) a map
$M\rightarrow\oplus_{\mathfrak{p}}{i_{\mathfrak{p}}}_{*}i_{\mathfrak{p}}^{!}M^{\prime}[1]$.
As $M$ is compact, it factors over a finite sum $\oplus_{\mathfrak{p}\in
T}{i_{\mathfrak{p}}}_{*}i_{\mathfrak{p}}^{!}M^{\prime}[1]$. Let
$j:S^{\prime}\rightarrow S$ be the complement of the points in $T$. The map
$M\rightarrow\eta_{*}\eta^{*}M^{\prime}$ factors over ${j}_{*}j^{*}M^{\prime}$
and gives a map $j^{*}M\rightarrow j^{*}M^{\prime}$ which extends
$\phi_{\eta}$. The first claim is shown.

For the unicity of the extension, we may assume that $\phi_{\eta}$ is zero,
and show that $\phi_{S^{\prime}}$ is zero for some suitable $S^{\prime}$. This
is the same argument as before: the map $M\rightarrow{j}_{*}j^{*}M^{\prime}$
constructed in the previous step factors over $\oplus_{\mathfrak{p}\in
S^{\prime}}{i_{\mathfrak{p}}}_{*}i_{\mathfrak{p}}^{!}M^{\prime}$, since
$M\rightarrow\eta_{*}\eta^{*}M^{\prime}$ is zero. By compacity of $M$, only
finitely many primes in the sum contribute to the map, discarding these yields
the claim.

If $\phi_{\eta}$ is an isomorphism, $\psi_{\eta}:=\phi_{\eta}^{-1}$ can be
extended to some $\psi_{S^{\prime}}$. As both
$\phi_{S^{\prime}}\circ\psi_{S^{\prime}}$ and $\mathrm{id}_{S^{\prime}}$
extend $\mathrm{id}_{\eta}$, they agree on some possibly smaller open
subscheme of $S$ and similarly with $\psi_{S^{\prime}}\circ\phi_{S^{\prime}}$.
∎

######  Remark 5.13.

The lemma shows the full faithfulness of the functor

$\varinjlim_{S^{\prime}\subset
S}\mathbf{DM}_{\mathrm{gm}}(S^{\prime})\stackrel{{\scriptstyle\eta^{*}}}{{{\longrightarrow}}}\mathbf{DM}_{\mathrm{gm}}(F).$

Its essential surjectivity is a consequence of Axiom 1.5, so we have an
equivalence. However, we will stick to the more basic language of colimits in
$\mathbf{DM}(S)$ instead of colimits of the categories of geometric motives.

######  Definition 5.14.

Let $M_{\eta}\in\mathbf{DM}_{\mathrm{gm}}(F)$ be a motive such that there
exists a generically smooth mixed motive $M$ over $S$ (Definition 5.8) with
$\eta^{*}M\cong M_{\eta}$. Then the _generic intermediate extension_
$\eta_{!*}M_{\eta}$ is defined as

$\eta_{!*}M_{\eta}:=j_{!*}j^{*}M$

where $j:S^{\prime}\rightarrow S$ is an open immersion such that $j^{*}M$ is
smooth.

This is independent of the choices of $j$ and $M$ (Lemmas 5.10, 5.12) and
functorial (5.12). For a mixed, non-smooth motive $M$, there need not be a map
$j_{!*}j^{*}M\rightarrow M$. Therefore, $\varinjlim j_{!*}j^{*}M$ does not
make sense unless there is some smoothness constraint on $M_{\eta}$.

###  5.5 Intermediate extension and $\ell$-adic realization

This subsection deals with the interplay of the (generic) intermediate
extension functor on mixed motives and the $\ell$-adic realization. In this
subsection, $S$ is an open subscheme of ${\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}[1/\ell]$. The following lemma is well-known.

######  Lemma 5.15.

Let $\mathcal{F}$ be an étale (honest) locally constant sheaf on $S$. Let
$\eta:{\mathrm{Spec}\text{ }}{F}\rightarrow{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}[1/\ell]$ be the generic point. Then the canonical map
$\mathcal{F}\rightarrow\mathrm{R}^{0}\eta_{*}\eta^{*}\mathcal{F}$ is an
isomorphism.

######  Lemma 5.16.

Let $M$ be a mixed motive over $S^{\prime}$. Let $j:S^{\prime}\rightarrow S$
be an open immersion. Then

$(j_{!*}M)_{\ell}=j_{!*}(M_{\ell}).$

Let $i$ be the complementary closed immersion to $j:S^{\prime}\rightarrow S$
and let $\eta^{\prime}$ and $\eta$ be the generic point of $S^{\prime}$ and
$S$, respectively. If $M$ is additionally smooth, one has

$(i^{*}j_{!*}M)_{\ell}=i^{*}j_{!*}M_{\ell}=i^{*}(\mathrm{R}^{0}\eta_{*}\eta^{\prime*}M_{\ell}[-1])[1].$

To clarify the statement, note that the $\ell$-adic realization of $M$ is a
perverse sheaf on $S^{\prime}$ by Axiom 4.8. Thus, $j_{!*}$ (Section 3) can be
applied to it.

######  Proof: .

The first statement follows from Axiom 2.1, the definition of $j_{!*}$ and the
exactness of ${\mathrm{R}}{\Gamma}_{\ell}$ (Axiom 4.8).

Let now $M$ be mixed and smooth over $S^{\prime}$. As $M_{\ell}$ is a perverse
sheaf by 4.8, there is an open immersion
$j^{\prime}:S^{\prime\prime}\rightarrow S^{\prime}$ such that
$j^{\prime*}M_{\ell}[-1]$ is a locally constant (honest) sheaf on
$S^{\prime\prime}$. As $M$ is smooth we know from Lemmas 5.5 and 5.10

$i^{*}j_{!*}M=i^{*}(j\circ j^{\prime})_{!*}j^{\prime*}M.$

By the interpretation of $(j\circ j^{\prime})_{!*}$ in terms of
$\mathrm{R}^{0}(j\circ j^{\prime})_{*}$ (Section 3) we have

$(i^{*}j_{!*}M)_{\ell}=i^{*}j_{!*}M_{\ell}=i^{*}(\mathrm{R}^{0}(j\circ
j^{\prime})_{*}j^{\prime*}M_{\ell}[-1])[1]\stackrel{{\scriptstyle\text{\ref{lemm_locconst}}}}{{=}}i^{*}(\mathrm{R}^{0}\eta_{*}\eta^{\prime*}M_{\ell}[-1])[1].$

∎

##  6 $f$-cohomology

###  6.1 $f$-cohomology via non-ramification

For any place $\mathfrak{p}$ of $F$, let $F_{\mathfrak{p}}$ be the completion,
$G_{\mathfrak{p}}$ the local Galois group. For finite places,
$I_{\mathfrak{p}}$ denotes the inertia group. For brevity, we will usually
write $\mathrm{H}^{*}(M)$ for $\mathrm{H}^{*}(S,M)$, where $M$ is any motive
over some base $S$.

######  Definition 6.1.

[BK90, Section 3] Let $V$ be a finite-dimensional $\ell$-adic vector space,
endowed with a continuous action of $G_{\mathfrak{p}}$, where $\mathfrak{p}$
is a finite place of $F$ not over $\ell$. Set

$\mathrm{H}^{i}_{f}(F_{\mathfrak{p}},V):=\left\\{\begin{array}[]{ll}\mathrm{H}^{0}(F_{\mathfrak{p}},V)&i=0\\\
\ker\mathrm{H}^{1}(F_{\mathfrak{p}},V)\rightarrow\mathrm{H}^{1}(I_{\mathfrak{p}},V)&i=1\\\
0&\text{else.}\end{array}\right.$

######  Remark 6.2.

If $\mathfrak{p}$ lies over $\ell$, the definition is completed by
$\mathrm{H}^{1}_{f}(F_{\mathfrak{p}},V):=\ker\mathrm{H}^{1}(F_{\mathfrak{p}},V)\rightarrow\mathrm{H}^{1}(F_{\mathfrak{p}},B_{crys}{\otimes}V)$,
where $B_{crys}$ denotes the ring of $\mathfrak{p}$-adic periods [FM87]. We
will disregard this case throughout.

######  Lemma 6.3.

Let $\eta_{\mathfrak{p}}:\mathrm{Spec}\text{
}F_{\mathfrak{p}}\rightarrow\mathrm{Spec}\text{
}{\mathcal{O}_{F_{\mathfrak{p}}}}$ be the generic point of the completion of
${\mathcal{O}_{F}}$ at $\mathfrak{p}$. Using the above notation, for
$\mathfrak{p}$ not over $\ell$, there is a canonical isomorphism
$\mathrm{H}^{1}_{f}(F_{\mathfrak{p}},V)\cong\mathrm{H}^{1}({\mathcal{O}_{F_{\mathfrak{p}}}},\mathrm{R}^{0}{\eta_{\mathfrak{p}}}_{*}V)$.
(The right hand side denotes $\ell$-adic cohomology over
${\mathcal{O}_{F_{\mathfrak{p}}}}$.)

######  Proof: .

Let $U_{n}$ be an $\ell^{n}$-torsion sheaf on $F_{\mathfrak{p}}$. In this
proof, we will write

$\mathrm{H}^{1}_{f}(F_{\mathfrak{p}},U_{n}):=\ker\mathrm{H}^{1}(F_{\mathfrak{p}},U_{n})\rightarrow\mathrm{H}^{1}(I_{\mathfrak{p}},U_{n}),$

i.e., the same definition as for ${\mathbb{Q}_{{\ell}}}$-sheaves is done. The
${\mathbb{Q}_{{\ell}}}$-sheaf $V$ is, by definition, of the form
$U{\otimes}_{\mathbb{Z}_{{\ell}}}{\mathbb{Q}_{{\ell}}}$, where $U=(U_{n})_{n}$
is a projective system of $\mathbb{Z}/\ell^{n}$-sheaves. By definition

$\mathrm{H}^{1}(F_{\mathfrak{p}},V)=\varprojlim_{n\in\mathbb{N}}\mathrm{H}^{1}(F_{\mathfrak{p}},U_{n}){\otimes}{\mathbb{Q}_{{\ell}}}$

and similarly for $\mathrm{H}^{1}(I_{\mathfrak{p}},V)$. Both $\varprojlim_{n}$
and $-{\otimes}_{\mathbb{Z}_{{\ell}}}{\mathbb{Q}_{{\ell}}}$ are left-exact
functors, so one has

$\mathrm{H}^{1}_{f}(F_{\mathfrak{p}},V)=\left(\varprojlim_{n}\mathrm{H}^{1}_{f}(F_{\mathfrak{p}},U_{n})\right){\otimes}{\mathbb{Q}_{{\ell}}}.$

Thus we may henceforth assume that $V$ is a $\mathbb{Z}/\ell^{n}$-sheaf.

Recall the description of étale sheaves on ${\mathcal{O}_{F_{\mathfrak{p}}}}$
from [Mil80, II.3.12, II.3.16]. Let $i:{\mathrm{Spec}\text{
}}{{\mathbb{F}_{\mathfrak{p}}}}\rightarrow\mathrm{Spec}\text{
}{\mathcal{O}_{F_{\mathfrak{p}}}}$ be the closed point. As
${\mathcal{O}_{F_{\mathfrak{p}}}}$ is a henselian ring [Mil80, Prop. I.4.5],
for any sheaf $\mathcal{F}$ on $\mathrm{Spec}\text{
}{\mathcal{O}_{F_{\mathfrak{p}}}}$, the global sections depend only on the
special fiber and

$\Gamma_{\mathrm{Spec}\text{ }F_{\mathfrak{p}}}=\Gamma_{\mathrm{Spec}\text{
}{\mathcal{O}_{F_{\mathfrak{p}}}}}\circ({\eta_{\mathfrak{p}}}_{*})=\Gamma_{\mathrm{Spec}\text{
}{\mathcal{O}_{F_{\mathfrak{p}}}}}\circ(i_{*}i^{*}{\eta_{\mathfrak{p}}}_{*}).$

These functors can be interpreted using group cohomology:
$\Gamma_{\mathrm{Spec}\text{ }{\mathcal{O}_{F_{\mathfrak{p}}}}}\circ
i_{*}=\Gamma_{{\mathbb{F}_{\mathfrak{p}}}}$ and
$(-)^{I_{\mathfrak{p}}}=i^{*}{\eta_{\mathfrak{p}}}_{*}$ (loc. cit.). The
Hochschild-Serre spectral sequence for
$(-)^{G_{\mathfrak{p}}}=(-)^{\mathrm{Gal}({\mathbb{F}_{\mathfrak{p}}})}\circ(-)^{I_{\mathfrak{p}}}$
can be translated to

$\mathrm{H}^{p}(\mathrm{Spec}\text{
}{\mathcal{O}_{F_{\mathfrak{p}}}},i_{*}i^{*}\mathrm{R}^{q}{\eta_{\mathfrak{p}}}_{*}V)\Rightarrow\mathrm{H}^{n}(F_{\mathfrak{p}},V).$

In addition we have the Leray spectral sequence

$\mathrm{H}^{p}(\mathrm{Spec}\text{
}{\mathcal{O}_{F_{\mathfrak{p}}}},\mathrm{R}^{q}{\eta_{\mathfrak{p}}}_{*}V)\Rightarrow\mathrm{H}^{n}(F_{\mathfrak{p}},V).$

The exact sequence of low degrees of the Hochschild-Serre sequence maps to the
sequence below:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(\mathrm{Spec}\text{
}{\mathcal{O}_{F_{\mathfrak{p}}}},\mathrm{R}^{0}{\eta_{\mathfrak{p}}}_{*}V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(F_{\mathfrak{p}},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{\mathrm{H}^{0}(\mathrm{Spec}\text{
}{\mathcal{O}_{F_{\mathfrak{p}}}},\mathrm{R}^{1}{\eta_{\mathfrak{p}}}_{*}V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}_{f}(F_{\mathfrak{p}},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(F_{\mathfrak{p}},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(I_{\mathfrak{p}},V)}$

As
$\mathrm{H}^{0}(\mathrm{Gal}({\mathbb{F}_{\mathfrak{p}}}),\mathrm{H}^{1}(I_{\mathfrak{p}},V))\subset\mathrm{H}^{1}(I_{\mathfrak{p}},V)$
and
$\Gamma_{\mathcal{O}_{F_{\mathfrak{p}}}}=\Gamma_{\mathcal{O}_{F_{\mathfrak{p}}}}\circ
i_{*}i^{*}$, the right hand map is injective, therefore there is a unique
isomorphism between the left hand terms making the diagram commutative. ∎

In order to proceed to a global level, the following definition is done:

######  Definition 6.4.

[FPR94, II.1.3] Given an $\ell$-adic continuous representation $V$ of
$G=\mathrm{Gal}(F)$, define $\mathrm{H}^{i}_{f}(F,V)$ to be such that the
following diagram is cartesian. In the lower row, $V$ is considered a
$G_{\mathfrak{p}}=\mathrm{Gal}(F_{\mathfrak{p}})$-module by restriction.

$\textstyle{\mathrm{H}^{i}_{f}(F,V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{i}(F,V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod\mathrm{H}^{i}_{f}(F_{\mathfrak{p}},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod\mathrm{H}^{i}(F_{\mathfrak{p}},V)}$

The product ranges over all finite places $\mathfrak{p}$ of $F$. We define
$\mathrm{H}^{i}_{f,\backslash\operatorname{crys}}(F,V)$ similarly, except that
in the lower row of the preceding diagram only places $\mathfrak{p}$ that do
not lie over $\ell$ occur.

######  Lemma 6.5.

Let $V$ be an $\ell$-adic étale sheaf on $\mathrm{Spec}\text{ }F$. Then there
is a natural isomorphism

$\mathrm{H}^{i}_{f,\backslash\operatorname{crys}}(F,V)\cong\mathrm{H}^{1}({\mathcal{O}_{F}}[1/\ell],\mathrm{R}^{0}{\eta}_{*}V).$

######  Proof: .

By the same argument as in the previous proof, we may assume that $V$ is a
sheaf of $\mathbb{Z}/\ell^{n}$-modules, since the isomorphism we are going to
establish is natural in $V$ and

$\mathrm{H}^{i}_{f,\backslash\operatorname{crys}}(F,V)=\ker\mathrm{H}^{i}(F,V)\rightarrow\prod_{\mathfrak{p}\nmid\ell}\left(\mathrm{H}^{i}(F_{\mathfrak{p}},V)/\mathrm{H}^{i}_{f}(F_{\mathfrak{p}},V)\right).$

Consider the following cartesian diagram ($\mathfrak{p}\nmid\ell$)

$\textstyle{\mathrm{Spec}\text{
}F_{\mathfrak{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{\mathfrak{p}}}$$\scriptstyle{b}$$\textstyle{\mathrm{Spec}\text{
}{\mathcal{O}_{F_{\mathfrak{p}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{\mathrm{Spec}\text{
}{\mathbb{F}_{\mathfrak{p}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{\mathfrak{p}}}$$\scriptstyle{=}$$\textstyle{\mathrm{Spec}\text{
}F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{\mathrm{Spec}\text{
}{\mathcal{O}_{F}}[1/\ell]}$$\textstyle{\mathrm{Spec}\text{
}{\mathbb{F}_{\mathfrak{p}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$

In the derived category of $\mathbb{Z}/\ell^{n}$-sheaves on
${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}[1/\ell]$, there is a triangle
$\mathrm{R}^{0}\eta_{*}V\rightarrow\mathrm{R}\eta_{*}V\rightarrow\mathrm{R}^{1}\eta_{*}[-1]V$.
Likewise,
$\mathrm{R}^{0}{\eta_{\mathfrak{p}}}_{*}b^{*}V\rightarrow{\mathrm{R}\eta_{\mathfrak{p}}}_{*}b^{*}V\rightarrow\mathrm{R}^{1}{\eta_{\mathfrak{p}}}_{*}b^{*}V[-1]$.
(We have used $\mathfrak{p}\nmid\ell$, since the inertia group has
cohomological dimension bigger than one for $\mathfrak{p}|\ell$.) This yields
exact horizontal sequences, the vertical maps are adjunction maps

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(\mathrm{Spec}\text{
}{\mathcal{O}_{F}}[1/\ell],{\eta}_{*}V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(F,V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{0}(\mathrm{Spec}\text{
}{\mathcal{O}_{F}}[1/\ell],\mathrm{R}^{1}\eta_{*}V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{\mathfrak{p}\nmid\ell}\mathrm{H}^{1}({\mathcal{O}_{F_{\mathfrak{p}}}},\mathrm{R}^{0}{\eta_{\mathfrak{p}}}_{*}b^{*}V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{\mathfrak{p}\nmid\ell}\mathrm{H}^{1}(F_{\mathfrak{p}},b^{*}V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{\mathfrak{p}\nmid\ell}\mathrm{H}^{0}({\mathcal{O}_{F_{\mathfrak{p}}}},\mathrm{R}^{1}{\eta_{\mathfrak{p}}}_{*}b^{*}V)}$

We will show that $\alpha$ is injective. Hence, the left square is cartesian
and by definition and Lemma 6.3 the claim is shown. Indeed, $\alpha$ factors
as

$\mathrm{H}^{0}({\mathcal{O}_{F}}[1/\ell],\mathrm{R}^{1}\eta_{*}V)\subset\prod_{\mathfrak{p}\nmid\ell}\mathrm{H}^{0}({\mathbb{F}_{\mathfrak{p}}},i_{\mathfrak{p}}^{*}\mathrm{R}^{1}\eta_{*}V)\rightarrow\prod_{\mathfrak{p}\nmid\ell}\mathrm{H}^{0}({\mathcal{O}_{F_{\mathfrak{p}}}},\mathrm{R}^{1}{\eta_{\mathfrak{p}}}_{*}b^{*}V)\left(\stackrel{{\scriptstyle\cong}}{{=}}\prod_{\mathfrak{p}\nmid\ell}\mathrm{H}^{0}({\mathbb{F}_{\mathfrak{p}}},i_{\mathfrak{p}}^{*}\mathrm{R}^{1}{\eta_{\mathfrak{p}}}_{*}b^{*}V)\right).$

using
$i^{*}\mathrm{R}^{1}\eta_{*}V=i_{\mathfrak{p}}^{*}a^{*}\mathrm{R}^{1}\eta_{*}V=i_{\mathfrak{p}}^{*}\mathrm{R}^{1}{\eta_{\mathfrak{p}}}_{*}b^{*}V$
(the latter isomorphism is found e.g. in [Mil08, Example 12.4]). ∎

######  Definition 6.6.

[Beĭ87a, Remark 4.0.1.b], [BK90, Conj. 5.3], [Fon92, Section 6.5], [FPR94,
III.3.1.3] Let $M_{\eta}$ be a mixed motive over $F$. Let, similarly to
Definition 6.4, $\mathrm{H}^{i}_{f}(M_{\eta})$ be defined such that the
following diagram, in which the bottom products are taken over all primes
$\ell$, is cartesian. As usual, ${M_{\eta}}_{\ell}$ is the $\ell$-adic
realization, seen as a $G$-module.

$\textstyle{\mathrm{H}^{i}_{f}(F,{M_{\eta}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{i}(F,{M_{\eta}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{\ell}\mathrm{H}^{i}_{f}(F,{M_{\eta}}_{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{\ell}\mathrm{H}^{i}(F,{M_{\eta}}_{\ell})}$

Again, to rid ourselves from crystalline questions at $\mathfrak{p}|\ell$, we
define $\mathrm{H}^{i}_{f,\backslash\operatorname{crys}}(F,{M_{\eta}})$ by
replacing $\prod_{\ell}\mathrm{H}^{i}_{f}(F,{M_{\eta}}_{\ell})$ in the bottom
row by
$\prod_{\ell}\mathrm{H}^{i}_{f,\backslash\operatorname{crys}}(F,{M_{\eta}}_{\ell})$.

We are now going to exhibit an interpretation of $f$-cohomology thus defined
in terms of the generic intermediate extension $\eta_{!*}$. Recall that we are
assuming in this section the axioms of Sections 1, 2, and 4. Mixed motives are
needed to even define $\eta_{!*}$. Moreover, for the comparison result, we
need to assume the following conjecture.

######  Conjecture 6.7.

(Beilinson) Let $X/{\mathbb{F}_{q}}$ be smooth and projective. Up to torsion,
numerical and rational equivalence agree on $X$.

As rational and numerical equivalence are the finest and the coarsest adequate
equivalence relation, respectively [And04, Section 3.2], this conjecture
implies that all adequate equivalence relations, in particular homological
equivalence agree.

######  Lemma 6.8.

Let $N$ be any mixed motive over ${\mathbb{F}_{\mathfrak{p}}}$. Conjecture 6.7
implies that the $\ell$-adic realization map
$\mathrm{H}^{0}({\mathbb{F}_{\mathfrak{p}}},N)\rightarrow\mathrm{H}^{0}({\mathbb{F}_{\mathfrak{p}}},N_{\ell}):=N_{\ell}^{\mathrm{Gal}({\mathbb{F}_{\mathfrak{p}}})}$
is injective.

######  Proof: .

We can assume that $N$ is pure of weight $0$, since both depend only on
$\operatorname{gr}_{0}^{W}N$ (Axiom 4.9). Under Conjecture 6.7 all adequate
equivalence relations agree, so that we may regard $N$ as a pure motive with
respect to any adequate equivalence relation. As the injectivity is stable
under taking direct summands, we may assume $N=h(X,n)$ for $X$ smooth and
projective over ${\mathbb{F}_{\mathfrak{p}}}$, by definition of pure motives
and Axiom 4.13. The left hand side is given by $\mathrm{CH}^{n}(X)$, so the
map is injective by Conjecture 6.7. ∎

######  Theorem 6.9.

Let $M$ be a generically smooth mixed motive over ${\mathcal{O}_{F}}$
(Definition 5.8). Set $\eta^{*}M[-1]=:M_{\eta}$. Assuming Conjecture 6.7,
there is a natural isomorphism

$\mathrm{H}^{0}({\mathcal{O}_{F}},\eta_{!*}\eta^{*}M)\stackrel{{\scriptstyle\cong}}{{{\longrightarrow}}}\mathrm{H}^{1}_{f,\backslash\operatorname{crys}}(F,M_{\eta}).$

######  Proof: .

Notice that $\eta_{!*}\eta^{*}M$ is well-defined by the assumptions. We want
to show that there is a cartesian commutative diagram

$\textstyle{\mathrm{H}^{0}(\eta_{!*}\eta^{*}M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{0}(\eta_{*}\eta^{*}M)=\mathrm{H}^{1}(M_{\eta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{\ell}\mathrm{H}^{1}_{f,\backslash\ell}(F,{M_{\eta}}_{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{\ell}\mathrm{H}^{1}(F,{M_{\eta}}_{\ell})}$

Let $j:U\rightarrow{\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$ be any open
immersion such that $j^{*}M$ is smooth. We have
$\eta_{!*}\eta^{*}M=j_{!*}j^{*}M$. The left hand term of the exact sequence

$\oplus_{\mathfrak{p}\in
U}\mathrm{H}^{0}({i_{\mathfrak{p}}}_{*}i_{\mathfrak{p}}^{!}M)\rightarrow\mathrm{H}^{0}(j_{*}j^{*}M)\rightarrow\mathrm{H}^{0}(\eta_{*}\eta^{*}M)\rightarrow\oplus_{\mathfrak{p}\in
U}\mathrm{H}^{1}({i_{\mathfrak{p}}}_{*}i_{\mathfrak{p}}^{!}M)$

induced by (4) vanishes as $i_{\mathfrak{p}}^{!}M$ is concentrated in
cohomological degree $1$ for $\mathfrak{p}\in U$ (Lemma 5.9). Any
$a\in\mathrm{H}^{0}(\eta^{*}M)$ maps to a finite sub-sum of
$\oplus_{\mathfrak{p}\in{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}}\mathrm{H}^{1}({i_{\mathfrak{p}}}_{*}i_{\mathfrak{p}}^{!}M)$,
so letting $j$ be the open complement of these points, $a$ lies in (the image
of) $\mathrm{H}^{0}(j_{*}j^{*}M)$:

$\mathrm{H}^{0}(\eta^{*}M)=\varinjlim_{j:U\rightarrow{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}\atop j^{*}M\text{ smooth}}\mathrm{H}^{0}(j_{*}j^{*}M).$

By Lemma 6.10 below, the map
$\mathrm{H}^{0}(j_{!*}j^{*}M)\rightarrow\mathrm{H}^{0}(j_{*}j^{*}M)\rightarrow\mathrm{H}^{0}(\eta^{*}M)$
is injective. Therefore, taking the colimit over all $U$ such that $M|_{U}$ is
smooth, the exact localization sequence

$0\rightarrow\mathrm{H}^{0}(j_{!*}j^{*}M)\rightarrow\mathrm{H}^{0}(j_{*}j^{*}M)\rightarrow\oplus_{\mathfrak{p}\notin
U}\mathrm{H}^{0}({{{}^{\mathrm{p}}}\mathrm{H}}^{0}i_{\mathfrak{p}}^{*}j_{*}j^{*}M)$

stemming from (7) gives

$0\rightarrow\mathrm{H}^{0}(j_{!*}j^{*}M)\rightarrow\mathrm{H}^{0}(\eta_{*}\eta^{*}M)\rightarrow\oplus_{\mathfrak{p}}\mathrm{H}^{0}({{{}^{\mathrm{p}}}\mathrm{H}}^{0}i_{\mathfrak{p}}^{*}{j_{\mathfrak{p}}}_{*}j_{\mathfrak{p}}^{*}M).$

Here $j_{\mathfrak{p}}$ is the complementary open immersion to
$i_{\mathfrak{p}}$ and the direct sum is over all (finite) places
$\mathfrak{p}$ of ${\mathcal{O}_{F}}$. We have
$i_{\mathfrak{p}}^{*}\eta_{*}\eta^{*}M=i_{\mathfrak{p}}^{*}{j_{\mathfrak{p}}}_{*}j_{\mathfrak{p}}^{*}M$,
so the top sequence in the following commutative diagram is exact:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{0}(j_{!*}j^{*}M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(M_{\eta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\oplus_{\mathfrak{p}}\mathrm{H}^{0}({{{}^{\mathrm{p}}}\mathrm{H}}^{0}i_{\mathfrak{p}}^{*}\eta_{*}\eta^{*}M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{\ell}\mathrm{H}^{0}((j_{\ell}^{*}j_{!*}j^{*}M)_{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{\ell}\mathrm{H}^{1}({M_{\eta}}_{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{\ell}\prod_{\mathfrak{p}\nmid\ell}\mathrm{H}^{0}(({{{}^{\mathrm{p}}}\mathrm{H}}^{0}i_{\mathfrak{p}}^{*}\eta_{*}\eta^{*}M)_{\ell})}$
(8)

The lower row denotes $\ell$-adic cohomology over ${\mathcal{O}_{F}}[1/\ell]$,
$F$, and the various ${\mathbb{F}_{\mathfrak{p}}}$, respectively. Moreover,
$j_{\ell}:{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}[1/\ell]\rightarrow{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}$ is the open immersion. The remainder of the proof
consists in the following steps: we show that the diagram is commutative, that
the second row is exact, identify its lower leftmost term and show that the
rightmost vertical map is injective. This implies that the left square is
cartesian, hence the theorem follows.

We write $\iota$ and $\iota_{\ell}$ for the open immersions of
$U\cap{\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}[1/\ell]$ into
${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}[1/\ell]$ and $U$, respectively. By
Lemma 5.16 and the exactness of $j_{\ell}^{*}$ we have

$(j_{\ell}^{*}j_{!*}j^{*}M)_{\ell}=(\iota_{!*}\iota^{*}j_{\ell}^{*}M)_{\ell}=\iota_{!*}\iota^{*}(j_{\ell}^{*}M)_{\ell}.$

Thus (8) is commutative since every term at the bottom just involves the
$\ell$-adic realization of the motive above it, restricted to
${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}[1/\ell]$.

The exactness of the bottom row is shown separately for each $\ell$, so $\ell$
is fixed for this argument. By the characterization just mentioned,
$\iota_{!*}\iota^{*}(j_{\ell}^{*}M)_{\ell}$ does not change when shrinking
$U$, since $j_{!*}j^{*}M$ is independent of $U$ (as soon as $M$ is smooth over
$U$). On the other hand, by the exactness of the $\ell$-adic realization
functor (Axiom 4.8) $(j_{\ell}^{*}M)_{\ell}$ is a perverse sheaf on
${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}[1/\ell]$, so is a locally constant
sheaf (shifted into degree $-1$) on a suitable small open subscheme. Hence we
may assume that $\iota^{*}(j_{\ell}^{*}M)_{\ell}$ is a locally constant sheaf
in degree $-1$. By Section 3,
$\iota_{!*}\iota^{*}(j_{\ell}^{*}M)_{\ell}=(\mathrm{R}^{0}\iota_{*}\iota^{*}(j_{\ell}^{*}M)_{\ell}[-1])[+1]$,
so the lower row is the exact sequence belonging to the distinguished triangle
of sheaves on ${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}[1/\ell]$

$\mathrm{R}^{0}\eta_{\ell}{}_{*}(M_{\eta})_{\ell}\rightarrow\mathrm{R}\eta_{\ell}{}_{*}(M_{\eta})_{\ell}\rightarrow(\mathrm{R}^{1}\eta_{\ell}{}_{*}(M_{\eta})_{\ell})[-1].$

Here $\eta_{\ell}:{\mathrm{Spec}\text{ }}{F}\rightarrow{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}[1/\ell]$ is the generic point. By Lemma 6.5 and Lemma
5.15, the factors in the lower left-hand term of (8) agree with
$\mathrm{H}^{1}_{f,\backslash\operatorname{crys}}(F,\eta^{*}M_{\ell}[-1])$.

To show that the rightmost vertical map of (8) is an injection, let
$a=(a_{\mathfrak{p}})_{\mathfrak{p}\in{\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}}}$ be a nonzero element of the rightmost upper term. Only
finitely many $a_{\mathfrak{p}}$ are nonzero. Pick some $\ell$ not lying under
any of these indices $\mathfrak{p}$. Then the image of $a$ in
$\oplus_{\mathfrak{p}\nmid\ell}\mathrm{H}^{0}(({{{}^{\mathrm{p}}}\mathrm{H}}^{0}i_{\mathfrak{p}}^{*}\eta_{*}\eta^{*}M)_{\ell})\subset\prod_{\mathfrak{p}\nmid\ell}\mathrm{H}^{0}(({{{}^{\mathrm{p}}}\mathrm{H}}^{0}i_{\mathfrak{p}}^{*}\eta_{*}\eta^{*}M)_{\ell})$
is nonzero by Lemma 6.8, which assumes Conjecture 6.7. ∎

######  Lemma 6.10.

Let $M$ be a mixed motive over $S$ such that $j^{*}M$ is smooth for some open
immersion $j:U\rightarrow S$. Then both maps
$\mathrm{H}^{0}(j_{!*}j^{*}M)\rightarrow\mathrm{H}^{0}(j_{*}j^{*}M)\rightarrow\mathrm{H}^{0}(\eta^{*}M)$
are injective.

######  Proof: .

Indeed the kernels are
$\mathrm{H}^{-1}({{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{*}j_{*}j^{*}M)=0$ and
$\oplus_{\mathfrak{p}\in U}\mathrm{H}^{0}(i_{\mathfrak{p}}^{!}M)$, which
vanishes since $i_{\mathfrak{p}}^{!}M$ sits in cohomological degree $+1$, for
$M$ is smooth around $\mathfrak{p}\in U$ (Lemma 5.9). ∎

###  6.2 $f$-cohomology via $K$-theory of regular models

######  Definition 6.11.

Let $X_{\eta}$ be a smooth and projective variety over $F$. Let
$X/{\mathcal{O}_{F}}$ be any projective model $X/{\mathcal{O}_{F}}$, i.e.,
$X{\times}_{\mathcal{O}_{F}}F=X_{\eta}$. Then we define

$\mathrm{H}^{i}(X_{\eta},n)_{\mathcal{O}_{F}}:=\operatorname{im}(\mathrm{H}^{i}(X,n)\rightarrow\mathrm{H}^{i}(X_{\eta},n)).$

Recall that we are assuming the axioms of Sections 1, 2, and 4; the full force
of mixed motives will be made use of in the sequel.

######  Theorem 6.12.

The above is well-defined, i.e., independent of the choice of the model $X$.
More precisely we have natural isomorphisms:

$\mathrm{H}^{0}(\eta_{!*}\mathrm{h}^{i-1}(X_{\eta},n)[1])=\left\\{\begin{array}[]{cl}\mathrm{H}^{i}(X_{\eta},n)_{{\mathcal{O}_{F}}}&i<2n\\\
\mathrm{CH}^{n}(X_{\eta})_{\mathbb{Q},\mathrm{hom}}&i=2n\end{array}\right.$

Moreover

$\mathrm{H}^{-1}(\eta_{!*}\mathrm{h}^{i-1}(X_{\eta},n)[1])=\mathrm{H}^{0}(\mathrm{h}^{i-1}(X_{\eta},n)).$

When $X$ is regular, the definition and the statement are due to Beilinson
[Beĭ86, Lemma 8.3.1]. In this case one has
$\mathrm{H}^{i}(X_{\eta},n)_{{\mathcal{O}_{F}}}=\operatorname{im}K^{\prime}_{2j-i}(X)_{\mathbb{Q}}^{(j)}\rightarrow
K^{\prime}_{2j-i}(X_{\eta})_{\mathbb{Q}}^{(j)}$, but that expression does in
general depend on the choice of the model [dJ00, dJ08]. An extension of
Beilinson’s definition to all Chow motives over $F$ due to Scholl is discussed
in the theorem below. We first provide a preparatory lemma.

######  Lemma 6.13.

Let $M\in\mathbf{MM}({\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}})$ be a mixed,
generically smooth motive with strictly negative weights (Definition 5.8). Let
$j:U\rightarrow{\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$ be an open non-
empty immersion such that $M|_{U}$ is smooth. The natural map
$j_{!*}j^{*}M\rightarrow\eta_{*}\eta^{*}M$ gives rise to an isomorphism

$\mathrm{H}^{0}(j_{!*}j^{*}M)=\mathrm{im}\left(\mathrm{H}^{0}(M)\rightarrow\mathrm{H}^{0}(\eta_{*}\eta^{*}M)\right).$

######  Proof: .

By Lemma 6.10,
$\mathrm{H}^{0}(j_{*}j^{*}M)\rightarrow\mathrm{H}^{0}(\eta_{*}\eta^{*}M)$ is
injective. Hence it suffices to show
$\mathrm{H}^{0}(j_{!*}j^{*}M)=\operatorname{im}(\mathrm{H}^{0}M\rightarrow\mathrm{H}^{0}(j_{*}j^{*}M))$.
Let $i$ be the complement of $j$. From (6), (7), we have a commutative exact
diagram

---
$\textstyle{\mathrm{H}^{0}(j_{!}j^{*}M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathrm{H}^{0}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{0}(i_{*}i^{*}M)}$$\textstyle{0=\mathrm{H}^{-1}(i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{*}j_{*}j^{*}M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{0}(j_{!*}j^{*}M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{0}(j_{*}j^{*}M)}$$\textstyle{\mathrm{H}^{1}(i_{*}{{{}^{\mathrm{p}}}\mathrm{H}}^{-1}i^{*}j_{*}j^{*}M)=0}$

The indicated vanishings are because of $t$-structure reasons and Axiom 4.1,
respectively. It remains to show that $\alpha$ is surjective. As $i^{*}M$ is
concentrated in cohomological degrees $[-1,0]$ (Axiom 4.2), there is an exact
sequence

$0=\mathrm{H}^{1}({{{}^{\mathrm{p}}}\mathrm{H}}^{-1}i^{*}M)\rightarrow\mathrm{H}^{0}(i^{*}M)\rightarrow\mathrm{H}^{0}({{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{*}M).$

However $\mathrm{H}^{0}({{{}^{\mathrm{p}}}\mathrm{H}}^{0}i^{*}M)=0$ as $i^{*}$
preserves negative weights (Axiom 4.11) and by strictness of the weight
filtration and compatibility with the $t$-structure (Axiom 4.9). ∎

######  Proof: .

Let $j:U\rightarrow{\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$ be an open
nonempty immersion (which exists by smoothness of $X_{\eta}$) such that
$X_{U}$ is smooth over $U$. By definition of $\eta_{!*}$ and Lemmas 5.1 and
5.11, the left hand term in the theorem agrees with
$\mathrm{H}^{0}(j_{!*}\mathrm{h}^{i}(X_{U},n))$. In the sequel, we write
$M:=\mathrm{h}^{2n}(X,n)$ and
$M_{\eta}:=\eta^{*}[-1]M=\mathrm{h}^{i-1}(X_{\eta},n)$.

We first do the case $i<2n$. The spectral sequences

$\mathrm{H}^{a}(\mathrm{h}^{b}(X,n))\Rightarrow\mathrm{H}^{a+b}(X,n),\,\mathrm{H}^{a}(\mathrm{h}^{b}(X_{\eta},n))\Rightarrow\mathrm{H}^{a+b}(X_{\eta},n)$

resulting from repeatedly applying truncation functors of the motivic
$t$-structure converge since the cohomological dimension is finite (Axiom 4.1
over $F$, Lemma 5.2 over ${\mathcal{O}_{F}}$). By Lemma 5.2,
$\mathrm{H}^{i}(-)$, applied to mixed motives over ${\mathcal{O}_{F}}$, is
non-zero for $i\in\\{-1,0,1\\}$ only. We thus have to consider two exact
sequences. The exact functor $\eta^{*}[-1]$ maps to similar exact sequences
for motivic cohomology over $F$ (the indices work out properly, see Lemma
5.1):

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{i}(X,n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{-1}(\mathrm{h}^{i+1}(X,n))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\eta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{i}(X_{\eta},n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{0}(\mathrm{h}^{i}(X_{\eta},n))\stackrel{{\scriptstyle\ref{axio_weight},\ref{bsp_weightTate}}}{{=}}0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
(9)

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(\mathrm{h}^{i-1}(X,n))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{0}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{2}(\mathrm{h}^{i-2}(X_{\eta},n))\stackrel{{\scriptstyle\ref{axio_cohomdim}}}{{=}}0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\eta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(M_{\eta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
(10)

Here, $K$ and $K_{\eta}$ are certain $E_{3}$-terms of the spectral sequences
above. The rightmost vertical map in (9) is injective as one sees by combining
(4) with the left-exactness of $i_{\mathfrak{p}}^{!}$. Hence

$\displaystyle\mathrm{H}^{i}(X_{\eta},n)_{{\mathcal{O}_{F}}}$ $\displaystyle=$
$\displaystyle\operatorname{im}(\mathrm{H}^{i}(X,n)\rightarrow\mathrm{H}^{i}(X_{\eta},n))=\operatorname{im}(K\rightarrow
K_{\eta})$ $\displaystyle=$
$\displaystyle\operatorname{im}(\mathrm{H}^{0}(M)\rightarrow\mathrm{H}^{1}(M_{\eta}))$

The motive $M=\mathrm{h}^{i}(X,n)$ is a generically smooth (mixed) motive by
Lemma 5.11. (Recall that this uses the decomposition axiom 4.5 for smooth
projective varieties.) By Example 4.12, its weights are strictly negative.
Thus Lemma 6.13 applies and the case $i<2n$ is shown.

We now do the case $i=2n$. Now
${{{}^{\mathrm{p}}}\mathrm{H}}^{1}i^{!}j_{!*}j^{*}M={{{}^{\mathrm{p}}}\mathrm{H}}^{1}i^{!}j_{*}j^{*}M$
has strictly positive weights because of the choice of $j$, Example 4.12,
Lemma 5.6, Axiom 4.11, and the compatibility of weights and the motivic
$t$-structure, i.e.,
$\operatorname{wt}{{{}^{\mathrm{p}}}\mathrm{H}}^{1}(-)\subset\operatorname{wt}(-)+1$.
Therefore $\mathrm{H}^{0}({{{}^{\mathrm{p}}}\mathrm{H}}^{1}i^{!}E)=0$,
$E:=j_{!*}j^{*}M$. Here $i$ is any closed immersion. The localization triangle
(4) yields

$\mathrm{H}^{0}(E)\stackrel{{\scriptstyle\alpha}}{{\rightarrow}}\mathrm{H}^{0}(\eta_{*}\eta^{*}E)\stackrel{{\scriptstyle\text{\ref{axio_morphismsChow}}}}{{=}}\mathrm{CH}^{n}(X_{\eta})_{\mathbb{Q},\mathrm{hom}}\rightarrow\oplus_{\mathfrak{p}}\mathrm{H}^{1}(i_{\mathfrak{p}}^{!}E)=\oplus\mathrm{H}^{0}({{{}^{\mathrm{p}}}\mathrm{H}}^{1}(i^{!}E))=0.$

Therefore, $\alpha$ is surjective. The injectivity of $\alpha$ is Lemma 6.10.

To calculate $\mathrm{H}^{-1}(\eta_{!*}M_{\eta}[1])$, let
$j:U\rightarrow{\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$ be as above and let
$i$ be its complement. The natural map $\mathrm{H}^{-1}({\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}},j_{!*}j^{*}M)\rightarrow\mathrm{H}^{-1}(U,j^{*}M)$ is an
isomorphism by the exact cohomology sequence belonging to (7). Thus we have to
show

$\mathrm{H}^{-1}({\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}},j_{*}j^{*}M)=\mathrm{H}^{-1}({\mathrm{Spec}\text{
}}{{\mathcal{O}_{F}}},\eta_{*}\eta^{*}M).$

This follows from the localization axiom 1.10 and $i_{\mathfrak{p}}^{!}M$
being in cohomological degree $+1$ for all points $\mathfrak{p}$ in $U$ (Lemma
5.9), so that
$\mathrm{H}^{0}({\mathbb{F}_{\mathfrak{p}}},i_{\mathfrak{p}}^{!}M)=\mathrm{H}^{-1}({\mathbb{F}_{\mathfrak{p}}},i_{\mathfrak{p}}^{!}M)=0$.
∎

By a theorem of Scholl [Sch00, Thm. 1.1.6], there is a unique functorial and
additive (i.e., converting finite disjoint unions into direct sums) way to
extend the definition of $\mathrm{H}^{i}(X_{\eta},n)_{\mathcal{O}_{F}}$ as the
image of the $K$-theory of a _regular_ proper flat model (Definition 6.11) to
all Chow motives over $F$, in particular to ones of smooth projective
varieties $X_{\eta}/F$ that do not possess a regular proper model $X$. The
following theorem compares this definition with the one via intermediate
extensions.

######  Theorem 6.14.

Let $h_{\eta}$ be a direct summand in the category of Chow motives of
$h(X_{\eta},n)$ where $X_{\eta}/F$ is smooth projective. Let $i\in\mathbb{Z}$
be such that $i-2n<0$. Let
$\iota:\mathbf{M}_{\mathrm{rat}}(F)\rightarrow\mathbf{DM}_{\mathrm{gm}}(F)$ be
the embedding. Then, the group

$\mathrm{H}^{i}(h_{\eta})_{\mathcal{O}_{F}}:=\mathrm{H}^{0}(\eta_{!*}({{{}^{\mathrm{p}}}\mathrm{H}}^{i-2n-1}(\iota(h_{\eta}))[1])).$

is well-defined and agrees with the aforementioned definition by Scholl.

######  Proof: .

Recall
$\iota(h(X_{\eta},n))=\operatorname{M}(X_{\eta},n)[2n]\in\mathbf{DM}_{\mathrm{gm}}(F)$.
We first check that the group is well-defined: let $X/{\mathcal{O}_{F}}$ be a
projective model of $X_{\eta}$. By Lemma 5.12, there is some model
$M\in\mathbf{MM}({\mathcal{O}_{F}})$ of
${{{}^{\mathrm{p}}}\mathrm{H}}^{i-2n-1}\iota(h_{\eta})[1]$ and an open
subscheme $U$ of ${\mathrm{Spec}\text{ }}{{\mathcal{O}_{F}}}$ such that $M$ is
a direct summand of
${{{}^{\mathrm{p}}}\mathrm{H}}^{i-1}\operatorname{M}(X)(n)$ and such that
$X{\times}U$ is smooth over $U$. Then $\mathrm{h}^{i-1}(X,n)$ is a smooth
motive when restricted to $U$ (Lemma 5.11). Hence so is $M$. Thus $\eta_{!*}$
can be applied to
$({{{}^{\mathrm{p}}}\mathrm{H}}^{i-2n-1}\iota(h_{\eta}))[1]$.

The assignment
$h_{\eta}\mapsto\mathrm{H}^{0}(\eta_{!*}({{{}^{\mathrm{p}}}\mathrm{H}}^{i-2n-1}\iota(h_{\eta}))[1])$
is functorial and additive and $h(X_{\eta})(n)$ maps to

$\mathrm{H}^{0}(\eta_{!*}({{{}^{\mathrm{p}}}\mathrm{H}}^{i-1}\operatorname{M}(X_{\eta},n))[1])\stackrel{{\scriptstyle\text{\ref{theo_summaryH1f}}}}{{=}}\mathrm{H}^{i}(X_{\eta},n)_{\mathcal{O}_{F}}.$

Thus the two definitions agree by Scholl’s theorem. ∎

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