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\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}



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