text:
\begin{document}
\begin{abstract} I show that the cohomology of the generic points of algebraic
complex varieties becomes {\sl stable} birational invariant, when considered
`modulo the cohomology of the generic points of the affine spaces'.
\end{abstract}
\maketitle

These notes are concerned with certain birational invariants of smooth
algebraic varieties. All such invariants are {\sl dominant} sheaves,
cf. below; the dominant sheaves are characterized in Proposition
\ref{bir-A1-presheaf}.

Two classes of invariants are of special interest: (i) {\sl stable},
i.e., taking the same values on a variety and on its direct product with
an affine space, and (ii) constant on the projective spaces. Though the
latter class is a priori wider, there are no known examples of non-stable
invariants vanishing on the projective spaces. Here an attempt of
comparison is made. Namely, it is shown that the corresponding adjoint
functors coincide on the following types of invariants: (i) of `level 1',
cf. Proposition \ref{pic-as-adj} and also p.\pageref{coniveau-filtr},
(ii) `related to cohomology' (or to closed differential forms).

Differential forms play a very special r\^ole in the story, cf. e.g.
Conjecture \ref{irred-diff-for}. Moreover, all known examples of
{\sl simple} invariants (as objects of an abelian category) `come from'
differential forms: except for two invariants related to the multiplicative
and the additive groups ($Y\mapsto(k(Y)^{\times}/k^{\times})_{{\mathbb Q}}$
and $Y\mapsto k(Y)/k$, the logarithmic and the exact differentials, cf.
below), they are values of the functor ${\mathbb B}^0$ from
\S\ref{Irred-obj}. For these reasons the differential forms are studied in
detail. It is shown in Corollary \ref{log-diff-form} that the cohomology of
the generic points of algebraic (complex) varieties becomes {\sl stable}
birational invariant, when considered `modulo the cohomology of the generic
points of the affine spaces'.

The principal new results of \S\ref{Differential-forms} are
Propositions \ref{A1-quotient-of-closed} and \ref{1-forms}. It is shown in
Proposition \ref{A1-quotient-of-closed} that (i) the quotient $V^{\bullet}$
of the sheaf of algebras of closed differential forms by the ideal generated
by the exact 1-forms and the logarithmic differentials is stable and
(ii) $V^{\bullet}$ is the maximal stable quotient of the sheaf of closed
differential forms. Proposition \ref{1-forms} gives a complete description
of the sheaf of closed 1-forms.

Depending on what is more convenient, we shall consider our `invariants'
either as dominant sheaves, or as representations, cf. \S\ref{Summ-equiv}.
E.g., the simplicity is more natural in the context of representations.

\section{Dominant presheaves and sheaves} \label{pre-sheaves}
{\bf Notations.} From now on we fix an algebraically closed field
$k$ of characteristic zero, and denote by $E$ a variable
coefficient field of characteristic zero.
Denote by $\text{Vec}_E$ the category of $E$-vector spaces.

I am interested in birational invariants of (or ``presheaves on'')
$k$-varieties. More precisely, let ${\mathcal S}m_k'$ be the category,
whose objects are smooth $k$-varieties and the morphisms are smooth
$k$-morphisms. Define the pretopology on ${\mathcal S}m_k'$ by saying
that the covers are dominant morphisms. Recall, that a presheaf is a
sheaf if the following diagram is an equalizer for any covering $Y\to X$:
\begin{equation}\label{opr-puchka}{\mathcal F}(X)\to{\mathcal F}(Y)
\rightrightarrows{\mathcal F}(Y\times_XY).\end{equation}

The category of the sheaves of $E$-vector spaces on this site
is denoted by $\Sm_G(E)$ and $\Sm_G:=\Sm_G({\mathbb Q})$.

{\it Example.} For each irreducible smooth $k$-variety $X$ and integer
$0\le q\le\dim X$ let $\Psi_{X,q}:Y\mapsto Z^q(k(X)\otimes_kk(Y))
_{{\mathbb Q}}$ (${\mathbb Q}$-linear combinations of irreducible
subvarieties on $X\times_kY$ of codimension $q$ dominant over $X$ and
$Y$.) This is a sheaf. Set $\Psi_X:=\Psi_{X,\dim X}$. The sheaves
$\Psi_X$ for all $X$ form a system of generators of $\Sm_G$.

{\it Definition.} 1. A presheaf ${\mathcal F}$ is
${\mathbb A}^1$-{\sl invariant} (or {\sl stable}) if
${\mathcal F}(X)\stackrel{\sim}{\longrightarrow}
{\mathcal F}(X\times{\mathbb A}^1)$ for all $X$.

2. Let ${\mathcal S}$ be a collection of dominant morphisms in
${\mathcal S}m_k'$ with connected fibres. Assume that ${\mathcal S}$ is
stable under base changes of its arbitrary element by itself: ${\rm pr}_1:
X\times_YX\to X$ belongs to ${\mathcal S}$ if $X\to Y$ belongs to
${\mathcal S}$. A presheaf ${\mathcal F}$ is called an ${\mathcal S}$-presheaf
if ${\mathcal F}(Y)\stackrel{\sim}{\longrightarrow}{\mathcal F}(X)$
for all $(X\to Y)\in{\mathcal S}$.

Denote by $\Sm_G^{{\mathcal S}}$ the full subcategory in $\Sm_G$
consisting of ${\mathcal S}$-sheaves. More particularly, denote by
${\mathcal I}_G$ the full subcategory in $\Sm_G$ consisting of
${\mathbb A}^1$-invariant sheaves. Under assumptions of
\S\ref{Properties-of}, $\Sm_G^{{\mathcal S}}\subseteq{\mathcal I}_G$.

For any dominant presheaf ${\mathcal F}$ denote
by $\underline{{\mathcal F}}$ its dominant sheafification.

For each smooth $k$-variety $Y$, we denote by $\overline{Y}$ a smooth
compactification of $Y$.

\subsection{Examples of ${\mathbb A}^1$-invariant presheaves}
In this section we consider some examples of dominant presheaves with values
in various abelian categories. They come either from algebro-geometric
constructions, or from a cohomology theory $H^*$ (with
coefficients in a commutative ${\mathbb Q}$-algebra $B$). As Example 5
suggests, those of these examples that are ${\mathbb A}^1$-invariant sheaves,
are related. This is one of motivations for Conjecture \ref{equiv-B}.

An effective pure motive is a pair consisting of a smooth projective
variety and a projector in the algebra of correspondences modulo
numerical equivalence. Morphisms of {\sl co(ntra)variant} pure motives
are defined by correspondences modulo numerical equivalence so that
they behave as action on the (co)homology.

Denote by ${\mathcal M}_k$ the category of covariant pure $k$-motives
(and by ${\mathcal M}_k^{\text{op}}$ its opposite, the category of
contravariant pure $k$-motives). By a well-known result of U.Jannsen, these
two categories are abelian and semisimple. A simple effective pure motive
is called {\sl primitive} if it is ``not divisible by the Lefschetz motive'',
the motive $({\mathbb P}^1,\pi)$, where $\pi$ induces 0 on the 0-th and the
identity on the second (co)homology.

Denote by $\overline{Y}^{\text{prim}}$ the sum in the motive of
$\overline{Y}$ of all its primitive submotives; $CH^q$ is the (Chow)
group of codimension $q$ cycles modulo rational equivalence.
We also use notations and identifications of \S\ref{obozn-predst}.

\begin{tabular}{rlcc} &{\bf Invariant} of a connected $Y$
(dominant presheaf) & {\bf Values} &{\bf stable} \\
1&$K_q(Y)_{{\mathbb Q}}$ for $q\ge 0$/ its sheafification&
$\text{Vec}_{{\mathbb Q}}$& yes/only for $q=0$ \\
2&$H^q(Y)$ for $q\ge 0$/ its sheafification $\underline{H}^q$&
$B$-mod & yes/only for $q=0$ \\
3& $\Gamma(\overline{Y},\bigotimes^{\bullet}_{{\mathcal O}_{\overline{Y}}}
\Omega^1_{\overline{Y}|k})$ / its
sheafification $\Gamma(\bigotimes^{\bullet}_F\Omega^1_{F|k})$,
cf. Remark on p.\pageref{glok-por} & $\text{Vec}_k$ & yes/no \\
4&$\Phi^pCH^q(X\times_kk(Y))_{{\mathbb Q}}$ for a smooth $X$ and
$\lefteqn{\text{a ``universal'' filtration $\Phi^{\bullet}$ on the
Chow groups}}$ \\
&(e.g., $A(k(Y))_{{\mathbb Q}}$ for an abelian $k$-variety $A$)&
$\text{Vec}_{{\mathbb Q}}$& yes \\
5&$\overline{Y}^{\text{prim}}=\bigoplus_M\overline{Y}^{\text{prim}}_M$
(multiplicity-one sheaf, by Proposition \ref{funktor-B})&
${\mathcal M}_k^{\text{op}}$ & yes \\
6&$Z^{\dim Y}(F\otimes_kk(Y))_{{\mathbb Q}}$&
$\Sm_G^{\text{op}}$& no \\
7&$Z^q(Y\times_kF)_{{\mathbb Q}}$ for $q\ge 0$ / its sheafification
$Z^q(F\otimes_kk(Y))_{{\mathbb Q}}$& $\Sm_G$& only for $q=0$ \\
&(a) composition with the evaluation functor on $X$, i.e., $\Psi_{X,q}$&
$\text{Vec}_{{\mathbb Q}}$ & only for $q=0$ \\
8&$C_{k(Y)}:={\mathcal I}\Psi_X$, cf. \S\ref{Properties-of},
(and its quotient $CH_0(\overline{Y}_F)_{{\mathbb Q}}$)
& ${\mathcal I}_G^{\text{op}}$& yes \\
&(a) composition with $\Hom_{\Sm_G^{\text{op}}}(\underline{H}^q_c,-)$: \\
& $H^q(\overline{Y})/N^1=:\underline{H}^q_c(Y)$ for any $q\ge 0$
(subsheaf of the sheaf $\underline{H}^q$) & $B$-mod & yes \\
&(a$'$) $\lefteqn{\text{$k={\mathbb C}$: the image in
$\underline{H}^{2q}(-({\mathbb C});{\mathbb Q})(Y)$ of the maximal Hodge
substructure of $H^{2q}(\overline{Y}({\mathbb C}),{\mathbb Q})$ in}}$ \\
& $F^1$, cf. \S\ref{Differential-forms}, p.\pageref{Differential-forms},
(its vanishing is equivalent to the Hodge's conjecture) &
$\text{Vec}_{{\mathbb Q}}$ & yes \\
&(b) composition with $\Hom_{\Sm_G^{\text{op}}}(\Omega_{F|k}^{\bullet},-)$: \\
& $\Gamma(\overline{Y},\Omega^{\bullet}_{\overline{Y}|k})$ (subsheaf of
the sheaf $\underline{H}^{\bullet}_{\text{dR}|k,c}$)& $\text{Vec}_k$ & yes
\end{tabular}

Except for $\Gamma(\overline{Y},\bigotimes^{\bullet}
_{{\mathcal O}_{\overline{Y}}}\Omega^1_{\overline{Y}|k})$,
all these invariants have Galois descent property.
Except for $Z^q(Y\times_kF)_{{\mathbb Q}}$ for $q>0$,
$K_q(Y)_{{\mathbb Q}}$ for $q\ge 0$ and $H^q(Y)$ for $q>0$,
all these invariants are birational. ($N^1$ in example 8 (a) denotes
the first term of the coniveau filtration on $H^*$.)

Some of the above presheaves are defined using a compactification
$\overline{Y}$. To show that each of such presheaves is in fact well-defined
(and therefore, birationally invariant), one can use the facts that (i)
any birational map is a composition of blow-ups and blow-downs with smooth
centres, cf. \cite{akmw}, and (ii) the cohomology (resp., motive) of a
blow-up is the direct sum of the cohomology of the original variety and
of the Gysin image (resp., Tate twist) of the cohomology (resp., motive)
of the subvariety which is blown up. Such a presheaf is
${\mathbb A}^1$-invariant, since the cohomology (resp., motive) of the
product of a proper variety $X$ with the projective line is the direct sum
of the pull-back of the cohomology (resp., motive) of $X$ and of the Gysin
image (resp., Tate twist) of the cohomology (resp., motive)
of $X\times\{0\}\cong X$.

To conclude that a birational ${\mathbb A}^1$-invariant presheaf
is a sheaf, one checks that it has the Galois descent property,
so Proposition \ref{bir-A1-presheaf} can be applied.

\begin{lemma} \label{example-4} For an arbitrary commutative
$k$-group $A$, let ${\mathcal H}^A_1$ be the presheaf
$Y\mapsto\bigoplus_{y\in Y^0}(A(k(y))/A(k))_{{\mathbb Q}}$. Then
${\mathcal H}^A_1$ is a sheaf; it is simple (=irreducible) for simple $A$.
Let a presheaf ${\mathcal F}$ be the composition of the Picard
functor $Y\mapsto\Pic^0(\overline{Y})$ with an additive functor
on the category of abelian $k$-varieties, e.g.
$\Pic^{\circ}_{{\mathbb Q}}:Y\mapsto\Pic^0(\overline{Y})_{{\mathbb Q}}$,
$\underline{H}^1_c:Y\mapsto H^1(\overline{Y})$,
or $\Omega^1_{|k,\text{{\rm reg}}}:Y\mapsto
\Gamma(\overline{Y},\Omega^1_{\overline{Y}|k})$. Then ${\mathcal F}$
is a sheaf and ${\mathcal F}=\bigoplus_A{\mathcal F}(\tilde{A})
\otimes_{\End(\tilde{A})}{\mathcal H}^{\tilde{A}}_1$, where $A$ runs through
the isogeny classes of simple abelian $k$-varieties and $\tilde{A}$
is a representative of $A$. \end{lemma}
Thus, such sheaves ${\mathcal F}$ are direct sums of copies
of simple sheaves ${\mathcal H}^A_1$.

{\it Proof.} Suppose that ${\mathcal A}$ is an abelian category. Then any
semisimple object $N\in{\mathcal A}$ splits {\sl canonically} into the direct
sum over the isomorphism classes $M$ of simple objects in ${\mathcal A}$ of
its $M$-isotypical parts $N_M$.
Clearly, for any representative $\tilde{M}$ of the isomorphism class $M$ the
natural morphism
$\Hom_{{\mathcal A}}(\tilde{M},N)\otimes_{\End(\tilde{M})}\tilde{M}\to
N_M$ by $\varphi\otimes a\mapsto\varphi(a)$. This is an isomorphism.
Applying an additive functor ${\mathfrak F}:{\mathcal A}\to
{\mathcal B}^{\text{op}}$ to the above isotypical decomposition of
$N$, we get a canonical isomorphism $\prod_M{\mathfrak F}(\tilde{M})
\otimes_{\End(\tilde{M})}\Hom_{{\mathcal A}}(N,\tilde{M})
\stackrel{\sim}{\longrightarrow}{\mathfrak F}(N)$,
$f\otimes l\mapsto l^{\ast}f$, where the following duality is used:
$\Hom_{\text{mod-$\End(\tilde{M})$}}(\Hom_{{\mathcal A}}
(\tilde{M},N),\End(\tilde{M}))_{{\mathbb Q}}\stackrel{\sim}{\longrightarrow}
\Hom_{{\mathcal A}}(N,\tilde{M})$. (It is induced by the composition
pairing $\Hom_{{\mathcal A}}(\tilde{M},N)\otimes
\Hom_{{\mathcal A}}(N,\tilde{M})\to\End(\tilde{M})$.)

As ${\mathcal A}$, we take either the category of abelian $k$-varieties
with morphisms $\otimes{\mathbb Q}$, or the bigger category
${\mathcal M}_k^{\text{op}}$.
In the case of abelian varieties, the isomorphism classes of simple
objects are the isogeny classes of simple abelian $k$-varieties,
whereas the existence of the isotypical decomposition corresponds to
the fact that for any abelian $k$-variety $B$ the natural morphism
$\bigoplus_A\Hom_{\text{ab.$k$-var}}(\tilde{A},B)\otimes_{\End(\tilde{A})}
\tilde{A}\to B$, $\varphi\otimes a\mapsto\varphi(a)$,
where $A$ runs through the isogeny classes of simple abelian $k$-varieties
and $\tilde{A}$ is a representative of $A$, is an isogeny.

Thus, any sheaf ${\mathcal F}$ with semisimple values in ${\mathcal A}$ also
splits {\sl canonically} into the direct sum over the isomorphism classes $M$
of simple objects in ${\mathcal A}$ of its $M$-isotypical parts
${\mathcal F}_M$.

When ${\mathcal A}={\mathcal M}_k^{\text{op}}$ and ${\mathcal F}$ is
the dominant sheaf $Y\mapsto\overline{Y}^{\text{prim}}$, we get that
${\mathcal F}$ splits canonically into the direct sum of its $M$-isotypical
parts $Y\mapsto\overline{Y}^{\text{prim}}_M$. By Proposition \ref{funktor-B},
the $M$-isotypical part $Y\mapsto\overline{Y}^{\text{prim}}_M$ is a simple
sheaf.

If $B={\rm Alb}(\overline{Y})$ (the Albanese variety) then
$\Hom_{\text{ab.$k$-var}}(B,\tilde{A})=\tilde{A}(k(Y))/\tilde{A}(k)$, and thus,
${\mathcal F}(Y)={\mathcal F}(B)\stackrel{\sim}{\longrightarrow}\bigoplus_A
{\mathcal F}(\tilde{A})\otimes_{\End(\tilde{A})}(\tilde{A}(k(Y))/\tilde{A}(k))$.
It is quite evident that ${\mathcal H}^{\tilde{A}}_1$ is a sheaf. By
Proposition \ref{bir-A1-presheaf}, in the case of abelian variety $\tilde{A}$,
it suffices to check the Galois descent property, which is equivalent to the
following one: for any abelian $k$-variety $\tilde{A}$ and any finite group
$H$ of its automorphisms such that $H_0(H,\tilde{A})=0$ one has
${\mathcal F}(\tilde{A})^H=0$. Clearly, this property holds. The simplicity
of the sheaf ${\mathcal H}^{\tilde{A}}_1$ follows from the fact that for any
algebraically closed field extension $K|k(\tilde{A})$ and for any subvariety
$Z$ of $A$ of positive dimension there are no proper subgroups of
$\tilde{A}(K)$ containing all generic $K$-points of $Z$. (Any point of
$\tilde{A}$ is a sum of generic points of $\tilde{A}$; any sum of $\dim A$
generic $K$-points of $Z$ in sufficiently general position is a generic
point of $Z$). This argument works more naturally in the context of
representations, cf. \S\ref{Alternative-descr}. \qed

\vspace{4mm}

{\it Remark.} For an abelian $k$-variety $A$, the sheaf
$A_{{\mathbb Q}}:Y\mapsto A(k(Y))_{{\mathbb Q}}$ factors through the
Albanese functor, but considered as a functor to the category of torsors
over abelian $k$-varieties, so additive functors do not make sense
and Lemma \ref{example-4} is not applicable to this sheaf.
In particular, it is not semisimple.

\vspace{4mm}

Propositions \ref{1-forms} and \ref{semisimpl-top} suggest that
(i) the isomorphism classes of irreducible subquotients of
$\underline{H}^{\bullet}_c$ are the same as that of
$\Omega^{\bullet}_{|k,{\rm reg}}:Y\mapsto
\Gamma(\overline{Y},\Omega^{\bullet}_{\overline{Y}|k})$,
(ii) they can be naturally identified with the irreducible
effective primitive motives, and
(iii) the isomorphism classes of irreducible subquotients
of $\underline{H}^{\bullet}$ are related to more general irreducible
effective motives, such as the Tate motive ${\mathbb Q}(-1)$ in the case
of $\underline{H}^1_{{\rm dR}/k}$.

\begin{lemma} Any dominant sheaf ${\mathcal F}$ with values in an abelian
category with objects of finite length (e.g., in a category of
finite-dimensional vector spaces) is ${\mathbb A}^1$-invariant. \end{lemma}
{\it Proof.} Any smooth morphism of connected smooth $k$-varieties is covering,
so $X\times({\mathbb A}^1\times{\mathbb A}^1\smallsetminus\Delta)\stackrel{p}
{\longrightarrow}X\times({\mathbb A}^1\times{\mathbb A}^1\smallsetminus\Delta)/
{\mathfrak S}_2$ is a cover for any $X$. On the other hand, it is the
coequalizer of $X\times({\mathbb A}^1\times{\mathbb A}^1\smallsetminus\Delta)
\stackrel{id,(12)}\rightrightarrows
X\times({\mathbb A}^1\times{\mathbb A}^1\smallsetminus\Delta)$.
Therefore, ${\mathcal F}(X\times({\mathbb A}^1\times{\mathbb A}^1
\smallsetminus\Delta)/{\mathfrak S}_2)\stackrel{p^*}{\longrightarrow}
{\mathcal F}(X\times({\mathbb A}^1\times{\mathbb A}^1\smallsetminus\Delta))$
(i) is injective, (ii) factors through the ${\mathfrak S}_2$-invariants.
As $({\mathbb A}^1\times{\mathbb A}^1\smallsetminus\Delta)/{\mathfrak S}_2
\cong{\mathbb A}^1\times{\mathbb A}^1\smallsetminus\Delta$($\cong
{\mathbb A}^1\times{\mathbb G}_m$), the source and the target of $p^*$ are
isomorphic. As they are of finite length, the inclusion $p^*$ is an
isomorphism. This implies that the involution $(12)$ is identical on
${\mathcal F}(X\times({\mathbb A}^1\times{\mathbb A}^1\smallsetminus\Delta))$,
so in the exact sequence, defining the sheaf condition for the cover
$X\times{\mathbb A}^1\longrightarrow X$, $0\to{\mathcal F}(X)\to
{\mathcal F}(X\times{\mathbb A}^1)\rightrightarrows{\mathcal F}
(X\times{\mathbb A}^1\times{\mathbb A}^1)$ the double arrow consists
of equal morphisms, i.e. ${\mathcal F}(X)\stackrel{\sim}{\longrightarrow}
{\mathcal F}(X\times{\mathbb A}^1)$. \qed

\subsection{Properties of $\Sm_G^{{\mathcal S}}$} \label{Properties-of}
Clearly, a subsheaf of an ${\mathcal S}$-sheaf is an ${\mathcal S}$-sheaf:
if ${\mathcal G}$ is a subsheaf of an ${\mathcal S}$-sheaf ${\mathcal F}$
then for any $(Y\to X)\in{\mathcal S}$ the parallel arrows in the upper
line in the commutative diagram
$$\begin{array}{ccccc}{\mathcal F}(X)&\longrightarrow&
{\mathcal F}(Y)&\rightrightarrows&{\mathcal F}(Y\times_XY)\\
\bigcup&&\bigcup&&\bigcup\\
{\mathcal G}(X)&\longrightarrow&{\mathcal G}(Y)&
\rightrightarrows&{\mathcal G}(Y\times_XY)\end{array}$$
coincide, so the parallel arrows in the lower line also coincide,
i.e. ${\mathcal G}$ is an ${\mathcal S}$-sheaf.

Assume that there are generically non-finite morphisms in ${\mathcal S}$
{\sl with arbitrary targets}. Thus as before, ${\mathcal I}_G$ is a
particular case of $\Sm_G^{{\mathcal S}}$. Moreover, as restriction of any
morphism $X\stackrel{f}{\longrightarrow}Y$ to an open dense subset $U$
of $X$ factors through $U\stackrel{(f,\phi)}{\longrightarrow}
Y\times{\mathbb A}^1\stackrel{{\rm pr}_Y}{\longrightarrow}Y$, one has
$\Sm_G^{{\mathcal S}}\subseteq{\mathcal I}_G$.

1. The categories $\Sm_G^{{\mathcal S}}$ and $\Sm_G$ are abelian, complete,
cocomplete and have enough injectives. (This is standard.)

2. The section functors $\Hom_{\Sm_G}(\Psi_Y,-):
{\mathcal F}\mapsto{\mathcal F}(Y)$ are exact on $\Sm_G^{{\mathcal S}}$
for all smooth $k$-varieties $Y$. As a consequence, quotients of
${\mathcal S}$-sheaves by their subsheaves coincide with their quotients
as presheaves: if ${\mathcal F}\in\Sm_G^{{\mathcal S}}$ and ${\mathcal G}$
is a subsheaf of ${\mathcal F}$ then
$({\mathcal F}/{\mathcal G})(Y)={\mathcal F}(Y)/{\mathcal G}(Y)$.

3. A sheaf is an ${\mathcal S}$-sheaf if and only if all its irreducible
subquotients are ${\mathcal S}$-sheaves.

[Proof of the ``only if'' part. As it was shown above, a subsheaf
${\mathcal G}$ of ${\mathcal F}\in\Sm_G^{{\mathcal S}}$ is an
${\mathcal S}$-sheaf. By property 2, $({\mathcal F}/{\mathcal G})(Y)=
{\mathcal F}(Y)/{\mathcal G}(Y)$, which implies that the quotient
${\mathcal F}/{\mathcal G}$ is also an ${\mathcal S}$-sheaf. The ``if''
part is shown in Proposition \ref{harakteriz-S-objektov} (in the
language of representations); cf. also Theorem \ref{irr-subq-hom-inv}.]

4. The inclusion ${\mathcal I}_G\hookrightarrow\Sm_G$
admits a left adjoint ${\mathcal I}$ and a right adjoint.

Examples of calculation of these adjoint functors are given
in Propositions \ref{wedge} and \ref{A1-quotient-of-closed}.

5. The sheaves $C_{k(X)}:={\mathcal I}\Psi_X$ form a system of projective
generators of ${\mathcal I}_G$. [This follows from 2 and 4.]

({\it Remark.} There are no projective objects in $\Sm_G$.)

\subsection{Irreducible objects of ${\mathcal I}_G$} \label{Irred-obj}
{\it Examples.} Let $M$ be a simple effective primitive pure covariant
motive. Then
$${\mathbb B}^0(M):Y\mapsto\Hom_{\{\text{{\rm pure $k$-motives}}\}}
(\overline{Y},M)$$ is a well-defined sheaf of finite-dimensional
${\mathbb Q}$-vector spaces (\cite{repr}).

A particular case of this example is the sheaf ${\mathcal H}_1^A$,
corresponding to the motive ``$H_1(A)$'' for any simple abelian
$k$-variety $A$.
\begin{proposition}[\cite{repr}] \label{funktor-B} ${\mathbb B}^0$
gives rise to a fully faithful functor ${\mathbb B}^{\bullet}$:
$$\{\mbox{{\rm pure $k$-motives}}\}\longrightarrow
\{\mbox{{\rm semisimple sheaves of finite length of
finite-dimensional graded ${\mathbb Q}$-vector spaces}}\}.$$
\end{proposition}
\begin{conjecture}[\cite{repr}] \label{equiv-B} This is an equivalence of
categories. {\rm (In other words, any irreducible sheaf of finite-dimensional
${\mathbb Q}$-vector spaces is isomorphic to ${\mathbb B}^0(M)$
for a primitive irreducible effective pure motive $M$.)}
\end{conjecture}

This can be complemented by the following conjecture, which I consider
as one of the principal problems on ${\mathbb A}^1$-invariant sheaves.

\begin{conjecture}[\cite{pgl}] \label{irred-diff-for} Any simple
${\mathbb A}^1$-invariant sheaf can be embedded into the sheaf
$\SOm^{\bullet}_{|k}:Y\mapsto\Omega^{\bullet}_{k(Y)|k}$. \end{conjecture}

This conjecture is rather strong: it implies the Bloch's conjecture:
\begin{conj-corollary}[\cite{pgl}] \label{gip-bloha}
Suppose that a rational map $f:Y\dasharrow X$ of smooth proper
$k$-varieties induces an injection
$\Gamma(X,\Omega^{\bullet}_{X|k})\hookrightarrow
\Gamma(Y,\Omega^{\bullet}_{Y|k})$.\footnote{{\it Example.} Let $r\ge 1$
be an integer and $X$ be a smooth proper $k$-variety with
$\Gamma(X,\Omega^j_{X|k})=0$ for all $r<j\le\dim X$. Let $Y$ be
a sufficiently general $r$-dimensional plane section of a smooth projective
variety $X'$ birational to $X$. Then, as all considered invariants
are birational, the inclusion $Y\hookrightarrow X'$ induces
an injection $\Gamma(X,\Omega^{\bullet}_{X|k})\hookrightarrow
\Gamma(Y,\Omega^{\bullet}_{Y|k})$.} Then $f$ induces a surjection
$CH_0(Y)\to CH_0(X)$.

If $\Gamma(X,\Omega^{\ge 2}_{X|k})=0$ then the Albanese map induces
an isomorphism $CH_0(X)^0\stackrel{\sim}{\longrightarrow}{\rm Alb}(X)(k)$,
where $CH_0(X)^0$ is the Chow group of 0-cycles of degree 0 and
${\rm Alb}(X)$ is the Albanese variety of $X$.
{\rm (The converse, due to Mumford, is well-known.)}
In that case $C_{k(X)}=CH_0(X_F)_{{\mathbb Q}}$.
\end{conj-corollary}
{\it Proof.} Let $C$ be the cokernel of
$\alpha:CH_0(Y_F)_{{\mathbb Q}}\to CH_0(X_F)_{{\mathbb Q}}$. Then the kernel
of the homomorphism $\alpha^{\ast}:\Hom_G(CH_0(X_F),
\Omega^{\bullet}_{F|k})\to\Hom_G(CH_0(Y_F),\Omega^{\bullet}_{F|k})$ is
$\Hom_G(C,\Omega^{\bullet}_{F|k})$. By Proposition \ref{wedge}, the
homomorphism $\alpha^{\ast}$ coincides with the pull-back under $f^{\ast}
:\Gamma(X,\Omega^{\bullet}_{X|k})\to\Gamma(X,\Omega^{\bullet}_{Y|k})$.
As the latter is injective, we conclude that
$\Hom_G(C,\Omega^{\bullet}_{F|k})=0$. If $C\neq 0$ then it
is cyclic, and thus, admits an simple quotient, and therefore,
a non-zero morphism to $\Omega^{\bullet}_{F|k}$.
This contradiction implies that $C=0$.

As the objects ${\mathbb Q}$ and ${\rm Alb}X(F)_{{\mathbb Q}}$ of
${\mathcal I}_G$ are projective (\cite[\S6.2]{repr}), the natural
surjections $\deg:C_{k(X)}\to{\mathbb Q}$ and ${\rm Alb}_F:\ker\deg\to
{\rm Alb}X(F)_{{\mathbb Q}}$ are split, so the cyclic $G$-module
$C_{k(X)}$ is isomorphic to a direct sum of type ${\mathbb Q}\oplus
{\rm Alb}X(F)_{{\mathbb Q}}\oplus\ker{\rm Alb}_F$. Thus,
$\Hom_G(C_{k(X)},\Omega^{\bullet}_{F|k})\cong\Hom_G({\mathbb Q}\oplus
{\rm Alb}X(F),\Omega^{\bullet}_{F|k})\oplus\Hom_G(\ker{\rm Alb}_F,
\Omega^{\bullet}_{F|k})$. By Proposition \ref{wedge},
$\Hom_G(C_{k(X)},\Omega^{\bullet}_{F|k})=\Gamma(X,\Omega^{\bullet}_{X|k})$
and $\Hom_G({\mathbb Q}\oplus{\rm Alb}X(F),\Omega^{\bullet}_{F|k})=
\Gamma(X,\Omega^{\le 1}_{X|k})$. If $\Gamma(X,\Omega^{\ge 2}_{X|k})=0$ this
means that $\Hom_G(C_{k(X)},\Omega^{\bullet}_{F|k})=\Hom_G({\mathbb Q}\oplus
{\rm Alb}X(F),\Omega^{\bullet}_{F|k})$. Therefore,
the $G$-module $\ker{\rm Alb}_F$ should be zero, as otherwise it
is cyclic, thus admits a non-zero simple quotient, and (by Conjecture
\ref{irred-diff-for}) a non-zero morphism to $\Omega^{\bullet}_{F|k}$.
It remains to take the $G$-invariants of $\ker\deg
\stackrel{\sim}{\longrightarrow}CH_0(X_F)^0_{{\mathbb Q}}
\stackrel{\sim}{\longrightarrow}{\rm Alb}X(F)_{{\mathbb Q}}$;
the torsion is controlled by Roitman's theorem. \qed

\vspace{4mm}

Also this would imply that any irreducible ${\mathbb A}^1$-invariant
sheaf is a sheaf of finite-dimensional vector spaces.

{\it Example.} Let ${\mathcal F}$ be a simple ${\mathbb A}^1$-invariant
sheaf and suppose that it is of level 1, i.e. it is non-constant and
${\mathcal F}(Y)\neq 0$ for a curve $Y$, cf. also
p.\pageref{coniveau-filtr}. Then, by \cite[Corollary 6.22]{repr},
${\mathcal F}\cong{\mathcal H}^A_1$ for a simple abelian variety $A$.
Now any non-zero $\eta\in\Gamma(A,\Omega^1_{A|k})$ gives an embedding
${\mathcal F}\hookrightarrow\Omega^1_{|k}$ by $[x:{\mathcal O}(U)\to k(Y)]
\mapsto x(\eta)\in\Omega^1_{Y|k}(Y)$ ($U\subset A$ is an affine open subset).

\begin{proposition} \label{bir-A1-presheaf} A dominant presheaf
${\mathcal F}$ is a sheaf if and only if the following three conditions hold:
{\rm (i)} the sequence ${\mathcal F}(X)\to{\mathcal F}(X\times{\mathbb A}^1)
\rightrightarrows{\mathcal F}(X\times{\mathbb A}^2)$ is exact for any
smooth $k$-variety $X$,\footnote{E.g., any ${\mathbb A}^1$-invariant presheaf
${\mathcal F}$ satisfies the condition (i).} {\rm (ii)} ${\mathcal F}$ is
birationally invariant, {\rm (iii)} it has the Galois descent property,
i.e. ${\mathcal F}(X)={\mathcal F}(Y)^{{\rm Aut}(Y|X)}$
for any Galois covering $Y\to X$. \end{proposition}
{\it Proof.} The conditions (i)--(iii) are particular cases of the equalizer
diagram (\ref{opr-puchka}) for coverings by (i) projections
$X\times{\mathbb A}^s\to X$, (ii) open dense $U\subset X$, (iii) \'etale
Galois covers $Y\to X$, respectively. Galois descent property for any
sheaf is clear, since \'etale morphisms with dense images are covering
and $U\times_XU=\coprod_{g\in{\rm Aut}(Y|X)}U_g$ for a Zariski open
${\rm Aut}(Y|X)$-invariant $U\subset Y$, where $U_g\cong U$ is
the image of the embedding $(id_U,g):U\hookrightarrow U\times_XU$.

Conversely, it is clear that any Galois-separable presheaf ${\mathcal F}$
satisfying (i) and (ii) is separable: if $Y\to X$ is a cover, i.e.
a smooth dominant morphism, then for any sufficiently general dominant map
$\varphi:Y\dasharrow{\mathbb A}^{\delta}$ (where $\delta=\dim Y-\dim X$)
we can choose a dominant \'etale morphism $\widetilde{Y}\to Y$ so that the
composition $\widetilde{Y}\to Y\dasharrow X\times{\mathbb A}^{\delta}$
is Galois with the group denoted by $H$, and therefore, the composition
$$\begin{array}{rccc}{\mathcal F}(X)
\stackrel{\text{(i)}}{\hookrightarrow}
{\mathcal F}(X\times{\mathbb A}^1)
\stackrel{\text{(i)}}{\hookrightarrow}
\dots\stackrel{\text{(i)}}{\hookrightarrow}&
{\mathcal F}(X\times{\mathbb A}^{\delta})&\longrightarrow&
{\mathcal F}(Y)\\ &\downarrow\lefteqn{\text{injective}}&&\downarrow\\
&{\mathcal F}(\widetilde{Y})^H&\hookrightarrow&
{\mathcal F}(\widetilde{Y})\end{array}$$ is injective.
Then in the commutative diagram
\begin{equation}\label{red-to-aff}
\begin{array}{ccccc}{\mathcal F}(X)&\to&{\mathcal F}(\widetilde{Y})
&\rightrightarrows&{\mathcal F}(\widetilde{Y}\times_X\widetilde{Y})\\
\|&&\uparrow&&\uparrow\\
{\mathcal F}(X)&\to&{\mathcal F}(Y)&\rightrightarrows&
{\mathcal F}(Y\times_XY)\\
\|&&\uparrow&&\uparrow\\
{\mathcal F}(X)&\to&{\mathcal F}(X\times{\mathbb A}^{\delta})&
\rightrightarrows&
{\mathcal F}(X\times{\mathbb A}^{\delta}\times{\mathbb A}^{\delta})
\end{array}\end{equation} all arrows are injective, so it suffices
to show the exactness of the upper row.

Let $f$ be an element of ${\mathcal F}(\widetilde{Y})$. The image of
$f$ in ${\mathcal F}(\widetilde{Y}\times_X\widetilde{Y})$ under the
projection to the first factor is fixed by $\{1\}\times H$; the image
of $f$ in ${\mathcal F}(\widetilde{Y}\times_X\widetilde{Y})$ under
the projection to the second factor is fixed by $H\times\{1\}$. Now
if $f$ is an element of the equalizer of ${\mathcal F}(\widetilde{Y})
\rightrightarrows{\mathcal F}(\widetilde{Y}\times_X\widetilde{Y})$ then
the two images coincide, so they are fixed by the group $H\times H$.
The injectivity of both parallel arrows in the upper row of the diagram
(\ref{red-to-aff}) implies that $f\in{\mathcal F}(\widetilde{Y})^H$.
By (iii) and the injectivity of the vertical arrow, $f$ comes from the
equalizer of the bottom row of the diagram (\ref{red-to-aff}).
Finally, the bottom row of the diagram (\ref{red-to-aff}) is exact by
Lemma \ref{red-to-A1}, and thus, $f$ comes from ${\mathcal F}(X)$. \qed

\begin{lemma} \label{red-to-A1} Let ${\mathcal V}$ be a category of
schemes such that for any $X\in{\mathcal V}$: {\rm (i)} the projection
$X\times{\mathbb A}^1\to X$ is a morphism in ${\mathcal V}$,
{\rm (ii)} any linear automorphism of any affine space ${\mathbb A}$
induces an automorphism of $X\times{\mathbb A}$ in ${\mathcal V}$.
Let ${\mathcal F}$ be a presheaf on this category such that
the sequence ${\mathcal F}(X)\to{\mathcal F}(X\times{\mathbb A}^1)
\rightrightarrows{\mathcal F}(X\times{\mathbb A}^2)$ is exact
for any $X\in{\mathcal V}$. Then the sequence ${\mathcal F}(X)
\to{\mathcal F}(X\times{\mathbb A}^s)\rightrightarrows
{\mathcal F}(X\times{\mathbb A}^{2s})$ is exact for any
$X\in{\mathcal V}$ and any $s\ge 1$. \end{lemma}
{\it Proof.} We proceed by induction on $s$, the case $s=1$ being trivial.

Denote by $\mathrm{pr}_1,\mathrm{pr}_2:X\times{\mathbb A}^{2s}
\rightrightarrows X\times{\mathbb A}^s$ the two projections.

For any $f\in{\mathcal F}(X\times{\mathbb A}^s)$ the element
$\mathrm{pr}_1^{\ast}f$ is fixed by $\Phi^{\ast}\in\End
{\mathcal F}(X\times{\mathbb A}^s\times{\mathbb A}^s)$
for any linear automorphism $\Phi(u,v)=(u,\varphi(u,v))$ of
${\mathbb A}^s\times{\mathbb A}^s$.
Similarly, $\mathrm{pr}_2^{\ast}f$ is fixed by $\Psi^{\ast}\in\End
{\mathcal F}(X\times{\mathbb A}^s\times{\mathbb A}^s)$
for any linear automorphism $\Psi(u,v)=(\psi(u,v),v)$.

Let now $f\in{\mathcal F}(X\times{\mathbb A}^s)$ be in the
equalizer of $\mathrm{pr}_1^{\ast}$ and $\mathrm{pr}_2^{\ast}$.
Then $\mathrm{pr}_1^{\ast}f=\mathrm{pr}_2^{\ast}f$ is fixed by the
group, generated by $\Phi^{\ast}$ and $\Psi^{\ast}$ as above.
Clearly, such automorphisms $\Phi$ and $\Psi$ generate the group
consisting of all linear automorphisms $\alpha$. Then
$\mathrm{pr}_1^{\ast}f=\alpha^{\ast}\mathrm{pr}_1^{\ast}f$.

Applying the induction assumption in the case where $\alpha$ is
identical on one of the first $s$ coordinates and interchanges
$i$-th and $(s+i)$-th for other $1\le i\le s$, we get that
$f$ belongs to the image of ${\mathcal F}(X\times{\mathbb A}^1)\to
{\mathcal F}(X\times{\mathbb A}^s)$ under morphism induced
by the projection ${\mathbb A}^s\to{\mathbb A}^1$ to one
of the copies of ${\mathbb A}^1$. Then the case $s=1$ implies
that $f$ comes from  ${\mathcal F}(X)$.  \qed

\vspace{4mm}

{\sc Examples.} 1.  A stable birationally invariant dominant presheaf
with the Galois descent is a sheaf.

2. Example of a birationally invariant presheaf ${\mathcal F}$ with
the Galois descent property which is not a sheaf. Let ${\mathcal G}$
be a dominant sheaf and $I\subsetneq\{0,1,2,\dots\}$ be a non-empty
(finite or infinite) interval. Assume that ${\mathcal G}(X)\neq 0$
for some $X$ with $\dim X\not\in I$. Then the presheaf ${\mathcal F}:
U\mapsto\left\{\begin{array}{ll}{\mathcal G}(U)&\text{if $\dim U\in I$}\\
0&\text{if $\dim U\not\in I$}\end{array}\right.$ (with the restriction maps
of ${\mathcal G}$, whenever possible, otherwise zero) is birationally
invariant and has the Galois descent property, but it is not a sheaf. The
sheafification of ${\mathcal F}$ is ${\mathcal G}$ if $I$ is infinite
and 0 otherwise.

\vspace{4mm}

Now, what are the projective generators of ${\mathcal I}_G$ from
\S\ref{Properties-of}, Property 5?
\begin{conjecture} \label{C=CH} For any smooth proper $k$-variety $X$,
the sheaf $C_{k(X)}$ coincides with $Y\mapsto CH_0(X_{k(Y)})_{{\mathbb Q}}$.
\end{conjecture}

{\it Remarks.} 1. This is known, e.g., if $X$ is a curve, cf.
\cite[Cor.6.21]{repr} and Proposition \ref{pic-as-adj} for a stronger
statement. Conjecture \ref{C=CH} would imply that ${\mathcal I}_G$ is a
tensor category under the operation $({\mathcal F},{\mathcal G})\mapsto
{\mathcal I}({\mathcal F}\otimes_{\Sm_G}{\mathcal G})=:{\mathcal F}
\otimes_{{\mathcal I}}{\mathcal G}$, where $\otimes_{\Sm_G}$
denotes the sheafification of the tensor product presheaf.
Moreover, the ``K\"{u}nneth formula'' holds:
$C_{k(X)}\otimes_{{\mathcal I}}C_{k(Y)}=C_{k(X\times_kY)}$.

2. It is shown in \cite[Proposition 6.17]{repr} that, roughly speaking,
$C_{k(X)}$ is the quotient of generic 0-cycles on $X$ by those divisors
of rational functions on generic curves on $X$ which are generic,
and thus, Conjecture \ref{C=CH} should be considered as a moving lemma.

3. Conjecture \ref{C=CH} and the motivic conjectures imply
conjectures \ref{equiv-B} and \ref{irred-diff-for}.

\section{Alternative descriptions of ${\mathbb A}^1$-invariant sheaves}
\label{Alternative-descr}
Now I want to introduce the language of representations and to use
it to explain some results and conjectures of \S\ref{pre-sheaves},
especially Conjecture \ref{C=CH}.

\subsection{Smooth representations and non-degenerate modules over
algebras of measures}
For any totally disconnected Hausdorff group\footnote{cf.
\cite[Appendix A]{glatt}} $H$ an $H$-set (group, etc.) is called
{\sl smooth} if the stabilizers are open.

Any smooth representation $W$ of $H$ over $E$ can be considered as
a module over the associative algebra ${\mathbb D}_E(H):=\prlim_UE[H/U]$
of the ``oscillating'' measures on $H$ (for which all open subgroups and
their translates are measurable): ${\mathbb D}_E(H)\times W\to W$ is
defined by $(\alpha,w)\mapsto\beta w$ for any $\beta\in E[H]$ with the
same image $E[H]$ as $\alpha$, where $w\in W^U$ for some open subgroup
$U$ of $H$.

Passing to the inverse limit, we get the algebra structure
on ${\mathbb D}_E(H)$ from ${\mathbb D}_E(H)\times E[H/U]\to E[H/U]$.

\label{nul-annul-sod-vse} If the annihilator of $W\in\Sm_H(E)$ in
${\mathbb D}_E(H)$ vanishes then the restriction of $W$ to any compact
subgroup $U$ contains each smooth irreducible representation of $U$.
(Otherwise, if $W$ does not contain a smooth irreducible representation
$\rho$ of $U$ then the natural projector in ${\mathbb D}_E(H)$ to the
$\rho$-isotypical part would annihilate $W$.)

\subsection{A representation theoretic setting for (${\mathbb A}^1$-invariant)
sheaves} \label{gomotopich-invar-prdst} In this section, for a group $H$
as in \S\ref{nul-annul-sod-vse} and a collection $S$ of pairs of its
subgroups, we study the category $\Sm_H^S(E)$ of smooth $E$-representations
$W$ of $H$, satisfying $W^{U_1}=W^{U_2}$ for all $(U_1,U_2)\in S$.

Theorem \ref{sheafif-i-equiv} explains the consistence
of this notation with that of \S\ref{pre-sheaves}.

Collections $S$ and $S'$ are called {\sl equivalent}
if they define the same subcategory of $\Sm_H:=\Sm_H^{\emptyset}$.

For any subgroup $U\subset H$ the functor $H^0(U,-)$ on the category of
smooth $H$-sets (or modules, etc.) coincides with $\indlim H^0(V,-)$, where
the limit is taken over the open
subgroups $V$ of $H$ containing $U$. Therefore, one can assume that the
subgroups $U_1,U_2$ are intersections of open ones, and in particular, that
they are closed. Further, as $W^{U_1}\cap W^{U_2}=W^{\langle U_1,U_2\rangle}$
for any $H$-module $W$ and $U_1,U_2\subseteq\langle U_1,U_2\rangle$, one can
assume that the pairs $(U_1,U_2)\in S$ are ordered: $U_1\subset U_2$.

\begin{lemma} \label{podfaktory-dlq-komp}
Assume that for any pair $(U_1\subset U_2)\in S$ the functor $H^0(U_1,-)$
is exact on $\Sm_H$. Then the category $\Sm_H^S(E)$ is stable under passing
to the subquotients in $\Sm_H(E)$, and in particular, it is abelian.
The inclusion functor $\Sm_H^S(E)\hookrightarrow\Sm_H(E)$
admits a left adjoint\footnote{The diagrams $\begin{array}{ccc}\Sm_H(E)&
\stackrel{{\mathcal I}_S}{\longrightarrow}&\Sm^S_H(E)\\
\otimes_EE'\downarrow\phantom{\otimes_EE'}&&
\phantom{\otimes_EE'}\downarrow\otimes_EE'\\
\Sm_H(E')&\stackrel{{\mathcal I}_S}{\longrightarrow}&\Sm^S_H(E')\end{array}
\quad\mbox{and}\quad\begin{array}{ccc}\Sm_H(E)&
\stackrel{{\mathcal I}_S}{\longrightarrow}&\Sm^S_H(E)\\
{\rm for}\uparrow\phantom{{\rm for}}&&\phantom{{\rm for}}\uparrow{\rm for}\\
\Sm_H(E')&\stackrel{{\mathcal I}_S}{\longrightarrow}
&\Sm^S_H(E')\end{array}$ are commutative for any field extension $E'|E$,
so omitting $E$ from the notation does not lead to a confusion.}
$W\longmapsto{\mathcal I}_SW$. \end{lemma}
{\it Proof.} If a sequence $0\to W_1\to W\to W_2\to 0$ in $\Sm_H$
is exact then the sequences $0\to W_1^{U_1}\to W^{U_1}\to W_2^{U_1}\to 0$
and $0\to W_1^{U_2}\to W^{U_2}\to W_2^{U_2}$ are also exact. If
$W\in\Sm_H^S$, i.e. $W^{U_1}=W^{U_2}$, then
$W_1^{U_1}=W_1\cap W^{U_1}=W_1\cap W^{U_2}=W_1^{U_2}$ and
$W^{U_2}\to W_2^{U_2}$ is surjective (since $W^{U_2}=W^{U_1}\to W_2^{U_1}$
is surjective and factors through $W_2^{U_2}\subseteq W_2^{U_1}$).
This means that $\Sm_H^S$ is stable under taking subquotients in $\Sm_H$.

The existence of the functor ${\mathcal I}_S$ can be deduced from the
special adjoint functor theorem, cf. \cite[\S5.8]{maclane}. However, we
construct it ``explicitly'', which enables us to relate the generators
of the category ${\mathcal I}_G$ to the Chow groups of 0-cycles.

Let $W'\in\Sm^S_H$. Any $H$-homomorphism
$W\stackrel{\alpha}{\longrightarrow}W'$ factors through the object
$\alpha(W)$ of $\Sm^S_H$. We may, therefore, assume that
$\alpha$ is surjective. Let $(U_1\subset U_2)\in S$. As the functor
$H^0(U_1,-)$ is exact on $\Sm_H$, the morphism $\alpha$
induces a surjection $W^{U_1}\longrightarrow(W')^{U_1}$. As
$(W')^{U_2}=(W')^{U_1}$, the subgroup $U_2$ acts on $(W')^{U_1}$ trivially,
and therefore, the subrepresentation
$W_{U_1\subset U_2}=\langle\sigma w-w~|~\sigma\in U_2,~w\in W^{U_1}\rangle_H$
of $H$ is contained in the kernel of $\alpha$. It follows that $\alpha$
factors through ${\mathcal I}_SW:=W/\sum_{(U_1\subset U_2)\in S}
W_{U_1\subset U_2}$.

The representation ${\mathcal I}_SW$ of $H$ is smooth, so the map
$W^{U_1}\longrightarrow({\mathcal I}_SW)^{U_1}$, induced by the projection,
is surjective, and therefore, any element $\overline{w}\in
({\mathcal I}_SW)^{U_1}$ can be lifted to an element $w\in W^{U_1}$.
Then $\sigma\overline{w}-\overline{w}$ coincides with the projection
of the element $\sigma w-w$ for any $\sigma\in U_2$. Notice that
$\sigma w-w\in W_{U_1\subset U_2}$, so its projection is zero, and
therefore, $\sigma\overline{w}=\overline{w}$ for any $\sigma\in U_2$.
As $({\mathcal I}_SW)^{U_2}\subseteq({\mathcal I}_SW)^{U_1}$, this means
that $({\mathcal I}_SW)^{U_2}=({\mathcal I}_SW)^{U_1}$, and thus,
${\mathcal I}_SW\in\Sm^S_H$.

One has $\Hom_{\Sm^S_H}({\mathcal I}_SW,W')=\Hom_{\Sm_H}(W,W')$ for any
$W\in\Sm_H$ and $W'\in\Sm^S_H$, i.e. the functor ${\mathcal I}_S$ is left
adjoint to the inclusion functor $\Sm^S_H\hookrightarrow\Sm_H$. \qed

\vspace{4mm}

{\it Remark.} The functor ${\mathcal I}_S$ generalizes the coinvariants,
since ${\mathcal I}_S=H_0(H,-)$ if $S=\{(\{1\}\subset H)\}$.

{\sc Examples.} 1. The functor $H^0(U_1,-)$ is exact on $\Sm_H$ if,
e.g., the subgroup $U_1$ is compact.

2. Suppose that $H$ is the automorphism group of an algebraically closed
field extension $F|k$ of countable transcendence degree and $U_1$ is the
subgroup of automorphisms of $F$ over a fixed subextension of $k$ in $F$
of infinite transcendence degree. Though $U_1$ need not be
compact, the functor $H^0(U_1,-)$ is exact on $\Sm_H$.

\begin{proposition} \label{harakteriz-S-objektov}
Let $H$ be a totally disconnected group and $S$ be such a collection of
pairs of its subgroups $(U_1\subset U_2)$ that
\begin{enumerate}\item \label{cond-1}
for any pair $(U_1\subset U_2)\in S$ there exists
an element $\sigma\in U_2$ such that
{\rm (i)} $(U_1\cap\sigma U_1\sigma^{-1}\subset U_1)\in S$; {\rm (ii)}
$U_1$ and $\sigma U_1\sigma^{-1}$ generate $U_2$, at least topologically.
\item there exists an equivalent collection of pairs of its subgroups
$(U_1\subset U_2)$, where all $U_1$ are compact. \end{enumerate}
Then an object of $\Sm_H(E)$ belongs to $\Sm_H^S(E)$ if
and only if all its irreducible subquotients are in $\Sm_H^S(E)$.
In particular, $\Sm_H^S(E)$ is a Serre subcategory of
$\Sm_H(E)$. \end{proposition}
{\it Proof.} Suppose that $W\not\in\Sm_H^S$, whereas all its
irreducible subquotients are in $\Sm_H^S$. Then $W^{U_1}\neq
W^{U_2}$ for some pair $(U_1\subset U_2)\in S$, that is there exist a
vector $v\in W^{U_1}\smallsetminus W^{U_2}$. Choose an element
$\sigma\in U_2$ as in condition (\ref{cond-1}) of the statement for the
pair $(U_1\subset U_2)\in S$. Then $\sigma v-v=:u\neq 0$, since $U_1$ and
$\sigma$ generate a dense subgroup in $U_2$.

One may replace $W$ by its quotient by a maximal subrepresentation not
containing $u$. Then the subrepresentation $\langle u\rangle$, generated
by $u$, becomes irreducible, and thus, an object of $\Sm_H^S$.

By definition, $u\in W^{U_1}+W^{\sigma U_1\sigma^{-1}}\subseteq
W^{U_1\cap\sigma U_1\sigma^{-1}}$. As $\langle u\rangle\in\Sm_H^S$
and $(U_1\cap\sigma U_1\sigma^{-1}\subset U_1)\in S$, we conclude that
$u\in W^{U_1}$. This implies that $\sigma v\in W^{U_1}$. On the other hand,
$\sigma v\in W^{\sigma U_1\sigma^{-1}}$, so $\sigma v\in W^{U_1}\cap
W^{\sigma U_1\sigma^{-1}}$. The latter vector space coincides with $W^{U_2}$,
and thus, $v\in W^{U_2}$, contradicting our assumptions.

The converse follows from Lemma \ref{podfaktory-dlq-komp}. \qed

\subsection{More notations and compatibility of notations of
\S\ref{gomotopich-invar-prdst} and \S\ref{pre-sheaves}:
the sheafification and smooth representations} \label{obozn-predst}
From now on we fix the following notations: $F|k$ is an algebraically
closed field extension of countably infinite transcendence degree,
and $G=G_{F|k}$ is the automorphism group of the extension $F|k$.

Consider connected smooth $k$-varieties $U$ endowed with a generic
$F$-point, i.e., with a $k$-field embedding $k(U)\overk F$.
For any presheaf ${\mathcal F}$ on ${\mathcal S}m_k'$ we can form the
direct limit ${\mathcal F}(F):=\indlim{\mathcal F}(U)$ over such $U$.
The group $G={\rm Aut}(F|k)$ acts naturally on ${\mathcal F}(F)$.
\begin{theorem}[\cite{glatt}]\label{sheafif-i-equiv} \begin{itemize}\item
${\mathcal F}\mapsto{\mathcal F}(F)$ gives an equivalence of the categories
$\Sm_G^{{\mathcal S}}(E)$ (of \S\ref{pre-sheaves}) and $\Sm_G^S(E)$
(of \S\ref{gomotopich-invar-prdst}), where $S$ is the collection
of pairs $G_{F|k(X)}\subseteq G_{F|k(Y)}$ for all morphisms
$(X\to Y)\in{\mathcal S}$.
\item For any presheaf ${\mathcal F}$, the sheaf corresponding to
${\mathcal F}(F)$ is the sheafification of ${\mathcal F}$.
\end{itemize} \end{theorem}

\subsection{An example: birational invariants constant on the projective
spaces} Let $S$ consist of a single pair $K\subset G$ such that $K$ is a
`maximal' compact subgroup, i.e., any compact subgroup is conjugate to a
subgroup of $K$. Then $S$ is equivalent to the collection consisting of a
single pair $K'\subset G$, where $K'$ is the pointwise stabilizer of some
transcendence base of $F|k$, and also to the collection $S'$ of pairs
$U\subset G$ such that $U$ is the pointwise stabilizer of a finite subset
of a fixed transcendence base of $F|k$. The collection $S'$ satisfies
the assumptions of Proposition \ref{harakteriz-S-objektov}.

\begin{lemma} \label{nepriv-tenz-proizv} Let $E'|E$ be an extension
of fields, $H$ be a group and $(\rho,W_2)$ be an irreducible
$E'$-representation of $H$. Let $W_1$ be an $E$-representation of $H$,
absolutely irredicible even in restriction to $\ker\rho$.\footnote{i.e.,
irredicible and with ${\rm End}_{E[\ker\rho]}(W_1)=E$: otherwise, if
${\rm End}_{E[\ker\rho]}(W_1)\neq E$ and $E'|E$ is a non-trivial
field extension in the division $E$-algebra ${\rm End}_{E[\ker\rho]}(W_1)$
then the action of $E'$ on $W_1$ gives a non-injective surjection of
$E'$-representations $W_1\otimes_EE'\longrightarrow W_1$.} Then the
$E'$-representation $W_1\otimes_EW_2$ of $H$ is irredicible. \end{lemma}
{\it Proof.} Let $\xi\in W_1\otimes_EW_2$ be a non-zero vector.
It suffices to check that the $E'[H]$-span of $\xi$ contains
$W_1\otimes v$ for any $v\in W_2$.
Any non-zero $E[\ker\rho]$-submodule in $W_1^m$ is isomorphic to $W_1^{m'}$
for some $1\le m'\le m$, and therefore, the $E[\ker\rho]$-submodule
in $W_1\otimes_EW_2$ spanned by $\xi$ (which is in fact a submodule
in $\oplus_{i=1}^mW_1\otimes v_i\cong W_1^m$ for some $m\ge 1$ and
$E$-linearly independent $v_1,\dots,v_m$) contains
a $E[\ker\rho]$-submodule $W_1'$ isomorphic to $W_1$.
As the endomorphisms of the $E[\ker\rho]$-module $W_1$ are scalar,
there exists a non-zero $m$-tuple $(a_1,\dots,a_m)\in E^n$ such that
$W_1'=\{a_1w\otimes v_1+\dots+a_mw\otimes v_m~|~w\in W_1\}$.
In other words, $W_1'=W_1\otimes v'$, where $v':=a_1v_1+\dots+ a_mv_m$
is a non-zero vector in $W_2$.

As any vector $v$ of $W_2$ is an $E'$-linear combination of several
elements in the $H$-orbit of $v'$, we may assume that $v=hv'$ for some
$h\in H$. Then $u\otimes v=h(h^{-1}u\otimes v')$ for any $u\in W_1$. \qed

\begin{lemma} \label{max-v-tenz-proizv} Let $E'|E$ be an extension
of fields, $H$ be a group and $(\rho,W_2)$ be an irreducible
$E'$-representation of $H$. Let $W_1$ be an $E$-representation of $H$
such that {\rm (i)} the sum $\Sigma$ of all proper $E$-subrepresentations
of $\ker\rho$ in $W_1$ is proper\footnote{$\Sigma$ is $H$-invariant:
as $\ker\rho$ is a normal subgroup of $H$, the group $H$ permutes the
$\ker\rho$-submodules in $W_1$, while $\Sigma$ is the maximal proper
$\ker\rho$-submodule in $W_1$.} and {\rm (ii)} $W_1/\Sigma$ is
absolutely irredicible in restriction to $\ker\rho$ and its restriction
to the pointwise stabilizer $\Xi$ of $\Sigma$ in $\ker\rho$ is non-trivial.
Then any proper $E'$-subrepresentation of $H$ in $W_1\otimes_EW_2$ is
contained in $\Sigma\otimes_EW_2$. \end{lemma}
{\it Proof.} Let $\xi\in W_1\otimes_EW_2$ be a vector, which is not in
$\Sigma\otimes_EW_2$. It suffices to check that the $E'[H]$-span $V$ of
$\xi$ contains $W_1\otimes v$ for some non-zero $v\in W_2$, as then $V$
coincides with $W_1\otimes_EW_2$: any vector of $W_2$ is an $E'$-linear
combination of several elements in the $H$-orbit of $v$ and
$W_1\otimes hv=h(W_1\otimes v)$ for any $h\in H$.

It follows from Lemma \ref{nepriv-tenz-proizv} that $V$ is surjective over
$(W_1/\Sigma)\otimes_EW_2$. In particular, $V$ contains an element of type
$\sum_{i=1}^ma_i\otimes b_i$ for some $a_1\in W_1\smallsetminus\Sigma$, whose
projection to $W_1/\Sigma$ is not fixed by $\Xi$, for some $a_2,\dots,a_m\in
\Sigma$ and for some $E'$-linearly independent $b_1,\dots,b_m\in W_2$.
Then there exists an element $h\in\Xi$ such that $ha_1-a_1\in
W_1\smallsetminus\Sigma$, and therefore, $V$ contains an element of type
$\sum_{i=1}^ma\otimes b_1$ for some $a\in W_1\smallsetminus\Sigma$. \qed

\begin{proposition} \label{I-S-polulin} Let $W\in\Sm_G(E)$ be an object.
For any open subgroup $U$ of $G$, denote by $W_{(U)}$ the sum of all proper
subrepresentations of $U$ in $W$; and by $\Xi_U$ the pointwise stabilizer of
$W_{(U)}$ in $U$. Suppose that for any open subgroup $U$ of $G$: {\rm (i)}
the $E$-representation $W/W_{(U)}$ of $U$ is absolutely irreducible and
non-trivial in restriction to $\Xi_U$\footnote{In particular, $W$ is absolutely
indecomposable. Any non-zero quotient of $A(F)$ for an absolutely simple
algebraic $k$-group $A$ is an example of such $W$. (Indeed, any open subgroup
$U\subset G$ contains $G_{F|L}$ for a finitely generated $L$ in $F|k$, so
any $t\in A(F)\smallsetminus A(\overline{L})$ is a cyclic vector of $A(F)$,
considered as $U$-module. Here $\overline{L}$ is the algebraic closure of
$L$ in $F$. If the transcendence degree of $L|k$ is minimal then, by
\cite{max}, $\overline{L}$ is $U$-invariant, so $A(F)_{(U)}=A(\overline{L})$.
\qed)} and {\rm (ii)} any irreducible smooth representation of $K$ can be
embedded into $W$ so that its image does not meet $W_{(U)}$. Then
${\mathcal I}_S$ annihilates any quotient of $W\otimes_EV$ for any
$V\in\Sm_G(E)$. \end{proposition}
{\it Proof.} It suffices to check the vanishing of
${\mathcal I}_S(W\otimes_EV)$. Extending the coefficients if needed,
we may assume that $E$ is big enough (i.e., algebraically closed and
$\# E>\# k$), so that any smooth irreducible $E$-representation of
any open subgroup of $G$ is absolutely irreducible.\footnote{Schur's
lemma=\cite[Claim 2.11]{bz}: {\it Let $H$ be a totally disconnected
group and $E$ be a field of cardinality greater than the cardinality of
$H/U$ for any open subgroup $U$ of $H$. Then the endomorphisms of the
smooth irreducible $\overline{E}$-representation of $H$ are scalar.}}

The vanishing holds if the $G$-module $W\otimes_EV$ is spanned by the
elements $g\xi-\xi$ for all $\xi\in(W\otimes_EV)^K$ and all $g\in G$.
Equivalently, as the restriction of $V$ to $K$ is semisimple, the $G$-span
of such elements $g\xi-\xi$ contains $W\otimes_E\rho$ for any irreducible
$E$-subrepresentation $\rho$ of $K$ in $V$.
By (ii), $W$ contains a $E$-subrepresentation of $K$ which is
(a) dual to $\rho$ and (b) outside of $W_{(U)}$, where
$U\subset G$ is the pointwise stabilizer of $\rho$.
Then there is an element $\xi\in(W\otimes_E\rho)^K$, which is not in
$W_{(U)}\otimes_E\rho$.

As the $\Xi_U$-module $W/W_{(U)}$ is non-trivial, there exists an element
$u\in\Xi_U$ such that $\eta:=u\xi-\xi$ is not in $W_{(U)}\otimes_E\rho$.
Denote by $\widetilde{U}$ the subgroup in $G$ generated by $U$ and $K$.
Then $\widetilde{U}$ contains $U$ as a normal subgroup of finite index;
$\widetilde{U}$ acts on $W_{(U)}$; $\rho$ can be viewed as a representation
of $\widetilde{U}$ via the identification $\widetilde{U}/U=K/U\cap K$.
By Lemma \ref{max-v-tenz-proizv} (with $H=\widetilde{U}$), the element
$\eta$ generates the $E[\widetilde{U}]$-module $W\otimes_E\rho$. \qed

\begin{lemma}[A source of representations of $G$ containing all
irreducible smooth representations of $K$] \label{primery-soderzh-vse}
If a subrepresentation $W$ of $G$ in
$\bigotimes^{\bullet}_F\Omega^1_{F|k}$ does not contain regular
forms,\footnote{Examples of such $W$ are subrepresentations of
${\rm Sym}^2_F\Omega^1_{F|k}$, of $\Omega^{\bullet}_{F|k,\text{exact}}$,
or of the image in $\Omega^j_{F|k}$ of $\wedge^j\Omega^1_{F|k,\log}$,
where $d\log:F^{\times}/k^{\times}\stackrel{\sim}{\longrightarrow}
\Omega^1_{F|k,\log}$, for any $j\ge 1$. It follows directly
from Hilbert's Satz 90 that the representation $F$ (and therefore, the
irreducible representation $d:F/k\stackrel{\sim}{\longrightarrow}
\Omega^1_{F|k,\text{exact}}$) of $G$ contains all irreducible smooth
(and thus, finite-dimensional) representations of $K$.} i.e.,
forms from $\Gamma(X,\Omega^{\bullet}_{X|k})$ for a smooth proper
$k$-variety $X$ with $k(X)\subset F$, then $W$ contains each
irreducible smooth representation $\rho$ of $K$. \end{lemma}

As mentioned in \S\ref{nul-annul-sod-vse}, if no non-zero element
of ${\mathbb D}_E(G)$ annihilates a smooth
representation $W$ then $W$ contains all irreducible smooth
representations of $K$. The vanishing of the annihilators of
$F/k$ and $F^{\times}/k^{\times}$ is shown in \cite[Prop.4.2]{repr}.
Assume for simplicity that $F^K|k$ is purely transcendental.

{\it Proof.} Let $p_{\rho}$ be the central projector
in the group algebra of $Q=K/\ker\rho$ onto the $\rho$-isotypical part.
As explained in \cite[Prop.7.6]{pgl}, $W$ contains a non-zero element
$\omega$ fixed by the group $G_{F|k({\mathbb P}^M)}$ for an appropriate
$M\ge 1$ and an embedding $k({\mathbb P}^M)\hookrightarrow F$.
The finite field extension $F^{\ker\rho}|F^K$ can be considered as a purely
transcendental extension of a function field extension $k(Y)|k(Y)^Q$ of
smooth projective $k$-varieties of dimension $\ge M$. Consider $\omega$ as
a differential form with poles on ${\mathbb P}^M_k$. Fix a sufficiently
general finite morphism $f:Y\longrightarrow{\mathbb P}^M_k$, unramified
above the poles of $\omega$, and such that the poles of $f^{\ast}\omega$
pass through a fixed point of $Y$, but not through another point of its
$Q$-orbit. Then, as $Q$ acts freely on the set of `sufficiently general'
divisors on $Y$, the form $p_{\rho}f^{\ast}\omega$ is non-zero, and thus,
$p_{\rho}f^{\ast}\omega$ spans a $K$-submodule in $W$ isomorphic to $\rho$.
\qed

\vspace{4mm}

{\it Remark.} \label{vanish-I-S-semilin} The vanishing of ${\mathcal I}_S$
on any smooth semilinear representation $V$ of $G$ is evident: Let $L$
be the function field of an affine $k$-space embedded into $F$. For any
$v\in V^{G_{F|L}}$ and any $x\in F$ transcendental over $L$ the vector $xv$
belongs to $V^{G_{F|L(x)}}$, so its image ${\mathcal I}_SV$ should be fixed
by $G$. In particular, the image of $xv$ in ${\mathcal I}_SV$ coincides
with the image of $2xv$, and thus, $xv$ becomes $0$ in ${\mathcal I}_SV$.
Such vectors $xv$ span $V$, so ${\mathcal I}_SV=0$.

\begin{cor} \label{log-diff-form} For any $k$-variety $U$ and any
rational closed form $\eta$ on $U\times{\mathbb A}^1$ there exist an
affine variety $Y$, dominant morphisms $\pi:Y\to U\times{\mathbb A}^1$,
$\pi_1,\dots,\pi_m:Y\to{\mathbb A}^N_k$ and rational closed forms
$\eta_1,\dots,\eta_m$ on ${\mathbb A}^N_k$ and $\eta_0$ on $U$
such that $\pi^{\ast}\eta=({\rm pr}_U\circ\pi)^{\ast}\eta_0
+\pi_1^{\ast}\eta_1+\dots+\pi_m^{\ast}\eta_m$. \end{cor}
{\it Proof.} We consider $\eta$ as a section of the sheaf
$\SOm^{\bullet}_{|k,\text{closed}}:X\mapsto
\Omega^{\bullet}_{k(X)|k,\text{closed}}$ over $U\times{\mathbb A}^1$.
Proposition \ref{A1-quotient-of-closed} describes the kernel of
$\SOm^{\bullet}_{|k,\text{closed}}\stackrel{\alpha}{\longrightarrow}
{\mathcal I}\SOm^{\bullet}_{|k,\text{closed}}$ as the ideal generated by the
exact and the logarithmic differentials. By Proposition \ref{I-S-polulin},
applied to $W=F^{\times}/k^{\times}$, ${\mathcal I}_S$ annihilates
the kernel of $\alpha$. Thus, modulo closed forms coming from
projective spaces, $\eta$ comes from $U$. \qed

\vspace{4mm}

Let $X$ be a smooth proper $k$-variety and $W:={\mathbb Q}[\{k(X)\overk F\}]$
be the module of generic 0-cycles on $X$. The space $W^K$ is the image of
the projector defined by the Haar measure of $K$. As the generators of $W$
are generic points of $X$, the space $W^K$ is spanned by the 0-cycles of
type $p_{\ast}\pi^{\ast}q$ for all diagrams of dominant $k$-morphisms
$X\stackrel{p}{\leftarrow\hspace{-3mm}\longleftarrow}Y
\stackrel{\pi}{\longrightarrow\hspace{-3mm}\to}{\mathbb P}^N_k$, where
$\pi$ is generically finite, and all generic points $q\in{\mathbb P}^N(F^K)$.
(Indeed, for any generic $F$-point $\sigma:k(X)\overk F$ of $X$ the orbit
$K\sigma$ is finite, so the compositum $L_1$ of the images of the elements
of $K\sigma$ is finitely generated over $k$. Let $L_0\subset F^K$ be a
finitely generated and purely transcendental extension of $k$ containing
$L_1^K$. Let $Y$ be a $K$-equivariant smooth $k$-model of $L_0L_1$. Then
$p$ and $\pi$ are induced by the inclusions $k(X)\subset k(Y)\supset L_0$.)
Thus, the module ${\mathcal I}_SW$ is the quotient of $W$ by the
${\mathbb Q}$-span of 0-cycles of type
$p_{\ast}\pi^{\ast}q_1-p_{\ast}\pi^{\ast}q_2$ for all dominant $k$-morphisms
$p:Y\to X$, generically finite $k$-morphisms $\pi:Y\to{\mathbb P}^N_k$ and
all generic points $q_1,q_2\in{\mathbb P}^N(F)$.

\begin{lemma} \label{cyclic-1} Let $X$ be a smooth proper curve over $k$
of genus $g$. Then the $G$-module $Z^{{\rm rat}}_0(k(X)\otimes_kF):=\ker[
Z_0(k(X)\otimes_kF)\longrightarrow CH_0(X\times_kF)]$ is
generated by $w_N=\sum^N_{j=1}\sigma_j-\sum^N_{j=1}\tau_j$ for all $N>g$,
where $(\sigma_1,\dots,\sigma_N;\tau_1,\dots,\tau_N)$ is a generic
$F$-point of the fibre over $0$ of the map
$X^N\times_kX^N\stackrel{p_N}{\longrightarrow}\Pic^0X$
sending $(x_1,\dots,x_N;y_1,\dots,y_N)$ to the class of
$\sum^N_{j=1}x_j-\sum^N_{j=1}y_j$. \end{lemma}
{\it Proof.} Let $\gamma_1,\dots,\gamma_s:k(X)\overk F$ and
$\delta_1,\dots,\delta_s:k(X)\overk F$ be
generic points of $X$ such that $\sum^s_{j=1}\gamma_j-\sum^s_{j=1}\delta_j$
is the divisor of a rational function on $X_F$.

We need to show that $\sum^s_{j=1}\gamma_j-\sum^s_{j=1}\delta_j$
belongs to the $G$-submodule in $Z^{{\rm rat}}_0(k(X)\otimes_kF)$
generated by $w_N$'s.

There is a collection $\alpha_1,\dots,\alpha_g:k(X)
\overk F$ of generic points of $X$ such
that the class of $\sum^s_{j=1}\gamma_j+\sum^g_{j=1}\alpha_j$ in
$\Pic^{s+g}X$ is a generic point. Then there is a collection
$\xi_1,\dots,\xi_{s+g}:k(X)\overk F$
of generic points of $X$ in general position such that
$\sum^s_{j=1}\gamma_j+\sum^g_{j=1}\alpha_j-\sum^{s+g}_{j=1}\xi_j$
is divisor of a rational function on $X_F$ (so the same holds
also for $\sum^s_{j=1}\delta_j+\sum^g_{j=1}\alpha_j-
\sum^{s+g}_{j=1}\xi_j$). We may, thus, assume that
$\delta_1,\dots,\delta_s$ are in general position.

Fix a collection $\{\varkappa_{ij}\}_{1\le i\le g,1\le j\le s}$
of generic points of $X$ in general position, also with respect
to $\gamma_1,\dots,\gamma_s$ and to $\delta_1,\dots,\delta_s$,
such that the classes of $\gamma_1+\sum^g_{i=1}\varkappa_{i1},
\dots,\gamma_s+\sum^g_{i=1}\varkappa_{is}$ in $\Pic^{g+1}X$
are generic points in general position. Then one can choose
a collection $\{\xi_{ij}\}_{0\le i\le g,1\le j\le s}$ of generic
points of $X$ in general position such that
$\gamma_j+\sum^g_{i=1}\varkappa_{ij}-\sum^g_{i=0}\xi_{ij}$
is divisor of a rational function on $X_F$ (so the same holds
also for $\sum^s_{j=1}\sum^g_{i=0}\xi_{ij}-\left(\sum^s_{j=1}
\delta_j+\sum^s_{j=1}\sum^g_{i=1}\varkappa_{ij}\right)$).
We may, thus, assume that both $\gamma_1,\dots,\gamma_s$
and $\delta_1,\dots,\delta_s$ are in general position.

Then there is a collection of generic points $\xi_1,\dots,\xi_s:
k(X)\overk F$ such that the points
$(\gamma_1,\dots,\gamma_s;\xi_1,\dots,\xi_s)$ and
$(\delta_1,\dots,\delta_s;\xi_1,\dots,\xi_s)$ are generic
on $p^{-1}_s(0)$. Then $\sum^s_{j=1}\gamma_j-\sum^s_{j=1}\xi_j$
and $\sum^s_{j=1}\delta_j-\sum^s_{j=1}\xi_j$ are divisors
of rational functions on $X_F$. Clearly, such elements
belong to the $G$-orbit of $w_s$. \qed

\vspace{4mm}

{\it Remark.} The $G$-module $Z^{{\rm rat}}_0(k(X)\otimes_kF)$ from
Lemma \ref{cyclic-1} is generated by $w_{g+1}$. {\it Proof.} There exists
an effective divisor $D$ (of degree $g$) in the linear equivalence class of
$\sum^N_{j=2}\sigma_j-\sum^N_{j=g+2}\tau_j$, so $w_N=
[\sum^N_{j=2}\sigma_j-D-\sum^N_{j=g+2}\tau_j]+
[\sigma_1+D-\sum^{g+1}_{j=1}\tau_j]$ is a sum of an element in the
$G$-orbit of $w_{N-1}$ and an element in the $G$-orbit of $w_{g+1}$. \qed

\begin{proposition} \label{pic-as-adj}
${\mathcal I}_S{\mathbb Q}[\{k(X)\overk F\}]=
\Pic(X_F)_{{\mathbb Q}}$ for any smooth proper curve $X$ over $k$.
\end{proposition}
{\it Proof.} By Lemma \ref{cyclic-1}, it suffices to show that the images
of the generators $w_N$ in ${\mathcal I}_SW$ are zero. Denote by $g\ge 0$
the genus of $X$, by $\psi_N$ a generic effective divisor on $X$ of degree
$N$ with a special class in $\Pic^NX$. Then $w_N=\sigma\psi_{N+g}-
\tau\psi_{N+g}$ for some $\sigma,\tau\in G$, so it suffices to show that
the images of $\psi_N$'s in ${\mathcal I}_SW$ are fixed by $G$.
Denote by $X^N\stackrel{s}{\longrightarrow}\Sigma^NX\stackrel{r}
{\longrightarrow}\Pic^N(X)$ the natural morphisms and set
$Y:=(rs)^{-1}(\ast)$. Let $p:Y\subseteq X^N\longrightarrow X$ be the
projection to the first multiple; set $\pi=s|_Y:Y\longrightarrow r^{-1}(\ast)$.
The projection to the first $N-g$ multiples $Y\longrightarrow X^{N-g}$
is generically finite of degree $g!$. If $N\ge 2g-1$ then
$r^{-1}(\ast)\cong{\mathbb P}^{N-g}$. Assume also that $N\ge g+1$
(i.e. $N\ge\max(2g-1,g+1)$). As $s$ is generically finite
of degree $N!$, one has $(N-1)!\psi_N=p_{\ast}\pi^{\ast}q$
for a generic point $q$ of $r^{-1}(\ast)$. \qed

\vspace{4mm}

Denote by ${\mathcal I}_G$ the full subcategory in $\Sm_G$ of
``homotopy invariant'' representations: $W^{G_{F|L'}}=W^{G_{F|L}}$ for
any purely transcendental subextension $L'|L$ in $F|k$.
\begin{theorem} \label{irr-subq-hom-inv}
A dominant sheaf is ${\mathbb A}^1$-invariant if and only if all its
simple subquotients are. \end{theorem}
{\it Proof.} Let $S$ be the collection of pairs of type $(G_{F|L(x)}\subset
G_{F|L})$ for all subfields $L$ in $F|k$ of finite type and elements
$x\in F$ transcendental over $L$. The following conditions are equivalent:
\begin{enumerate} \item \label{poopr} a smooth representation $W$ of $G$ is
``homotopy invariant''; \item \label{S-dlq-kon-por} $W^{U_1}=W^{U_2}$
for all pairs $(U_1\subset U_2)\in S$; \item \label{S-komp}
$W^{G_{F|L}}=W^{G_{F|L'}}$ for all subfields $L$ in $F|k$ of finite type
and purely transcendental extensions $L'|L$ in $F$ such that $F$ is
algebraic over $L'$. \end{enumerate}
(\ref{poopr})$\Leftrightarrow$(\ref{S-komp}) and
(\ref{poopr})$\Leftrightarrow$(\ref{S-dlq-kon-por}) are evident;
(\ref{S-dlq-kon-por})$\Leftrightarrow$(\ref{poopr}) is proved in
\cite[Corollary 6.2]{repr}. This verifies the condition (2) of
Proposition \ref{harakteriz-S-objektov}. For each pair $(G_{F|L(x)}\subset
G_{F|L})\in S$ fix some $\sigma\in G_{F|L}$ with $x$ and $\sigma x$
algebraically independent over $L$. Then the condition (1)(i) is obvious:
$G_{F|L(x)}\cap G_{F|L(\sigma x)}=G_{F|L(x,\sigma x)}$ and
$(G_{F|L(x,\sigma x)}\subset G_{F|L(x)})\in S$; the condition (1)(ii)
follows from \cite[Lemma 2.16]{repr}: the subgroups $G_{F|L(x)}$ and
$G_{F|L(\sigma x)}$ generate $G_{F|L}$. \qed

\subsection{Summary of equivalences} \label{Summ-equiv}
The following categories are equivalent:
\begin{enumerate} \item the category of dominant ${\mathbb A}^1$-invariant
sheaves of $E$-vector spaces; \item the category of dominant
${\mathbb A}^1$-invariant presheaves of $E$-vector spaces with
the Galois descent property;
\item the category $\Sm_G^S(E)$, where $S$ consists of the pairs of type
$(G_{F|L'}\subset G_{F|L})$ with purely transcendental $L'|L$ in $F|k$.
\end{enumerate}
These equivalences restrict to equivalences of corresponding subcategories:
(1) of sheaves of finite-dimensional spaces, (2) of presheaves of
finite-dimensional spaces,
(3) of admissible representations of $G$.\footnote{A representation of a
totally disconnected group is {\sl admissible} if it is smooth and the fixed
subspaces of all open subgroups are finite-dimensional.}

Consider the following properties of a smooth representation $W$ of $G$:
\begin{enumerate} \item \label{A1-inv-predst} $W\in\Sm_G^S(E)$, where $S$
consists of the pairs of type $(G_{F|L'}\subset G_{F|L})$ with purely
transcendental $L'|L$ in $F|k$;
\item \label{ne-sod-vse} the restriction of $W$ to a compact subgroup
$U$ does not contain all smooth irreducible representations of $U$;
\item \label{nenul-ann} the annihilator of $W$ in the algebra
${\mathbb D}_{{\mathbb Q}}(G)$ is non-zero. \end{enumerate}
One has (\ref{A1-inv-predst})$\Rightarrow$(\ref{ne-sod-vse})$\Rightarrow$
(\ref{nenul-ann}).
[(\ref{ne-sod-vse})$\Rightarrow$(\ref{nenul-ann}) is explained
in \S\ref{nul-annul-sod-vse}.
(\ref{A1-inv-predst})$\Rightarrow$(\ref{ne-sod-vse}):
If $F^U$ is purely transcendental over $k$, there are many irreducible
smooth representations of $U$, entering in no object of $\Sm_G^S(E)$.
Any non-trivial smooth irreducible representation $\tau$ of $U$ such that
$F^{\ker\tau}$ is unirational (e.g., purely transcendental) over $k$ is an
example of such representation. Clearly, for any such $\tau$ the natural
projector $p_{\tau}\in{\mathbb D}_{{\mathbb Q}}(G)$ onto the $\tau$-isotypical
part belongs to the common annihilator of the objects of $\Sm_G^S(E)$.]

{\it Remark.} \label{glok-por} For a discrete valuation $v$ of rank $1$ on
$F$, trivial on $k$,\footnote{By definition, this means that any maximal
system of elements of $F^{\times}$ with independent images in the valuation
group, should be a transcendence base of $F$ over a lift of a subfield of
the residue field.} and a smooth representation $W$ of $G$ set
$W_v:=\sum_LW^{G_{F|L}}\subseteq W$, where $L$ runs over the subfields in
the valuation ring of $v$. The intersection $\Gamma(W):=\bigcap_vW_v$ over
all such $v$'s is again in $\Sm_G$. As shown in \cite[Cor.4.7]{max}, the
property (\ref{A1-inv-predst}) for $W$ implies that $W=W_v$ (and also
$W=\Gamma(W)$, since all $v$'s as above form a $G$-orbit, cf. \cite{max}).

\section{Differential forms} \label{Differential-forms}
Let $H^{\bullet}=\bigoplus_{q\ge 0}H^q$ be a cohomology theory, considered as
a dominant ${\mathbb A}^1$-presheaf. Denote by $\underline{H}_c^{\bullet}$
the dominant ${\mathbb A}^1$-sheaf $X\mapsto H^{\bullet}(X)/N^1$ for smooth
proper $k$-varieties $X$, which is a subsheaf of $\underline{H}^{\bullet}$,
e.g., $\underline{H}^1_c:X\mapsto H^1(\overline{X})$. Clearly,
$\underline{H}^{\bullet}_c$ is a sheaf of finite $H^{\bullet}(k)$-modules.
It would follow from the standard semisimplicity conjecture that the sheaf
$\underline{H}^{\bullet}_c$ is semisimple if $H^{\bullet}(k)$ is a field.

We shall be interested in the case of de Rham cohomology $H^{\bullet}
=H^{\bullet}_{{\rm dR}/k}:X\mapsto H^{\bullet}_{{\rm dR}/k}(X):=
{\mathbb H}^{\bullet}(X,\Omega^{\bullet}_{X|k})$, where $H^{\bullet}(k)=k$,
cf. \cite{Grothendieck}. Clearly, $\underline{H}^q_{{\rm dR}/k}=
\SOm^q_{|k,\text{{\rm closed}}}/\SOm^q_{|k,\text{{\rm exact}}}$, where
$\Omega^q_{|k,\text{{\rm closed}}}:Y\mapsto
\ker(d|\Gamma(Y,\Omega^q_{Y|k}))$ and $\Omega^q_{|k,\text{{\rm exact}}}:
Y\mapsto d\Gamma(Y,\Omega^{q-1}_{Y|k})$, so $d:
{\mathcal H}^{{\mathbb G}_a}_1\stackrel{\sim}{\longrightarrow}
\SOm^1_{|k,\text{{\rm exact}}}$. The sheaf
$\underline{H}^1_{{\rm dR}/k,c}$ is semisimple.
It is described in Lemma \ref{example-4}.

\subsection{Maximal ${\mathbb A}^1$-subsheaf and the ${\mathbb A}^1$-quotient
of (closed) forms}
Recall (\S\ref{Properties-of}) that the inclusion functor
${\mathcal I}_G\to\Sm_G$ admits a right adjoint $W\mapsto W^{(0)}$,
the maximal subobject in ${\mathcal I}_G$. The following fact points out
once more the cohomological nature of the objects of ${\mathcal I}_G$.
\begin{proposition}[\cite{pgl}, Prop.7.6] \label{wedge} The maximal
subobject in ${\mathcal I}_G$ of the sheafification of
$\bigotimes^{\bullet}_{{\mathcal O}}\Omega^1_{|k}$
is $\Omega^{\bullet}_{|k,\text{{\rm reg}}}$.

For any smooth proper $k$-variety $Y$ there are the following
canonical isomorphisms \begin{equation} \label{ch-0-forms}
\Hom_{{\mathcal I}_G}(C_{k(Y)},\Omega^q_{|k,\text{{\rm reg}}})=
\Hom_{\Sm_G}(\Psi_Y,\Omega^q_{|k,\text{{\rm reg}}})
\stackrel{\sim}{\longleftarrow}\Gamma(Y,\Omega^q_{Y|k})
\stackrel{\sim}{\longrightarrow}\Hom_{\Sm_G}(CH_0(Y_F),
\Omega^q_{|k,\text{{\rm reg}}}).\end{equation}
The first isomorphism is functorial with respect to the dominant morphisms
$Y\longrightarrow Y'$, the second one is functorial with respect to
arbitrary morphisms $Y\longrightarrow Y'$. \end{proposition}

\begin{lemma} \label{reduction-exact} Let $L$ be an algebraically closed
extension of $k$ and $x$ be an indeterminant. Then there are isomorphisms
$id+\sum_{\alpha\in L}\wedge\frac{d(x-\alpha)}{x-\alpha}:(L(x)
\otimes_L\Omega^q_{L|k})/\Omega^q_{L|k,\text{{\rm exact}}}\oplus\bigoplus
_{\alpha\in L}(\Omega^{q-1}_{L|k}/\Omega^{q-1}_{L|k,\text{{\rm exact}}})
\stackrel{\sim}{\longrightarrow}\Omega^q_{L(x)|k}/
\Omega^q_{L(x)|k,\text{{\rm exact}}}$ and
$d+\sum_{\alpha\in L}\wedge\frac{d(x-\alpha)}{x-\alpha}:(L(x)
\otimes_L\Omega^q_{L|k})/\Omega^q_{L|k,\text{{\rm closed}}}\oplus\bigoplus
_{\alpha\in L}\Omega^q_{L|k,\text{{\rm exact}}}\stackrel{\sim}{\longrightarrow}
\Omega^{q+1}_{L(x)|k,\text{{\rm exact}}}$ for any $q\ge 1$. The former
isomorphism restricts to an isomorphism $H^q_{{\rm dR}/k}(L)\oplus
\bigoplus_{\alpha\in L}H^{q-1}_{{\rm dR}/k}(L)
\stackrel{\sim}{\longrightarrow}H^q_{{\rm dR}/k}(L(x))$. \end{lemma}
{\it Proof.} As $\Omega^q_{L(x)|k}=L(x)\otimes_L\Omega^q_{L|k}\oplus
L(x)\otimes_L\Omega^{q-1}_{L|k}\wedge dx$, for any $\omega\in
\Omega^q_{L(x)|k}$ one has $\omega\equiv\eta\wedge dx
\pmod{L(x)\otimes_L\Omega^q_{L|k}}$ for a unique
$\eta\in L(x)\otimes_L\Omega^{q-1}_{L|k}$. Using partial fraction
decomposition of rational functions in $L(x)$, we get a presentation
$\eta=\sum_{j\ge 0}x^j\eta_j+\sum_{\alpha\in L,~j\ge 1}
\frac{\eta_{j,\alpha}}{(x-\alpha)^j}$, where
$\eta_j,\eta_{j,\alpha}\in\Omega^{q-1}_{L|k}$. Then $\eta\wedge dx\equiv
\sum_{\alpha\in L}\eta_{1,\alpha}\wedge\frac{d(x-\alpha)}{x-\alpha}
\pmod{L(x)\otimes_L\Omega^q_{L|k}+\Omega^q_{L(x)|k,\text{{\rm exact}}}}$,
so $\omega\equiv\sum_i\phi_i(x)\eta_i+\sum_{\alpha\in L}\eta_{1,\alpha}\wedge
\frac{d(x-\alpha)}{x-\alpha}\pmod{\Omega^q_{L(x)|k,\text{{\rm exact}}}}$,
and thus, $d\omega=\sum_id\phi_i(x)\wedge\eta_i+\sum_i\phi_i(x)d\eta_i+
\sum_{\alpha\in L}d\eta_{1,\alpha}\wedge\frac{d(x-\alpha)}{x-\alpha}\equiv
\sum_i\phi'_i(x)dx\wedge\eta_i+\sum_{\alpha\in L}d\eta_{1,\alpha}\wedge
\frac{dx}{x-\alpha}\pmod{L(x)\otimes_L\Omega^q_{L|k}}$ for some
$\phi_i(x)\in L(x)$ and $\eta_i\in\Omega^q_{L|k}$ (and we may assume that
$\eta_i$ are $L$-linearly independent). Using partial fraction decomposition
of the rational functions $\phi_i\in L(x)$, we see that if $\omega$ is
closed then $d\eta_{1,\alpha}=0$, $\phi_i\in L$ and
$\sum_i\phi_i\eta_i\in\Omega^q_{L|k}$ is closed. \qed

\begin{proposition} \label{A1-quotient-of-closed} Let $M_q$ be the
sheaf associated with the presheaf $\Omega^q_{|k,\text{{\rm exact}}}+
\Omega^{q-1}_{|k,\text{{\rm closed}}}\wedge d\log{\mathbb G}_m\subset
\Omega^q_{|k}$ for any $q\ge 1$. Then {\rm (i)}
$\Omega^q_{k(X\times{\mathbb A}^n)|k,\text{{\rm closed}}}
=\Omega^q_{k(X)|k,\text{{\rm closed}}}+M_q(X\times{\mathbb A}^n)$ for any
$n\ge 1$; {\rm (ii)} $M_q$ is the kernel of the natural projection $\pi_q:
\SOm^q_{|k,\text{{\rm closed}}}\to V^q:={\mathcal I}(\SOm^q_{|k,
\text{{\rm closed}}})={\mathcal I}(\underline{H}^q_{{\rm dR}|k})$;
{\rm (iii)} for $q\ge 2$, $M_q$ is the sheaf associated with the presheaf
$\Omega^{q-1}_{|k,\text{{\rm closed}}}\wedge d\log{\mathbb G}_m$
and $d+d\log:{\mathcal H}^{{\mathbb G}_a}_1\oplus
k\otimes{\mathcal H}^{{\mathbb G}_m}_1\to M_1$ is an isomorphism.
\end{proposition}
In particular, the natural projections $\SOm^{\bullet}_{|k,\text{{\rm closed}}}
\stackrel{p_1}{\to}\underline{H}^{\bullet}_{{\rm dR}|k}\stackrel{p_2}{\to}
V^{\bullet}:={\mathcal I}(\underline{H}^{\bullet}_{{\rm dR}|k})
={\mathcal I}(\SOm^{\bullet}_{|k,\text{{\rm closed}}})$ are morphisms of
sheaves of supercommutative $k$-algebras. (The kernel of $p_1$, i.e.
$\SOm^{\bullet}_{|k,\text{{\rm exact}}}$, is the ideal generated by
$\SOm^1_{|k,\text{{\rm exact}}}$,\footnote{More generally, let
$\omega\in\Omega^{\ge i}_{F|k}\smallsetminus\Omega^{\ge i+1}_{F|k}$
be a closed form for some $i\ge 0$. Then any ideal in
$\SOm^{\bullet}_{|k,\text{{\rm closed}}}$ containing the $G$-orbit
of $\omega$ contains $\SOm^{\ge i+1}_{|k,\text{{\rm exact}}}$. {\it Proof.}
By \cite[Lemma 7.7]{pgl}, the semilinear representation $\Omega^j_{F|k}$
is irreducible for any $j\ge 0$. In particular, $F$-linear envelope of the
$G$-orbit of $\omega$ is the direct sum of $\Omega^j_{F|k}$ over all $j\ge i$
such that the homogeneous component of $\omega$ of degree $j$ is non-zero.
Then $dz\wedge\sigma\omega=d(z\cdot\sigma\omega)$ for all $z\in F$ and all
$\sigma\in G$ span the direct sum of $\SOm^j_{|k,\text{{\rm exact}}}$
over all $j\ge i$ as above. \qed} the kernel of $p_2$
is the ideal generated by $d\log{\mathbb G}_m$.)
They are surjective even as morphisms of presheaves.

{\it Proof.}
Let us show that $\ker\pi_q$ contains $M_q$. For any irreducible smooth
$k$-variety $X$, any $\eta\in\Omega^{q-1}_{k(X)|k,\text{{\rm closed}}}$
and a generator $t$ of the field $k(X\times{\mathbb G}_m)$ over $k(X)$ the
closed $q$-forms $\omega_m=\eta\wedge d\log t$ and $\omega_a=\eta\wedge dt$
are sections of the sheaf $\Omega^q_{|k,\text{{\rm closed}}}$ over
$X\times{\mathbb G}_m$, so their images in
${\mathcal I}(\Omega^q_{|k,\text{{\rm closed}}})$
should be sections over $X$. As there are endomorphisms $g_m,g_a$ of
$X\times{\mathbb G}_m|X$ such that $g_mt=t^2$ and $g_at=2t$ (so
$g_?\omega_?=2\omega_?$), the images of $\omega_?$ in
${\mathcal I}(\Omega^q_{|k,\text{{\rm closed}}})$ should be zero. The elements
of type $\eta\otimes d\log t$ (resp., $\eta\otimes dt$) span the sheaf
$\Omega^{q-1}_{|k,\text{{\rm closed}}}\otimes{\mathcal H}^{{\mathbb G}_m}_1$
(resp., $\Omega^{q-1}_{|k,\text{{\rm closed}}}\otimes
{\mathcal H}^{{\mathbb G}_a}_1$, which is surjective over
$\Omega^q_{|k,\text{{\rm exact}}}$).

By \cite[Lemma 6.3, p.200]{repr}, to show that $\ker\pi_q=M_q$
it suffices to check that, for any algebraically
closed extension $F'|k$ in $F$ and any $t\in F\smallsetminus F'$,
any $\omega\in\Omega^q_{F'(t)|k,\text{{\rm closed}}}$
belongs in fact to $\Omega^q_{F'|k,\text{{\rm closed}}}+M_q$.

By Lemma \ref{reduction-exact}, $\omega\equiv
\xi+\sum_{\alpha\in F'}\eta_{\alpha}\wedge\frac{d(t-\alpha)}{t-\alpha}
\pmod{\Omega^q_{F'(t)|k,\text{{\rm exact}}}}$, where
$\xi\in\Omega^q_{F'|k,\text{{\rm closed}}}$ and
$\eta_{\alpha}\in\Omega^{q-1}_{F'|k,\text{{\rm closed}}}$, which
means that $\omega\in\Omega^q_{F'|k,\text{{\rm closed}}}+M_q$. \qed

\begin{conjecture} The sheaf $V^{\bullet}$ is semisimple. \end{conjecture}

\vspace{4mm}

{\it Remarks.} 1. It follows from Proposition \ref{A1-quotient-of-closed}
that the natural morphism $\underline{H}^{\bullet}_{{\rm dR}/k,c}\to
V^{\bullet}$ is injective.

2. As explained in Remark on p.\pageref{vanish-I-S-semilin},
${\mathcal I}V=0$ for any semilinear smooth representation $V$:
if $v\in V^{G_{F|L}}$ and $f\in F$ is transcendental over $L$ then
$v=fv-(f-1)v$ becomes zero in any quotient of $V$ in ${\mathcal I}_G$.

3. For an algebra $A\in\Sm_G$ it is not always true that the kernel
$A^{\circ}$ of the projection $A\to{\mathcal I}A$ is an ideal. E.g., let
$A=A_{\bullet}$ be the (graded) tensor, symmetric or skew-symmetric algebra
of $A_1={\mathbb Q}[F\smallsetminus k]$. Then ${\mathcal I}A_1={\mathbb Q}$,
so $A_1^{\circ}\otimes A_1+A_1\otimes A_1^{\circ}$ consists of all
sums in ${\mathbb Q}[(F\smallsetminus k)\times(F\smallsetminus k)]$ of degree
0. On the other hand, ${\mathcal I}(A_1\otimes A_1)=\bigoplus_{x\in
\mathop{\mathbf{Spec}}(k({\mathbb P}^1)\otimes_kk({\mathbb P}^1))}C_{k(x)}$,
and therefore, $A_1^{\circ}\otimes A_1+A_1\otimes A_1^{\circ}$ is
strictly bigger than $A_2^{\circ}$.

\subsection{The semisimplicity of the regular forms of top degree}
\label{semisimpl-top-sect}
Let $L$ be an algebraically closed extension of $k$ with
$1\le q=\mathop{\mathrm{tr.deg}}(L|k)<\infty$. Define a representation
$\Omega^q_{L|k,\text{{\rm reg}}}$ as the union in $\Omega^q_{L|k}$ of
all spaces $\Gamma(X,\Omega^q_{X|k})$ over all smooth proper varieties
$X$ over $k$ with the function field embedded into $L$ over $k$.

The na\"{\i}ve truncation filtration on $\Omega^{\bullet}_{\overline{U}|k}$
gives the descending Hodge filtration $F^{\bullet}$ on
$H^q_{{\rm dR}/k}(\overline{U})$. The Hodge filtrations on
$H^q_{{\rm dR}/k}(\overline{U})$ for all $U$'s induce a canonical
filtration $F^{\bullet}$ on $\underline{H}^q_{{\rm dR}/k,c}$
by subsheaves of $k$-vector spaces with associated graded quotients
$H^{p,q-p}_{|k}:Y\mapsto{\rm coker}[\bigoplus_D
H^{p-1}(D,\Omega^{q-p-1}_{D|k})\longrightarrow
H^p(\overline{Y},\Omega^{q-p}_{\overline{Y}|k})]$, where
$D\to\overline{Y}$ runs over all resolutions of the divisors on
$\overline{Y}$. In particular,
$H^{q,0}_{|k}=F^q\underline{H}^q_{{\rm dR}/k,c}=\Omega^q_{|k,\text{reg}}:
Y\mapsto\Gamma(\overline{Y},\Omega^q_{\overline{Y}|k})$ is the dominant
subsheaf of $\underline{H}^q_{{\rm dR}/k,c}$ consisting of regular
differential $q$-forms.

\begin{proposition}[\cite{glatt}] \label{semisimpl-top}
Suppose that the cardinality of $k$ is at most continuum.
The representation $H^q_{{\rm dR}/k,c}(L)$ $($and therefore,
$\Omega^q_{L|k,\text{{\rm reg}}})$ of $G_{L|k}$ is semisimple. Any
embedding $\iota:k\hookrightarrow{\mathbb C}$ into the field of complex
numbers determines \begin{itemize} \item a ${\mathbb C}$-antilinear
isomorphism $H^{s,t}_{L|k}\otimes_{k,\iota}{\mathbb C}\cong
H^{t,s}_{L|k}\otimes_{k,\iota}{\mathbb C}$, \item a positive definite
$G_{L|k}$-equivariant hermitian form $({\mathbb C}\otimes_{k,\iota}
H^q_{{\rm dR}/k,c}(L))\otimes_{id,{\mathbb C},\sigma}({\mathbb C}
\otimes_{k,\iota}H^q_{{\rm dR}/k,c}(L))\longrightarrow
{\mathbb C}(\chi)$, where $\sigma$ is the complex conjugation and
$\chi$ is the modulus of $G_{L|k}$. \end{itemize} There exists a
non-canonical ${\mathbb Q}$-linear isomorphism $H^{s,t}_{L|k}\cong
H^{t,s}_{L|k}$. \end{proposition}
{\it Proof.} For any smooth projective $k$-variety $X$ the complexified
projection $F^pH^{p+q}_{{\rm dR}/k}(X)\to H^q(X,\Omega^p_{X|k})$
identifies $F^pH^{p+q}_{{\rm dR}/k}(X)\otimes_{k,\iota}{\mathbb C}\cap
\overline{F^qH^{p+q}_{{\rm dR}/k}(X)\otimes_{k,\iota}{\mathbb C}}$
with $H^q(X,\Omega^p_{X|k})\otimes_{k,\iota}{\mathbb C}$. This gives
a decomposition ${\mathbb C}\otimes_{k,\iota}H^q_{{\rm dR}/k,c}(L)=
\bigoplus_{s+t=q}{\mathbb C}\otimes_{k,\iota}H^{s,t}_{L|k}$. Then the
complex conjugation on $H^{p+q}(X_{\iota}({\mathbb C}),{\mathbb C})=
H^{p+q}(X_{\iota}({\mathbb C}),{\mathbb R})\otimes_{{\mathbb R}}{\mathbb C}$
identifies $H^q(X,\Omega^p_{X|k})\otimes_{k,\iota}{\mathbb C}$ with
$H^p(X_{\iota}({\mathbb C}),\Omega^q_{X_{\iota}({\mathbb C})})=
H^p(X,\Omega^q_{X|k})\otimes_{k,\iota}{\mathbb C}$.

The semisimplicity of the $k$-representation $H^q_{{\rm dR}/k,c}(L)$
of $G_{L|k}$ is equivalent to the semisimplicity of its complexification.
For the latter note that there is a positive
definite $G_{L|k}$-equivariant hermitian form $({\mathbb C}\otimes_{k,\iota}
H^{s,t}_{L|k})\otimes_{id,{\mathbb C},\sigma}({\mathbb C}\otimes_{k,\iota}
H^{s,t}_{L|k})\longrightarrow{\mathbb C}(\chi)$, given by $(\omega,\eta)=
\int_{X_{\iota}({\mathbb C})}i^{q^2+2t}\omega\wedge\overline{\eta}\cdot
[G_{L|k(X)}]$ for any $\omega,\eta\in H^{s,t}_{{\rm prim}}(X_{\iota}
({\mathbb C}))=H^t_{{\rm prim}}(X,\Omega^s_{X|k})\otimes_{k,\iota}
{\mathbb C}\subset{\mathbb C}\otimes_{k,\iota}H^{s,t}_{L|k}$.
Here $H^{s,t}_{{\rm prim}}(X_{\iota}({\mathbb C}))$ denotes the subspace
orthogonal to the sum of all Gysin maps $H^{s-1,t-1}(D)\longrightarrow
H^{s,t}(X_{\iota}({\mathbb C}))$ for all desingularizations $D$ of all
divisors on $X_{\iota}({\mathbb C})$, as in the definition of
$\Omega^q_{L|k,\text{{\rm reg}}}$, $X$ runs over all smooth proper
$k$-varieties with the function field embedded into $L|k$. \qed

\subsection{Structure of closed 1-forms}
Let ${\rm Div}^{\circ}_{{\mathbb Q}}:Y\mapsto{\rm Div}_{{\rm alg}}
(\overline{Y})_{{\mathbb Q}}$ be the presheaf of algebraically trivial
divisors. It is a sheaf.
\begin{lemma} \label{surj-res} The residue homomorphism ${\rm Res}_Y:
H^1_{{\rm dR}/k}(k(Y))\to k\otimes{\rm Div}(\overline{Y})$,
$\omega\mapsto({\rm res}_x\omega)_{x\in\overline{Y}^1}$, defines
a morphism of sheaves ${\rm Res}:\underline{H}^1_{{\rm dR}/k}\to
k\otimes{\rm Div}^{\circ}_{{\mathbb Q}}$. The short sequence
$0\to\underline{H}^1_{{\rm dR}/k,c}\to\underline{H}^1_{{\rm dR}/k}
\stackrel{{\rm Res}}{\longrightarrow}{\rm Div}^{\circ}_{{\mathbb Q}}
\otimes k\to 0$ is exact, even as a sequence of presheaves. \end{lemma}
{\it Proof.} As ${\rm Res}$ commutes with the restriction to any
sufficiently general curve $C$,
${\rm Res}_X(\omega)\cdot C={\rm Res}_C(\omega |_C)\in CH_0(X)$,
$\deg({\rm Res}_X(\omega)\cdot C)=0$ by Cauchy theorem,
the pairing ${\rm NS}(X)_{{\mathbb Q}}\otimes CH_1(X)_{{\mathbb Q}}/hom
\longrightarrow{\mathbb Q}$ is non-degenerate (by Lefschetz hyperplane
section theorem), the class of ${\rm Res}_X(\omega)$ in
${\rm NS}(X)_{{\mathbb Q}}$ is zero. Thus, ${\rm Res}_X$
factors through  the algebraically trivial divisors on $X$.

Clearly, the kernel of ${\rm Res}$ coincides with $\underline{H}^1
_{{\rm dR}/k,c}$, cf. \cite{rosenlicht}.\footnote{If the residues of
$\omega\in H^1_{{\rm dR}/k}(k(X))$ are zero then integration along a loop
depends only on its homology class in $H_1(X,{\mathbb Q})$. There is an
element $\eta$ of $H_{{\rm dR}/k}^1(X)$ with the same periods as $\omega$,
so integration of $\omega-\eta$ along a path joining a fixed
(rational) point with the variable one is independent of a
chosen path, and defines a meromorphic (i.e. rational) function.} Then
it remains to show that any algebraically trivial divisor on $X$ is
the residue of a closed 1-form. Any algebraically trivial divisor can
be written as $D_1-D_2$ for a pair $D_1,D_2$ of algebraically equivalent
effective divisors on $X$. There is a smooth projective curve $C$, and
an effective divisor $D$ on $X\times C$, such that ${\rm pr}_X:D\to X$
is generically finite and $D_P-D_Q=D_1-D_2$ for some points $P,Q\in C$.
By Riemann--Roch theorem for curves, there exists a 1-form
$\omega_{P,Q}\in\Omega^1_C(P+Q)$ such that ${\rm Res}_C(\omega_{P,Q})=P-Q$:
there is a non-holomorphic 1-form with simple poles in the set $\{P,Q\}$,
since $\dim_k\Gamma(C,\Omega^1_C(P+Q))=\dim_k\Gamma(C,\Omega^1_C)+1$;
there are no 1-forms with precisely one simple pole, since
$\Gamma(C,\Omega^1_C(P))=\Gamma(C,\Omega^1_C(Q))=\Gamma(C,\Omega^1_C)$.
Then ${\rm Res}_X({\rm pr}_{X\ast}(({\rm pr}_C^{\ast}\omega_{P,Q})|_D))
=D_1-D_2$. \qed

\begin{proposition} \label{1-forms}
\begin{itemize} \item The maximal semisimple subsheaf of
$\SOm^1_{|k,\text{{\rm closed}}}$ is canonically isomorphic to the direct
sum $\bigoplus_A\Gamma(A,\Omega^1_{A|k})^{A(k)}\otimes_{\End(A)}
{\mathcal H}^A_1={\mathcal H}^{{\mathbb G}_a}_1\oplus
k\otimes{\mathcal H}^{{\mathbb G}_m}_1\oplus\Omega^1_{|k,\text{{\rm reg}}}$,
where $A$ runs over the set of isogeny classes of simple commutative
algebraic $k$-groups; $\Gamma(A,\Omega^1_{A|k})^{A(k)}=\Hom_k({\rm Lie}(A),k)$
denotes the space of translation invariant 1-forms on $A$. The projection
$\SOm^1_{|k,\text{{\rm closed}}}\to\SOm^1_{|k,\text{{\rm closed}}}/
\Omega^1_{|k,\text{{\rm reg}}}$ is split (but not canonically).
\item The maximal semisimple subsheaf of $\underline{H}^1_{{\rm dR}/k}$
is canonically isomorphic to $\bigoplus_AH^1_{{\rm dR}/k}(A)
\otimes_{\End(A)}{\mathcal H}^A_1=k\otimes{\mathcal H}^{{\mathbb G}_m}_1
\oplus\underline{H}^1_{{\rm dR}/k,c}$, where $A$ runs over the set of
isogeny classes of simple commutative algebraic $k$-groups (with the zero
summand corresponding to ${\mathbb G}_a$). The projection
$\underline{H}^1_{{\rm dR}/k}\to\underline{H}^1_{{\rm dR}/k}/
\underline{H}^1_{{\rm dR}/k,c}$ is split (but not canonically).
\item The sheaf $V^1:Y\mapsto H^1_{{\rm dR}/k}(k(Y))/
k\otimes(k(Y)^{\times}/k^{\times})$ from Proposition
\ref{A1-quotient-of-closed} is canonically isomorphic to
$\bigoplus_AV^1(A)\otimes_{\End(A)}{\mathcal H}^A_1$, where $A$ runs
over the set of isogeny classes of simple abelian $k$-varieties.
\end{itemize} For any integer $q\ge 1$, the representation
$\Omega^1_{L|k,\text{{\rm closed}}}$ of the group $G_{L|k}$
admits similar description (cf. \S\ref{semisimpl-top-sect}).
\end{proposition}
{\it Proof.} In notation of Lemma \ref{surj-res}, the sheaf
${\rm Div}^{\circ}_{{\mathbb Q}}$ admits a natural surjective
morphism onto the Picard sheaf ${\rm Pic}^{\circ}_{{\mathbb Q}}=
{\rm coker}[{\mathcal H}^{{\mathbb G}_m}_1
\stackrel{{\rm div}}{\longrightarrow}{\rm Div}^{\circ}_{{\mathbb Q}}]:
Y\mapsto{\rm Pic}^0(\overline{Y})_{{\mathbb Q}}$ with the irreducible
kernel ${\mathcal H}^{{\mathbb G}_m}_1$. The Picard sheaf
${\rm Pic}^{\circ}_{{\mathbb Q}}$ is semisimple and
it is described in Lemma \ref{example-4}.

According to Lemma \ref{example-4}, for any simple abelian variety $A$ over
$k$, any non-zero element $\xi$ of ${\rm Pic}^0(A)(k)_{{\mathbb Q}}$ provides
an embedding of ${\mathcal H}^A_1$ into ${\rm Pic}^{\circ}_{{\mathbb Q}}$.
Let us show that the natural extension $0\to{\mathcal H}^{{\mathbb G}_m}_1
\to{\rm Div}^{\circ}_{{\mathbb Q}}\to{\rm Pic}^{\circ}_{{\mathbb Q}}\to 0$
does not split, even after restricting to ${\mathcal H}^A_1$ via $\xi$.

All elements of ${\rm Pic}^{\circ}_{{\mathbb Q}}(A):=
{\rm Pic}^0(A)_{{\mathbb Q}}$ are fixed by translations of $A$ by torsion
elements in $A(k)$. However, as the torsion subgroup in $A(k)$ is Zariski
dense, it cannot fix a non-zero element of ${\rm Div}^{\circ}_{{\mathbb Q}}(A)
:={\rm Div}_{\text{alg}}(A)_{{\mathbb Q}}$.

This implies that ${\mathcal H}^{{\mathbb G}_m}_1$ is the maximal
semisimple subsheaf of ${\rm Div}^{\circ}_{{\mathbb Q}}$, which proves,
by Lemma \ref{surj-res}, the second assertion. It follows also that the
simple subquotients of $V^1$ are isomorphic to ${\mathcal H}^A_1$ for
simple abelian $k$-varieties $A$. There are no extensions between
${\mathcal H}^A_1$ and ${\mathcal H}^B_1$ for abelian $k$-varieties $A$
and $B$, since ${\mathcal I}_G$ is a Serre subcategory of $\Sm_G$ by
Proposition \ref{harakteriz-S-objektov} and $Y\mapsto A(k(Y))_{{\mathbb Q}}$
is a projective object of ${\mathcal I}_G$ by property 5 of
\S\ref{Properties-of} and Proposition \ref{pic-as-adj}. This
means that $V^1$ is semisimple, which proves the third assertion.

Once we know the simple subquotients of $\SOm^1_{|k,\text{{\rm closed}}}$,
the first assertion follows from Proposition \ref{wedge} and Lemma
\ref{example-4}. To see that the projections $\SOm^1_{|k,\text{{\rm closed}}}
\to\SOm^1_{|k,\text{{\rm closed}}}/\Omega^1_{|k,\text{{\rm reg}}}$ and
$\underline{H}^1_{{\rm dR}/k}\to\underline{H}^1_{{\rm dR}/k}/
\underline{H}^1_{{\rm dR}/k,c}$ are split, it is enough to notice that $V^1$
is semisimple, and therefore, the compositions $\Omega^1_{|k,\text{{\rm reg}}}
\hookrightarrow\SOm^1_{|k,\text{{\rm closed}}}\to V^1$ and
$\underline{H}^1_{{\rm dR}/k,c}\hookrightarrow\underline{H}^1_{{\rm dR}/k}\to
V^1$ admit splittings. \qed

\vspace{4mm}

\label{coniveau-filtr}
Given a subfield $L$ in $F$, define the filtration $N_{\bullet}^{(L)}$ on
the $G_{F|L}$-modules $W$ by $N_j^{(L)}W=\sum_{F'}W^{G_{F|F'}}$, where $F'$
runs over the subfields in $F|L$ of transcendence degree $j$. (Clearly,
$N_0^{(L)}W=W^{G_{F|\overline{L}}}$ and $N_{\bullet}^{(L)}=
N_{\bullet}^{(\overline{L})}\subseteq N_{\bullet}^{(L')}$ if
$L\subset L'\subset F$.) In particular, define the {\sl level}
filtration on the $G$-modules by $N_{\bullet}:=N_{\bullet}^{(k)}$.

It is conjectured in \cite[Conj.6.9]{repr} that the graded pieces
of $N_{\bullet}$ on the objects of ${\mathcal I}_G$ are semisimple.

If $U$ is an open subgroup of $G$, contained between $G_{F|\overline{L}}$
and the normalizer of $G_{F|\overline{L}}$ in $G$ then $N_{\bullet}^{(L)}$
is a filtration by $U$-submodules. The forgetful functor $\Sm_G\to\Sm_U$
does not preserve the irreducibility (or the semisimplicity). E.g.,
for any commutative simple algebraic $k$-group $A$, the restriction to $U$
of the irreducible $G$-module $A(F)/A(k)$ is a non-split extension of
the irreducible $U$-module $A(F)/A(\overline{L})$ by the $U$- (in fact,
$(U/U\cap G_{F|\overline{L}})$- ) module $A(\overline{L})/A(k)$.

{\it Questions.} Let $W\in\Sm_G$ be irreducible
and $W=N_qW$. Is it true that the representation $W/N_{q-1}^{(L)}W$ of $U$
is irreducible (or zero)?

Clearly, $N_j\Omega^i_{F|k}=\Omega^i_{F|k}$ for any $j>i$,
$N_j\Omega^i_{F|k}=0$ for any $j<i$ and
$N_j\Omega^j_{F|k}\subseteq\Omega^j_{F|k,\text{closed}}$.
\begin{conjecture} $N_j\Omega^j_{F|k}=N_j\Omega^j_{F|k,\text{{\rm closed}}}
=\Omega^j_{F|k,\text{{\rm closed}}}$. \end{conjecture}
A ``weak'' version,
$N_j\Omega^j_{F|k,{\rm reg}}=\Omega^j_{F|k,{\rm reg}}$,
follows from Grothendieck's diagonal decomposition conjecture.

The Conjecture obviously holds true for $j=0$. The case $j=1$ follows
from (i) Proposition \ref{1-forms}, (ii) the fact (\cite[Cor.3.8]{glatt})
that $F/k$ and $F^{\times}/k^{\times}$
are acyclic, so $\Omega^1_{L|k,\text{closed}}\to\hspace{-3mm}\to H^0(G_{F|L},
H^1_{\text{dR}/k}(F)/k d\log(F^{\times}/k^{\times}))$,
(iii) $N_1(A(F)/A(k))=A(F)/A(k)$ for any commutative $k$-group $A$. \qed

\vspace{5mm}

\noindent
{\sl Acknowledgements.} {\small The project originates from my stay at
the Max-Planck-Institut in Bonn, it has reached its present state at
the Institute for Advanced Study in Princeton, and it has reached the
final form at the I.H.E.S. in Bures-sur-Yvette. I am grateful to these
institutions for their hospitality and exceptional working conditions.

}

\vspace{5mm}

config: arxiv


TITLE: Evaluate $\int \frac{1-x}{(1+x)\sqrt{x+x^2+x^3}}dx$
QUESTION [8 upvotes]: Evaluate $$\int \frac{1-x}{(1+x)\sqrt{x+x^2+x^3}}dx$$ i used substitution $x=\tan^2 y$ so $dx=2\tan y \sec^2 y dy$ so the integral becomes
$$I=\int\frac{2\cos 2y\: \tan y\: \sec^2 y \:dy}{\sqrt{\tan^2 y+\tan^4 y+\tan^6 y}}=\int\frac{2\cos 2y  \:\sec^2 y\: dy}{\sqrt{1+\tan^2 y+\tan^4 y}}$$ I was stuck here

REPLY [5 votes]: Let $$I = \int\frac{1-x}{(1+x)\sqrt{x+x^2+x^3}}dx =\int\frac{(1-x^2)}{(x^2+2x+1)\sqrt{x+x^2+x^3}}dx$$
Now again Reaaranging we get $$I = -\int\frac{\left(1-\frac{1}{x^2}\right)}{\left(x+\frac{1}{x}+2\right)\sqrt{x+\frac{1}{x}+1}}dt$$
Now put $\displaystyle x+\frac{1}{x}+1=u^2\;,$ Then $\displaystyle \left(1-\frac{1}{x^2}\right)dt = 2udu$
So we get $$I = -2\int\frac{1}{u^2+1}du = -2\tan^{-1}(u)+\mathcal{C} = -2\tan^{-1}\left(\sqrt{x+\frac{1}{x}+1}\right)+\mathcal{C}.$$


TITLE: Special cases of the embedding problem
QUESTION [1 upvotes]: Embedding problem. Let $I$ be the ideal of polynomial algebra $A=K^{[n]}$, such that $A/I$ is also a polynomial algebra with smaller number $k$ of variables. Is it true that $I$ is generated by $n-k$ variables of $A$.
Definition. Call an ideal of polynomial algebra coordinate-like if it us generated by some coordinates of a polynomial algebra.
I am interested in the following particular cases of the embedding problem.
Problem 1. Consider any polynomial algebra $A=K^{[n]}$ and its $k$ polynomials $f_1,..., f_k$. For a polynomial algebra $B=K[y_1,..., y_{n+k}]$ consider the morphism sending $ y_i$ to $n$ coordinates of $A$ for $i\leq n$ and $y_j$ to $f_{j-n}$ for $j>n$. It is obvious that thr kernel of this morphism is the ideal $I$ with $B/I\cong A$. So is it coordinate-like?
Problem 2. If $I\subset J$ are two coordinate-like ideals of polynomial algebra $A$, is it true that $J/I$ is coordinate-like ideal of $A/I$?
It is obvious that these problems follow from the embedding problem, but are they true?

REPLY [1 votes]: First, the embedding problem has a partial affirmative answer if $k\geq n-2$ or if $n\geq 2k+2$. In these cases, $I$ is generated by $n-k$ elements, not necessarily variables.
Problem 1 has an affirmative answer. $I$ is generated by $y_j-f_{j-n}(y_1,\ldots, y_n)$ for $j>n$ and these are coordinates of $B$.


TITLE: MNTILE SPOJ Tiling patterns
QUESTION [4 upvotes]: We have tiles of size 2 * 1. We need to arrange the tiles to get the floor size of m * n.
For m = 3, we get arrangement like this:

My question is: In how many different ways can we arrange the tiles?
For m = 3, we get recurrence relation as:
T(n) = 4*T(n-2) - T(n-4)

For m = 4, we get
T(n) = T(n−1) + 5*T(n−2) + T(n−3) − T(n−4)

How to solve it for n=5 and m = n? Is there any generalized solution?
Problem Definitions :
1) http://www.spoj.com/problems/M5TILE/
2) http://www.spoj.com/problems/MNTILE/

REPLY [3 votes]: Here is a link to a presentation (by some undergraduate student) about the computation of those numbers:
http://math.cmu.edu/~bwsulliv/domino-tilings.pdf
It also contains references to Kastelyn's original paper and other interesting material.


TITLE: Find $B $ s.t. $B=\lbrace y\in s \mid \sum x_ky_k \text{ converges for all }x\in A\rbrace$ for $A=l_\infty$
QUESTION [0 upvotes]: let $A$ be a sunbset of $s$,
and let,
$$B=\left\lbrace y\in s \mid \sum x_ky_k\text{ converges for all }x\in A\right\rbrace$$
Could you please find $B$ if (i) $A=s$ and (ii) $A=l_\infty$
we worked on them for an hour. not much for (i). for (ii) we found the answer $s$th like $l_1 - \text{alternating series}$
it is from the book: Maddox, Elements of Functioanl Analysis.

REPLY [1 votes]: In case (i), $A = s$, $B$ consists of the sequences with only finitely many nonzero terms.
It is clear that for a sequence $(y_k)$ where $y_k \neq 0$ for only finitely many $k$, and any sequence $(x_k)$, the sum
$$\sum x_k y_k$$
converges, since it has only finitely many nonzero terms. On the other hand, if $y_k \neq 0$ for infinitely many $k$, then consider
$$x_k = \begin{cases}\:\; 1 &, y_k = 0\\ \dfrac{1}{\lvert y_k\rvert} &, y_k \neq 0.\end{cases}$$
Then $x_ky_k$ does not converge to $0$, and a fortiori $\sum x_ky_k$ does not converge.
In case (ii), $A = l_\infty$, we have $B = l_1$. For if $y \in l_1$ and $x\in l_\infty$, we have
$$\sum \lvert x_k y_k\rvert \leqslant \sum \lVert x\rVert_\infty \lvert y_k\rvert = \lVert x\rVert_\infty \lVert y\rVert_1 < \infty,$$
so the sum is absolutely convergent, and a fortiori convergent.
And if $y \notin l_1$, consider
$$x_k = \begin{cases}\;\: 1 &, y_k = 0\\ \frac{\overline{y_k}}{\lvert y_k\rvert} &, y_k \neq 0. \end{cases}$$
Then $\lvert x_k\rvert = 1$ for all $k$, so $x \in l_\infty$, and for every $k$ we have $x_k y_k = \lvert y_k\rvert$, hence
$$\sum x_k y_k = \sum \lvert y_k\rvert = +\infty.$$