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\DeclareMathOperator{\Pic}{Pic} \DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\Quot}{Quot} \DeclareMathOperator{\Alb}{Alb} %\DeclareMathOperator{\Div}{Div} \providecommand{\Mod}{\ensuremath{\mathbf{Mod}}} \providecommand{\KM}{\ensuremath{\mathrm{K}^\mathrm{M}}} \providecommand{\Htate}{\ensuremath{\hat{\H}}} \providecommand{\What}{\ensuremath{\widehat{W}}} \providecommand{\Ihat}{\ensuremath{\hat{I}}} \providecommand{\Acal}{\ensuremath{\mathscr{A}}} \providecommand{\Bcal}{\ensuremath{\mathscr{B}}} \providecommand{\AM}{\ensuremath{\mathcal{A}^\mathrm{M}}} \renewcommand{\P} {\ensuremath{\mathrm{P}}} \providecommand{\Het}{\ensuremath{\H_\mathrm{\acute{e}t}}} \newcommand{\tors} {\ensuremath{\mathrm{tors}}} \newcommand{\Tors} {\ensuremath{\mathrm{Tors}}} \newcommand{\Div} {\ensuremath{\mathrm{Div}}} %\swapnumbers \newtheorem{theorem}{Theorem}[subsection] %\newtheorem{Theorem}[theorem]{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition}%[section] \newtheorem{proposition}[theorem]{Proposition}%[section] \theoremstyle{definition} \newtheorem{conjecture}[theorem]{Conjecture}%[section] \newtheorem{example}[theorem]{Example}%[section] \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{remark}[theorem]{Remark} \usepackage{cleveref} \crefname{theorem}{Theorem}{Theorems} \crefname{lemma}{Lemma}{Lemmata} \crefname{corollary}{Corollary}{Corollaries} \crefname{proposition}{Proposition}{Propositions} \crefname{definition}{Definition}{Definitions} \crefname{conjecture}{Conjecture}{Conjectures} \crefname{example}{Example}{Examples} \crefname{algorithm}{Algorithm}{Algorithms} \crefname{remark}{Remark}{Remarks} \numberwithin{equation}{subsection} \addto\captionsngerman{% \renewcommand{\refname}{Bibliography}% \renewcommand{\bibname}{Bibliography}% } \allowdisplaybreaks[1] \fancyhead[L]{} %{\footnotesize \rightmark} \fancyhead[C]{\footnotesize\textsc \leftmark} \fancyhead[R]{\footnotesize \thepage} \fancyfoot[C]{} %\renewcommand\sectionmark[1]{\markright{#1}{}} %\renewcommand\sectionmark[1]{\markleft{#1}} \renewcommand{\headrulewidth}{0pt} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%% document %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \hyphenation{} \begin{document} \nonfrenchspacing \section*{Introduction} In~\cite{WeilI}, in the following denoted by I, we proved the Weil conjecture on the complex absolute values of the eigenvalues of the Frobenius acting on the cohomology of a smooth projective variety over a finite field. We studied there the cohomology with values in a sheaf; it is about to pass the punctual properties of a sheaf with those of its cohomology. [TODO] So let $\X_0$ be a scheme of finite type over $\F_q$ and $\Fcal_0$ a $\bar{\Q}_\ell$-sheaf on $\X_0$. Fix an algebraic closure $\F$ of $\F_q$, and denote by omitting of the index $0$ the base change from $\F_q$ to $\F$ (cf.~\ref{(0.7)}). For a closed point $x_0 \in |\X_0|$ of $\X_0$ and $x \in \X(\F)$ over it, denote the Frobenius automorphism $\Frob_{x_0}^*$ of the fibre $\Fcal_x$ of $\F$ in $x$ (I, (1.11), (1.13), (1.18)). One calls it eigenvalues the \emph{eigenvalues} of $\Frob_{x_0}$ \emph{on} $\Fcal_0$. On says that $\Fcal_0$ is \emph{punctually pure} of weights $n$ if, for any $x_0 \in |\X_0|$, the eigenvalues of $\Frob_{x_0}$ on $\Fcal_0$ are algebraic integers all of whose complex conjugates have absolute values $\Norm(x_0)^{n/2}$. On says that $\Fcal_0$ is \emph{mixed} if it is an iterated extension of punctually pure sheaves; its weights are the weights of $\Fcal_0$. Our main result is the \begin{theorem}[\ref{(3.3.1)}] Let $f: \X_0 \to \Srm_0$ be a morphism of finite type of schemes over $\F_q$, and $\Fcal_0$ a sheaf mixed of weights $\leq n$ on $\X$. Then, for every $i$, the sheaf $\R^if_!\Fcal_0$ on $\Srm_0$ is mixed of weights $\leq n + i$. \end{theorem} For $\Srm_0 = \Spec(\F_q)$, the theorem says that, for every eigenvalue $\alpha$ of the Frobenius on $\H_c^i(\X, \Fcal)$, there is an integer $m \leq n + i$ (the weights of $\alpha$) such that the complex conjugates of $\alpha$ are of absolute value $q^{m/2}$, Poincaré duality permits sometimes to complete the majorisations with minorisations (\ref{(3.3.5)}). For example, if $\X_0$ is proper and smooth, and the sheaf $\Fcal_0$ is lisse (= constant tordu, in another terminology) and punctually pure of weights $n$, then the eigenvalues of the Frobenius on $\H^i(\X,\Fcal)$ are all of weight $n + i$ such that we can say that $H^i(\X,\Fcal)$ is pure of weight $n + i$. For $\Fcal_0 = \Q_\ell$ (of weight $0$), one recovers the main result of I (with the hypothesis projective and smooth replaced by proper and smooth). An easy reduction, parallel to the proof of the finiteness theorem for the $\R^if_!$ (cf.~SGA~4, XIV, 1) reduces theorem 1 to the following theorem, and to a local study at infinity of lisse punctually pure sheaves on a curve (in the following $\Crm$). \begin{theorem}[cf.~\ref{(3.2.3)}] Let $\X_0$ be a proper smooth curve over $\F_q$, $j: \Urm_0 \to \X_0$ an open dense inclusion, and $\Fcal_0$ a lisse sheaf punctually pure of weights $n$ on $\Urm_0$. Then, $H^i(\X, j_*\Fcal)$ is pure of weight $n + i$. \end{theorem} Here are the main steps of the proof. A) Cleaning up. (i) If $u: \X_0' \to \X_0$ is a finite surjective morphism with source a proper smooth curve $\X_0'$, and if one denotes by $'$ the base change by $u$, the $\H^i(\X,j_*\Fcal)$ are direct factors of the $\H^i(X',j'_*\Fcal')$. This argument permits us to reduce to the case where the local monodromy of $\Fcal$ in the points of $\X - \Urm$ is unipotent. (ii) A duality theorem assures us that $\H^i(\X,j_*\Fcal)$ and $\H^{2-i}(X,j_*(\Fcal^\vee))$ are in perfect duality, with values in $\bar{\Q}_\ell(-1)$. This reduces us to verify the complex conjugates $\alpha'$ of the eigenvalues $\alpha$ of the Frobenius on $\H^i(\X,j_*\Fcal)$ are of absolute values $|\alpha'| \leq q^{(n+i)/2}$. The difficult case is $\H^1$. \end{document}