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\begin{document}
\title{Galois theory of periods}
\author{Annette Huber}
\address{Math. Institut\\ Ernst-Zermelo-Str. 1\\ 79104 Freiburg\\ Germany}
\email{annette.huber@math.uni-freiburg.de}

\date{\today}
\maketitle
\tableofcontents
\section{Introduction}
We are interested in the period algebra $\Per$, a countable subalgebra of $\mathbb{C}$. Its elements are roughly of the form
\[ \int_\sigma\omega\]
where $\omega$ is a closed differential form over $\mathbb{Q}$ or ${\overline{\Q}}$ and
$\sigma$ a suitable domain of integration also defined over $\mathbb{Q}$ or ${\overline{\Q}}$.
All algebraic numbers are periods in this sense. Moreover, many interesting transcendental number like $2\pi i$, $\log(2)$, or $\zeta(3)$ are also period numbers.

In this mostly expository note, we want to explain how the work of Nori extends Galois theory from the field extension ${\overline{\Q}}/\mathbb{Q}$ to $\Per/\mathbb{Q}$, at least conjecturally. This realises and generalises a vision of Grothendieck.

As explained in more detail below, we should formulate Galois theory as saying that the natural operation
\[ \mathrm{Gal}({\overline{\Q}}/\mathbb{Q})\times \mathrm{Spec}({\overline{\Q}})\to \mathrm{Spec}({\overline{\Q}})\]
is a torsor, see Section~\ref{ssec:galois} for more details.
By the period conjecture (in Kontsevich's formulation in \cite{kontsevich}), we also have a torsor structure
\[ G_\mathrm{mot}(\mathbb{Q})\times \mathrm{Spec}(\Per)\to\mathrm{Spec}(\Per)\]
where $G_\mathrm{mot}(\mathbb{Q})$ is Nori's motivic Galois group of $\mathbb{Q}$. This is a pro-algebraic group attached to the rigid tensor category of mixed motives over $\mathbb{Q}$.
By Tannaka theory, rigid tensor subcategories of the category of motives correspond to quotients of $G_\mathrm{mot}(\mathbb{Q})$.  The case of Artin motives  gives back ordinary Galois theory. In particular, there is a natural exact sequence
\[ 0\to G_\mathrm{mot}({\overline{\Q}})\to G_\mathrm{mot}(\mathbb{Q})\to\mathrm{Gal}({\overline{\Q}}/\mathbb{Q})\to 0.\]
It corresponds to the inclusion ${\overline{\Q}}\subset\Per$.

It is a nice feature of Nori's theory (and a key advantage over the older approach) that it works for any abelian category of motives, even if it is not closed under tensor product. We get a weaker structure that we call semi-torsor, see Definition~\ref{defn:semi}. Hence, we also have a (conjectural) generalisation of Galois theory to
sub-vector-spaces of $\Per$. It is unconditonal in the case of $1$-motives.

While not bringing us closer to a proof of the period conjecture in general, the theory has structural consequences: for example the period conjectures for intermediate fields $\mathbb{Q}\subset\mathbb{K}\subset {\overline{\Q}}$ are equivalent. Another direct consequence
is the fully faithfulness of the functor attaching to a Nori motive $M$ the pair
$(H_\mathrm{dR}(M),H_\mathrm{sing}(M),\phi)$ of its realisations linked by the period isomorphism.
This observation connects the period conjecture to fullness conjectures in the spirit of the Hodge or Tate conjecture. One such was recently formulated by
Andreatta, Barbieri-Viale and Bertapelle, see \cite{ABB}. As explained below, their conjecture has overlap with the period conjecture, but there is no implication in either direction.

Actual evidence for the period conjecture is weak. The $0$-dimensional case
amounts to Galois theory. The $1$-dimensional case has now been completed by the author and W\"ustholz in \cite{huber-wuestholz}, generalising work by Lindemann, Gelfond, Schneider, Siegel, Baker and W\"ustholz.

The idea of the present paper was conceived after my talk at the 2017 workshop
''Galois Theory of Periods and Applications'' at MSRI. It does not aim at proving new results, but rather at making key ideas of the long texts \cite{period-buch} and \cite{huber-wuestholz} more accessible. We also take the opportunity to
discuss the relation between the various period conjectures in the literature.

\subsection*{Structure of the paper} We start by giving several, equivalent definitions of the period algebra. We then turn to the period conjecture. We explain Kontsevich's formal period algebra $\Perf$ in terms of explicit generators and relations. The period conjecture simply asserts that the obvious surjective map $\Perf\to\Per$ is also injective.
We then give a reinterpretation of $\Perf$ as a torsor. Thus the main aim of the note is achieved: a (conjectural) Galois theory of periods.

The following sections make the connections to Grothendieck's version of the period conjecture and the more recent cycle theoretic approach of
Bost/Charles and Andreatta/Barbieri-Viale/Bertapelle.

The last section inspects some examples of special period spaces studied in the literature: Artin motives (${\overline{\Q}}$), mixed Tate motives (multiple zeta-values) and $1$-motives (periods of curves).

\subsection*{Acknowledgements}I thank my coauthors Stefan M\"uller-Stach and
Gisbert W\"ustholz for many insights and stimulating discussions. My ideas on the period conjecture were also heavily influenced by a discussion with Yves Andr\'e during a visit to Freiburg. The aspect of a Galois theory of periods
is also stressed in his survey articles \cite{andre1}, \cite{andre3}.

\section{Period spaces}
When we talk about periods, we mean a certain countable subalgebra  $\Per$ of
$\mathbb{C}$. It has several, surprisingly different, descriptions. Roughly
a period is a number of the form
\[ \int_\sigma\omega\]
where both $\omega$ and $\sigma$ are something defined over $\mathbb{Q}$ or
${\overline{\Q}}$.

There are many possible ways to specify what $\omega$ and $\sigma$ to use. We will go through a list of possible choices:
nc-periods (or just periods) in Section~\ref{ssec:first}, Kontsevich-Zagier periods in Section~\ref{ssec:KZ}, cohomological periods in Section~\ref{ssec:coh},
and periods of motives in Section~\ref{ssec:mot_period}.
In most cases, we will
define the subalgebra $\Per^\mathrm{eff}$ of effective periods and then
pass to $\Per:=\Per^\mathrm{eff}[\pi^{-1}]$. In particular, $\pi$ will be an element
of $\Per^\mathrm{eff}$.

Good to know:

\begin{metathm}All definitions describe the same set.
\end{metathm}

Throughout, we fix an embedding ${\overline{\Q}}\to\mathbb{C}$. Let $\tilde{\Q}={\overline{\Q}}\cap \mathbb{R}$.
We denote by $\mathbb{K}$ a subfield of ${\overline{\Q}}$.

\subsection{First definition}\label{ssec:first}
Let $X$ be a smooth algebraic variety over $\mathbb{K}$ of dimension $d$, $D\subset X$ a divisor with normal crossings (also defined over $\mathbb{K}$),
 $\omega\in\Omega^d_X(X)$
a global algebraic differential form, $\sigma=\sum a_i\gamma_i$ a
differentiable relative chain, i.e., $a_i\in\mathbb{Q}$, $\gamma_i:\Delta^d\to X^\mathrm{an}$
a $C^\infty$-map on the $d$-simplex such that $\partial \sigma$ is a chain
on $D^\mathrm{an}$.
Then
\[ \int_\sigma\omega:=\sum a_i\int_{\Delta^d}\gamma_i^*\omega^\mathrm{an}\]
is a complex number.

\begin{defn}The set of these complex numbers is called the set of
effective periods $\Per^\mathrm{eff}$.
\end{defn}
They appear as nc (for normal crossings) periods in \cite[Section~11.1]{period-buch}.
\begin{ex}The algebraic number $\alpha$ satisfying the non-trivial polynomial $P\in \mathbb{K}[t]$ is an effective period as
the value of the integral
\[ \int_0^\alpha dt\]
with $X=\mathbb{A}^1$, $D=V(tP)$, $\omega=dt$, $\sigma=\gamma$ the standard
path from $0$ to $\alpha\in D(\mathbb{C})$ in $\mathbb{A}^1(\mathbb{C})=\mathbb{C}$.
\end{ex}

\begin{ex}[Cauchy's integral] The number $2\pi i$ is an effective period as the
value of the integral
\[ \int_{S^1}\frac{dz}{z}\]
with $X=\mathbb{G}_m$, $D=\emptyset$, $\omega= dz/z$, $\sigma=\gamma:[0,1]\to \mathbb{C}^*$ given by
$\gamma(t)=\exp(2\pi i t)$.
\end{ex}
\begin{lemma}The set $\Per^\mathrm{eff}$ is a ${\overline{\Q}}$-algebra. It is independent
of the choice of $\mathbb{K}$.
\end{lemma}
\begin{proof}\cite[Proposition~11.1.7]{period-buch}.\end{proof}
In particular, $\pi \in \Per^\mathrm{eff}$.

\begin{defn}\label{defn:first}
\[ \Per:=\Per^\mathrm{eff}[\pi^{-1}].\]
\end{defn}

\begin{rem}
\begin{enumerate}
\item
In the above definition, it even suffices to restrict to affine $X$.
\item We may also relax the assumptions: $X$ need not be smooth, $D$ only
a subvariety, $\omega\in\Omega^n(X)$ not necessarily of top degree but only closed.
\item We may also allow $\omega$ to have poles on $X$. We then need to impose the condition that the integral is absolutely convergent. This is the version given by Kontsevich and Zagier in \cite{kontsevich_zagier}.
\end{enumerate}
These implications are not obvious, but see \cite[Theorem~12.2.1]{period-buch}.
\end{rem}

\subsection{Semi-algebraic sets}\label{ssec:KZ}
The easiest example of a differential form is $dt_1\wedge\dots\wedge dt_n$
on $\mathbb{A}^n$ with coordinates $t_1,\dots,t_n$. Actually, this is the only one we need.

\begin{defn}\label{defn:KZ}An \emph{effective Kontsevich-Zagier period} is complex number whose real and imaginary part can be written as difference between the volumes of
$\mathbb{Q}$-semi-algebraic subsets of $\mathbb{R}^n$ (of finite volume).
\end{defn}
Note that a subset of $\mathbb{R}^n$ is $\mathbb{Q}$-semi-algebraic if and only if it is $\tilde{\Q}$-semi-algebraic.

\begin{prop}\label{prop:all_KZ} The set of effective Kontsevich-Zagier periods agrees with
$\Per^\mathrm{eff}$.
\end{prop}
\begin{proof}\cite[Prop.~12.1.6]{period-buch}, combined with \cite[Theorem~12.2.1]{period-buch}.
\end{proof}

\subsection{Cohomological periods}\label{ssec:coh}
The definition becomes more conceptual when thinking in terms of cohomology.
Chains define classes in (relative) singular homology and differential forms define classes in (relative) algebraic de Rham cohomology. The definition of algebraic de Rham cohomology is very well-known in the case of a single smooth algebraic variety. There are different methods of extending this to the singular and to the relative case, see \cite[Chapter~3]{period-buch}. Singular cohomology and algebraic de Rham cohomology become isomorphic after base change to the complex numbers. By adjunction this defines
the \emph{period pairing}.

Let $X/\mathbb{K}$ be an algebraic variety, $Y\subset X$ a subvariety. Let
\[ \int: H^i_\mathrm{dR}(X,Y)\times H_i^\mathrm{sing}(X,Y;\mathbb{Q})\to \mathbb{C}\]
be the period pairing.

\begin{defn}\label{defn:coh_period}The numbers in the image of $\int$ (for varying $X$, $Y$, $i$)
are called \emph{effective cohomological periods}.
\end{defn}

\begin{prop}[{\cite[Theorem~11.4.2]{period-buch}}]\label{prop:all_coh}The set of effective cohomological periods agrees with
$\Per^\mathrm{eff}$.
\end{prop}

\subsection{Periods of motives}\label{ssec:mot_period}
The insistance that we need to invert $\pi$ in order to get a good object becomes
clear from the motivic point  of view.

Two kinds of motives are relevant for our discussions:
\begin{enumerate}
\item Voevodsky's triangulated categories of (effective) geometric motives
$\DM_{\gm}^\mathrm{eff}(\mathbb{K},\mathbb{Q})$ and $\DM_{\gm}(\mathbb{K},\mathbb{Q})$;
\item Nori's abelian category $\mathcal{MM}_{\mathrm{Nori}}(\mathbb{K},\mathbb{Q})$ of motives over $\mathbb{K}$ with respect to
singular cohomology and its subcategory of effective motives.
\end{enumerate}

\begin{thm}[Nori, Harrer {\cite[Theorem~7.4.17]{harrer}}] There is a natural contravariant trianguled tensor-functor
\[ \DM_{\gm}^\mathrm{eff}(\mathbb{K},\mathbb{Q})\to D^b(\mathcal{MM}_{\mathrm{Nori}}^\mathrm{eff}(\mathbb{K},\mathbb{Q}))\]
compatible with the singular realisation. It extends to non-effective motives.
\end{thm}

All motives have singular and de Rham cohomology and hence periods. We formalise as in \cite[Section~11.2]{period-buch}.

\begin{defn}\label{defn:VV}
Let $(\mathbb{K},\mathbb{Q})\Vect$ be the category of tuples $(V_\mathbb{K},V_\mathbb{Q},\phi)$ where
$V_\mathbb{K}$ is a  finite dimensional $\mathbb{K}$-vector space, $V_\mathbb{Q}$ a finite dimensional
$\mathbb{Q}$-vector space and $\phi:V_\mathbb{K}\otimes_\mathbb{K}\mathbb{C}\to V_\mathrm{sing}\otimes_\mathbb{Q}\mathbb{C}$ an isomorphism.
Morphisms are given by pairs of linear maps such that the obvious diagram commutes.

Given $V\in (\mathbb{K},\mathbb{Q})\Vect$, we define the set of periods of $V$ as
\[ \Per(V)=\mathrm{Im} (V_\mathbb{K}\times V_\mathbb{Q}^\vee\to \mathbb{C}),\hspace{2ex} (\omega,\sigma)\mapsto \sigma_\mathbb{C}(\phi(\omega))\]
and the space of periods $\Per\langle H\rangle$ as the additive group generated by it.
If $\mathcal{C}\subset (\mathbb{K},\mathbb{Q})\Vect$ is a subcategory, we put
\[ \Per(\mathcal{C})=\bigcup_{V\in(\mathbb{K},\mathbb{Q})\Vect}\Per(V).\]
\end{defn}

\begin{rem}
\begin{enumerate}
\item If $\mathcal{C}$ is additive, then $\Per(\mathcal{C})$ is a $\mathbb{K}$-vector space.
In particular, $\Per\langle V\rangle$ agrees with $\Per(\langle V\rangle)$ where
$\langle V\rangle$ is the full abelian subcategory of $(\mathbb{K},\mathbb{Q})\Vect$ closed under subquotients generated by $V$.

\item If $\mathcal{C}$ is an additive tensor category, then $\Per(\mathcal{C})$ is a $\mathbb{K}$-algebra.
\end{enumerate}
\end{rem}

By the universal property of Nori motives, there is a faithful exact functor
\[ \mathcal{MM}_{\mathrm{Nori}}(\mathbb{K})\to (\mathbb{K},\mathbb{Q})\Vect.\]
This defines periods of Nori motives, and by composition with
$M\mapsto \bigoplus H^i_\mathrm{Nori}(M)$ also periods of geometric motives.

\begin{defn}\label{defn:mot_period}A complex number is called an \emph{(effective) motivic period}
if is the period of an (effective) motive.
\end{defn}
Every Nori motive is a subquotient of the image of a geometric motive, hence is
does not matter whether we think of geometric or Nori motives in the above definition.

\begin{prop}[{\cite[Proposition~11.5.3]{period-buch}}]\label{prop:all_mot}The set of effective motivic period agrees with $\Per^\mathrm{eff}$. The
set of all motivic periods agrees with $\Per$. In particular, both are independent of the choice of $\mathbb{K}$.
\end{prop}

As a consequence, we see that both $\Per^\mathrm{eff}$ and $\Per$ are ${\overline{\Q}}$-algebras.
Of course this is also easy to deduce directly from any of the other descriptions.

There is a big advantage of the motivic description: any type of structural result on motives has consequences for periods. One such are the bounds on the spaces of multiple zeta-values by Deligne-Goncharov. Another example are the transcendence results for $1$-motives as discussed below.

In particular, the motivic point of view allows us (or rather Nori) to give a structural reinterpretation of the period conjecture.

\subsection{Comparison}
We conclude by putting all information together:

\begin{thm}[{Proposition~\ref{prop:all_KZ}}, \ref{prop:all_coh}, \ref{prop:all_mot}]The following sets of complex numbers agree:
\begin{enumerate}
\item the set of periods in the sense of Definition~\ref{defn:first};
\item the set of Kontsevich-Zagier periods, see Definition~\ref{defn:KZ};
\item the set of cohomological periods, see Definition~\ref{defn:coh_period};
\item the set of all motivic periods, see Definition~\ref{defn:mot_period}.
\end{enumerate}
Moreover, they are independent of the field of definition
$\mathbb{Q}\subset\mathbb{K}\subset{\overline{\Q}}$ and form a countable ${\overline{\Q}}$-algebra.
\end{thm}

\section{Formal periods and the period conjecture}

In reverse order to the historic development, we start with Kontsevich's formulation and discuss the Grothendieck conjecture afterwards.

\subsection{Kontsevich's period conjecture}

We take the point of view of cohomological periods. At this point we restrict to $\mathbb{K}={\overline{\Q}}$ for simplicity. There are obvious  relations between periods:

\begin{itemize}
\item (bilinearity) Let $\omega_1,\omega_2\in H^i_\mathrm{dR}(X,Y)$, $\alpha_1,\alpha_2\in{\overline{\Q}}$, $\sigma_1,\sigma_2\in H_i^\mathrm{sing}(X,Y;\mathbb{Q})$, $a_1,a_2\in\mathbb{Q}$. Then
\[ \int_{\sigma_1+\sigma_2}(\alpha_1\omega_1+\alpha_2\omega_2)=\sum_{i,j=1}^2\alpha_j\int_{\sigma_i}\omega_j.\]
\item (functoriality) For $f:(X,Y)\to (X',Z)$, $\omega'\in H^i_\mathrm{dR}(X',Y')$,
$\sigma\in H_i^\mathrm{sing}(X,Y;\mathbb{Q})$ we have
\[ \int_{f_*\sigma}\omega'=\int_\sigma f^*\omega'.\]
\item (boundaries) For $Z\subset Y\subset X$, $\partial:H^i_\mathrm{dR}(Y,Z)\to H^{i+1}(X,Y)$, $\delta:H_{i+1}^\mathrm{sing}(X,Y;\mathbb{Q})\to H_i^\mathrm{sing}(Y,Z;\mathbb{Q})$, $\omega\in H^i_\mathrm{dR}(Y,Z)$ and $\sigma\in H_{i+1}^\mathrm{sing}(X,Y;\mathbb{Q})$ we have
\[ \int_{\delta\sigma}\omega=\int_\sigma\partial\omega.\]
\end{itemize}

\begin{defn}[Kontsevich]\label{defn:perf}The space of \emph{formal effective periods} $\Perf^\mathrm{eff}$ is defined
as the $\mathbb{Q}$-vector space generated by symbols
\[ (X,Y,i,\omega,\sigma)\]
with $X$ an algebraic variety over ${\overline{\Q}}$, $Y\subset X$ a closed subvariety,
$i\in\mathbb{N}_0$, $\omega\in H^i_\mathrm{dR}(X,Y)$, $\sigma\in H_i^\mathrm{sing}(X,Y;\mathbb{Q})$ modulo
the space of relations generated by bilinearity, functoriality and boundary maps as above.

It is turned into a $\mathbb{Q}$-algebra with the product using the external product
induced by the K\"unneth decomposition
\[ H^*(X,Y)\otimes H^*(X',Y')\cong H^*(X\times X',Y\times X'\cup Y'\times X).\]

Let $2\tilde{\pi}i$ be the class of the symbol
$(\G_m,\{1\},dz/z,S^1)$. The space of \emph{formal periods} $\Perf$ is
defined as the localisation
\[ \Perf=\Perf^\mathrm{eff}[\tilde{\pi}^{-1}].\]
\end{defn}

\begin{rem}
\begin{enumerate}
\item $\Perf^\mathrm{eff}$ and $\Perf$ are even ${\overline{\Q}}$-algebras. We define the scalar multiplication by $\alpha(X,Y,i,\omega,\sigma)=(X,Y,i,\alpha\omega,\sigma)$ for
$\alpha\in{\overline{\Q}}$.
\item We do not know if it suffices to work with $X$ smooth, $D$ a divisor with normal crossings as Kontsevich does in \cite{kontsevich}. As explained in
 \cite[Remark~13.1.8]{period-buch} these symbols generate the algebra, but is is not clear if they also give all relations. Indeed, Kontsevich only imposes relations in an even more special case.
\item It is not at all obvious that the product is well-defined. This fact  follows from Nori's theory, see \cite[Lemma~14.1.3]{period-buch}.
\end{enumerate}
\end{rem}

There is a natural evaluation map
\[ \Perf^\mathrm{eff}\to\mathbb{C},\quad (X,Y,i,\omega,\sigma)\mapsto \int_\sigma\omega.\]
By definition its image is the set of effective cohomological periods, hence equal to $\Per^\mathrm{eff}$.
\begin{conj}[Kontsevich]\label{conj:kont}
 the evaluation map
\[ \Perf\to\Per\]
is injective. In other words,
all ${\overline{\Q}}$-linear relations between
periods are induced by the above trivial relations.
\end{conj}

\subsection{Categorical and motivic description}\label{ssec:form_cat}
The above description is ``minimalistic'' in using a small set of generators and relations. The downside is that it is somewhat arbitrary. We now explain alternative descriptions. Let again $\mathbb{K}\subset{\overline{\Q}}$ be a subfield. Recall that we have fixed an embedding ${\overline{\Q}}\subset\mathbb{C}$.

\begin{defn}Let $\mathcal{C}$ be an additive category, $T:\mathcal{C}\to(\mathbb{K},\mathbb{Q})\Vect$ an additive functor. We define the \emph{space of formal periods of $\mathcal{C}$} as the
$\mathbb{Q}$-vector space $\Perf(\mathcal{C})$ generated by symbols
$(M,\sigma,\omega)$ with $M\in\mathcal{C}$, $\sigma\in T(M)_\mathbb{Q}^\vee$, $\omega\in T(M)_\mathbb{K}$ with relations
\begin{itemize}
\item bilinearity in $\sigma$ and $\omega$,
\item functoriality in $M$.
\end{itemize}
\end{defn}

We are going to apply this mainly to subcategories of
$\mathcal{MM}_{\mathrm{Nori}}(\mathbb{K},\mathbb{Q})$ or $\DM_{\gm}(\mathbb{K},\mathbb{Q})$.

\begin{thm}\label{thm:indep}
\[ \Perf(\DM_{\gm}(\mathbb{K},\mathbb{Q}))=\Perf(\mathcal{MM}_{\mathrm{Nori}}(\mathbb{K},\mathbb{Q}))=\Perf,\]
independent of $\mathbb{K}$.
\end{thm}
\begin{proof} This is only implicit in  \cite{period-buch}. It is a consequence of Galois theory of periods as discussed in the next section. We defer the proof.
\end{proof}

\begin{rem}In particular, the validity of the period conjecture does not depend on $\mathbb{K}\subset {\overline{\Q}}$. This is the reason that it was safe to
consider only $\mathbb{K}={\overline{\Q}}$ in Definition~\ref{defn:perf}.
\end{rem}

The evaluation map $(\mathbb{K},\mathbb{Q})\Vect\to\mathbb{C}$ induces linear maps
\[ \Perf(\mathcal{C})\to\mathbb{C}.\]
We denote the image by $\Per(\mathcal{C})$.

\begin{defn}We say that the period conjecture holds for $\mathcal{C}$ if
the map $\Perf(\mathcal{C})\to\Per(\mathcal{C})$ is injective.
\end{defn}
In the case $\mathcal{C}=\mathcal{MM}_{\mathrm{Nori}}(\mathbb{K},\mathbb{Q})$, we get back Kontsevich's conjecture \ref{conj:kont}.
We are going to study the relation between the period conjecture for $\mathcal{C}$ and subcategories of $\mathcal{C}$ below using a Galois theory for formal periods.

\section{Galois theory of (formal) periods}

	\subsection{Torsors and semi-torsors}
\begin{defn}\label{defn:semi}Let $k$ be a field.
\begin{enumerate}
\item
 A \emph{torsor} in the category of affine $k$-schemes is an affine
scheme $X$ with an operation of an affine $k$-group $G$
\[ G\times X\to X\]
such that there is an affine $S$ over $k$ and a point $x\in X(S)$ such that the
morphism
\[ G_S\to X_S,\quad g\mapsto gx\]
induced by the operation
is an isomorphism.
\item An \emph{semi-torsor} over $k$ is a $k$-vector space $V$ together with a coalgebra $A$ and a right co-operation
\[ V\to V\otimes A\]
such there is a a commutative $k$-algebra $K$ and a $k$-linear map $V\to K$ such that the map \[ V_K\to A_K\]
induced by the comodule structure
is an isomorphism.
\end{enumerate}
\end{defn}

\begin{rem}\label{rem:stor}
\begin{enumerate}
\item
The isomorphism makes $X$ a torsor in the fpqc-to\-po\-logy because $k\to K$ is faithfully flat. See also \cite[Section~1.7]{period-buch} for an extended discussion of different notions of torsors.
\item \label{it:stor_tor}
The second notion is newly introduced here. A torsor gives rise to a semi-torsor by taking global sections. A semi-torsor is a torsor, if, in addition,
$A$ is a Hopf algebra, $V$ a $k$-algebra and the structure map is a ring homomorphism. Hence a semi-torsor only has half of the information of a torsor. Note that an operation $G\times X\to X$ that gives rise to a semi-torsor, is already a torsor because an algebra homomorphism which is a vector space isomorphism is also an algebra isomorphism.
\item If $B$ is a finite dimensional $k$-algebra, $N$ a left-$B$-module that is fpqc-locally free of rank $1$ (i.e., here is  a commutative $k$-algebra $K$ such that $K\otimes_kN$ is free of rank $1$), then $N^\vee:=\mathrm{Hom}_k(N,k)$ is a semi-torsor over the coalgebra $B^\vee$.
\item Once there is one $x$ satisfying the torsor condition, all elements of $X(S)$ for all $S$ will satisfy the condition. This is not the case for semi-torsors.
\end{enumerate}
\end{rem}

\subsection{Galois theory revisited}\label{ssec:galois}
As promised in the introduction, we want to formulate Galois theory in torsor language.

Let $\Gamma=\mathrm{Gal}({\overline{\Q}}/\mathbb{Q})$. It is profinite, hence proalgebraic where we view abstract finite groups as algebraic groups over $\mathbb{Q}$. Indeed, we identify a
finite group $G$ with the algebraic group given by $\mathrm{Spec}( \mathbb{Q}[G]^\vee)$.
Let $X_0=\mathrm{Spec}({\overline{\Q}})$. We view it as a pro-algebraic variety over $\mathbb{Q}$.
There is a natural operation
\[ \Gamma\times X_0\to X_0.\]
We describe it on finite level. Let $K/\mathbb{Q}$ be Galois. Then
\[ \mathrm{Gal}(K/\mathbb{Q})\times \mathrm{Spec} (K)\to \mathrm{Spec}( K)\]
is given by the ring homomorphism
\[ \mathbb{Q}[\mathrm{Gal}(K/\mathbb{Q})]^\vee\otimes K\leftarrow K\]
adjoint to
\[ K\leftarrow \mathbb{Q}[\mathrm{Gal}(K/\mathbb{Q})]\otimes K\]
induced by the defining operation of the abstract group $G(K/\mathbb{Q})$ on $K$.

\begin{thm}[Galois theory]
The space $X_0$ is a torsor under $\Gamma$. More precisely,
the choice $\mathrm{id}\in X_0({\overline{\Q}})$ induces an isomorphism
\[ \Gamma_{\overline{\Q}}\to (X_0)_{\overline{\Q}}.\]
\end{thm}
\begin{proof}
We check the claim on the isomorphism on finite level.
Let $K/\mathbb{Q}$ be a finite Galois extension. The projection $X_0\to \mathrm{Spec}(K)$
amounts to fixing an embedding $K\subset{\overline{\Q}}$.

We want to study the base change of
\[\mathrm{Gal}(K/\mathbb{Q})\times \mathrm{Spec}(K)\to\mathrm{Spec}(K)\]
to ${\overline{\Q}}$. On rings, this is
\begin{equation}\label{eq1} {\overline{\Q}}[\mathrm{Gal}(K/\mathbb{Q})]^\vee\otimes_{\overline{\Q}} (K\otimes{\overline{\Q}})\leftarrow K\otimes{\overline{\Q}}.\end{equation}
On the other hand, consider the map
\[ K\otimes {\overline{\Q}}\to \mathrm{Map}(\mathrm{Hom}(K,{\overline{\Q}}),{\overline{\Q}})\cong {\overline{\Q}}[\mathrm{Hom}(K,{\overline{\Q}})]^\vee,\quad \lambda\otimes a\mapsto (\sigma\mapsto a\sigma(\lambda)).\]
It is Galois equivariant. As $K/\mathbb{Q}$ is separable, it is an isomorphism.
Hence our map (\ref{eq1}) can be written as
\begin{equation}\label{eq2}
{\overline{\Q}}[\mathrm{Gal}(K/\mathbb{Q})]^\vee\otimes {\overline{\Q}}[\mathrm{Hom}(K,{\overline{\Q}})]^\vee\leftarrow {\overline{\Q}}[\mathrm{Hom}(K,{\overline{\Q}})]^\vee.\end{equation}
It is obtained by duality from the right operation
\begin{equation} \mathrm{Gal}(K/\mathbb{Q})\times\mathrm{Hom}(K/{\overline{\Q}})\to\mathrm{Hom}(K/{\overline{\Q}}).\end{equation}
By Galois theory, this operation is simply transitive. By evaluating
on the fixed inclusion $K\subset {\overline{\Q}}$, we get a bijection
\[  \mathrm{Gal}(K/\mathbb{Q})\to\mathrm{Hom}(K,{\overline{\Q}})\]
translating to the isomorphism
\[ {\overline{\Q}}[\mathrm{Gal}(K/\mathbb{Q}]^\vee\leftarrow K\otimes {\overline{\Q}}\]
that we needed to show.
\end{proof}

\subsection{Nori's semi-torsor}
We return to the situation of Section~\ref{ssec:form_cat}. We want to explain how formal periods are a semi-torsor or even torsor.

Let $\mathbb{K}\subset \mathbb{C}$,
$\mathcal{C}$ an additive category,
\[ T:\mathcal{C}\to (\mathbb{K},\mathbb{Q})\Vect,\quad M\mapsto (T(M)_\mathbb{K}, T(M)_\mathbb{Q},\phi_M)\]
an additive functor. From now on we write
\[ T_\mathrm{dR}(M):= T(M)_\mathbb{K}, \hspace{2ex}T_\mathrm{sing}(M):=T(M)_\mathbb{Q}.\]
Both $T_\mathrm{dR}$ and $T_\mathrm{sing}$ are additive functors to $\mathbb{K}\mathrm{-Vect}$ and $\mathbb{Q}\mathrm{-Vect}$, respectively. In the Tannakian spirit we call
them \emph{fibre functors}.

We first give an alternative description of the formal period space. For a fixed $M$, the $\mathbb{Q}$-vector space generated by the symbols $(M,\sigma,\omega)$ with the relation of bilinearity is nothing but
\begin{multline*}
 T_\mathrm{dR}(M)\otimes_\mathbb{Q} T_\mathrm{sing}(M)^\vee\cong T_\mathrm{dR}(M)\otimes_\mathbb{K} T_\mathrm{sing}^\vee(M)_\mathbb{K}\\
\cong \mathrm{Hom}_\mathbb{K}(T_\mathrm{dR}(M),T_\mathrm{sing}(M)_\mathbb{K})^\vee.
\end{multline*}
Hence we have an alternative description of the space of formal periods
\[ \Perf(\mathcal{C})=\left(\bigoplus_{M\in\mathcal{C}} \mathrm{Hom}_{\overline{\Q}}(T_\mathrm{dR}(M),T_\mathrm{sing}(M))^\vee\right)/\text{functoriality}.\]
Note that the $\mathbb{Q}$-sub vector space of relations induced by functoriality is
even a $\mathbb{K}$-vector space. This makes clear that $\Perf(\mathcal{C})$ is a $\mathbb{K}$-vector space.

In the next step, we introduce the coalgebra (or even affine $\mathbb{K}$-group) under which the
formal period space is a semi-torsor (or even torsor). Actually, its definition is parallel to the definition of the formal period space, but with one fibre functor instead of two.

\begin{defn}For $M\in\mathcal{C}$ put
\[ \mathcal{A}(M)=\mathrm{End}_\mathbb{Q}(T_\mathrm{sing}(M))^\vee\]
and
\[ \mathcal{A}(\mathcal{C})=\left(\bigoplus_{M\in\mathcal{C}}\mathcal{A}(M)\right)/\text{functoriality}.\]
\end{defn}
Note that $\mathcal{A}(M)$ and $\mathcal{A}(\mathcal{C})$ are $\mathbb{Q}$-coalgebras.

\begin{rem}This object is at the very heart of Nori's work. It is an
 explicit description of the coalgebra that Nori  attaches to the additive functor $T_\mathrm{sing}:\mathcal{C}\to  \mathbb{Q}\mathrm{-Vect}$.
\end{rem}

We extend scalars to
$\mathbb{K}$:
\[ \mathcal{A}(\mathcal{C})_\mathbb{K}=\left(\bigoplus_{M\in\mathcal{C}}\mathrm{End}_\mathbb{K}(T_\mathrm{sing}(M)_\mathbb{K})\right)^\vee/\text{functoriality}.\]

The left-module structure
\begin{multline*}
 \mathrm{End}_\mathbb{K}(T_\mathrm{sing}(M)_\mathbb{K},T_\mathrm{sing}(M)_\mathbb{K})\times \mathrm{Hom}_\mathbb{K}(T_\mathrm{dR}(M),T_\mathrm{sing}(M)_\mathbb{K})\\
\longrightarrow \mathrm{Hom}_\mathbb{K}(T_\mathrm{dR}(M),T_\mathrm{sing}(M)_\mathbb{K})
\end{multline*}
induces a right-comodule structure
\[ \mathrm{Hom}_\mathbb{K}(T_\mathrm{dR}(M),T_\mathrm{sing}(M)_\mathbb{K})^\vee\to \mathcal{A}(M)_\mathbb{K}\otimes \mathrm{Hom}_\mathbb{K}(T_\mathrm{dR}(M),T_\mathrm{sing}(M)_\mathbb{K})^\vee \]
and hence
\[ \Perf(\mathcal{C})\to\mathcal{A}(\mathcal{C})_\mathbb{K}\otimes_\mathbb{K}\Perf(\mathcal{C})\cong \mathcal{A}(\mathcal{C})\otimes_\mathbb{Q}\Perf(\mathcal{C}).\]
Recall that $T$ takes values in $(\mathbb{K},\mathbb{Q})\Vect$, hence we have a distinguished
isomorphism of functors $\phi_T:T_\mathrm{dR}(M)_\mathbb{C}\to T_\mathrm{sing}(M)_\mathbb{C}$. It is an element
of $\mathrm{Hom}_\mathbb{C}(T_\mathrm{dR}(M)_\mathbb{C},T_\mathrm{sing}(M)_\mathbb{C})$ or equivalently a linear map
\[ \mathrm{Hom}_\mathbb{K}(T_\mathrm{dR}(M),T_\mathrm{sing}(M)_\mathbb{K})^\vee \to\mathbb{C}.\] Hence, this is precisely the data that we need for a semi-torsor. Indeed, because $\phi_T$ is an isomorphism,
 it induces
an isomorphism
\[ \Perf(\mathcal{C})_\mathbb{C}\to \mathcal{A}(\mathcal{C})_\mathbb{C}.\]
We have shown:
\begin{prop}The space $\Perf(\mathcal{C})$ is a $\mathbb{K}$-semi-torsor over $\mathcal{A}(\mathcal{C})_\mathbb{K}$ in the sense of Definition~\ref{defn:semi}.
\end{prop}

\begin{defn}Let $\CDTarg{\mathcal{C}}{T_\mathrm{sing}}$ be the category of $\mathcal{A}(\mathcal{C})$-comodules finite dimensional over $\mathbb{Q}$.
\end{defn}
This is nothing but Nori's diagram category, see \cite[Section~7.1.2]{period-buch}.

\begin{thm}[Nori, see {\cite[Theorem~7.1.13]{period-buch}}] There is a natural factorisation
\[ T_\mathrm{sing}:\mathcal{C}\to\CDTarg{\mathcal{C}}{T_\mathrm{sing}}\xrightarrow{\tilde{T}_\mathrm{sing}}\mathbb{Q}\mathrm{-Vect}\]
with $\CDTarg{\mathcal{C}}{T_\mathrm{sing}}$ abelian and $\mathbb{Q}$-linear, the functor $\tilde{T}$ faithful and exact, and the category is universal with this property.
\end{thm}

\begin{rem}\label{rem:univ}In particular, $\mathcal{C}$ is equivalent to $\CDTarg{\mathcal{C}}{T_\mathrm{sing}}$ if
$\mathcal{C}$ is itself abelian and $T_\mathrm{sing}$ faithful and exact.
\end{rem}
\begin{prop}\label{prop:make_ab}We have
\[ \Perf(\mathcal{C})\cong\Perf(\CDTarg{\mathcal{C}}{T_\mathrm{sing}}).\]
\end{prop}
\begin{proof}We use the semi-torsor structure to reduce the claim to
the comparison of the diagram algebra. It remains to check that the natural map
\[ \mathcal{A}(\mathcal{C})\to\mathcal{A}(\CDTarg{\mathcal{C}}{T_\mathrm{sing}})\]
is an isomorphism. This is only implicit in \cite[Section~7.3]{period-buch}.
Here is the argument: By construction, $\mathcal{A}(\mathcal{C})=\lim_i A_i$ for
coalgebras finite dimensional over $\mathbb{Q}$ of the form $A_i=E_i^*$ for a
$\mathbb{Q}$-algebra $E_i$. We also have $\CDTarg{\mathcal{C}}{T_\mathrm{sing}}=\bigcup_i E_i\mathrm{-Mod}$, the category of finitely generated $E_i$-modules.
Hence it suffices to show that
\[ E_i\cong \mathcal{A}(E_i\mathrm{-Mod})^\vee.\]
The category $E_i\mathrm{-Mod}$ has the projective generator $E_i$. By \cite[Lemma~7.3.14]{period-buch}, we have $\mathcal{A}(E_i\mathrm{-Mod})^\vee\cong\mathrm{End}(T_\mathrm{sing}|_{E_i})$. The latter means all $\mathbb{Q}$-linear maps $E_i\to E_i$ commuting with all $E_i$-morphisms
$E_i\to E_i$. We have $\mathrm{End}_{E_i\mathrm{-Mod}}(E_i)=E_i^\circ$ (the opposite algebra of
$E_i$) and hence $\mathrm{End}(T_\mathrm{sing}|_{E_i})=E_i$ as claimed.
\end{proof}

\subsection{The Tannakian case}
Assume now that, in addition, $\mathcal{C}$ is a rigid tensor category and
$T$ a faithfully exact tensor functor. Using $T_\mathrm{sing}$ as fibre functor, this makes $\mathcal{C}$ a \emph{Tannakian} category. By Tannaka duality, it is
equivalent to the category of finite dimensional representations of a
pro-algebraic group $G(\mathcal{C})$. There is a second fibre functor
given by $T_\mathrm{dR}$. Again by Tannaka theory, the comparison of the two
fibre functors defines a pro-algebriac affine scheme $X(\mathcal{C})$ over $\mathbb{K}$ such that
\[ X(S)=\left\{\Phi:T_\mathrm{sing}(\cdot)_S\to T_\mathrm{dR}(\cdot)_S| \text{isom. of tensor functors}\right\}.\]
It is a torsor under $G(\mathcal{C})$
\[ G(\mathcal{C})\times X(\mathcal{C})\to X(\mathcal{C}).\]
When passing to the underlying semi-torsor, these objects are identical to the ones considered before.

\begin{prop}[{\cite[Section~7.1.4]{period-buch}}]
There are natural isomorphisms
$\mathcal{O}(X)\cong \Perf(\mathcal{C})$ and $\mathcal{O}(G(\mathcal{C}))=\mathcal{A}(\mathcal{C})$.
\end{prop}
\begin{proof}The case of the group is \cite[Theorem~7.1.21]{period-buch}.
The same arguments also applies  to $X(\mathcal{C})$, see also \cite[Remark~8.2.11]{period-buch}.
\end{proof}

\subsection{The case of motives}\label{ssec:motive_torsor}
A particularly interesting case ist $\mathcal{C}=\mathcal{MM}_{\mathrm{Nori}}(\mathbb{K})$. Put
\begin{align*}
 G_\mathrm{mot}(\mathbb{K})&:=\mathrm{Spec}\left(\mathcal{A}(\mathcal{MM}_{\mathrm{Nori}}(\mathbb{K}))\right),\\
 X(\mathbb{K})&:=\mathrm{Spec}\left(\Perf(\mathcal{MM}_{\mathrm{Nori}}(\mathbb{K}))\right).
\end{align*}
Then
\[ G_\mathrm{mot}(\mathbb{K})_\mathbb{K}\times_\mathbb{K} X(\mathbb{K})\to X(\mathbb{K})\]
is a torsor. We use this to give the proof of Theorem~\ref{thm:indep} that was left open.

\begin{proof}[Proof of Theorem~\ref{thm:indep}]
Fix $\mathbb{K}\subset{\overline{\Q}}$. Let $\mathcal{A}_\mathrm{Nori}(\mathbb{K})$ be the coalgebra defined
using the same data as in Definition~\ref{defn:perf}, but with $H^i_\mathrm{dR}$ replaced by $H^i_\mathrm{sing}$ and with base field $\mathbb{K}$ instead of ${\overline{\Q}}$.

The considerations of the present section also apply to this case and show
that $\mathcal{A}_\mathrm{Nori}(\mathbb{K})$ is nothing but Nori's diagram coalgebra for the diagram
of pairs, see \cite[Definition~9.1.1]{period-buch}. By definition, see
\cite[Definition~9.1.3]{period-buch}, the category $\mathcal{MM}_{\mathrm{Nori}}(\mathbb{K})$ is the category
of representations of $G_\mathrm{mot}(\mathbb{K})=\mathrm{Spec}\left(\mathcal{A}_\mathrm{Nori}(\mathbb{K})\right)$.
Hence $\mathcal{A}_\mathrm{Nori}(\mathbb{K})\cong\mathcal{A}(\mathcal{MM}_{\mathrm{Nori}}(\mathbb{K}))$.

By \cite[Corollary~10.1.7]{period-buch}
we also have $\mathcal{A}_\mathrm{Nori}(\mathbb{K})\cong\mathcal{A}(\DM_{\gm}(\mathbb{K}))$. By Galois theory of periods, i.e., the torsor property, this also implies
\[ \Perf(\mathbb{K})\cong\Perf(\mathcal{MM}_{\mathrm{Nori}}(\mathbb{K}))\cong\Perf(\DM_{\gm}(\mathbb{K})).\]

We now have to compare different $\mathbb{K}$'s. Let $K/k$ be an algebraic extension of
subfields of ${\overline{\Q}}$. The comparison
\[ \Perf(K)\cong\Perf(k)\]
is claimed in \cite[Proposition~3.1.11]{period-buch}. Unfortunately, the argument is not complete. It only shows that the map induced by base change
\[ \Per(k)\to\Perf(K)\]
is surjective. We are going to complete the argument now. It suffices to consider the case $K/k$ finite and Galois. In this case, there is a natural short exact sequence
\[ 0\to G_\mathrm{mot}(K)\to G_\mathrm{mot}(k)\to\mathrm{Gal}(K/k)\to 0\]
by \cite[Theorem 9.1.6]{period-buch}. Moreover, we have torsor structures
\begin{gather*}
G_\mathrm{mot}(k)_K\times_K X(k)_K\to X(k)_K,\\
G_\mathrm{mot}(K)_K\times_K X(K)\to X(K).
\end{gather*}
We have ${\overline{\Q}}\subset\Perf(K)$ as formal periods of zero-dimensional varieties.
Hence the structure morphism $X(k)\to \mathrm{Spec}(k)$ factors via
$\mathrm{Spec}(K)$. We write $X'(k)$ for $X(k)$ viewed as $K$-scheme. Then
\[ X(k)_K=X(k)\times_kK\cong X'(k)\times_K(\mathrm{Spec}(K)\times_k\mathrm{Spec}(K))\cong\coprod_{\sigma\in\mathrm{Gal}(K/k)}X'(k).\]
Comparing with the structure of $G_\mathrm{mot}(k)$, we deduce the torsor
\[ G_\mathrm{mot}(K)_K\times_KX'(k)\to X'(k).\]
Hence the natural map $X'(k)\to X(K)$ is an isomorphism.
\end{proof}

\section{Consequences for the period conjecture}
\subsection{Abstract considerations}
As before,
let $\mathcal{C}$ be an additive category and $T:\mathcal{C}\to (\mathbb{K},\mathbb{Q})\Vect$ an additive functor
and $\CDTarg{\mathcal{C}}{T_\mathrm{sing}}$ the diagram category.
\begin{prop}
The period conjecture for $\mathcal{C}$ is equivalent to the period conjecture for
$\CDTarg{\mathcal{C}}{T_\mathrm{sing}}$.
\end{prop}
\begin{proof}The formal period algebras agree by
Proposition~\ref{prop:make_ab}.
\end{proof}
Hence we only need to consider abelian categories and faithful exact functors
$T$ when analysing the period conjecture.

\begin{prop}\label{prop:inj}Let $\mathcal{C}$ be an abelian, $\mathbb{Q}$-linear category. Let $\mathcal{C}'\subset \mathcal{C}$ be an abelian subcategory such that the inclusion is exact. The following are equivalent:
\begin{enumerate}
\item \label{it:1}the category $\mathcal{C}'$ is closed under subquotients in $\mathcal{C}$;
\item \label{it:2}the map $\Perf(\mathcal{C}')\to\Perf(\mathcal{C})$ is injective.
\end{enumerate}
\end{prop}
\begin{proof}Injectivity can be tested after base change to $\mathbb{C}$, hence the second assertion of equivalent to the injectivity of $\mathcal{A}(\mathcal{C}')\to\mathcal{A}(\mathcal{C})$.
The implication from (\ref{it:1}) to (\ref{it:2}) is \cite[Proposition~7.5.9]{period-buch}.

For the converse, consider $K=\ker(\mathcal{A}(\mathcal{C}')\to\mathcal{A}(\mathcal{C})$. It is an $\mathcal{A}(\mathcal{C})$-comodule, but not a $\mathcal{A}(\mathcal{C}')$-comodule.
By construction
$\mathcal{A}(\mathcal{C}')$ is a direct limit of coalgebras finite over $\mathbb{Q}$. Hence
$K$ is a direct limit of $\mathcal{A}(\mathcal{C})$-comodules of finite dimension over $\mathbb{Q}$.
The cannot all be $\mathcal{A}(\mathcal{C}')$-comodules, hence we have found objects of
$\mathcal{C}$ that are not in $\mathcal{C}'$.
\end{proof}
\begin{cor}\label{cor:faithful}Suppose $\mathcal{C}$ is abelian and $T$ faithful and exact. If the period conjecture holds
for $\mathcal{C}$, then $T:\mathcal{C}\to (\mathbb{K},\mathbb{Q})\Vect$ is fully faithful with image closed
under subquotients.
\end{cor}
\begin{proof}We view $\mathcal{C}$ as a subcategory of $(\mathbb{K},\mathbb{Q})\Vect$ via $T$.
Let $\tilde{\mathcal{C}}$ be its closure under
subquotients. Note that the periods of $\mathcal{C}$ agree with the periods
of $\tilde{\mathcal{C}}$. By assumption, the composition
\[ \Perf(\mathcal{C})\to \Perf(\tilde{\mathcal{C}})\to\Per(\mathcal{C})\]
is injective. Hence the first map injective. By Proposition~\ref{prop:inj} this implies
that $\mathcal{C}$ is closed under subquotients in $({\overline{\Q}},\mathbb{Q})\Vect$. Let $X,Y\in\mathcal{C}$ and
$f:X\to Y$ a morphism in $({\overline{\Q}},\mathbb{Q})\Vect$. Then its graph $\Gamma\subset X\times Y$
is a subobject in $({\overline{\Q}},\mathbb{Q})\Vect$, hence in $\mathcal{C}$. This implies that $f$ is in
$\mathcal{C}$. Hence $\mathcal{C}$ is a full subcategory.
\end{proof}

\begin{cor}\label{cor:full}Let $\mathcal{C}'\subset\mathcal{C}$ be a full abelian subcategory closed under subquotients. If the period conjecture holds for $\mathcal{C}$, then it holds for $\mathcal{C}'$.
\end{cor}
\begin{proof}We have $\Perf(\mathcal{C}')\subset\Perf(\mathcal{C})\to\mathbb{C}$. If the second
map is injective, then so is the composition.
\end{proof}

\subsection{The case of motives}
We now specialise to $\mathcal{C}=\mathcal{MM}_{\mathrm{Nori}}(\mathbb{K},\mathbb{Q})$.

\begin{defn}Let $M\in \mathcal{MM}_{\mathrm{Nori}}(\mathbb{K},\mathbb{Q})$. We denote by $\langle M\rangle$ the full
abelian category of $\mathcal{MM}_{\mathrm{Nori}}(\mathbb{K},\mathbb{Q})$ closed under subquotients generated by $M$.
\end{defn}

\begin{prop}\label{prop:equiv1}The following are equivalent:
\begin{enumerate}
\item
The period conjecture holds for $\mathcal{MM}_{\mathrm{Nori}}(\mathbb{K},\mathbb{Q})$.
\item  The period conjecture holds for $\langle M\rangle $ for all $M\in\mathcal{MM}_{\mathrm{Nori}}(\mathbb{K},\mathbb{Q})$.
\end{enumerate}
\end{prop}
Note that $\Per(\langle M\rangle)=\Per\langle M\rangle$ in the notation of
 Definition~\ref{defn:VV}.
\begin{proof}We have $\Perf(\langle M\rangle)\subset\Perf(\mathcal{MM}_{\mathrm{Nori}})$ because the subcategory is full and closed under subquotients. Hence
\[ \Perf=\Perf(\mathcal{MM}_{\mathrm{Nori}})=\bigcup_M\Perf(\langle M\rangle).\]
The map $\Perf\to\mathbb{C}$ is injective if and only if this is true for all
$\Perf(\langle M\rangle)$.
\end{proof}

\subsection{Grothendieck's period conjecture}
In the previous section, we did not make use of the tensor product on motives.
Indeed, by Section~\ref{ssec:motive_torsor} we have the torsor
\[ G_\mathrm{mot}({\overline{\Q}})_{\overline{\Q}}\times_{\overline{\Q}} X({\overline{\Q}})\to X({\overline{\Q}}).\]

\begin{cor}[Nori]\label{galois}If the period conjecture holds, then $\Per$ is a $G_\mathrm{mot}(\mathbb{K})_\mathbb{K}$-torsor.
\end{cor}

\begin{defn}Let $M\in\mathcal{MM}_{\mathrm{Nori}}({\overline{\Q}})$. We put $\langle M\rangle^{\otimes, \vee}\subset \mathcal{MM}_{\mathrm{Nori}}({\overline{\Q}})$ the full abelian rigid tensor subcategory closed under subquotients generated by $M$. We denote $G(M)$ its Tannaka dual.
\end{defn}

\begin{prop}\label{prop:equiv2}The following are equivalent:
\begin{enumerate}
\item the period conjecture holds for $\mathcal{MM}_{\mathrm{Nori}}({\overline{\Q}})$;
\item the period conjecture holds for $\langle M\rangle^{\otimes,\vee}$ for
all $M\in\mathcal{MM}_{\mathrm{Nori}}({\overline{\Q}})$.
\end{enumerate}
\end{prop}
\begin{proof}Same proof as for Proposition~\ref{prop:equiv1}\end{proof}

The scheme $X(M):=\mathrm{Spec}(\Perf\langle M\rangle^{\otimes,\vee})$
is a $G(M)$-torsor. Hence it inherits all properties of the algebraic group $G(M)$ that can be tested after a faithfully flat base change. In particular it is smooth.

\begin{conj}[Grothendieck's period conjecture]
Let $M\in\mathcal{MM}_{\mathrm{Nori}}$. Then $X(M)$ is
connected and
\[ \dim G(M)=\mathrm{trdeg} \mathbb{Q}( \Per(M)).\]
\end{conj}
In the case of pure motives, this is the formulation of Andr\'e in \cite[Chapitre~23]{andre2}.

\begin{prop}[{\cite[Conjecture~13.2.6]{period-buch}}]The following are equivalent:
\begin{enumerate}
\item the period conjecture holds for $\langle M\rangle^{\otimes,\vee}$;
\item the point $\mathrm{ev}\in X(M)(\mathbb{C})$ is generic and $X(M)$ is connected;
\item Grothendieck's period conjecture holds for $M$.
\end{enumerate}
\end{prop}
\begin{proof}
See the proof of equivalence in \cite{period-buch}. The crucial input is
that $X(M)$ is smooth. Hence being connected makes it integral, so that there is only one generic point. Moreover, $X(M)$ and $G(M)$ have the same dimension.
\end{proof}

\section{Cycles and the period conjecture}
We now discuss the relation between conjectures on fullness of cycle class maps
and the period conjecture.
\begin{prop}Let $\mathcal{C}\subset \mathcal{MM}_{\mathrm{Nori}}(\mathbb{K})$ be a full abelian category closed under subquotients. If the period conjecture holds
for $\mathcal{C}$, then $H:\mathcal{C}\to (\mathbb{K},\mathbb{Q})\Vect$ is fully faithful with image closed
under subquotients.
\end{prop}
\begin{proof}This is a special case of Corollary \ref{cor:full}.
\end{proof}
In the Tannakian case this is already formulated in \cite[Proposition~13.2.8]{period-buch}. Indeed, it seems well-known.

By definition $H$ is faithful, hence the interesting part about this is
fullness. This point is taken up in a version of the period conjecture
formulated by Bost and Charles in \cite{bost-charles} and generalised
by Andreatta, Barbieri-Viale and Bertapelle in \cite{ABB}. We  formulate the latter version.
Recall from \cite{harrer} the realisation functor
\[ \DM_{\gm}(\mathbb{K})\to D^b(\mathcal{MM}_{\mathrm{Nori}}(\mathbb{K})).\]
By composition with $\mathcal{MM}_{\mathrm{Nori}}(\mathbb{K})\to (\mathbb{K},\mathbb{Q})\Vect$ this gives us a functor
\[ H^0:\DM_{\gm}(\mathbb{K})\to (\mathbb{K},\mathbb{Q})\Vect.\]

\begin{defn}We say that the fullness conjecture holds
for $\mathcal{C}\subset \DM_{\gm}(\mathbb{K})$ if
\[ H^0:\mathcal{C}\to(\mathbb{K},\mathbb{Q})\Vect\]
is full.
\end{defn}

\begin{conj}[ {\cite[Section~1.3]{ABB}}]\label{conj:ABB}
Let $\mathcal{C}\subset \DM_{\gm}(\mathbb{K})$ be a full additive subcategory. Then the fullness conjecture holds for $\mathcal{C}$.
\end{conj}
\begin{rem}
\begin{enumerate}
\item
Andreatta, Barbieri-Viale and Bertapelle also consider refined integral information. We do not go into this here.
\item This terminology deviates from \cite{ABB} where the term period conjecture is used.
\end{enumerate}
\end{rem}

Arapura established in \cite{arapura} (see also  \cite[Theorem~10.2.7]{period-buch}) that the category of pure Nori motives is equivalent to Andr\'es's abelian category $\mathrm{AM}$ of motives via motivated cycles.

\begin{prop}If the period conjecture holds for pure Nori motives and
the K\"unneth and Lefschetz standard conjectures are true, then the fullness conjecture
holds for Chow motives.
\end{prop}
\begin{proof}
The functor from Chow motives to Grothendieck motives is full by definition.
The K\"unneth and Lefschetz standard conjectures together  imply that the category of
Grothendieck motives is abelian and agrees with $\mathrm{AM}$, hence with the
category of pure Nori motives. By the period conjecture, the functor
to $({\overline{\Q}},\mathbb{Q})\Vect$ is full. Together this proves fullness of the composition.
\end{proof}

A similar result actually holds for the full category of geometric motives. Its assumptions are very strong, but are indeed expected to be true.

\begin{prop}\label{prop:full_gm}We assume
\begin{enumerate}
\item the period conjecture holds for all Nori motives;
\item the category $\DM_{\gm}(\mathbb{K})$ has a $t$-structure compatible with the singular realisation;
\item  the singular realisation is conservative;
\item the category $\DM_{\gm}(\mathbb{K})$ is of cohomological dimension $1$.
\end{enumerate}
Then the fullness conjecture holds
for $\DM_{\gm}(\mathbb{K})$.
\end{prop}
\begin{proof}The period conjecture gives again fullness of $\mathcal{MM}_{\mathrm{Nori}}(\mathbb{K})\to(\mathbb{K},\mathbb{Q})\Vect$.
It remains to check surjectivity of
\begin{equation}\label{eq:surj}\tag{*} \mathrm{Hom}_{\DM_{\gm}(\mathbb{K})}(M_1,M_2)\to \mathrm{Hom}_{\mathcal{MM}_{\mathrm{Nori}}(\mathbb{K})}(H^0_\mathrm{Nori}(M_1),H^0_\mathrm{Nori}(M_2))\end{equation}
for all $M_1,M_2\in\DM_{\gm}(\mathbb{K})$.

Let $\mathcal{MM}$ be the heart of the $t$-structure. By conservativity, $H^0$ is faithful. By the universal property of Nori motives, this implies $\mathcal{MM}\cong \mathcal{MM}_{\mathrm{Nori}}(\mathbb{K})$. As the cohomological dimension is $1$, this implies
$\DM_{\gm}(\mathbb{K})\cong D^b(\mathcal{MM}_{\mathrm{Nori}}(\mathbb{K}))$.
It also implies that every object of $\DM_{\gm}(\mathbb{K})$ is (uncanonically) isomorphic to the direct sum of its cohomology objects. Hence it suffices to consider
$M_1=N_1[i_1]$, $M_2=N_2[i_2]$ for $N_1,N_2\in\mathcal{MM}$, $i_1,i_2\in\mathbb{Z}$. We have $H^0(M_k)=0$ unless
$0=i_1,i_2$. Hence surjectivity in (\ref{eq:surj}) is trivial for $i_1\neq 0$ or $i_2\neq 0$.
In the remaining case $i_1=i_2=0$, we have equality in equation (\ref{eq:surj}), so again, the map is surjective.
\end{proof}

\begin{rem}The converse is not true, the fullness conjecture does \emph{not} imply
the period conjecture. An explicit example is given in \cite[Remark~5.6]{huber-wuestholz}. See also the case of mixed Tate motives, Proposition~\ref{prop:mt}.

On the other hand, the condition on the cohomological dimension is necessary if we want to conclude from fully faithfulness on the level of abelian categories to the triangulated category. Note that $(\mathbb{K},\mathbb{Q})\Vect$ has cohomological dimension $1$.
\end{rem}

\section{Examples}
\subsection{Artin motives}
The category of Artin motives over $\mathbb{K}$ can be described as the full subcategory of
$\mathcal{MM}_{\mathrm{Nori}}(\mathbb{K})$ generated by motives of $0$-dimensional varieties. The Tannakian
dual of this category is nothing but $\mathrm{Gal}(\bar{\mathbb{K}}/\mathbb{K})$. The space of
periods is ${\overline{\Q}}$. The torsor structure is the one described in Section~\ref{ssec:galois}. For more details see also \cite[Section~13.3]{period-buch}.

\begin{rem}The period conjecture and the fullness conjecture hold true for Artin motives.
\end{rem}

\subsection{Mixed Tate motives}
Inside the triangulated category $\DM_{\gm}(\mathrm{Spec}(\mathbb{Q}),\mathbb{Q})$ let $\mathsf{DMT}$ the full thick rigid tensor triangulated subcategory generated by the Tate motive $\mathbb{Q}(1)$. Equivalently, it is the full thick triangulated subcategory generated by the set of $\mathbb{Q}(i)$ for $i\in\mathbb{Z}$.
By deep results of Borel it carries a $t$-structure, so that we get a well-defined \emph{abelian} category $\mathcal{MT}$ of mixed Tate motives over $\mathbb{Q}$, see \cite{LeMTM}. Deligne and Gochanarov find a smaller abelian tensor subcategory $\MT^f$ of
\emph{unramified} mixed Tate motives, see \cite{deligne-goncharov}. Its distinguishing feature is that
\[ \mathrm{Ext}^1_{\MT^f}(\mathbb{Q}(0),\mathbb{Q}(1))=\mathbb{Z}^*\otimes_\mathbb{Z}\mathbb{Q}=0\subsetneq \mathrm{Ext}^1_{\mathcal{MT}}(\mathbb{Q}(0),\mathbb{Q}(1))=\mathbb{Q}^*\otimes_\mathbb{Z}\mathbb{Q}.\]
By \cite{deligne-goncharov}, the periods of $\MT^f$ are precisely the famous multiple zeta values. The Tannaka dual of $\MT^f$ is very well-understood.
Using Galois theory of periods this information translates into information
on $\Perf(\MT^f)$. The surjectivity of $\Perf(\MT^f)\to \Per(\MT^f)$ then yields
upper bounds on the dimensions of spaces of multiple zeta values. For more
details see \cite{deligne-goncharov} or \cite[Chapter~15]{period-buch}.

\begin{rem}
The period conjecture and the
fullness conjecture are open for $\MT^f$.
\end{rem}
Indeed, the period conjecture for $\MT^f$ implies statements like the algebraic independence of the values of Riemann $\zeta$-function $\zeta(n)$ for $n\geq 3$ odd that are wide open. The fullness conjecture is a lot  weaker.

\begin{prop}\label{prop:mt}The fullness conjecture holds for $\mathsf{DMT}$ if and only if $\zeta(n)$ is irrational for $n\geq 3$ odd.
\end{prop}
\begin{proof}Assume the irrationality condition. By \cite{deligne-goncharov}, the category $\mathcal{MT}$ has cohomological dimension $1$ and $D^b(\mathcal{MT})$ is equivalent to $\mathsf{DMT}$. Hence we can use the same reasoning as in the proof of Proposition~\ref{prop:full_gm}. It suffices to show that $\mathcal{MT}\to (\mathbb{Q},\mathbb{Q})\Vect$ is fully faithful. We address this issue with the same argument as in the proof of \cite[Proposition~2.14]{deligne-goncharov} treating the functor from $\MT^f$ to mixed realisations. It remains to check injectivity of
\[ \mathrm{Ext}^1_{\mathcal{MT}}(\mathbb{Q},\mathbb{Q}(n))\to \mathrm{Ext}^1_{(\mathbb{Q},\mathbb{Q})\Vect}(\mathbb{Q},\mathbb{Q}(n))\]
for all $n\in\mathbb{Z}$. Here $\mathbb{Q}(n)$ denotes the object of rank $1$ given by
$\mathbb{Q}(n)_\mathrm{dR}=\mathbb{Q}, \mathbb{Q}(n)_\mathrm{sing}=(2\pi i)^n\mathbb{Q}$ and the natural comparison isomorphism.

By Borel's work, the left hand side is known to vanish for $n\leq 0$ and $n\geq 2$ even, hence it suffices to consider $n=2m-1$ for $m\geq 1$. By \cite[Proposition~4.3.2 and Proposition~4.2.3]{huber_1604}
\[ \mathrm{Ext}^1_{(\mathbb{Q},\mathbb{Q})\Vect}=\mathrm{coker}\left( \mathbb{Q}(n)_\mathrm{dR}\oplus \mathbb{Q}(n)_\mathrm{sing}\to \mathbb{C}\right)=\mathbb{C}/\langle 1,(2\pi i)^n\rangle_\mathbb{Q} .\]
This is a quotient of
\begin{equation}\label{eq:ext} \mathrm{Ext}^1_{\mathrm{MHS}_\mathbb{Q}}(\mathbb{Q},\mathbb{Q}(n))\cong \mathbb{C}/(2\pi i)^n\mathbb{Q}.\end{equation}
We first consider $n=m=1$, the Kummer case. The left hand side is
$\mathbb{Q}^*\otimes\mathbb{Q}$ and the map is the Dirichlet regulator $u\mapsto \log|u|$.
The prime numbers give a basis of $\mathbb{Q}^*\otimes\mathbb{Q}$. By Baker's theorem, their
logarithms are ${\overline{\Q}}$-linearly independent. They are also real whereas $2\pi i$ is imaginary. Hence the regulator map (\ref{eq:ext})
is injective.

Now let $m\geq 2$.
From the well-known explicit computation of the Deligne regulator, we know that the image of a generator of
$\mathrm{Ext}^1_{\mathcal{MT}}(\mathbb{Q},\mathbb{Q}(2m-1))$ is given by $\zeta(2m-1)\in \mathbb{C}/\langle 1,(2\pi i)^{2m-1}\rangle$. It is real hence $\mathbb{Q}$-linearly independent of $(2\pi i)^{2m-1}\in i\mathbb{R}$. If it is irrational, then it is $\mathbb{Q}$-linearly independent of $1$ as well.

Now assume conversely that the fullness conjecture holds for $\mathsf{DMT}$. This implies that $\mathcal{MT}\to (\mathbb{Q},\mathbb{Q})\Vect$ is fully faithful. We claim that (\ref{eq:ext}) is injective. Take an element on the right, i.e., a short exact sequence
\[ 0\to \mathbb{Q}(n)\to E\to \mathbb{Q}(0)\to 0\]
in $\mathcal{MT}$ whose image in $(\mathbb{Q},\mathbb{Q})\Vect$ is split. This means that there is a splitting morphism $\mathbb{Q}(0)\to E$ in $(\mathbb{Q},\mathbb{Q})\Vect$. By fullness it comes from a morphism in $\mathcal{MT}$. This shows injectivity of (\ref{eq:ext}). By the explicit computation this implies
that $\zeta(2m-1)$ is $\mathbb{Q}$-linearly independent of $1$, i.e., irrational.
\end{proof}

We see that the fullness statement is much weaker that than the period conjecture. There are even partial results due to Ap\'ery and Zudilin.
\subsection{The case of $1$-motives}
The situation for $1$-motives is quite similar to the case of $0$-motives:
there is an abelian category $1\mathrm{-Mot}_\mathbb{K}$ of iso-$1$-motives over $\mathbb{K}$. It was originally defined by Deligne, see \cite{hodge3}. Its derived category
is equivalent the subcategory of $\DM_{\gm}(\mathbb{K})$ generated by motives
of varieties of dimension at most $1$. This result is due to Orgogozo, see \cite{orgogozo}, see also Barbieri-Viale and Kahn \cite{BVK}. There is a significant difference to the $0$-dimensional case, though: $1\mathrm{-Mot}_\mathbb{K}$ is not closed under tensor products, so it does not make sense to speak about torsors. However, our notion of a semi-torsor is built to cover this case: $\Perf(1\mathrm{-Mot}_\mathbb{K})$ is semi-torsor
under the coalgebra $\mathcal{A}(1\mathrm{-Mot}_\mathbb{K})_\mathbb{K}$.

The main result of \cite{huber-wuestholz} settles the period conjecture:

\begin{thm}[{\cite{huber-wuestholz}}]\label{thm:1-mot}
The evalution map $\Perf(1\mathrm{-Mot}_{\overline{\Q}})\to\Perf(1\mathrm{-Mot}_{\overline{\Q}})$ is injective, i.e., the period conjecture holds for $1\mathrm{-Mot}_{\overline{\Q}}$.
\end{thm}

This is first and foremost a result in transcendence theory. With a few extra arguments, it implies:

\begin{cor}[{\cite[Theorem~5.10]{huber-wuestholz}}]Let $X$ be  an algebraic curve over ${\overline{\Q}}$, $\omega\in\Omega^1_{{\overline{\Q}}(X)}$ an algebraic differential form, $\sigma=\sum_{i=1}^na_i\gamma_i$ a singular chain avoiding the poles of $\omega$ and with $\partial\sigma$ a divisor
on $X({\overline{\Q}})$. Then the following are equivalent:
\begin{enumerate}
\item the number $\int_\sigma\omega$ is algebraic;
\item we have $\omega= df +\phi$ with $\int_\phi\phi=0$.
\end{enumerate}
\end{cor}

This contains the previously known results on transcendence of $\pi$ or
values of $\log$ in algebraic numbers as well as periods of differential forms over closed paths. The really new case concerns differential forms of the third kind (so with non-vanishing residues) and non-closed paths. This settles
questions of Schneider that had been open since the 1950s.
Here is a very explicit example:

\begin{ex}[{\cite[Theorem~7.6]{huber-wuestholz}}]Let $E/{\overline{\Q}}$ be an elliptic curve. Recall the Weierstrass $\wp$-, $\zeta$- and $\sigma$-function for $E$. Let
$u\in\mathbb{C}$ such that $\wp(u)\in{\overline{\Q}}$ and $\exp_E(u)$ is non-torsion in $E({\overline{\Q}})$. Then
\[ u\zeta(u)-2\log(\sigma(u))\]
is transcendental.
\end{ex}
In the spirit of Baker's theorem,
Theorem~\ref{thm:1-mot} also allows us to give an explicit formula for
$\dim_{\overline{\Q}}\Per\langle M\rangle$ for all $1$-motives $M$. It is not easy to state, so we simply refer to \cite[Chapter~6]{huber-wuestholz}.

On the other hand, Theorem~\ref{thm:1-mot} has consequences for the various conjectures discussed before.
In \cite{ayoub-barbieri}, Ayoub and Barbieri-Viale show that $1\mathrm{-Mot}_k$
is a full subcategory closed under subquotients in $\mathcal{MM}_{\mathrm{Nori}}^\mathrm{eff}(k)$. They also
give descpription of $\mathcal{A}(1\mathrm{-Mot}_k)$ in terms of generators and relations.
Together with Theorem~\ref{thm:1-mot} this implies Kontsevich's period conjecture for curves.

In detail: Following \cite[Definition~4.6]{huber-wuestholz} we call a period number
 of \emph{curve type} if it is a period of a motive of the form
$H^1(C,D)$ with $C$ a smooth affine curve over ${\overline{\Q}}$ and
$D\subset C$ a finite set of ${\overline{\Q}}$-valued points.

\begin{cor}[{\cite[Theorem~5.3]{huber-wuestholz}}]
All relations between periods of curve type are induced by bilinearity and functoriality of pairs $(C,D)\to (C',D')$ with $C, C'$ smooth affine curves and
$D\subset C$, $D'\subset C'$ finite sets of points.
\end{cor}

In a different direction:
\begin{cor}[{\cite[Theorem~5.4]{huber-wuestholz}}, \cite{ABB}]
The functor $1\mathrm{-Mot}_{\overline{\Q}}\to ({\overline{\Q}},\mathbb{Q})\Vect$ is fully faithful.
\end{cor}
\begin{proof}
This is Corollary~\ref{cor:full}. An independent proof was given by Andreatta, Barbieri-Viale and Bertapelle.
\end{proof}

On the other hand, Orgogozo \cite{orgogozo} and Barbieri-Viale, Kahn \cite{BVK}
showed that the derived category of $1\mathrm{-Mot}_k$ is a full subcategory of
$\DM_{\gm}(k)$.

\begin{cor}
The fullness conjecture holds for $d_{\leq 1}\DM_{\gm}({\overline{\Q}})$.
\end{cor}
\begin{proof}
By \cite[Proposition~3.2.4]{orgogozo}, the category
$d_{\leq 1}\DM_{\gm}({\overline{\Q}})\cong D^b(1\mathrm{-Mot}_{\overline{\Q}})$ has cohomological dimension
$1$. We can now use the same argument as in the proof of Proposition~\ref{prop:full_gm}, unconditionally.
\end{proof}

\begin{cor}[{\cite[Theorem~5.4]{huber-wuestholz}}]
The natural functors $1\mathrm{-Mot}_{\overline{\Q}}\to \mathcal{MM}_{\mathrm{Nori}}({\overline{\Q}})$ (non-effective Nori motives) and $1\mathrm{-Mot}_{\overline{\Q}}\to\mathrm{MHS}_{\overline{\Q}}$ (mixed Hodge structures over ${\overline{\Q}}$)
are full.
\end{cor}
\begin{proof}Their composition with the functor to $({\overline{\Q}},\mathbb{Q})\Vect$ is full.
\end{proof}

A much stronger version of the above was recently shown by Andr\'e by comparing the Tannakian dual of the tensor category generated by $1$-motives to the Mumford Tate group.
\begin{thm}[Andr\'e {\cite[Theorem~2.0.1]{andre4}}]
Let $k\subset \mathbb{C}$ be algebraically closed. Let $1\mathrm{-Mot}_k^\otimes\subset \mathcal{MM}_{\mathrm{Nori}}(k)$ be the full Tannakian subcategory closed under subquotients generated by
$1\mathrm{-Mot}_k$.
Then the functor
\[ 1\mathrm{-Mot}_k^\otimes\to \mathrm{MHS}\]
is fully faithful.
\end{thm}
In the case $k={\overline{\Q}}$, this means that the ${\overline{\Q}}$-structure can actually be recovered from the Hodge- and weight filtration -- and conversely. His result also gives fullness of $1\mathrm{-Mot}_k\to \mathcal{MM}_{\mathrm{Nori}}(k)$ for all algebraically closed $k\subset \mathbb{C}$ (actually he has to show this on the way).

\bibliographystyle{alpha}

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Rabbi Paul M. Kurland
Rabbi Kurland has always been involved and deeply committed to Jewish education.
Having received his Masters degree in Jewish Education from the Jewish Theological Seminary he has served as Educational Director in congregations in Pennsylvania and New York. He is also well known for his active involvement in Camp Ramah in the Berkshires for decades.
Rabbi Kurland returned to The Jewish Theological Seminary for Rabbinical training in & made the shift into pulpit work when he took the position of Rabbi at the Croton Jewish Center. He moved to Rockland County in 1997 to serve the Nanuet Hebrew Center, and now serves the new Congregation Shir Shalom, founded July 1, 2021 which is a consolidation of Nanuet Hebrew Center and Montebello Jewish Center.
Nanuet Hebrew Center experienced significant growth during his 24 years as spiritual leader; and we are proud that Rabbi Kurland will continue to serve Congregation Shir Shalom. His involvement with children through his commitment to our Makom Religious School and Youth Programs plus his involvement with all facets of the congregation have been our blessings. We have also been entertained by his creative YouTube videos that help teach prayers to young children (such as http://www.youtube.com/watch?v=ineQdUIjG80)
Rabbi Kurland was asked to take on the presidency of the Lower Hudson Valley Region of the Rabbinical Assembly and assumed those duties in the Summer 2014; and in March 2017, he was honored by the Jewish Theological Seminary with Convocation of an Honorary Doctor of Divinities Degree.
Rabbi Kurland and his wife Gerri have 4 sons, Yoni, Noam, Eytan and Ami.
"Identity Evropa", a neo-Nazi hate group as identified by the ADL, has been putting flyers up in Rockland County and other locations throughout the Hudson Valley, and also posting them on social media.
Rabbi Kurland, along with State Senator David Carlucci and spiritual leaders from different faiths gathered at Rock Apostolic Church in Spring Valley to send the message: There's no room for bigotry in Rockland County.
We invite you to click here for a short video on the meeting, which was aired on the local Channel 12 cable network news, and includes an interview with Rabbi Kurland: http://westchester.news12.com/story/39149357/hate-group-posts-flyers-throughout-hudson-valley
We are pleased to provide Rabbi and Gerri Kurland's Sabbatical Greetings during their recent stay in Israel.
Please click here for their 1st Sabbatical Greetings from Jerusalem: www.nanuethc.org/wp-content/uploads/2018/03/2018-Sabbatical-Greetings-from-Jerusalem-1.pdf
Please click here for their 2nd Sabbatical Greetings from Jerusalem: www.nanuethc.org/wp-content/uploads/2018/04/2018-Sabbatical-Greetings-from-Jerusalem-2.pdf .
Please click here for their 3rd Sabbatical Greetings from Jerusalem: www.nanuethc.org/wp-content/uploads/2018/04/Sabbatical-Greetings-from-Jerusalem-3.pdf
Please click here for their 4th Sabbatical Greetings from Jerusalem: www.nanuethc.org/wp-content/uploads/2018/04/Sabbatical-Greetings-from-Jerusalem-4.pdf
A Selection of Programs Started by Rabbi Kurland for Nanuet Hebrew Center and Congregation Shir Shalom:
Sukkah Decorating Gatherings
Havdallah in the Rabbi's Sukkah
Havdeli & Havdairy Programs
Summer Torah Reading Project
Vacation Havdallah & Pizza
Friday Night Tot Shabbat
Tu B'shevat Seders
Passover Zing, Zest & Pizzazz
Pre-Passover Fast/Feast of the 1st Born
Yom Ha'atzma'ut Parties
Musical Coffee Houses (Pre-Selichot, "Warm up the Winter" & Family Fun)
Weekly Torah Portion Study
"Learn to Lead" Skills Workshops
Bnei Torah Project for Adult beginners
Shabbat Learning (and Lunch & Learns)
Scholar-in Residence Weekends
Involvement in the Greater Community
Rockland Jewish Academy – Rabbinic volunteer
JCC Seniors Group – Speaker
Keep Rockland Beautiful Board Member
Local Singles Groups – Speaker
Hadassah (Bergen County & West Hartford, Ct) – Speaker
Reuben Gittelman Hebrew Day School – Rabbinics Teacher & former Middle School Prayer Coordinator & Supervisor (Thank you RGHDS for everything you meant to the Jewish Community)
RGHDS Committee on Education – former Committee member
Rockland Board of Rabbis – member – former Treasurer, Vice President & President
Rockland Federation – former Board Member
Rockland JCC – former Board Member
Helping Hands – Board Member
Rabbinical Assembly & R.A. Hudson Valley Region – President
Clarkstown Interfaith Clergy Group
Camp Ramah in the Berkshires – staff member since 1970
Conversion Advisor


Discussion in 'General' started by Leainna, Feb 25, 2003.
I am finally at 200 posts! Celebrate!!! Yei!! *dances* So it's not so big... I'm still happy.
I'm now banned from the library for doing the 200 dance! It is still done with the trousers down isnt it?
It is still done with the trousers down isnt it?
I didn't get that memo...have I been doing it wrong all this time?
Man, I've been doing it all wrong. You mean to tell me that you don't do a dance with your pants down for every post you make? Damn, I guess I've made a fool of myself now. No wonder all of my friends have started looking at me funny.
sorry the pants don,t come down till the 1000 post ,lol.
A little late but it is here!
Oh man, can't wait till my 1000! I'm expecting to see all you there!


text:
Aleksandar Vukic was the defending champion but chose not to defend his title.

Max Purcell won the title after defeating James Duckworth 3–6, 7–5, 7–6(7–5) in the final.

Seeds

Draw

Finals

Top half

Bottom half

References

External links
Main draw
Qualifying draw

Bengaluru Open - 1
2023 Singles

redpajama_set_name: RedPajamaWikipedia


The new Mac OS X Panther is ready to pounce, but the story on this cat starts a long time back. With the release of Panther just a few days out, it's a great time to look at the story of NeXT, the company that developed the technologies Mac OS X is largely based upon.
We'll start by looking at how NeXT got started, what they developed, why Apple bought them and then what Apple has done with the technology since the NeXT acquisition.
Small, fresh companies have the luxury of clean drawing boards and are motivated to achieve great things merely to survive. After achieving success, those small nimble efforts more often than not morph into cubicles of corporate stagnation, where "achieving greatness" becomes a threat to sustained profitability.
Apple started out with an outrageous success in the Apple II, a product Apple still credits itself with in every press release for 'igniting the personal computer revolution in the 1970's.' As the Apple II line grew outdated, Apple struggled through a couple failures before again hitting upon success with the Macintosh in 1984. While the Mac continued to develop incrementally, many of those involved in Mac development at Apple moved on to other projects in search of the next new thing.
After cofounder Steve Jobs lost a power struggle within Apple in 1985, he left to pursue other interests as well. For the next two years, Jobs and several former Mac engineers laid out plans for developing the computing environment of the future. Initially called NeXT Inc., the group determined that no existing system was capable of running the tools they envisioned. They changed their name to NeXT Computer, and developed their own advanced hardware that in many ways built upon Apple's Mac. The software running NeXT, however, would be very different.
Rather than attempting to write an entire operating system from scratch, NeXT built upon a academic project called Mach. Since Mach was designed to only handle extremely low level functions, NeXT paired it with a 'user space' or personality of BSD Unix. Above that, NeXT developed a series of object oriented frameworks for developers, along with a graphic user environment that used the same PostScript display language popular in print imaging, ensuring that what was displayed on the screen was exactly what would be printed.
This OS package, called NeXTSTEP, built upon existing work and layered functionality into manageable tasks. Mach supplied the core machine functions and BSD Unix the middle OS functions, such as working with files and networking. This left NeXT to focus on refining their user interface and developing the frameworks to make development of NeXT software fast, easy and beautiful.


Travel News, Travel Stories, Guides and Tips | Find travel news, pictures and video from this blog; Learn about the best destinations, the best ways to get there, the travel business, vacation tips from Travel blog done by Anil Sahoo.
Find travel news, pictures and video from this blog; Learn about the best destinations, the best ways to get there, the travel business, vacation tips from Travel blog done by Anil Sahoo.
Friend it's already April, which means summer is just down the block. Have you made your lovely summer vacation plans yet?
Are you travelling? Do you have a place you go to year after year? Or do you stay home and enjoy what's nearby?
With the economy improving, gas & Petrol prices all over the map, airfares high and some vacation areas still recovering from Sandy, we'd like to know what's in your vacation future.
Please share your thoughts on same. I am waiting for your comments.
This is the world's famous tourist destination Darjeeling is nestled in the Lesser Himalayan range of West Bengal and renowned for tea estates and UNESCO World Heritage Site 'Darjeeling Himalayan Railway'. The town was developed under British Raj as a military depot and a sanatorium.
The pristine land Ladakh is a desert in higher altitude placed between Karakoram mountain ranges in the northwest and Himalayan ranges in the southwest. Indus is like vein that make vital fluid water runs through the desert land and Leh is its main city. Among travellers Ladakh is famous as adventurous destination that provides range of activities to do.
The north eastern state of Arunachal Pradesh is housing several hill stations and Tawang is one of those. The spellbinding hill station Tawang has derived name from Tawang monasteries and sits at an elevation of 3048 meters above sea level.
Set amidst the rare evergreen forests in Satara district of Indian State Maharashtra, Mahabaleshwar occupies the space of low lying Western Ghats and had served as summer capital of Bombay Province during colonial period.
Wow, awesome blog layout! How long have you been blogging for? you make blogging look easy. The overall look of your web site is magnificent, as well as the content!. Thanks For Your article about Travel News, Travel Stories, Guides and Tips | Find travel news, pictures and video from this blog; Learn about the best destinations, the best ways to get there, the travel business, vacation tips from Travel blog done by Anil Sahoo. .
Wow, incredible blog layout! How long have you been blogging for? you made blogging look easy. The overall look of your website is excellent, as well as the content!. Thanks For Your article about Travel News, Travel Stories, Guides and Tips | Find travel news, pictures and video from this blog; Learn about the best destinations, the best ways to get there, the travel business, vacation tips from Travel blog done by Anil Sahoo. .
Wow, wonderful blog layout! How long have you been blogging for? you made blogging look easy. The overall look of your website is magnificent, as well as the content!. Thanks For Your article about Travel News, Travel Stories, Guides and Tips | Find travel news, pictures and video from this blog; Learn about the best destinations, the best ways to get there, the travel business, vacation tips from Travel blog done by Anil Sahoo. .
Wow, incredible blog layout! How long have you been blogging for? you make blogging look easy. The overall look of your web site is excellent, let alone the content!. Thanks For Your article about Travel News, Travel Stories, Guides and Tips | Find travel news, pictures and video from this blog; Learn about the best destinations, the best ways to get there, the travel business, vacation tips from Travel blog done by Anil Sahoo. .
business, vacation tips from Travel blog done by Anil Sahoo.
. I do have some questions for you if you do not mind.
they are written by brain dead people? 😛 And, if you are posting at additional online sites, I'd like to follow you. Would you make a list of the complete urls of your shared sites like your linkedin profile, Facebook page or twitter feed?


We aim to deliver orders to mainland UK addresses within 3-5 working days. For embroidered or printed jobs it will be 10-12 working days.
Please note during the peak ordering months (July - August) delivery times will increase.
We will e-mail you when your order has been received and again on dispatch.
SL Clothing will inform you as soon as possible if we are unable to deliver your order within the suggested time. If you are ordering more than one item, your uniform may be sent to you in instalments if certain items are out of stock.
We will communicate the estimated delivery time for delayed items. At this stage you may cancel your order if you wish. In no event shall we be liable for any damages or penalty for any delay in delivery.
For School Delivery information please look for 'Delivery' in 'School Uniform' section.
For Account customers please log in and refer to the delivery information on your uniform page.
The courier service may speed up the delivery time, however, please note the despatch time will remain the same of 2 to 3 days for undecorated items.
The courier company will attempt delivery and will leave you a card if you are out. After the second failed attempt your uniform order will be returned to us and we will contact you to advise.
We accept payment by Credit Card, Debit Card or cheque. We shall not process any orders until we receive confirmation of payment or cheque clearance.
The prices are exclusive of VAT. Delivery charges are as set out in the checkout process. Delivery is free on orders with a purchase value of £60 and above.


text:
Casino Glitch Frustrates Harrah's Gamblers
A casino glitch at a local slot parlor had some gamblers seeing red on Monday.
They were frustrated when computer troubles at the brand new Harrah's kept them from cashing in, at least right away.
Not only is it freezing outside but some casino customers said they were left out in the cold when their slot machines froze up.
Some people hoping to hit the jack pot had no such luck at Harrah's in Chester, Delaware County.
The glitch happened while the casino was rebooting its system. Customers say about 60-percent of the machines went down while people were playing and that the frozen machines would not allow any pay outs.
"I have been here four hours, can I at least get a meal, you can't even get a meal," said Cecilia Quarles.
Some customers left without their money, while others took their chances on waiting things out.
Dina Parenti stayed two hours for her pay out of USD 20.
"Fights broke out, people were banging the machines, getting really mad," said Parenti.
Parenti took a picture of another customer's slot machine showing the customer was owed USD 242.
"It was a woman sitting next to me, she waited 2 hours and 10 minutes," said Parenti.
Others waited even longer, CBS 3 spoke on the phone to a woman who refused to leave her slot machine for more than four hours.
"If they close, they will have to throw me out, I will fight I want my USD 58 and I will never come back," said Thomasine McLucas.
Harrah's did release a statement saying even though the machines froze; the casino was still able to capture all the activity on them.
They also said that all jackpots will be verified and paid out.


suggested that you print out the Preferred Drug List (PDL) …. Basic Provisions.
medical team determines that you need short-term inpatient.
completing a paper FAFSA, you can only list four colleges in the.
print your confirmation …. page 9 for a list of benefits eligible to be.
Oct 26, 2018 … Beginning in fiscal year 2019, designation of. Federal … List of Sections Affected.
Related Supplies. 8.9 …. Nonpersonal Health Care Services. 37.5.
enrolled providers, and explain how to use your Medicaid benefits in general. ….
Medical Facts …. *For those desiring printed copies, current pricing is available.
DEPARTMENT of HEALTH and HUMAN. SERVICES. Fiscal Year. 2019.
first and the health plan pays second. …. this guide, to request printed.


Experience world-class art for absolutely no cost.
Plan an adrenaline-fueled weekend that won't break the bank.
Appreciate the freedom of Oklahoma's natural landscape.
Kick the thought of saving money by staying home straight to the curb. Adventure doesn't have to come with a hefty price tag; seek it out with an open mind, and you (and your wallet) are sure to be pleasantly surprised.
Now's the time to get creative with those weekend plans. Road trip for a concert in the park, make a list of free museums or gather up handful of must-see street festivals. Throw the budget out the window, you won't need it.
Grab your mat for a $10 Tuesday night yoga class in the fresh air at Myriad Botanical Gardens.
Peruse fine art at Exhibit C and the Red Earth Art Center for absolutely no cost.
Stop in at Bedré Fine Chocolate for a free sample of something sweet.
Get away from the material world for a weekend camping trip in Chickasaw Country.
Rent an electric scooter and zip through Downtown Oklahoma City.
Everything's better when it's free (or nearly).
Need some music for the road? Listen to a Nearly Free Adventure playlist here.


text:
Local Government & Politics
Bobby Scott pushes for more accurate criminal history records
By Dave Ress
House Education and the Workforce Committee Chair Bobby Scott, D-Va., walks through the Hall of Columns at the Capitol as House Democratic chairs gather for a meeting with Majority Leader Steny Hoyer, D-Md., in Washington, Wednesday, March 27, 2019. (AP Photo/J. Scott Applewhite) (J. Scott Applewhite / AP)
What he describes as serious problems with the FBI crime database are prompting Rep. Robert C. "Bobby" Scott, D-Newport News, to introduce legislation to tighten rules for the criminal records potential employers can see.
The bill bars the release of records or information about an arrest if it does not say what the result was — that is, whether the arrested person was convicted, acquitted or if the charge was dropped. It also bars the release of any information that's not clearly about and arrest and its disposition, as well the research of information about such minor offenses as drunkenness, vagrancy, disturbing the peace and traffic violations
It says the subject of such a record must consent to its release, and should have the ability to challenge its accuracy.
"Far too often, many face an additional barrier created by faulty criminal background records released by the FBI," Scott said.
"Local law enforcement agencies routinely report arrest records to the FBI; however, these same groups often fail to report important information on the final disposition of a criminal case. Frequently, mistakes in the records cause qualified job applicants to be denied employment or cause significant delays while the mistakes are corrected," he said. "This bill will ensure that job seekers are treated fairly."
Newport News' Hilton Village is getting a beach


text:
The 1985–86 Saint Joseph's Hawks men's basketball team represented Saint Joseph's University as a member of the Atlantic-10 Conference during the 1985–86 NCAA Division I men's basketball season. Led by 2nd year head coach Jim Boyle, the Hawks finished with an overall record of 26–6 (16–2 in A-10 play). Saint Joseph's won both regular season and A-10 Tournament titles, and received an automatic bid to the NCAA tournament as No. 6 seed in the East Regional. The team defeated Richmond before losing to No. 14 seed Cleveland State in the second round.

Roster

Schedule and results

|-
!colspan=9 style=| Regular season

|-
!colspan=9 style=| A-10 Tournament

|-
!colspan=9 style=| NCAA Tournament

Rankings

References

Saint Joseph's
Saint Joseph's
Saint Joseph's Hawks men's basketball seasons


So, I knew this is the wrong place to make this thanks.
2. Your Documentation is much better then this of Blue.
And So I would only say thank to your all.
I'm very happy with Cabbage.
My documentation might look better than Blue, but I'm not sure it's actually better But then, they are very different systems. I'm happy you got it sorted. It also gave me a chance to look into creating deb package, which I think I will now offer to all users on debian based Linux distros.
I too would be very happy to know that there will be a debian package.
I often use an Xubuntu machine (16.04LTS) , so it would be very nice to get Cabbage working on that PC too without too much hassle with compiling, unresolved dependencies, etc.
And, Yes it works finer with this System to prepare CSound Programms.
The Widgets all what I need.


Gorgeous Surrender bodycon dress, featuring water drop hole at the centre bust with wide halter strap and black transparent lines at front. Cross straps at back, fully lined with invisible zipper at the centre back. Available in Black with 4 sizes including XS, S, M, L.


text:
package com.googlecode.jmapper;

public interface DestinationFactory<D> {

	public D make();
}

redpajama_set_name: RedPajamaGithub


text:
How We Work Toward Conflict-Free
By Suzanne Hilker July 23, 2015
Everyone has a cell phone these days, but have you thought about what goes into making it and where the parts come from?
Making products such as cell phones involves countless people, including workers around the world who mine and process the raw materials. Take tungsten, tin, tantalum and gold (known as 3TG), for example. These minerals are used in myriad products, including consumer electronics, and are mined from deposits found across the globe, with reserves in the Democratic Republic of the Congo and adjoining countries.
Occasionally, these minerals are illegally sourced and traded by armed groups who are responsible for human rights violations. As such, these minerals are known as "conflict minerals."
Enter Hamlin Metzger, Director of Corporate Responsibility & Sustainability at Best Buy. It's his job to help Best Buy be a responsible manufacturer of our private label products. It begins by knowing the ins and outs of our supply chain, from mine to manufacturer to shelf, and ensuring the materials used in our products are responsibly sourced. Smelters — where 3TG ore is processed to remove impurities — are a critical link in the chain. While there are tens of thousands of mines in the world and even more manufacturers, there are just a few hundred smelters. If we can ensure the smelters are sourcing from legitimate mines, then we know our products are "conflict free."
For Best Buy, we work with industry peers to put pressure on the smelters to be audited to ensure responsible sourcing. That's why we are a member of the Conflict Free Sourcing Initiative (CFSI), where we partner with more than 30 companies, from some of our branded vendors to those in other industries.
Hamlin recently traveled with members of CFSI to visit smelters in China. He met with several smelters and discussed the expectations of indirect customers, the importance of conducting due diligence on the source of their raw materials, and how they can be certified as conflict-free through a CFSI audit. In just two years since we have been involved in this work, we've seen phenomenal progress, with a significant increase in the percentage of smelters in our private label supply chain certified as conflict-free. However, we still have a lot of work to do, and it will take the entire industry working together to get the remaining smelters to go through an audit and ensure they have responsible sourcing practices.
It's meaningful work for our investors, our employees, our communities and our customers.