\section{Introduction}\label{sec1}

The main result of this paper is the following arithmetic Riemann-Roch theorem for ${\overline{\Spec\Z}}$ (see Theorem~\ref{rrspzb})

\begin{theorem*} \label{rrspzbintro} 
Let $D$ be an Arakelov divisor  on ${\overline{\Spec\Z}}$. Then 
\begin{equation}\label{rrforq1intro}
\dim_{{\sss[\pm 1]}}H^0(D)-\dim_{{\sss[\pm 1]}}H^1(D)=\bigg\lceil \frac{\deg D+ \log 2}{\log 3}\bigg\rceil'	-\mathbf 1_L.
\end{equation}
\end{theorem*}
\noindent On the (topological) right-hand  side of \eqref{rrforq1intro}, $\lceil x \rceil'$ denotes the odd function on ${\mathbb R}$ that agrees with the ceiling function on positive reals. The set $L\subset {\mathbb R}$ is the union, for $k\geq 0$, of the intervals $(\log \frac{3^k}{2},\log \frac{3^k+1}{2})$:  $L$ has finite Lebesgue measure. The symbol $\mathbf 1_L$ denotes the (degree) indicator function:  this is zero unless   $\deg D\in L$ in which case it is  $1$.   The numbers $\dim_{{\sss[\pm 1]}}H^i(D)$ on the (cohomological) left-hand  side   are integers providing the dimension of the cohomology of  $D$, as a module over the spherical extension ${\sss[\pm 1]}$ of the absolute base ${\S}$   (see section~\ref{sec2}). Before to introduce the definitions of the cohomology and their dimension,   we comment on the new geometric perspective  introduced by Theorem~\ref{rrspzbintro}  in comparison with the  asymptotic  Riemann-Roch formula of S. Lang \cite{Lang} (foreseen by A. Weil in \cite{Weil39}) comparing $\log \# H^0(D)$ and  $\deg D+ \log 2$, when $\deg D\to \infty$.  The first novelty  is that unlike $\log \# H^0(D)$,   $\dim_{{\sss[\pm 1]}} H^i(D)$  are non-negative integers. Secondly, the cohomological side of \eqref{rrforq1intro}  implements explicitly (the dimension of) $H^1(D)$, thus it goes  well beyond the arithmetic Riemann-Roch formula for ${\mathbb Q}$,  where $H^1(D)$ is either undefined or set to equate $H^0(K-D)$, for an expected Serre duality. All the more:   \eqref{rrforq1intro} is an exact formula and not an asymptotic statement. All these advancements  derive from the new understanding of  ${\overline{\Spec\Z}}$  as a curve over the absolute ring ${\sss[\pm 1]}$.  The third relevant fact  is that on the topological side of the Riemann-Roch formula  the traditional sum  $\deg D+ \log 2$ is now divided by $\log 3\sim 1.09861$, and there also appears an exceptional set $L\subset {\mathbb R}$  of finite Lebesgue measure $\vert L\vert=\log \prod (1+3^{-n})\sim 1.14099$. The Euler characteristic of $D$ is  a non-decreasing function of  $\deg D$, while the topological side in \eqref{rrforq1intro} drops by $1$ if $\deg D\in L$. Except for the  set $L$, this topological side  is invariant under Serre's  duality (Theorem~\ref{serredualintro}) which replaces $D$ with $K-D$ ($\deg K=-2\log 2$), since the function $\lceil x\rceil'$ is odd.

 In \cite{CCprel, schemeF1} we have developed the fundamentals of a geometric theory for Segal's $\Gamma$-rings : this means that the initial ring ${\mathbb Z}$ of the category of rings is replaced by the sphere spectrum  ${\S}$. In particular, we have defined the structure sheaf of the one-point  compactification ${\overline{\Spec\Z}}$ of the algebraic spectrum of the integers   by suitably extending, at the archimedean place, the structure sheaf of ${{\rm Spec\,}{\mathbb Z}}$ as a subsheaf of the constant sheaf ${\mathbb Q}$. The global sections of this sheaf determine the ${\S}$-algebra  ${\sss[\pm 1]}$. In  \cite{CCgromov} we extended the notion of homology for simplicial complexes, when group coefficients are replaced by ${\S}$-modules. In \cite{CCAtiyah} we implemented homological algebra in this context by showing a generalization of the Dold-Kan correspondence which associates a $\Gamma$-space to a short complex of ${\S}$-modules. In this paper we apply all these results  to adapt, for ${\overline{\Spec\Z}}$,  the  adelic proof  of the Riemann-Roch formula for function fields given by Weil in \cite{Weil-basic}. 
 
 Next, we introduce  the cohomologies and their dimensions.
  An Arakelov divisor $D=\sum_j a_j\{p_j\} + a\{\infty\}$ on ${\overline{\Spec\Z}}$ determines a compact subset ${\mathcal O}(D)\subset {\mathbb A}_{\mathbb Q}$ of the adeles of ${\mathbb Q}$. This is the product, indexed by the places of ${\mathbb Q}$, of the abelian group ${\mathbb Z}_p\subset {\mathbb Q}_p$ for each finite  $p\notin \{p_j\}_j$, the  abelian groups  $p_j^{-a_j}{\mathbb Z}_{p_j}$, and the interval $[-e^a,e^a]\subset {\mathbb R}$. This last component is implemented by a sub-module of the Eilenberg-MacLane ${\sss[\pm 1]}$-module $H{\mathbb R}$  which functorially encodes the additive structure of the group ${\mathbb R}$. The morphisms of abelian groups used by Weil in the development of the adelic geometry of function fields are meaningful in this absolute set-up since they are viewed as  morphisms of ${\sss[\pm 1]}$-modules. In particular, we shall focus on the morphism
 \begin{equation}\label{psiintro}
 \psi: {\mathbb Q}\times {\mathcal O}(D) \to {\mathbb A}_{\mathbb Q}\quad \psi(q,a) = q+ a\quad \forall q\in {\mathbb Q},~a\in {\mathcal O}(D),
 \end{equation}
 where ${\mathbb Q}$ is embedded diagonally  in the adeles. The $\Gamma$-space ${\bf H}(D)$   associated by the Dold-Kan correspondence to \eqref{psiintro} ($\psi$ is viewed here as a short complex of ${\sss[\pm 1]}$-modules: see Appendix~\ref{secA}) provides the absolute incarnation of the Riemann-Roch problem for the divisor $D$. In fact, by implementing the multiplicative action of ${\mathbb Q}^\times$  on ${\mathbb A}_{\mathbb Q}$, one is reduced to consider only the case   $D=a\{\infty\}$, for which  $\ker(\psi)$ is   the ${\sss[\pm 1]}$-module $H^0(D)=\| H{\mathbb Z}\|_{e^a}$ that associates to a finite pointed set $X$ the ${\mathbb Z}$-valued  divisors $\sum_i n_ix_i$, $x_i\in X$, which fulfill the condition  $\sum_i \vert n_i\vert\leq e^a$ ($\vert\cdot\vert=$  euclidean absolute value).  By definition, $H^0(D)$ is a covariant functor $\Gamma^{o}\longrightarrow \frak{ Sets}_*$ from the small category $\Gamma^{o}$ (a skeleton of the category of finite pointed sets) to pointed sets  that keeps track of the partially defined addition in $I=\| H{\mathbb Z}\|_{e^a}(1_+)=[-e^a,e^a]\cap {\mathbb Z}$, so that for $n_i\in I$, $\sum n_i$ is meaningful provided $\sum_i \vert n_i\vert \leq e^a$.
Its dimension $\dim_{{\sss[\pm 1]}}H^0(D)$ is  defined to be the smallest number of generators of $I$, where by definition, a subset $F\subset I$ generates if and only if for every $m\in I$ there exist coefficients $\alpha(f)\in \{-1,0,1\}={\sss[\pm 1]}(1_+)$ such that $m=\sum \alpha(f)f$, $f\in F$, and $\sum \vert\alpha(f)f\vert\leq e^a$ (see section~\ref{sec4}). The ${\sss[\pm 1]}$-module $H^0(D)$  only depends upon the integer part $n=\lfloor e^a\rfloor$ of $e^a$ and Proposition \ref{dimhzn} determines its dimension to be\footnote{For the  values $n=4,13,40,\ldots, (3^k-1)/2$, every element of $I$ is uniquely written in terms of the generating set $F=\{3^i\mid 0\leq i<k\}$ (Proposition \ref{dimhzn} $(iii)$).}
\begin{equation}\label{dimhznintro}
\dim_{{\sss[\pm 1]}}(\| H{\mathbb Z}\|_n)=\bigg\lceil \frac{\log(2n+1)}{\log 3}\bigg\rceil.
\end{equation}
The cokernel of $\psi$, that is $H^1(D)$, is  the couple of the ${\sss[\pm 1]}$-module $H{\mathbb A}_{\mathbb Q}$ and a suitable tolerance relation ${\mathcal R}$ on it (Proposition~\ref{setup1}). This relation is defined by pairs whose difference belongs to the image of $\psi$. More specifically, $H^1(D)$ defines an object of the category $\Gamma{\mathcal T}_*$ of tolerance ${\S}$-modules (see section~\ref{sec3}). Its dimension $\dim_{{\sss[\pm 1]}}H^1(D)$ is defined to be the smallest number of generators in a generating set, where a subset $F\subset H{\mathbb A}_{\mathbb Q}(1_+)={\mathbb A}_{\mathbb Q}$ generates if and only if for every $x\in H{\mathbb A}_{\mathbb Q}(1_+)$ there exists coefficients $\alpha(f)\in \{-1,0,1\}$ such that the pair $(x,\sum \alpha(f)f)$ belongs to the relation ${\mathcal R}$ (see section~\ref{sec5}). For a divisor  $D=a\{\infty\}$, the dimension of $H^1(D)$ is the same as the dimension of the pair $(H({\mathbb R}/{\mathbb Z}),{\mathcal R})$, where the relation ${\mathcal R}$ on $H({\mathbb R}/{\mathbb Z})(1_+)={\mathbb R}/{\mathbb Z}$ is given by
 \begin{equation}\label{tolerrelat}
(x,y)\in {\mathcal R} \iff d(x,y)\leq e^a,
\end{equation}
 for $d$ the translation invariant Riemannian metric of length $1$ on ${\mathbb R}/{\mathbb Z}$.  In Proposition \ref{caseh1} we show that this dimension is 
 \begin{equation}\label{dimh1intro}
\dim_{{\sss[\pm 1]}}(H({\mathbb R}/{\mathbb Z}),{\mathcal R})=\bigg\lceil \frac{-a- \log 2}{\log 3}\bigg\rceil.	
\end{equation}
Theorem~\ref{rrspzbintro}    follows from  \eqref{dimhznintro} when  $\deg D \geq -\log 2$, since in this case   $\dim_{{\sss[\pm 1]}}H^1(D)=0$. The case $\deg D<-\log 2$ is proved using \eqref{dimh1intro}.

The symmetry of the term  $\lceil \frac{\deg D+ \log 2}{\log 3}\rceil'$ under the replacement of  $D$ by the divisor $K-D$, with $K=-2 \{2\}$, is the numerical evidence of a geometric duality holding on the curve ${\overline{\Spec\Z}}$ over ${\sss[\pm 1]}$ (see section~\ref{sec6}) which is  precisely stated by the following  (see Theorem~\ref{serredual})
\begin{theorem*}\label{serredualintro}
Let $D$ be an Arakelov divisor  on ${\overline{\Spec\Z}}$ and $K=-2 \{2\}$, with $\deg K=-2 \log 2$. Then there is an isomorphism of ${\sss[\pm 1]}$-modules
\begin{equation}\label{dualdual}
H^0(K-D)\simeq {\underline \Hom}_{\Gamma{\mathcal T}_*}(H^1(D),U(1)_{\frac 14}),
\end{equation}
where $U(1)_{\frac 14}$ coincides with $H^1(K)$,  playing the role of the dualizing module in Pontryagin duality.
\end{theorem*}
\noindent The isomorphism $H^0(K-D)\simeq {\underline \Hom}_{\Gamma{\mathcal T}_*}(H^1(D),H^1(K))$ is the analogue of Grothendieck's version of Serre's duality \cite{GS} (12.15.1955).
The duality  \eqref{dualdual} derives from   Pontryagin's duality holding for tolerant ${\sss[\pm 1]}$-modules  (Proposition~\ref{pontrj}), in analogy with   the corresponding result proven by Weil for curves over  fields of positive characteristic. 


\section{The  cohomology $H^\cdot(D)$}\label{sec2}

We recall from \cite{CCAtiyah} the construction of the $\Gamma$-space ${\bf H}(D)$   naturally associated to an Arakelov divisor $D$ on ${\overline{\Spec\Z}}$. This space  is the  absolute homological incarnation of the Riemann-Roch problem for the divisor $D$. We formulate this construction  in terms of the adeles of ${\mathbb Q}$.

Let $D=\sum_j a_j\{p_j\} + a\{\infty\}$ be an Arakelov divisor on ${\overline{\Spec\Z}}$. This is a formal finite sum with $a_j\in{\mathbb Z}$ and $a\in{\mathbb R}$, where $p_j$ are rational primes in ${\mathbb Z}$ and the symbol $\infty$ stands for the restriction $\vert\cdot\vert: {\mathbb Q}\stackrel{}{\to} {\mathbb R}$ of the  euclidean absolute value $\vert\cdot\vert_\infty$. Let  $\Sigma_{\mathbb Q}$ denote the full set of places $\nu$ of ${\mathbb Q}$.   To  $D$ is naturally associated the following idele $\exp(D)$ of ${\mathbb Q}$ 
\[
\exp(D)_\nu:=\begin{cases}p_j^{-a_j}& \text{if $\nu=\nu_j$}\\
1& \forall \nu\neq \nu_j,\, \forall j\\
e^a& \text{if $\nu=\infty$}.	
\end{cases}
\] 
The equality $\exp(D)(\nu)=\vert \exp(D)_\nu\vert_\nu$ 
 defines a map $\exp(D):\Sigma_{\mathbb Q}\to {\mathbb R}_+^*$ such that $\exp(D)(\nu)\in  {\rm mod}({\mathbb Q}_\nu)$, $\forall \nu \in \Sigma_{\mathbb Q}$, and $\exp(D)(\nu)=1$,  for   $ \nu\neq \infty,  \nu_j, \forall j$.  To the divisor $D$ corresponds the compact subset  of the adeles of ${\mathbb Q}$
\[
{\mathcal O}(D):=\big\{(a_\nu)\in{\mathbb A}_{\mathbb Q}\mid\vert a_\nu\vert_\nu\le \exp(D)(\nu), \forall \nu\in\Sigma_{\mathbb Q}\big\}\subset {\mathbb A}_{\mathbb Q}.
\]
By definition  ${\mathcal O}(D)={\mathcal O}(D)_f \times {\mathcal O}(D)_\infty=\prod_\nu {\mathcal O}(D)_\nu$ with
\[
{\mathcal O}(D)_\nu=\begin{cases}p_j^{-a_j}{\mathbb Z}_{p_j}& \text{if $\nu=\nu_j$}\\
{\mathbb Z}_\nu & \forall \nu\neq \nu_j,\, \forall j,~ \nu<\infty\\
[-e^a, e^a]& \text{if $\nu=\infty$}.
\end{cases}
\]
In particular, the product ${\mathcal O}(D)_f=\prod_{\nu\neq \infty} {\mathcal O}(D)_\nu$ is a compact abelian group. The archimedean component on the other hand, {\it i.e.\/}\  the real interval $[-e^a, e^a]$, is not, and for this reason we implement the functorial viewpoint to suitably understand ${\mathcal O}(D)$.  Indeed, to ${\mathcal O}(D)_\infty$  we associate the  covariant functor 
\[
\| H{\mathbb R}\|_{e^a}: \Gamma^{o}\longrightarrow \frak{ Sets}_*\qquad \| H{\mathbb R}\|_{e^a}(F):=\bigg\{\phi\in H{\mathbb R}(F)\mid \sum_{F\setminus \{*\}} \vert\phi(x)\vert\leq e^a\bigg\}
\]
from the opposite of the Segal category (see {\it e.g.\/}\  \cite{DGM} Chpt. 2 and \cite{CCprel})  to pointed sets, which maps a finite pointed set $F$ to the pointed set of ${\mathbb R}$-valued divisors on $F$ vanishing on the base point and whose total mass $\sum_{F\setminus \{*\}} \vert\phi(x)\vert$ is bounded by $e^a$. Covariant functors $\Gamma^{o}\longrightarrow \frak{ Sets}_*$ and their natural transformations determine the category $\Gamma{\Se_*}$ of $\Gamma$-sets (aka ${\S}$-modules). In particular, the Eilenberg-MacLane functor $H$  encodes an abelian group $A$ as the covariant functor $HA:\Gamma^{o}\longrightarrow \frak{ Sets}_*$  that associates to  $F$ the pointed set of $A$-valued divisors on $F$ vanishing on the base point. Functoriality holds by taking  sums  on the inverse images of a point. The functor $HA$ encodes the addition on $A=HA(1_+)$. In general, let $\sigma\in {\rm{Hom}}_{\Gamma^{o}}(k_+,1_+)$ with $\sigma(\ell)=1$ $\forall \ell\neq *$ and
\begin{equation}\label{partmaps}
\delta(j,k)\in {\rm{Hom}}_{\Gamma^{o}}(k_+,1_+), \quad \delta(j,k)(\ell):=\begin{cases} 1 &\text{if $\ell=j$}\\ * &\text{if $\ell\neq j$}.
\end{cases}
\end{equation} 
Given an ${\S}$-module ${\mathcal F}$ and  elements $x,x_j\in {\mathcal F}(1_+)$, $j=1,\ldots, k$, one writes 
\begin{equation}\label{defnsum}
x=\sum_j x_j\iff \exists z	\in {\mathcal F}(k_+)~\text{s.t.}~  {\mathcal F}(\sigma)(z)=x,~ {\mathcal F}(\delta(j,k))(z)=x_j,\ \forall j.
\end{equation}
  For $A={\mathbb R}$ the key point is that this general construction respects the bound on the total mass while encoding the addition. One easily sees that $\| H{\mathbb R}\|_{e^a}$ is an ${\sss[\pm 1]}$-module (an ``(${\sss[\pm 1]}\wedge -)$-algebra'' as in  \cite{DGM} 2.1.5), where  ${\sss[\pm 1]}:  \Gamma^{o}\longrightarrow \frak{ Sets}_*$, ${\sss[\pm 1]}(F):=\{-1,0,1\}\wedge F$ is the spherical monoid algebra of the multiplicative monoid $\{\pm 1\}$. In particular, the inclusion functor $\| H{\mathbb R}\|_{e^a}\longrightarrow H{\mathbb R}$ determines $\| H{\mathbb R}\|_{e^a}$ as  a sub ${\sss[\pm 1]}$-module of $H{\mathbb R}$, where ${\S}: \Gamma^{o}\longrightarrow \frak{ Sets}_*$ is the identity functor. 

Given two ${\S}$-modules ${\mathcal F}_j$, $j=1,2$, one defines their  product as the functor
\[
{\mathcal F}_1 \times {\mathcal F}_2:\Gamma^{o}\longrightarrow \frak{ Sets}_*, \qquad ({\mathcal F}_1 \times {\mathcal F}_2)(F)={\mathcal F}_1(F)\times {\mathcal F}_2(F)
\]
 with the base point of ${\mathcal F}_1(F)\times {\mathcal F}_2(F)$ taken to be $(*,*)$. For abelian groups $A,B$ one has a canonical isomorphism $H(A\times B)\simeq HA\times HB$. In particular, 
 the morphism of  addition in adeles $\alpha:{\mathbb Q}\times {\mathbb A}_{\mathbb Q} \to {\mathbb A}_{\mathbb Q}$, $\alpha(q, a) = q+ a$     (using the diagonal embedding of ${\mathbb Q}$ in ${\mathbb A}_{\mathbb Q}$) determines a morphism of ${\sss[\pm 1]}$-modules $H\alpha: H{\mathbb Q}\times H{\mathbb A}_{\mathbb Q} \longrightarrow H{\mathbb A}_{\mathbb Q}$. Next proposition shows how this functorial construction is well combined with the adelic formalism.
 
\begin{proposition} \label{setup}
\begin{enumerate}
\item[(i)] The ${\sss[\pm 1]}$-module 
\[
H{\mathcal O}(D):=H({\mathcal O}(D)_f)\times \| H{\mathbb R}\|_{e^a}
\]
determines, canonically, a sub ${\sss[\pm 1]}$-module $\iota:H{\mathcal O}(D)\longrightarrow H{\mathbb A}_{\mathbb Q}$.
\item[(ii)] The restriction  of $H\alpha$ to $H{\mathbb Q}\times H{\mathcal O}(D)$ defines a morphism of ${\sss[\pm 1]}$-modules \begin{equation}\label{psi}
 \psi: H{\mathbb Q}\times H{\mathcal O}(D) \longrightarrow H{\mathbb A}_{\mathbb Q}.
 \end{equation} 	
\end{enumerate}
\end{proposition}
\proof $(i)$~With ${\mathbb A}_{\mathbb Q}={\mathbb A}_f\times {\mathbb R}$, let $\iota_f:{\mathcal O}(D)_f\to {\mathbb A}_f$ be the inclusion of abelian groups.  Then  $\iota:H{\mathcal O}(D)\to H{\mathbb A}_{\mathbb Q}$ is the product of $H\iota_f$ and the inclusion $\| H{\mathbb R}\|_{e^a}\to H{\mathbb R}$.\newline
 $(ii)$~By composition one obtains $\psi= H\alpha \circ (H{\rm id}_{\mathbb Q} \times \iota)$. \endproof 
 
 The morphism $\psi$ in \eqref{psi} is obtained by restricting  $H\alpha$  to the (adelic) divisor $D$, where $\alpha$ is a morphism of abelian groups.   The theory developed in \cite{CCAtiyah}  associates to   $\psi$ a $\Gamma$-space ${\bf H}(D)$, by implementing the Dold-Kan correspondence in the special case of a short complex of abelian groups and its restriction to a sub-${\S}$-module. The $\Gamma$-space ${\bf H}(D)$  supplies the absolute cohomology of the divisor $D$. We recall this construction in Appendix \ref{secA}. Moreover, we review how the generalized homotopy $\pi^{\cT}_\ast({\bf H}(D))$, where ${\mathcal T}$ is the category of tolerance relations defined in \ref{gamma-t},  is meaningful in this context supplying, with $\pi^{\cT}_1({\bf H}(D))$   and $\pi^{\cT}_0({\bf H}(D))$,  the (tolerant) ${\S}$-modules descriptions  of {\it resp.\/}\   the  kernel and the cokernel  of $\psi$. In other words, one obtains the following result

\begin{proposition} \label{setup1}
\begin{enumerate}
\item[(i)] The kernel  of $\psi$ is the  ${\sss[\pm 1]}$-module:
\begin{equation}\label{kerpsi}
H^0(D) =(H{\rm id}_{\mathbb Q} \times \iota)^{-1}( H\ker( \alpha))\simeq \iota^{-1} H{\mathbb Q} =\Vert H^0({\Spec\Z} ,{\mathcal O}(D)_f)\Vert_{e^a}. 
\end{equation}
\item[(ii)] The cokernel of $\psi$ is the  ${\sss[\pm 1]}$-module $H^1(D) = (H{\mathbb A}_{\mathbb Q}, {\mathcal R})$  endowed with the relations
\begin{equation}\label{cokerpsi}
 {\mathcal R}_k:=\{ (x,y)\in H{\mathbb A}_{\mathbb Q}(k_+)\times  H{\mathbb A}_{\mathbb Q}(k_+)\mid x-y\in {\rm Image}\, \psi(k_+)\}.
\end{equation}
\item[(iii)] Both $H^0(D)$ and $H^1(D)$ only depend, up to isomorphism, on the linear equivalence class of the divisor $D$.
\end{enumerate}
\end{proposition}
\proof We refer to Appendix \ref{sectproofprop}.
\endproof  


\section{Tolerance ${\S}$-modules}\label{sec3}

 The construction of the category $\Gamma{\mathcal S}_*$ of $\Gamma$-spaces (see Appendix~\ref{secA}) can be broadly generalized by considering in place of  the category ${\mathcal S}_*$ of simplicial pointed sets any  pointed category ${\mathcal C}$ with initial and final object $*$. In this way, one obtains a category $\Gamma{\mathcal C}$ of pointed covariant functor ${\Gamma^{o}} \longrightarrow {\mathcal C}$. We shall apply this formal construction to  the category of tolerance relations and introduce the notion of tolerant ${\sss[\pm 1]}$-modules which plays a central role in the development  of the absolute Riemann-Roch problem.
We start with the following general fact
 
 \begin{lemma}\label{catcat} Let ${\mathcal C}$ be a pointed category with initial and final object $*$. Then $\Gamma{\mathcal C}$ is naturally enriched in ${\S}$-modules. More  precisely, the following formula endows the internal ${\underline \Hom}_{\Gamma{\mathcal C}}(A,B)$ with a structure of ${\S}$-module defined by
 \begin{equation}	\label{catcat1}
 	{\underline \Hom}_{\Gamma{\mathcal C}}(A,B)(k_+):={\rm{Hom}}_{\Gamma{\mathcal C}}(A,B(k_+\wedge -))\qquad k\in{\mathbb N}.
 \end{equation}
 \end{lemma}
\proof Let $\phi\in {\rm{Hom}}_{\Gamma^{o}}(k_+,\ell_+)$. For every object $F$ of ${\Gamma^{o}}$ the morphism $\phi \wedge {\rm id}\in {\rm{Hom}}_{\Gamma^{o}}(k_+\wedge F,\ell_+\wedge F)$ gives, by functoriality of $B:{\Gamma^{o}} \longrightarrow{\mathcal C}$, a morphism $B(\phi \wedge {\rm id})\in {\rm{Hom}}_{\mathcal C}(B(k_+\wedge F),B(\ell_+\wedge F))$. These morphisms define a natural transformation of functors $B(\phi\wedge -)\in {\rm{Hom}}_{\Gamma{\mathcal C}}(B(k_+\wedge -), B(\ell_+\wedge -))$, and one obtains the functoriality on the right hand side of \eqref{catcat1} using the left composition 
\[
\psi\in {\rm{Hom}}_{\Gamma{\mathcal C}}(A,B(k_+\wedge -))\mapsto B(\phi\wedge -)\circ \psi \in 
{\rm{Hom}}_{\Gamma{\mathcal C}}(A,B(\ell_+\wedge -)).
\]
 \endproof 
 
  \subsection{The category $\Gamma{\mathcal T}_*$}\label{gamma-t}

 A tolerance relation ${\mathcal R}$ on a set $X$ is a reflexive and symmetric relation on  $X$.  Equivalently, ${\mathcal R}$ is a subset ${\mathcal R}\subset X\times X$ which is symmetric and containing the diagonal. We shall denote by ${\mathcal T}$ the category of tolerance relations $(X,{\mathcal R})$. Morphisms in  ${\mathcal T}$  are defined by \[ 
{\rm{Hom}}_{\mathcal T}((X,{\mathcal R}),(X',{\mathcal R}')):=\{	\phi: X \to X', \  \phi({\mathcal R})\subset {\mathcal R}'\}.
\]
We denote  ${\mathcal T}_*$ the pointed category under the object $\{\ast\}$ endowed with the trivial relation. One has the following

\begin{definition} \label{cT} A tolerant ${\S}$-module is  a pointed covariant functor ${\Gamma^{o}} \longrightarrow {\mathcal T}_*$.
We denote by $\Gamma{\mathcal T}_*$ the category of tolerant ${\S}$-modules.
\end{definition}

Next statement is an easy but useful fact

\begin{proposition}\label{adjoint} 
\begin{enumerate}
\item[(i)] The functor $\frak{ Sets}\longrightarrow {\mathcal T}$ which endows a set with the diagonal relation, embeds the category of sets as a full subcategory of ${\mathcal T}$, and consequently the category of ${\S}$-modules  as a full subcategory of the category $\Gamma{\mathcal T}_*$.
\item[(ii)] The forgetful functor is the right adjoint of the inclusion in $(i)$.	
\end{enumerate}
\end{proposition}

\subsection{The tolerant ${\S}$-module $(A,d)_\lambda$}

 A relevant example of tolerant ${\S}$-module is given by the following construction. Let $A$ be an additive abelian group. A translation invariant metric $d$ on $A$ is a metric on $A$ that satisfies $d(x,y)=d(x-y,0)$, so that the triangle inequality can be  read as $d(x+y,0)\leq d(x,0)+d(y,0)$ $\forall x,y \in A$. This fact implies that the inequality 
\begin{equation}\label{inequ}
d\bigg(\sum_I x_i,\sum_I y_i\bigg)\leq \sum_I d(x_i,y_i)
\end{equation}
holds for any finite index set $I$ and maps $x, y:I\to A$.

\begin{proposition}\label{defnA} Let $(A,d)$ be an abelian group endowed with a translation invariant metric $d$ and let $\lambda\in {\mathbb R}_{>0}$. The following relations   turn the Eilenberg?MacLane ${\S}$-module $HA$ into a tolerant ${\sss[\pm 1]}$-module $(A,d)_\lambda: {\Gamma^{o}}\longrightarrow {\mathcal T}_*$:
\begin{equation}\label{inequdef} \mathcal R_k= \{((x_i),(y_i))\in A^{\times k}\times A^{\times k}\mid \sum_{I=k_+} d(x_i,y_i)\leq \lambda \}\qquad \forall k_+,~k\in{\mathbb N}.
\end{equation}
\end{proposition}
\proof For any $\phi\in {\rm{Hom}}_{\Gamma^{o}}(k_+,\ell_+)$, the map $HA(\phi):HA(k_+)\to HA(\ell_+)$ fulfills $HA(\phi)^{\times 2}({\mathcal R}_k)\subset {\mathcal R}_\ell$. Indeed, this follows from \eqref{inequ} applied to the finite sets $I_j=\{i\in k_+\mid \phi(i)=j\}$ which label pairs of elements of $A$. \endproof 

 Let  $U(1)$ be the abelian group ${\mathbb R}/{\mathbb Z}$ endowed with the canonical metric $d$ of length $1$. We shall denote by $U(1)_\lambda$ the tolerant ${\S}$-module $(U(1),d)_\lambda$,  ($\lambda\in{\mathbb R}_{>0}$).

 
 \subsection{The dimension of a tolerant ${\sss[\pm 1]}$-module}
 
 In this part we  introduce a notion of dimension for a tolerant ${\sss[\pm 1]}$-module that naturally generalizes, in the absolute context, the  definition of dimension of a vector space. 
 
\begin{definition}\label{generator}
 Let $(E,{\mathcal R})$ be a tolerant ${\sss[\pm 1]}$-module. 
	A subset $F\subset E(1_+)$ generates $E(1_+)$ if the following two conditions hold
	\begin{enumerate}
	\item For $x,y\in F$, with $x\neq y ~\Longrightarrow~ (x,y)\notin {\mathcal R}$	
	\item For every   $x\in E(1_+)$ there exists  $\alpha_j\in \{-1,0,1\}$,  $j\in F$ and $y\in E(1_+)$ such that $y=\sum_F \alpha_j j\in E(1_+)$ in the sense of \eqref{defnsum},  and $(x,y)\in {\mathcal R}$.
	\end{enumerate}
  The dimension $\dim_{{\sss[\pm 1]}}(E,{\mathcal R})$ is defined as the minimal cardinality of a generating set $F$.
	\end{definition}
 
 
 To familiarize with this notion we  prove the following 
 
 \begin{proposition}\label{pullback} Let $p:A\to B$ be a  morphism of abelian groups and $(HB,{\mathcal R})$ a tolerant ${\sss[\pm 1]}$-module.
 \begin{enumerate}
\item[(i)] Consider the relation  $p^*({\mathcal R}_k):=\{(x,y)\in HA(k_+)\times HA(k_+)\mid (Hp(x),Hp(y))\in {\mathcal R}_k\}$. Then the pair $p^*(HB,{\mathcal R}):=(HA,p^*({\mathcal R}))$ is a tolerant ${\sss[\pm 1]}$ module.
\item[(ii)] If $p$ is surjective: $\dim_{{\sss[\pm 1]}}(p^*(HB,{\mathcal R}))=\dim_{{\sss[\pm 1]}}(HB,{\mathcal R})$.
\end{enumerate}
 \end{proposition}
 \proof $(i)$~Since $(HB,{\mathcal R})$ is a tolerant ${\sss[\pm 1]}$-module, for any $\phi\in {\rm{Hom}}_{\Gamma^{o}}(k_+,\ell_+)$ one has $HB(\phi)^{\times 2}({\mathcal R}_k)\subset {\mathcal R}_\ell$.  Also  
 \[
 HA(\phi)^{\times 2}(p^*{\mathcal R}_k)\subset p^*{\mathcal R}_\ell \qquad\forall \phi\in {\rm{Hom}}_{\Gamma^{o}}(k_+,\ell_+),
 \]
 which shows that $p^*(HB,{\mathcal R})=(HA,p^*({\mathcal R}))$ is a tolerant ${\sss[\pm 1]}$ module. \newline 
 $(ii)$~Let $F\subset HB(1_+)=B$ be a generating set for $(HB,{\mathcal R})$, and let $F'\subset HA(1_+)=A$ be a lift of $F$, with $\# F'=\# F$. Let us show that $F'$ is a generating set for $p^*(HB,{\mathcal R})=(HA,p^*({\mathcal R}))$. Let $a\in HA(1_+)=A$, then there exists coefficients $\alpha_j\in \{-1,0,1\}$,  $j\in F$, such that  $(p(a),\sum_F \alpha_j j)\in {\mathcal R}$. It follows that using the lifts $j'\in F'$ of $j\in F$ one has $(a,\sum_{F'} \alpha_j j')\in p^*({\mathcal R})$, hence $F'$ is a generating set for $p^*(HB,{\mathcal R})$. Thus $\dim_{{\sss[\pm 1]}}(p^*(HB,{\mathcal R}))\leq \dim_{{\sss[\pm 1]}}(HB,{\mathcal R})$. Conversely, let $F'\subset A$ be a generating set for $p^*(HB,{\mathcal R})$ and $F:=p(F')$. Then condition 1. of Definition \ref{generator} for $F'$ implies the same condition for $F$, thus one has $\# F'=\# F$. Let  $b\in HB(1_+)=B$ and $a\in A$ with $p(a)=b$. Then there exists coefficients $\alpha_j\in \{-1,0,1\}$,  $j\in F'$, such that  $(a,\sum_{F'} \alpha_j j')\in p^*({\mathcal R})$. This implies
 $(b,\sum \alpha_j j)\in {\mathcal R}$ so that $F$ is a generating set for $(HB,{\mathcal R})$. \endproof  
 
 Next, we apply this functorial machinery to the geometry of ${\overline{\Spec\Z}}$. We retain the notations of section~\ref{sec2}.
 
 \begin{proposition}\label{thedimh1} Let $D=\sum_j a_j\{p_j\} + a\{\infty\}$ be an Arakelov divisor on ${\overline{\Spec\Z}}$ and let $\pi:{\mathbb A}_{\mathbb Q}\to {\mathbb R}/L$ be the projection of the  adeles on their archimedean component modulo the lattice
 \begin{equation}\label{weil6}
 L= H^0({\rm Spec\,}{\mathbb Z},{\mathcal O}(D)_f):=\{q \in {\mathbb Q}\mid \vert q\vert_\nu \leq \exp(D)(\nu),~\forall \nu\neq \infty \}.
\end{equation}
\begin{enumerate}
\item[(i)] Let $d$ be the metric on ${\mathbb R}/L$ induced by the standard metric on ${\mathbb R}$ and set $\lambda=\exp a$. Then one has $H^1(D)=\pi^*(({\mathbb R}/L,d)_\lambda)$. 
\item[(ii)] 
 	$\dim_{{\sss[\pm 1]}}H^1(D)=\dim_{{\sss[\pm 1]}}({\mathbb R}/L,d)_\lambda$. 	
 	\end{enumerate}
 \end{proposition}
\proof 
$(i)$~Let $j: {\mathbb A}_{\mathbb Q}^f\to {\mathbb A}_{\mathbb Q}$, $j(x):= (x,0)$, be the embedding of  finite adeles in adeles. Using the ultrametric property of  the local norms at  the finite places one sees that $j({\mathcal O}(D)_f)={\mathcal O}(D)\cap j({\mathbb A}_{\mathbb Q}^f)\subset {\mathbb A}_{\mathbb Q}$ is a compact subgroup. Set $G={\mathbb Q}\times j({\mathcal O}(D)_f)$: one has $j({\mathcal O}(D)_f)\cap{\mathbb Q}=\{0\}$ since all the adeles in $j({\mathcal O}(D)_f)$ have archimedean component equal to $0$. Thus, the  restriction  of the morphism of  addition,  $\alpha:{\mathbb Q}\times {\mathbb A}_{\mathbb Q} \to {\mathbb A}_{\mathbb Q}$, $\alpha(q,a)=q+a$,  to $G$   determines an isomorphism of $G$   with  the  subgroup  $\alpha(G)={\mathbb Q}+j({\mathcal O}(D)_f)$ of ${\mathbb A}_{\mathbb Q}$. Note, in particular, that $\alpha(G)$  is closed  in ${\mathbb A}_{\mathbb Q}$, since ${\mathbb Q}$ is discrete  (hence closed) and $j({\mathcal O}(D)_f)$ is compact. In the following, we identify (set-theoretically)  ${\mathbb A}_{\mathbb Q}$ with the product ${\mathbb A}^f_{\mathbb Q}\times {\mathbb R}$  endowed with the two projection morphisms   $p_f: {\mathbb A}_{\mathbb Q}\to {\mathbb A}^f_{\mathbb Q}$ and $p_\infty: {\mathbb A}_{\mathbb Q}\to {\mathbb R}$. The subgroup $p_f({\mathbb Q})\subset {\mathbb A}^f_{\mathbb Q}$ is dense and the subgroup $p_f(j({\mathcal O}(D)_f))={\mathcal O}(D)_f\subset {\mathbb A}^f_{\mathbb Q}$ is open.  Hence   $p_f(\alpha(G))={\mathbb A}^f_{\mathbb Q}$. Thus  the projection $p_\infty$ induces   the isomorphism  of groups   $p:{\mathbb A}_{\mathbb Q}/\alpha(G)\stackrel{\sim}{\to} {\mathbb R}/L$, where $L=p_\infty(\ker(p_f\circ \alpha)) $. The kernel  of the composite $p_f\circ \alpha: G\to {\mathbb A}_{\mathbb Q}^f$  is the group of pairs $(q,a)\in {\mathbb Q}\times j({\mathcal O}(D)_f)$ such that $p_f(q)+p_f(a)=0$. Such pairs are determined by the value of $q$ and thus 
\begin{equation}\label{weil6}
L=p_\infty(H^0({\rm Spec\,}{\mathbb Z},{\mathcal O}(D)_f))=\{q \in {\mathbb Q}\mid \vert q\vert_\nu \leq \exp(D)(\nu), \quad\forall \nu\neq \infty \}.
\end{equation}
By Proposition \ref{setup1}, $H^1(D)$ is the tolerant  ${\sss[\pm 1]}$-module $H^1(D) = (H{\mathbb A}_{\mathbb Q}, {\mathcal R})$ where the relations are given by \eqref{cokerpsi}, {\it i.e.\/}\ 
\[
 {\mathcal R}_k:=\{ (x,y)\in H{\mathbb A}_{\mathbb Q}(k_+)\times  H{\mathbb A}_{\mathbb Q}(k_+)\mid x-y\in {\rm Image}\, \psi(k_+)\},
\]
and where $\psi$ is as in \eqref{psi}.
    By construction  one has ${\mathbb Q}\times {\mathcal O}(D)\simeq G\times [-e^a,e^a]$.
  After quotienting both sides of \eqref{psi} by $G={\mathbb Q}\times j({\mathcal O}(D)_f)$, the map $\psi$  becomes
 	 \begin{equation}\label{weil5}
\phi: \Vert H{\mathbb R} \Vert_{e^a}\to H({\mathbb R}/L), \quad  \phi(x)=x+L \in {\mathbb R}/L\quad\forall x\in [-e^a,e^a]\subset {\mathbb R},
\end{equation}
where $L\subset {\mathbb Q}\subset {\mathbb R}$ is the lattice \eqref{weil6}. Thus one obtains that the relation ${\mathcal R}$ is equal to the inverse image by the map $\pi:{\mathbb A}_{\mathbb Q}\to {\mathbb R}/L$ of the relation \eqref{inequdef}.\newline
$(ii)$~Follows from Proposition \ref{pullback} $(ii)$.\endproof 

 
\section{The dimension of $H^0(D)$}\label{sec4}
  
In this section we consider, for each integer $n>0$, the ${\sss[\pm 1]}$-module $\Vert H{\mathbb Z}\Vert_n$ 
\[
\| H{\mathbb Z}\|_{n}: \Gamma^{o}\longrightarrow \frak{ Sets}_*\qquad \| H{\mathbb Z}\|_{n}(F):=\{\phi\in H{\mathbb Z}(F)\mid \sum_{F\setminus \{*\}} \vert\phi(x)\vert\leq n\}
\]
and evaluate its dimension.  It follows from the ${\sss[\pm 1]}$-module structure, that \eqref{defnsum} holds for a sum of the form 
\[
\sum_F \alpha_j j\in [-n,n]\cap {\mathbb Z}=\Vert H{\mathbb Z}\Vert_n(1_+) \quad \forall \alpha_j\in \{-1,0,1\}~~\text{s.t.}~ \sum_F \vert \alpha_j j\vert \leq n.
\]
By applying Definition \ref{generator}, 	a subset $F\subset [-n,n]\cap {\mathbb Z}=\Vert H{\mathbb Z}\Vert_n(1_+)$ generates $\Vert H{\mathbb Z}\Vert_n(1_+)$ if and only if for every element  $x\in \Vert H{\mathbb Z}\Vert_n(1_+)$ there exists coefficients $\alpha_j\in \{-1,0,1\}$, $j\in F$, such that $x=\sum_F \alpha_j j$ and $\sum_F \vert \alpha_j j\vert \leq n$. The {\em dimension} $\dim_{{\sss[\pm 1]}}\Vert H{\mathbb Z}\Vert_n$ is the minimal cardinality of a generating set.
	
The explicit computation of  $\dim_{{\sss[\pm 1]}}\Vert H{\mathbb Z}\Vert_n$ follows from the next two statements.
	
\begin{lemma}\label{bijectivecase} For $k\in{\mathbb N}$,  let $n=(3^k-1)/2$. Consider the subset $F:=\{3^i,0\leq i\leq k-1\}\subset [-n,n]$. Then $F$ is a generating set  and the map 
\begin{equation}\label{maptheta}
	\theta:\{-1,0,1\}^{\times k}\to [-n,n]\cap {\mathbb Z}, \qquad \theta((\alpha_i)):=\sum_{i=0}^{k-1} \alpha_i 3^i
	\end{equation}
is bijective. Moreover $\sum_{i=0}^{k-1} \vert \alpha_i 3^i\vert \leq n$,	 for all $\alpha_i\in \{-1,0,1\}^{\times k}$.
	\end{lemma}
\proof First of all, note that for all $\alpha\in \{-1,0,1\}^{\times k}$ one has
\[
\sum_{i=0}^{k-1} \vert \alpha_i 3^i\vert \leq \sum_{i=0}^{k-1} 3^i= n.
\]
This shows that for any $\alpha \in \{-1,0,1\}^{\times k}$ one has $z=(\alpha_i 3^i)\in \Vert H{\mathbb Z}\Vert_n(k_+)$, and, as in \eqref{defnsum}, $\Vert H{\mathbb Z}\Vert_n(\sigma)(z)=\theta((\alpha_i))$.
 We prove the bijectivity of $\theta$ by induction on $k$. It is clear for $k=1$. For $k=2$, $n=4$ and one writes the set $[-4,4]\cap {\mathbb Z}$ as the disjoint union of three subsets as follows
\begin{align*}
[-4,4]\cap {\mathbb Z}&=\{-4,-3,-2\}\cup \{-1,0,1\}\cup \{2,3,4\}=\\
&=(-3+\{-1,0,1\})\cup \{-1,0,1\}\cup (3+\{-1,0,1\}).
\end{align*}
This shows  that  $\theta$ is bijective. Next, by assuming to have shown the bijectivity up to $k-1$, we deduce it for $k$. Indeed, with $n=(3^k-1)/2$ let $m=(3^{k-1}-1)/2$. The induction hypothesis ensures that every element of the set $[-m,m]\cap {\mathbb Z}$ can be uniquely written as a sum $\sum_{i=0}^{k-2} \alpha_i 3^i$. We  divide the set $[-n,n]\cap {\mathbb Z}$ into three disjoint parts of the form 
\[
J_{-1}:=[-n,-n+2m]\cap {\mathbb Z}, \quad J_0:=[-m,m]\cap {\mathbb Z}, \quad J_{1}:=[n-2m,n]\cap {\mathbb Z}.
\]
One has $n-2m=m+1$ and thus the union of these three disjoint sets is $[-n,n]\cap {\mathbb Z}$. Moreover one has $J_{1}=(n-m)+J_0$ with $n-m=3^{k-1}$, and similarly $J_{-1}=J_0-3^{k-1}$. Thus, given $x\in [-n,n]\cap {\mathbb Z}$ one determines uniquely the coefficients $\alpha_i\in \{-1,0,1\}$ such that $\theta((\alpha_i))=x$. Indeed,  one uses the above partition in three intervals to determine the coefficient $\alpha_{k-1}$ and then one applies the induction hypothesis to determine the  others. \endproof 
\begin{remark}\label{f3} \begin{enumerate}
\item The conceptual explanation of Lemma \ref{bijectivecase}  derives from the following peculiar property of the Teichm\"uller lift $\tau:{\mathbb F}_3\to {\mathbb Z}_3$. One has $\tau({\mathbb F}_3)=\{-1,0,1\}\subset {\mathbb Z}\subset {\mathbb Z}_3$ and $\tau$ extends to a canonical map $\tilde \tau :W_k({\mathbb F}_3)\to {\mathbb Z}$ from Witt vectors to ${\mathbb Z}$  
\[
\xi=(\xi_j)\in W_k({\mathbb F}_3)\mapsto \tilde \tau(\xi):= \sum_0^{k-1} \tau(\xi_j) 3^j\in {\mathbb Z}.
\]
This map coincides with $\theta$ in \eqref{maptheta} and  one can compute the partially defined sums \eqref{defnsum} in $\Vert H{\mathbb Z}\Vert_n$ using the addition in $W_k({\mathbb F}_3)$. The prime $p=3$ is the only prime such that the subring ${\mathbb Z}\subset {\mathbb Z}_p=W({\mathbb F}_p)$ coincides with  the ring of Witt vectors with finitely many non-zero components. When $p>3$ one has $\tau({\mathbb F}_p)\not\subset {\mathbb Z} $, while for $p=2$ the integer $-1\in {\mathbb Z}\subset {\mathbb Z}_2$ is  the Witt vector  whose components are all equal to $1$. In general the Witt vectors with finitely many non-zero components do not even form a subgroup of the additive group of Witt vectors.
\item The Teichm\"uller lift $\tau$, as a morphism of multiplicative pointed monoids, induces a morphism ${\sss[\pm 1]}\to H{\mathbb Z}$ of ${\S}$-algebras.
\item The ordering of the natural numbers encoded by  $\theta$ coincides with the  lexicographic ordering of the coefficients  $(\alpha_i)\in\{-1,0,1\}^{\times k}$ \[ \theta((\alpha_i))< \theta((\beta_i))\iff \exists j ~\text{s.t.}~\alpha_j<\beta_j~~\text{and}~~ \alpha_\ell=\beta_\ell \quad \forall \ell>j.
\]
\end{enumerate}
\end{remark}


We recall that the ceiling function $\lceil x\rceil$ associates to a positive real number $x$ the smallest integer $n\geq x$.

\begin{proposition}\label{dimhzn}
\begin{enumerate}
\item[(i)] For any integer $n \geq 0$ one has 
\begin{equation}\label{dimhzn1}
\dim_{{\sss[\pm 1]}}\Vert H{\mathbb Z}\Vert_n=\bigg\lceil \frac{\log(2n+1)}{\log 3}\bigg\rceil.
\end{equation}
\item[(ii)] For $n\notin \{2,5\}$, there exists a generating subset $F\subset \{1,\ldots,n\}$ of cardinality $\#F=\lceil \frac{\log(2n+1)}{\log 3}\rceil$, with $\sum_{j\in F} j=n$, such that the following map surjects onto $[-n,n]\cap {\mathbb Z}$
\[
	\theta:\{-1,0,1\}^{\times(\#F)}\to [-n,n]\cap {\mathbb Z}, \qquad \theta((\alpha_i)):=\sum_{j\in F} \alpha_j j.
\]
		\end{enumerate}
\end{proposition}
\proof $(i)$~Set $\kappa(n):=\dim_{{\sss[\pm 1]}}\Vert H{\mathbb Z}\Vert_n$. Since the cardinality of $[-n,n]\cap {\mathbb Z}$ is $2n+1$ and the cardinality of $\{-1,0,1\}^{\times\#F}$ is $3^{\#F}$, it follows that if $F$ is a generating set one must have $3^{\#F}\geq 2n+1$. Thus one has $\#F\geq \frac{\log(2n+1)}{\log 3} $, and since $\#F$ is an integer one gets $\#F\geq \lceil \frac{\log(2n+1)}{\log 3}\rceil $. This shows that $\kappa(n)\geq \lceil \frac{\log(2n+1)}{\log 3}\rceil $. It remains to prove that for each $n\ge 0$ one can find a generating set $F$ with $\# F=\lceil \frac{\log(2n+1)}{\log 3}\rceil $. One uses Lemma \ref{bijectivecase}, and distinguishes several cases starting with the easiest one. Set $F(k):=\{3^i,0\leq i\leq k-1\}$ and let $E\subset {\mathbb N}$ be the subset
\begin{equation}\label{exceptionale}
	E=\bigg\{3^\ell+\frac{1}{2} \left(3^m-1\right)\mid m\in {\mathbb N}, \ell<m \bigg\}.
\end{equation}
We list the first elements of $E$ as follows
\[
2,5,7,14,16,22,41,43,49,67,122,124,130,148,202,365,367,373,391,445,607\ldots
\]
 First, assume  that $n\notin E$. Then let $m$ be the largest integer such that $3^m\leq 2n+1$. If  $3^m= 2n+1$, then  Lemma \ref{bijectivecase} provides one with 
a generating set of cardinality $m=\frac{\log(2n+1)}{\log 3}$. Assume now that $3^m< 2n+1$. Then,  the map $\theta$ for $F(m)$  covers the set $[-q,q]\cap {\mathbb Z}$, where $q=\frac{1}{2} \left(3^m-1\right)$. Thus by adjoining to $F(m)$  the element $n-q$ one  covers $[-n,n]\cap {\mathbb Z}$. More precisely,  let 
$F=F(m)\cup \{n-\frac{1}{2} \left(3^m-1\right)\}$, then we show that $F$ fulfills the conditions in $(ii)$. One has $\# F = \# F(m)+1$ since the additional element $n-\frac{1}{2} \left(3^m-1\right)$ does not belong to $F(m)$ as, by hypothesis, $n\notin E$. Since $\# F(m)=m$, $\#F=m+1=\lceil \frac{\log(2n+1)}{\log 3}\rceil$. The sum of elements in $F(m)$ is $\frac{1}{2} \left(3^m-1\right)$, so the sum of elements in $F$ is equal to $n$. Next, we show that  $\theta$ is surjective. Its image contains the three sets $\theta(F(m))$, $\theta(F(m))+n-\frac{1}{2} \left(3^m-1\right)$, and $\theta(F(m))-(n-\frac{1}{2} \left(3^m-1\right))$.  Using Lemma \ref{bijectivecase} one obtains, with $q=\frac{1}{2} \left(3^m-1\right)$
\begin{align*}
&\theta F(m)=[-q,q]\cap {\mathbb Z}, \quad \theta F(m)+n-q=[n-2q,n]\cap {\mathbb Z},\\ &\theta F(m)-(n-q)=[-n,-n+2q]\cap {\mathbb Z}.
\end{align*}
One has $3(2q+1)=3^{m+1}>2n+1$. This inequality  prevents the three subsets from being disjoint, thus the upper limit $q$ of $\theta F(m)$ is greater or equal to the lower limit $n-2q$ of the interval $\theta F(m)+n-q$. Thus $\theta$ is surjective. \newline
The next case is for $n\in E$ of the form $n=1+\frac{1}{2} \left(3^m-1\right)$, for some $m$ ($\ell=0$ in \eqref{exceptionale}). We assume that $m>2$ (this excludes the cases $n=2$ for $m=1$, and $n=5$ for $m=2$). Let 
\[
F:=\{3^k\mid 0\leq k\leq m-2\}\cup \{2,3^{m-1}-1\}.
\]
(This choice avoids the repetition of the element $1=n-q$ while keeping the sum of elements of $F$ equal to $n$).
We show that $F$ fulfills the conditions in $(ii)$. We have $\# F=m-1+2=m+1$ since $3^{m-1}-1 \notin \{1,2\}$. Moreover $2n+1=3^m+2$,  thus $\#F=m+1=\lceil \frac{\log(2n+1)}{\log 3}\rceil$. The sum of terms in $F$ is equal to one plus the sum of the powers of $3$ up to $3^{m-1}$ and thus is equal to  $n$. Let us now show that the map $\theta$ is surjective. By construction $\theta(F)$ is the union of  nine intervals obtained, with $q'=\frac{1}{2} (3^{m-1}-1)$, by adding to the set $[-q',q']\cap {\mathbb Z}$ the terms in $\{2,3^{m-1}-1\}$ with coefficients in $\{-1,0,1\}$. With the term $3^{m-1}-1$ one covers the set $[-q+1,q-1]\cap {\mathbb Z}$, where $q=\frac{1}{2} \left(3^{m}-1\right)$ as the union of the three sets
\[
[-q',q']\cap {\mathbb Z}, \ \ ([-q',q']+3^{m-1}-1)\cap {\mathbb Z}, \ \ ([-q',q']-(3^{m-1}-1))\cap {\mathbb Z}.
\] 
One has $q'+3^{m-1}-1=n-2$, and thus it  remains to cover the two elements $n-1,n$ and their opposite. This is done using the set
$([-q',q']+3^{m-1}-1 +2)\cap {\mathbb Z}$ whose upper limit is $n$, and appealing to the fact that since $q'>1$ this interval contains $n-1$.\newline
Finally, we consider the case  $n\in E$  of the form $n=3^\ell+\frac{1}{2} \left(3^m-1\right)$, with $m\in {\mathbb N}, ~0< \ell<m $. Here, the additional element $n-\frac{1}{2} \left(3^m-1\right)$ does  belong to $F(m)$ so the proof of the first case considered above does not apply since $3^\ell$ appears twice in $F(m)\cup \{n-\frac{1}{2} \left(3^m-1\right)\}$. We replace this double occurrence of $3^\ell$ by the two distinct elements $3^\ell\pm 1$. This gives
\[
F=(F(m)\setminus \{3^\ell\})\cup \{3^\ell-1,3^\ell+1\}.
\]
By construction  $\# F(m) +1=m+1$. Furthermore one has:  
$n=3^\ell+\frac{1}{2} \left(3^m-1\right)$, $2n+1=3^m+2\times  3^\ell$ and since $0<\ell<m$ one gets $\#F=m+1=\lceil \frac{\log(2n+1)}{\log 3}\rceil$. The sum of elements of $F$ is the same as the sum of elements of $F(m)$ plus $3^\ell$ which is equal to $n$. It remains to show that, for such $F$,  $\theta$  is surjective. 
We first deal with the set $G(\ell):=F(\ell)\cup \{3^\ell-1,3^\ell+1\}$ and show that the map $\theta_{G(\ell)}$, for this set, surjects onto the interval $[-t(\ell),t(\ell)]\cap {\mathbb Z}$, where $t(\ell)=\sum_{G(\ell)}j=3^\ell+\frac{1}{2}(3^{\ell+1}-1)$. Lemma \ref{bijectivecase} shows that the image of $\theta_{F(\ell)}$ is the set $J\cap {\mathbb Z}$, with $J=[-\frac{1}{2}(3^\ell-1),\frac{1}{2} (3^\ell-1)]$. To show the surjectivity of $\theta_{G(\ell)}$ it is enough to show that the following intervals cover $[0,t(\ell)]$:
\[
J, \quad J+3^\ell-1, \quad J+ 3^\ell+1, \quad J+2\times 3^\ell.
\]
The upper limit of $J+2\times 3^\ell$ is $\frac{1}{2} (3^\ell-1)+2\times 3^\ell=t(\ell)$, its lower limit is equal to the upper limit of $J+ 3^\ell+1$. The lower limit of $J+3^\ell-1$ is $3^\ell-1-\frac{1}{2}(3^\ell-1)= \frac{1}{2}(3^\ell-1)$ which is the upper limit of $J$. Moreover, the length of $J$ is $3^\ell-1\geq 2$ so that two translates of $J$ of the form $J+a$, $J+a +2$ necessarily overlap. This shows that the map $\theta_{G(\ell)}$  surjects onto the interval $[-t(\ell),t(\ell)]\cap {\mathbb Z}$. Thus  $\theta_{G(\ell)}$ is the symmetric enlargement of the interval $\theta(F(\ell+1))$ obtained by adding $3^\ell$ to the upper limit. Then, as in the proof of Lemma \ref{bijectivecase}, one obtains by induction that when one adjoins the higher powers $3^{\ell'}$, where $\ell<\ell'<m$, one achieves the enlargement of the interval $\theta(F(m))$ obtained by adding $3^{\ell}$ to the upper limit. This shows that the map $\theta$ for $F$ is surjective. In fact the above arguments prove $(ii)$ and $(i)$, except that one has to take care of the special cases $n=2,5$. For $n=2$,  $\{1,2\}$ is a generating set but the sum of its terms is $>2$. For $n=5$,   $\{1,2,3\}$ is a generating set but the sum of its terms is $>5$.  \endproof 




We are now ready to determine the dimension of the ${\sss[\pm 1]}$-module $H^0(D)$, for any Arakelov divisor $D$. It is a non-decreasing function of $\deg D$ but the formula for this dimension drops by $1$ on an exceptional set $L$.   It is the open set defined as the union of open intervals as follows 
\begin{equation}\label{excL}
	L+ \log 2=\bigcup_{k\in {\mathbb N}} (k\log 3,k\log 3+\epsilon_k), \ \ \epsilon_k:=\log(1+3^{-k}).
\end{equation}
It has finite Lebesgue measure
\[
 \vert L\vert=\sum_{k\geq 0}  \epsilon_k=\log \left(-1;\frac{1}{3}\right)_{\infty }= 1.14099\ldots
\]
where $\left(-1;\frac{1}{3}\right)_{\infty }=\prod_{n=0}^\infty (1+3^{-n})$ is the $-1$ Pochhammer symbol at $\frac 13$.

\begin{theorem} 
Let $D$ be an Arakelov divisor  on ${\overline{\Spec\Z}}$. If $\deg D\geq -\log 2$, then
\begin{equation}\label{rrforq}
\dim_{{\sss[\pm 1]}}H^0(D)=\bigg\lceil \frac{\deg D+ \log 2}{\log 3}\bigg\rceil	-{\mathbf 1}_L
\end{equation}
where ${\mathbf 1}_L$ is the characteristic function of the open set $L$. 
\end{theorem}
\proof Let $a=\deg D$ and  $n=\lfloor\exp(\deg(D)\rfloor$, then  $H^0(D)=\Vert H{\mathbb Z}\Vert_n$.  By Proposition \ref{dimhzn} one gets 
\[
\dim_{{\sss[\pm 1]}}H^0(D)=\bigg\lceil \frac{\log(2n+1)}{\log 3}\bigg\rceil, \qquad n:=\lfloor\exp(a)\rfloor.
\]
Let us compare this formula with \eqref{rrforq}. Assume first that $a\in L$, thus there exists $k\in{\mathbb N}$ such that $a+\log 2\in (k\log 3,k\log 3+\epsilon_k)$. Then, $2 \exp a\in (3^k, 3^k+1) $ and with $n=\lfloor \exp a\rfloor$,  one has $n=\frac{1}{2}(3^k-1)$. Thus  $2n+1=3^k$ and 
 $\lceil \frac{\log(2n+1)}{\log 3}\rceil=k$. Moreover,  $\frac{a+\log 2}{\log 3}\in (k,k+\frac{\epsilon_k}{\log 3})$, where $0<\frac{\epsilon_k}{\log 3}<1$, hence one has $\lceil \frac{a+ \log 2}{\log 3}\rceil=k+1$, so that \eqref{rrforq} holds since ${\mathbf 1}_L(a)=1$. Now, assume that $a\notin L$,  equivalently that $\frac{a+\log 2}{\log 3}\in [k+\frac{\epsilon_k}{\log 3},k+1]$ for some integer $k\geq 0$ (since $a\geq- \log 2$ by hypothesis). Then  $\lceil \frac{a+ \log 2}{\log 3}\rceil=k+1$. One has 
 \[
 2 \exp a\in [\exp(k\log 3)(1+3^{-k}),\exp((k+1)\log 3)]=[3^k+1,3^{k+1}].
 \]
 In this case $n=\lfloor\exp a\rfloor\in[\frac{1}{2} (3^k+1),\frac{1}{2}(3^{k+1}-1)]$ and thus $2n+1\in [3^k+2,3^{k+1}]$.
 This implies that $\frac{\log(2n+1)}{\log 3}\in (k,k+1]$ and hence that  $\lceil \frac{\log(2n+1)}{\log 3}\rceil=k+1$. In this case \eqref{rrforq} holds since ${\mathbf 1}_L(a)=0$. \endproof 

\section{The dimension of $H^1(D)$}\label{sec5}
 
 Let $(A,d)$ be an abelian group endowed with a translation invariant metric $d$. For $\lambda\in{\mathbb R}_{>0}$, we shall refer to $(A,d)_\lambda$ as the associated object  of the category $\Gamma{\mathcal T}_*$ as in Proposition \ref{defnA}. The ${\S}$-module structure determines the addition in $A=(A,d)_\lambda(1_+)$, as well as the action of ${\sss[\pm 1]}$ on it, where for $x\in A=(A,d)_\lambda(1_+)$, $-x$ is the additive inverse of $x$. The metric $d$ determines the tolerance relation ${\mathcal R}$ on $A=(A,d)_\lambda(1_+)$.
 Next result computes the dimension of the tolerant  ${\sss[\pm 1]}$-module $({\mathbb R}/{\mathbb Z},d)_\lambda$.
 
 
 \begin{proposition}\label{caseh1}
 Let $\lambda\in{\mathbb R}_{>0}$, and  $U(1)_\lambda=(U(1),d)_\lambda$ where $U(1)$ is the abelian group ${\mathbb R}/{\mathbb Z}$ endowed with the canonical metric $d$ of length $1$.	Then 
 
 \begin{equation}\label{dimh1}
\dim_{{\sss[\pm 1]}}U(1)_\lambda=\begin{cases}\bigg\lceil \frac{-\log \lambda- \log 2}{\log 3}\bigg\rceil &\text{if $\lambda< \frac 12$}\\0 &\text{if $\lambda\geq \frac 12$.}	
\end{cases}
\end{equation}
 \end{proposition}
\proof For $\lambda\geq \frac 12$, any element of $U(1)_\lambda=({\mathbb R}/{\mathbb Z},d)_\lambda$ is at distance $\leq \lambda$ from $0$, thus one can take $F=\emptyset$ as generating set since, by convention, $\sum_\emptyset=0$. Thus $\dim_{{\sss[\pm 1]}}U(1)_\lambda=0$. Next, we assume $\lambda<\frac 12$. Let $F\subset U(1)$ be a generating set and let $k=\# F$. One easily sees that there are at most $3^k$ elements of the form $\sum_F \alpha_j j$. The subsets  $\{x\in U(1) \mid d(x,\alpha_j j)\leq \lambda\}$  cover $U(1)$, and since each of them has measure $2\lambda$ one gets the inequality $2 \lambda\cdot  3^k\geq 1$. Thus  $k\geq \frac{-\log \lambda- \log 2}{\log 3}$. For $\lambda=\frac 16$ one has $k\geq 1$ and the subset  $F=\{\frac 13\}$ generates, thus $\dim_{{\sss[\pm 1]}}U(1)_\lambda=1$.  When $\frac{-\log \lambda- \log 2}{\log 3}=m$ is an integer, one has $\lambda=\frac 12 3^{-m}$. Let $F(m)=\{\frac 13,\ldots,\frac{1}{3^{m}}\}$. The minimal distance between two elements of $F(m)$ is the distance between $3^{-m}$ and $3^{-m+1}= 3\cdot  3^{-m}$ which is $2 \cdot 3^{-m}=4 \lambda$.
Let us show that $F(m)$ is a generating set. By Lemma \ref{bijectivecase} any integer $q$ in the interval $[-N,N]$, for $N=\frac 12 (3^m-1)$ can be written as  $q=\sum_{i=0}^{m-1} \alpha_i 3^i$, with $\alpha_i\in\{-1,0,1\}$. One then gets 
\[
q \cdot 3^{-m}=\sum_{i=0}^{m-1} \alpha_i 3^{i-m}=\sum_{j=1}^{m}\alpha_{m-j}3^{-j}.
\]
 Let $x\in [-\frac 12,\frac 12]$, then $3^m  x\in [-\frac {3^m}{2},\frac {3^m}{2}]$ and there exists an integer $q\in [-N,N]$ such that $ \vert 3^m  x-q\vert \leq \frac 12$. 
 Hence $d(x,q \cdot 3^{-m})\leq \lambda$. This proves that $F(m)$ is a generating set (see Definition \ref{generator}) and one derives $\dim_{{\sss[\pm 1]}}U(1)_\lambda=m$. Assume now that $\frac{-\log \lambda- \log 2}{\log 3}\in(m,m+1)$, where $m$ is an integer. For any generating set $F$ of cardinality $k$ one has $k\geq \frac{-\log \lambda- \log 2}{\log 3}>m$ so that $k\geq m+1$. The subset $F(m+1)=\{\frac 13,\ldots,\frac{1}{3^{m+1}}\}$ fulfills the first condition of Definition \ref{generator} since the minimal distance between two elements of  $F(m+1)$ is $2 \cdot 3^{-m-1}$ which is larger than $\lambda< \frac 12 3^{-m}$. As shown above, the subset $F(m+1)$ is generating for $\lambda=\frac 12 3^{-m-1}$ and a fortiori for $\lambda>\frac 12 3^{-m-1}$ (as by assumption  
$\frac{-\log \lambda- \log 2}{\log 3}<m+1$). Thus one obtains $\dim_{{\sss[\pm 1]}}U(1)_\lambda=m+1$ and  \eqref{dimh1} is proven.\endproof 
\begin{remark}\label{triadic}
The proof of Proposition \ref{caseh1} is related to a definite advantage of the triadic expansion of numbers \cite{triadic}:  truncating a number is identical to rounding it. This property does not hold in the decimal system where rounding a number requires the knowledge of the next digit.
\end{remark}


We extend the ceiling function to negative values of the variable as follows
 \begin{equation}\label{ceilingprime}
\lceil x \rceil'	=\begin{cases} \lceil x \rceil\ \text{if}\ x\geq 0\\ 	-\lceil -x \rceil\ \text{if}\ x\leq  0.\end{cases}
\end{equation}
 
 We can now state and prove the absolute Riemann-Roch theorem for ${\overline{\Spec\Z}}$  over ${\sss[\pm 1]}$. With $L$ the exceptional set defined in \eqref{excL} one has
 
 \begin{theorem}\label{rrspzb} 
Let $D$ be an Arakelov divisor  on ${\overline{\Spec\Z}}$. Then 
\begin{equation}\label{rrforq1}
\dim_{{\sss[\pm 1]}}H^0(D)-\dim_{{\sss[\pm 1]}}H^1(D)=\bigg\lceil \frac{\deg D+ \log 2}{\log 3}\bigg\rceil'	-{\mathbf 1}_L.
\end{equation}
\end{theorem}
 
 \proof  By appealing to the invariance under linear equivalence (Proposition \ref{setup1} $(iii)$), one may assume that $D=a\{\infty\}$. Then it follows from Proposition \ref{thedimh1} $(ii)$ that $\dim_{{\sss[\pm 1]}}H^1(D)=\dim_{{\sss[\pm 1]}}U(1)_\lambda$ for $\lambda=\exp(\deg(D))$. Assume first that $\exp(\deg(D))\geq \frac 12$, then by Proposition \ref{caseh1}: $\dim_{{\sss[\pm 1]}}H^1(D)=0$, thus for $\deg(D)\geq -\log 2$ \eqref{rrforq1} follows from \eqref{rrforq}. Let us now assume that $\deg(D)< -\log 2$. Then $\dim_{{\sss[\pm 1]}}H^0(D)=0$ since $H^0(D)=\{\ast\}$  when $\deg D<0$. Moreover  $\deg D\notin L$ since $L$ is lower bounded by $-\log 2$. Thus \eqref{rrforq1} follows from \eqref{dimh1} which gives, using \eqref{ceilingprime}
 \[
 -\dim_{{\sss[\pm 1]}}H^1(D)=-\dim_{{\sss[\pm 1]}}U(1)_\lambda=-\bigg\lceil \frac{-\log \lambda- \log 2}{\log 3}\bigg\rceil=\bigg\lceil \frac{\deg D+ \log 2}{\log 3}\bigg\rceil'.	
 \]
 This ends the proof of \eqref{rrforq1}. \endproof
 
\section{Duality}\label{sec6}

In this section we prove an absolute analogue of  Serre's duality, namely the following isomorphism of ${\sss[\pm 1]}$-modules, for any divisor $D$ on ${\overline{\Spec\Z}}$: 
\[
H^0(D)\simeq {\underline \Hom}_{\Gamma{\mathcal T}_*}(H^1(K-D),U(1)_{\frac 14}).
\]
Here, the divisor $K=-2\{2\}$ plays the role of the canonical divisor. The choice of the  tolerant ${\sss[\pm 1]}$-module $(U(1),d)_{\frac 14}$ as dualizing module is motivated by  Pontryagin duality (see \ref{sectpontdual}). One has $\dim_{{\sss[\pm 1]}}(U(1)_{\frac 14})=1$. This equality, in fact,  also holds for the tolerant ${\sss[\pm 1]}$-module $U(1)_{\lambda} $ for $\frac 16\leq \lambda<\frac 12$ :  the specific choice $\lambda=\frac 14$ is dictated by the invariance of the Riemann-Roch formula \eqref{rrforq1} when one switches $H^0$ and $H^1$ and replaces $D$ by $K-D$ (ignoring the exceptional set $L$).

 \subsection{${\underline \Hom}_{\S}(\Vert H{\mathbb R}\Vert_\lambda,\Vert H{\mathbb R}\Vert_\mu)$}
 
 We start with the following general statement.  
 \begin{lemma}\label{homhom} Let $\lambda, \mu\in{\mathbb R}_{>0}$. The ${\S}$-algebra structure of $H{\mathbb R}$ induces an isomorphism of ${\S}$-modules
  	\begin{equation}\label{homhom1}
\Vert H{\mathbb R}\Vert_{\mu/\lambda}\simeq {\underline \Hom}_{\S}(\Vert H{\mathbb R}\Vert_\lambda,\Vert H{\mathbb R}\Vert_\mu).
\end{equation}
  \end{lemma}
\proof One starts by defining the morphism of ${\S}$-modules 
\begin{equation}\label{etamap}
 	\eta: \Vert H{\mathbb R}\Vert_{\mu/\lambda}\to {\underline \Hom}_{\S}(\Vert H{\mathbb R}\Vert_\lambda,\Vert H{\mathbb R}\Vert_\mu).
 \end{equation}
Precisely, the multiplication in the ${\S}$-algebra  $H{\mathbb R}$  determines  natural maps 
\[
m:H{\mathbb R}(X)\wedge H{\mathbb R}(Y)\to H{\mathbb R}(X\wedge Y)
\]
inducing, for a fixed pointed set $X$, the map $m_X: H{\mathbb R}(X)\to {\rm{Hom}}_{\S}( H{\mathbb R},H{\mathbb R}( X\wedge -))$.
The morphism  $\eta$ is defined as the natural transformation of functors which associates to  $X=k_+$, the map  $\eta_X:\Vert H{\mathbb R}\Vert_{\mu/\lambda}(X)\to {\rm{Hom}}_{\S}(\Vert H{\mathbb R}\Vert_\lambda,\Vert H{\mathbb R}\Vert_\mu( X\wedge -))$ defined as the restriction of $m_X$. This restriction is meaningful in view of  \cite{CCprel} (Proposition 6.1).\newline
Next, we show that $\eta$ is an isomorphism. First, we determine the ${\S}$-module ${\rm{Hom}}_{\S}(\Vert H{\mathbb R}\Vert_\lambda,\Vert H{\mathbb R}\Vert_\mu)$. For integers $1\leq j\leq k$ we let as in \eqref{partmaps}
\[
\delta(j,k)\in {\rm{Hom}}_{\Gamma^{o}}(k_+,1_+), \quad \delta(j,k)(\ell):=\begin{cases} 1 &\text{if $\ell=j$}\\ * &\text{if $\ell\neq j$}.	
\end{cases}
\]
 Let $\phi\in {\rm{Hom}}_{\S}(\Vert H{\mathbb R}\Vert_\lambda,\Vert H{\mathbb R}\Vert_\mu)$. By construction, the natural transformation $\phi$ reads, for each $k$, as a map $\phi(k_+):\Vert H{\mathbb R}\Vert_\lambda(k_+)\to \Vert H{\mathbb R}\Vert_\mu(k_+)$ and by naturality we have
 \begin{equation}\label{etamap1}
\Vert H{\mathbb R}\Vert_\mu(\delta(j,k))\circ \phi(k_+)=\phi(1_+)\circ \Vert H{\mathbb R}\Vert_\lambda(\delta(j,k)).
\end{equation}
 Since an element  $y\in\Vert H{\mathbb R}\Vert_\mu(k_+)$ is determined by its components $y_j\in {\mathbb R}$, $1\leq j\leq k$,
with $y_j=\Vert H{\mathbb R}\Vert_\mu(\delta(j,k))(y)$, \eqref{etamap1} shows that  $\phi(k_+)$ is uniquely determined by the map  $\phi(1_+)$ acting componentwise, {\it i.e.\/}\  
\begin{equation}\label{etamap2}
	\phi(k_+)((x_j))=\left(\phi(1_+)(x_j)\right).
\end{equation}
Moreover, the map $f=\phi(1_+):[-\lambda,\lambda]\to [-\mu, \mu]$  fulfills 
\begin{equation}\label{etamap3}
	f(x+y)=f(x)+f(y)\quad\forall x,y\  s. t. \ \vert x\vert +\vert y\vert \leq \lambda, 
\end{equation}
as one sees using the naturality of $\phi$ for the map $\sigma\in {\rm{Hom}}_{\Gamma^{o}}(2_+,1_+)$, {\it i.e.\/}\  using 
\[
(\Vert H{\mathbb R}\Vert_\mu(\sigma)\circ \phi(2_+))(x,y)=(\phi(1_+)\circ \Vert H{\mathbb R}\Vert_\lambda(\sigma))(x,y)\Longrightarrow f(x)+f(y)=f(x+y).
\]
By \eqref{etamap3} one has $f(x 2^{-n})= 2^{-n} f(x)$, for any $x\in [-\lambda,\lambda]$ and $n\geq 0$. Thus the `germ of map' $f$ uniquely extends to a map $\tilde f:{\mathbb R}\to {\mathbb R}$ defined by $\tilde f(x):=2^n f(x 2^{-n})$ for any $n$ such that $\vert x 2^{-n}\vert \leq \lambda$. Moreover, again by \eqref{etamap3}, the map $\tilde f$ is additive and since $\tilde f([-2^{-n}\lambda,2^{-n}\lambda])\subset [-2^{-n}\mu,2^{-n}\mu]$, it is also continuous and hence determined by the multiplication by a real number $r$. One has $\vert r \lambda \vert \leq \mu$ and hence $r\in \Vert H{\mathbb R}\Vert_{\mu/\lambda}(1_+)$. Thus one gets $\phi(1_+)(x)=r x$, $\forall x \in [-\lambda,\lambda]$. By  \eqref{etamap2} one obtains $\phi(k_+)((x_j))=(r x_j)$, $\forall (x_j)\in \Vert H{\mathbb R}\Vert_\lambda(k_+)$. This shows that for $X=1_+$, $\eta_X:\Vert H{\mathbb R}\Vert_{\mu/\lambda}(X)\to {\rm{Hom}}_{\S}(\Vert H{\mathbb R}\Vert_\lambda,\Vert H{\mathbb R}\Vert_\mu( X\wedge -))$ is bijective. The next step is to determine ${\rm{Hom}}_{\S}(\Vert H{\mathbb R}\Vert_\lambda,\Vert H{\mathbb R}\Vert_\mu( X\wedge -))$ for $X=\ell_+$. Let $\phi \in {\rm{Hom}}_{\S}(\Vert H{\mathbb R}\Vert_\lambda,\Vert H{\mathbb R}\Vert_\mu( \ell_+\wedge -))$. One has $\ell_+\wedge k_+=(\ell k)_+$ and an element 
$z\in \Vert H{\mathbb R}\Vert_\mu( \ell_+\wedge k_+)$ is determined by its components $z(i,j)\in {\mathbb R}$ for $1\leq i\leq \ell$, $1\leq j\leq k$ such that $\sum \vert z(i,j)\vert \leq \mu$. In particular, for each $j$, the $z(i,j)$, $1\leq i\leq \ell$, are the components of $z_j=\Vert H{\mathbb R}\Vert_\mu( {\rm id}\wedge \delta(j,k)))(z)\in \Vert H{\mathbb R}\Vert_\mu(\ell_+)$. This implies by applying the naturality of $\phi$,  that  
\begin{equation}\label{etamap4}
	\phi(k_+)(z)=\left(\phi(1_+)(z_j)\right).
\end{equation}
As shown above the map $f=\phi(1_+):[-\lambda,\lambda]\to \Vert H{\mathbb R}\Vert_\mu(\ell_+)$  fulfills 
\begin{equation}\label{etamap5}
	f(x+y)=f(x)+f(y)\quad\forall x,y\  s. t. \ \vert x\vert +\vert y\vert \leq \lambda 
\end{equation}
and it extends to an additive map $\tilde f: {\mathbb R} \to {\mathbb R}^\ell$ which is continuous since $\tilde f$ maps the interval $[-\lambda,\lambda]$ inside $\Vert H{\mathbb R}\Vert_\mu(\ell_+)$. Thus there exists 
real numbers  $r_i$, $1\leq i\leq \ell$, such that $\tilde f(x)=(r_i x)$ $\forall x\in {\mathbb R}$. One has $\sum \vert r_i \lambda  \vert \leq \mu$ and thus $r=(r_i)\in \Vert H{\mathbb R}\Vert_{\mu/\lambda}(\ell_+)$. Finally, using \eqref{etamap4} it follows that $\phi=\eta_X(r)$. This proves that  $\eta$ as in \eqref{etamap} is an isomorphism.\endproof  


\subsection{Pontryagin duality}\label{sectpontdual}


In order to formulate Pontryagin duality in this context we consider, for $\lambda>0$,  the ${\sss[\pm 1]}$-module  $U(1)_\lambda=(U(1),d)_\lambda$, where $U(1)$ is the abelian group ${\mathbb R}/{\mathbb Z}$ endowed with its canonical metric $d$ of length $1$. For a metric abelian group $(A,d)$, we denote by $\widehat A$  the abelian group of continuous characters, {\it i.e.\/}\  of  continuous group homomorphisms $A\to U(1)$, where $A$ is endowed with the topology associated to the metric $d$. 
We retain the notations of section~\ref{sec3}. Next statement is motivated by  Lemma~\ref{homhom}.


\begin{proposition}\label{pontrj} Let $(A,d)$ be an abelian group endowed with a translation invariant metric $d$. 
\begin{enumerate}
\item[(i)] For $\lambda, \mu\in{\mathbb R}_{>0}$, the ${\sss[\pm 1]}$-module ${\underline \Hom}_{\Gamma{\mathcal T}_*}((A,d)_\lambda,U(1)_\mu)$ is isomorphic to the sub ${\sss[\pm 1]}$-module of $H\widehat A$ which, on the set $k_+$, is given by k-tuples $(\chi_j)$, $1\leq j\leq k$ of continuous characters  
	$\chi_j\in \widehat A$ such that, with $\vert x\vert:=d(x,0)$, and for all finite collections $\{x_i\}\subset A$, fulfill
	\begin{equation}\label{inequ1}
	\sum_i \vert x_{i}\vert \leq \lambda~ \Rightarrow ~\sum_{i,j} \vert \chi_j(x_{i})\vert \leq \mu.
	\end{equation}
	\item[(ii)] Let $p:A'\to A$ be a surjective morphism of abelian groups and let $p^*(A,d)_\lambda$ be the pullback as in Proposition \ref{pullback}. One has the following canonical isomorphism 
	\begin{equation}\label{isoduals}
	{\underline \Hom}_{\Gamma{\mathcal T}_*}(p^*(A,d)_\lambda,U(1)_\mu)\simeq {\underline \Hom}_{\Gamma{\mathcal T}_*}((A,d)_\lambda,U(1)_\mu).
	\end{equation}
	\end{enumerate}
	\end{proposition}
	\proof $(i)$~Let $\phi\in{\underline \Hom}_{\Gamma{\mathcal T}_*}((A,d)_\lambda,U(1)_\mu)(1_+)={\rm{Hom}}_{\Gamma{\mathcal T}_*}((A,d)_\lambda,U(1)_\mu)$. By applying the forgetful functor ${\mathcal F}:{\mathcal T}\longrightarrow \frak{ Sets}$ which associates to $(X,{\mathcal R})$ the set $X$ (see Proposition~\ref{adjoint}), one obtains  an element ${\mathcal F}(\phi)\in {\rm{Hom}}_{\S}(HA,HU(1))$. Since the Eilenberg-MacLane functor   $H$ determines a full and faithful embedding of the category of abelian groups inside the category of ${\S}$-modules,  there exists a unique  group homomorphism $\chi:A\to U(1)$  such that $H(\chi)={\mathcal F}(\phi)$. The condition that $\phi$ preserves the relation ${\mathcal R}_k$  on the set $k_+$ means that for any $x_i,y_i\in A$, $1\leq i\leq k$ such that $\sum d(x_i,y_i)\leq \lambda$ one has $\sum d(\chi(x_i),\chi(y_i))\leq \mu$. By translation invariance of the metrics this condition is equivalent to 
	\begin{equation}\label{inequ2}
	\sum_i \vert x_{i}\vert \leq \lambda~ \Rightarrow ~\sum_{i} \vert \chi(x_{i})\vert \leq \mu.
	\end{equation}
	This shows that ${\rm{Hom}}_{\Gamma{\mathcal T}_*}((A,d)_\lambda,U(1)_\mu)$ consists exactly of the group homomorphisms $\chi:A\to U(1)$ fulfilling \eqref{inequ1}. Specializing \eqref{inequ2} to the case where all $x_i=x$, $i=1,\ldots, n$,  one obtains the implication $\vert x\vert \leq \lambda/n \Rightarrow \vert \chi(x)\vert \leq \mu/n $, and hence that $\chi$ is uniformly continuous. Let $\phi\in{\underline \Hom}_{\Gamma{\mathcal T}_*}((A,d)_\lambda,U(1)_\mu)(k_+)={\rm{Hom}}_{\Gamma{\mathcal T}_*}((A,d)_\lambda,U(1)_\mu(k_+\wedge -))$. The object $U(1)_\mu(k_+\wedge -)$ of $\Gamma{\mathcal T}_*$ is $(U(1)^{\times k},d_k)_\mu$, where the metric $d_k$ on the product group $(U(1)^{\times k}$ is defined by
	\[
	d_k((x_i),(y_i)):=\sum_1^kd(x_i,y_i)\quad\forall x_i,y_i \in U(1).
	\]
	Replacing $U(1)_\mu$ with $(U(1)^{\times k},d_k)_\mu$ in the first part of the proof one obtains that ${\mathcal F}(\phi)\in {\rm{Hom}}_{\S}(HA,HU(1)^{\times k})=H((\chi_j))$, where $(\chi_j)$, $1\leq j\leq k$ is a $k$-tuple of  characters of $A$ fulfilling \eqref{inequ1}. It follows that $\chi_j\in \widehat A$.\newline
	$(ii)$~Let $\phi\in{\underline \Hom}_{\Gamma{\mathcal T}_*}(p^*(A,d)_\lambda,U(1)_\mu)(1_+)={\rm{Hom}}_{\Gamma{\mathcal T}_*}(p^*(A,d)_\lambda,U(1)_\mu)$. As in the proof of $(i)$, there exists a group homomorphism $\chi': A'\to U(1)$ such that $H(\chi')={\mathcal F}(\phi)$. Moreover $\chi'$ preserves the relation ${\mathcal R}_k$ for any $k$, and this implies 
	\[
	 \sum_{i=1}^k \vert p(x'_i)\vert \leq \lambda ~\Rightarrow~ \sum_{i=1}^k \vert \chi'(x'_i)\vert \leq \mu, \qquad \forall (x'_i)\in (A')^{\times k}.
	\]
	In particular, taking all $x'_i =x'\in \ker(p)$ one obtains $\vert \chi'(x')\vert\leq \mu/k $ $\forall k$, and hence 
	$\chi'(\ker(p))=\{1\}$. This implies that there exists a group homomorphism $\chi:A\to U(1)$ such that $\chi'=\chi\circ p$. 
	\endproof 
	
We can now state and prove Serre's duality.
	
	
\begin{theorem}\label{serredual} Let $D=\sum_j a_j\{p_j\} + a\{\infty\}$ be an Arakelov divisor  on ${\overline{\Spec\Z}}$.  There is a canonical isomorphism of ${\sss[\pm 1]}$-modules 
\begin{equation}\label{serredual1}
H^0(K-D)\simeq {\underline \Hom}_{\Gamma{\mathcal T}_*}(H^1(D),U(1)_{\frac 14}),
\end{equation}
where $K$ is the divisor   $K=-2\{2\}$.
\end{theorem}
\proof By Proposition \ref{thedimh1},  with $\lambda=\exp a$, one has $H^1(D)=\pi^*(({\mathbb R}/L,d)_\lambda)$ and by  Proposition \ref{pontrj}, $(ii)$, one gets the isomorphism 
\[
{\underline \Hom}_{\Gamma{\mathcal T}_*}(H^1(D),U(1)_{\frac 14})\simeq {\underline \Hom}_{\Gamma{\mathcal T}_*}(({\mathbb R}/L,d)_\lambda,U(1)_{\frac 14}).
\]
In fact, we can assume that $D=a \{\infty\}$, with $a=\deg D$ so that $L={\mathbb Z}$. Then we apply Proposition \ref{pontrj} $(i)$, with $A={\mathbb R}/{\mathbb Z}$ and $\mu=\frac 14$. One has $\widehat A={\mathbb Z}$ and the characters $\chi_n\in \widehat A$ are given by multiplication by $n$, {\it i.e.\/}\  $\chi_n(s):=ns\in {\mathbb R}/{\mathbb Z}$, $\forall s\in {\mathbb R}/{\mathbb Z}$. Next, we need to determine the ${\sss[\pm 1]}$-submodule of $H\widehat A=H{\mathbb Z}$ which, on the set $k_+$, is given by $k$-tuples $(n_j)$, $1\leq j\leq k$ of  characters  
	$n_j\in {\mathbb Z}$ such that \eqref{inequ1} holds. This means, using the distance $d$ on ${\mathbb R}/{\mathbb Z}$, that
	\begin{equation}\label{inequ1for}
	\sum_i d (x_{i},0) \leq \lambda~ \Rightarrow ~\sum_{i,j} d( n_j x_{i},0) \leq \frac 14.
	\end{equation}
	The distance $d(x,0)$ is given, for any $x'\in {\mathbb R}$ in the class of $x$ by the distance between $x'$ and the closed subset ${\mathbb Z}\subset {\mathbb R}$. Thus for any integer $n$ one has: $d(nx,0)\leq \vert n\vert d(x,0)$. Assume that $\sum_j \vert n_j\vert \leq \frac{1}{4\lambda}$, then  \eqref{inequ1for} follows since
	\[
	\sum_{i,j} d( n_j x_{i},0)\leq \sum_{i,j} \vert n_j\vert d( x_{i},0)\leq \frac{1}{4\lambda}\sum_i d (x_{i},0).
	\]
	Conversely, assume  \eqref{inequ1for}. Then repeating $m$ times the same $x$,  gives  
	\[
	 d (x,0) \leq \frac\lambda m ~\Rightarrow ~\sum_{j} d( n_j x,0) \leq \frac {1}{4m}.
	\]
 Taking $m$ large enough and  $x=\frac\lambda m$ one obtains $\sum_{j} \vert n_j \vert x\leq \frac {1}{4m}$, and hence $\sum_j \vert n_j\vert \leq \frac{1}{4\lambda}$. This proves that the ${\sss[\pm 1]}$-submodule of $H\widehat A=H{\mathbb Z}$ determined by  \eqref{inequ1} is equal to $\Vert H{\mathbb Z}\Vert_{\frac{1}{4\lambda}}$ which gives \eqref{serredual1}.\endproof 
 








\begin{appendices}


\section{The $\Gamma$-space ${\bf H}(D)$}\label{secA}

Let $\phi: A \to B$ be a morphism of abelian groups. To $\phi$ one associates  the following (short) complex  $\mathfrak C=\{C_n,\phi_n\}$ of abelian groups indexed in  non-negative degrees
\begin{equation}\label{short}
\mathfrak C=\{C_1\stackrel{\phi_1}{\to} C_0\};\qquad C_0=B, \ C_1=A, \quad \phi_1(x):=\phi(x).\end{equation}
 
  The Dold-Kan correspondence associates to $\mathfrak C$ the  simplicial abelian group (see \cite{GJ} III.2, Proposition 2.2) 
  \begin{equation}\label{descriptshorto}
{\mathcal A}_n=\bigoplus_{{\mathcal F}(n,k)} C_k, \quad {\mathcal F}(n,k):=\{\sigma\in {\rm{Hom}}_\Delta([n],[k])\mid \sigma([n])=[k]\}
\end{equation}
where $\Delta$ denotes the simplicial category.
The direct sum in \eqref{descriptshorto} repeats the term $C_k$ of the chain complex as many times as  the number of elements of  the set ${\mathcal F}(n,k)$ of surjective morphisms $\sigma\in {\rm{Hom}}_\Delta([n],[k])$. For the short complex  $\mathfrak C$, the allowed values of $k$ are $k=0,1$. Therefore the set ${\mathcal F}(n,0)$ is  reduced to   ${\rm{Hom}}_\Delta([n],[0])=:\Delta_n^0$, {\it i.e.\/}\  to a single point.  A morphism  $\xi\in{\rm{Hom}}_\Delta([n],[1])=:\Delta^1_n$ is characterized by  $\xi^{-1}(\{1\})$ which is an hereditary subset of $[n]$. It follows that the  vertices in $\Delta^1_n$  are   labelled by the
\begin{equation}\label{identxi}
\xi_j, ~ 0\leq j\leq n+1, \quad \xi_j(k)=1\iff k\geq  j.
\end{equation}
 For each integer $n\geq 0$  the finite set $F(n):={\mathcal F}(n,1)$ of surjective elements  $\xi\in\Delta^1_n$  excludes $\xi_{0}$ and  $\xi_{n+1}$, thus the set $F(n)=\{\xi_j, \, 1\leq j\leq n \}$ has $n$ elements. This gives the  identification  
 \begin{equation}
{\mathcal A}_n= HB(1_+)\oplus HA(F(n)_+)\label{descriptshort}
\end{equation}  
We refer  to \cite{GJ} (III.2, pp. 160-161) for a detailed description of  the simplicial structure,   namely the definition  for each $\theta \in
 {\rm{Hom}}_\Delta([m],[n])$ of a map of sets ${\mathcal A}(\theta):{\mathcal A}_n\to {\mathcal A}_m$. 
 Next, we introduce some notations.
 
  We  identify the  opposite  ${\Delta^o}$  of the simplicial category with (the skeleton of) the category of finite intervals.   An interval  is  a totally ordered set with the smallest element distinct from the largest one.  A morphism of intervals is a non-decreasing map  that preserves the smallest and largest elements. The canonical contravariant functor  $\Delta\longrightarrow {\Delta^o}$   which identifies the opposite category of  $\Delta$  with  ${\Delta^o}$ (described by intervals as above),  maps the  finite ordinal object $[n]=\{0,1,\ldots,n\}$ in $\Delta$  to the interval $[n]^*=\{0, \ldots, n+1\}$. 

We denote by ${\Se_{2,*}}$    the category of pairs  of pointed sets $(X,Y)$, with $X\supset Y$. The morphisms are maps  of pairs of pointed sets. We let $c:{\Se_{2,*}}\longrightarrow {\Se_*}$ be the functor $(X,Y)\to X/Y$  of collapsing  $Y$ to the base point $\ast$.

Let  $\partial:{\Delta^o}\longrightarrow {\Se_{2,*}} $ be the functor that  replaces an interval $I$ with the pair $\partial I=(X,Y)$, where $X$ is the set $I$ pointed by its smallest element, and $Y\subset X$ is the subset formed by the smallest and largest elements of $I$.

Finally, we denote by $\gamma:{\Se_{2,*}} \longrightarrow\Gamma{\Se_{2,*}}$ (see section \ref{sec3}) the functor that associates to an object $(X,Y)$ of ${\Se_{2,*}}$ the covariant functor
\[
\gamma(X,Y):{\Gamma^{o}}\longrightarrow {\Se_{2,*}},\quad k_+:=\{0,\ldots,k\}\mapsto (X\wedge k_+,Y\wedge k_+).
\]
\vspace{.01in}

  The following formula defines a covariant functor that associates to a pair of pointed sets  an abelian group directly related to the morphism $\phi: A\to B$
 \begin{equation}\label{descript1}
  H_\phi:{\Se_{2,*}}\longrightarrow \mathfrak{ Ab},  \qquad	H_\phi(X,Y):=HB(Y)\times HA(X/Y).
 \end{equation}
 On morphisms	$f:(X,Y)\to (X',Y')$  in ${\Se_{2,*}}$ with $\alpha= (\alpha_Y,\alpha_{X/Y})\in H_\phi(X,Y)$,  the functor $H_\phi$ acts as follows
 \begin{align}\label{descript2}
 	H_\phi(f)(\alpha)&=\left(\alpha_{Y'}, HA(f)(\alpha_{X/Y})\right) \\\notag  \alpha_{Y'}(y')&:=HB(f)(\alpha_{Y})(y')+\sum_{x\in X\setminus Y, f(x)=y'}\phi(\alpha_{X/Y}(x)),\quad y'\neq *.\notag
 \end{align}
   By \cite{CCAtiyah} (Proposition~4.5), the Dold-Kan correspondence for the short complex \eqref{short}, {\it i.e.\/}\   the simplicial abelian group $\mathcal A_n$  in  \eqref{descriptshort}   is canonically isomorphic    to the composite functor $H_\phi\circ \partial: {\Delta^o}\longrightarrow \mathfrak{ Ab}$, with $H_\phi$ as in \eqref{descript1}. 
  By composing $H_\phi$  with the Eilenberg-MacLane functor $H$ one obtains  a covariant functor $HH_\phi=H\circ H_\phi :{\Se_{2,*}}\longrightarrow {\sss-{\rm{Mod}}}$  to the category of ${\S}$-modules,  which is naturally isomorphic  to the functor $((U\circ H_\phi)\times {\rm id})\circ \gamma$, where $U:\mathfrak{ Ab}\longrightarrow {\Se_*}$ is the forgetful functor (see {\it op.cit.\/}\  Lemma~4.6). Moreover, as shown in {\it op.cit.\/}\  (Theorem~4.7)  the $\Gamma$-space associated by the Dold-Kan correspondence to the complex $\mathfrak C$ is canonically isomorphic  to the functor 
 \begin{equation}\label{gammasp}((U\circ H_\phi)\times {\rm id})\circ \gamma\circ \partial : {\Delta^o} \longrightarrow {\sss-{\rm{Mod}}}.
\end{equation}
  Notice  that the above construction  involves the morphism $\phi:A\to B$  {\em only} through  the composite functor $U\circ H_\phi:{\Se_{2,*}} \longrightarrow {\Se_*}$. This latter functor  is still meaningful  when one restricts $\phi$ to a sub-${\S}$-module  $E$    of the ${\S}$-module $HA$ and it is given by 
  \begin{equation}\label{descript1sub}
   H_\phi\vert_E:{\Se_{2,*}}\longrightarrow {\Se_*},  \qquad	H_\phi(X,Y):=HB(Y)\times E(X/Y).
 \end{equation} 
 This  provides the following construction 
 
 \begin{definition}  Let $\phi:A\to B$ be  a morphism of abelian groups and $E$ a sub ${\S}$-module of the ${\S}$-module $HA$. We denote by $ \Gamma(\phi\vert_E)$ 
 	the $\Gamma$-space obtained as a sub-functor of \eqref{gammasp}
 \begin{equation}\label{descript1sub1}
 \Gamma(\phi\vert_E):= (H_\phi\vert_E\times {\rm id})\circ \gamma\circ \partial : {\Delta^o} \longrightarrow {\sss-{\rm{Mod}}}.
 \end{equation}
 \end{definition}
 
  When evaluated on the object $1_+=\{0,1\}$ of ${\Gamma^{o}}$, the $\Gamma$-space $\Gamma(\phi\vert_E)$  defines   a sub-simplicial set of the Kan simplicial set $\mathcal A_n$  in \eqref{descriptshort}.  However it is not in general a Kan simplicial set, thus one needs to exert care  when  considering its homotopy.
 We refer to \cite{CCgromov}  2.1,  for the definition of the homotopies  used there. The relation ${\mathcal R}$ of homotopy between $n$-simplices $(x,y)\in X_n\times X_n$  is defined as follows
\begin{equation}\label{Maydefn}
 (x,y)\in {\mathcal R} \iff \partial_j x=\partial_jy\, \forall j \, \&\,  \exists z~\text{s.t.}~ \partial_jz=s_{n-1}\partial_j x\, \forall j<n, \ \partial_nz=x,\, \partial_{n+1}z=y.
  \end{equation}

In general, the  relation ${\mathcal R}$   in  \eqref{Maydefn} fails to be an equivalence relation.  In place of the quotient we consider pairs of sets and relations.  We define $\pi^{\cT}_n$ to be the set  of spherical elements in $X_n$ ({\it i.e.\/}\  of $n$-simplices $x$, with $\partial_j x=*$ $\forall j$), endowed with the relation ${\mathcal R}$. Then we have the following result (we refer to section~\ref{sec3} for the notion of tolerant module)

\begin{proposition}\label{proppi0} Let $\phi:A\to B$ be  a morphism of abelian groups and let $E$ be a sub-${\sss[\pm 1]}$-module of the Eilenberg-MacLane ${\sss[\pm 1]}$-module $HA$ (where $(\pm 1)x:=\pm x$).
\begin{enumerate}
\item[(i)] The homotopy $\pi^{\cT}_1(\Gamma(\phi\vert_E))$ is the sub-${\sss[\pm 1]}$-module $\ker(H\phi\vert_E)$ of $E$ \begin{equation}\label{kerphi}
 \ker(H\phi\vert_E)(k_+)=\{x\in E(k_+)\mid H\phi(x)=*\}.
  \end{equation} 
\item[(ii)]The homotopy $\pi^{\cT}_0(\Gamma(\phi\vert_E))$ is the tolerant ${\sss[\pm 1]}$-module $HB$ endowed with the relations 
	\begin{equation}\label{relk}
 {\mathcal R}_k=\{(x,y)\in HB(k_+)\times HB(k_+)\mid x-y\in H\phi(E(k_+))\}\quad k\in{\mathbb N}.
  \end{equation}
  \end{enumerate}
\end{proposition}  
\begin{proof}
    By construction,  $\Gamma(\phi\vert_E)$ is the composite of  the functors $ H_\phi\vert_E:{\Se_{2,*}} \longrightarrow {\Se_*} $ of \eqref{descript1sub} and $\gamma\circ\partial:{\Delta^o} \longrightarrow{\Se_{2,*}}$.  One has $\partial[0]^*=(X,Y)$ where $X=Y=\{0,1\}$ with base point $0$. Thus  $\gamma\circ\partial[0]^*(k_+)=(X\wedge k_+,Y\wedge k_+)=(k_+,k_+)$ and by \eqref{descript1sub}
\[
H_\phi\vert_E(\gamma\circ\partial[0]^*)(k_+)=H_\phi\vert_E(X\wedge k_+,Y\wedge k_+)=HB(k_+).
\]
One has $\partial[1]^*=(X,Y)$, where $X=\{0,1,2\}$, $Y=\{0,2\}$ with base point $0$. Thus one has $\gamma\circ\partial[1]^*(k_+)=(X\wedge k_+,Y\wedge k_+)$ and $Y\wedge k_+=k_+$, while $(X\wedge k_+)/(Y\wedge k_+)=\{0,1\}\wedge k_+=k_+$ and by \eqref{descript1sub} it follows that
        \begin{equation}\label{pideg1}
 H_\phi\vert_E(\gamma\circ\partial[1]^*)(k_+)= E(k_+)\times HB(k_+).
\end{equation}
The boundaries $\partial_j: H_\phi\vert_E(\gamma\circ\partial[1]^*)\to H_\phi\vert_E(\gamma\circ\partial[0]^*)$ are obtained as in \cite{CCAtiyah} Proposition 4.11 
\begin{equation}\label{pi0}
\partial_0(\psi)=\psi_2, \ \ \partial_1(\psi)=H\phi(\psi_1)+\psi_2, \quad \forall \psi=(\psi_1,\psi_2)\in H_\phi\vert_E(\gamma\circ\partial[1]^*).
\end{equation}

$(i)$~The spherical condition $\partial_j(\psi)=*$ on  $\psi\in H_\phi\vert_E(\gamma\circ\partial[1]^*)(k_+)$ means that $\psi_2=0$ and $H\phi(\psi_1)=0$. Thus the solutions  correspond to $\ker(H\phi\vert_E)(k_+)$ as in \eqref{kerphi}. One shows as in \cite{CCAtiyah} Proposition 4.11 $(iii)$, that the relation of homotopy is the identity.\newline
$(ii)$~The relation ${\mathcal R}_k$ on elements of $H_\phi\vert_E(\gamma\circ\partial[0]^*)(k_+)$ is defined as follows
\begin{equation}\label{pizeroR}
(\alpha,\beta)\in {\mathcal R}_k\iff \exists 	\psi \in H_\phi\vert_E(\gamma\circ\partial[1]^*)(k_+)~\text{s.t.}~\partial_0(\psi)=\alpha \  \& \ \partial_1(\psi)=\beta.
\end{equation}
By \eqref{pi0} it coincides with \eqref{relk}.\vspace{.03in}

 \emph{Note} that this relation is a tolerance relation {\it i.e.\/}\  is  symmetric since $E$ is a sub ${\sss[\pm 1]}$-module of the ${\sss[\pm 1]}$-module $HA$, so that 
\[
x-y\in H\phi(E(k_+))\iff y-x\in H\phi(E(k_+)).
\]
\end{proof}

 \subsection{The $\Gamma$-space ${\bf H}(D)$} \label{secta3}
The above general construction applies in the geometric adelic context and by implementing the action of ${\mathbb Q}^\times$ on adeles, we obtain the following variant of  Proposition~4.9 in \cite{CCAtiyah}.

 \begin{proposition}\label{bfhd} Let  $D$ be an Arakelov divisor on ${\overline{\Spec\Z}}$. Let $A={\mathbb Q} \times {\mathbb A}_{\mathbb Q}$, $B={\mathbb A}_{\mathbb Q}$ and $\alpha:A \to B$, $\alpha(q,a) = q+ a$. Let  $E= H{\mathbb Q}\times H{\mathcal O}(D)$ be the sub ${\sss[\pm 1]}$-module of $HA$ as in Proposition \ref{setup}. 
 	Then the  functor  
\begin{equation}\label{propgam0}
{\bf H}(D):=\Gamma(\alpha\vert_E): {\Delta^o} \longrightarrow {\sss-{\rm{Mod}}}
\end{equation}
defines a $\Gamma$-space
  that depends only on the  linear equivalence class of $D$.
 \end{proposition}
 
 \subsection{Proof of Proposition \ref{setup1}} \label{sectproofprop}
 
 With the notations of section \ref{sec3} and Proposition \ref{bfhd}, one has $H^0(D):= \pi^{\cT}_1({\bf H}(D))=\pi^{\cT}_1(\Gamma(\alpha\vert_E))$. Proposition \ref{proppi0}, $(i)$, gives 
 \[
 H^0(D)(k_+)=\{x\in E(k_+)\mid H\alpha(x)=*\}.
 \]
 An element $x\in E(k_+)$ is  a $k$-tuple $(q_j,a_j)$ with $q_j\in {\mathbb Q}$ for all $j$, and  $(a_j)\in H{\mathcal O}(D)(k_+)$, where $H{\mathcal O}(D):=H{\mathcal O}(D)_f\times \Vert H{\mathbb R}\Vert_{e^a}$. The condition $H\alpha(x)=*$ means $q_j=-a_j$ for all $j$, so that $x$ is uniquely determined by the $k$-tuple $(q_j)$. Moreover, the allowed $k$-tuples are those for which   $\sum \vert q_j\vert \leq e^a$ and $q_j\in  {\mathcal O}(D)_f$ for all $j$. One has  
 \[
 {\mathbb Q}\cap {\mathcal O}(D)_f= H^0({\Spec\Z} ,{\mathcal O}(D)_f).
 \]
This proves $(i)$ of Proposition \ref{setup1}. The  statement $(ii)$ follows  from Proposition \ref{proppi0} $(ii)$.  Finally, $(iii)$ follows from Proposition \ref{bfhd}.



 
\bmhead{Acknowledgments}

The second author is partially supported by the Simons Foundation collaboration grant n. 691493 and thanks IHES for the hospitality,  where part of this research was done.




\end{appendices}




