# semicontinuity results for weights

Let $f:X\rightarrow Y$ a proper flat map between smooth schemes over a finite field $k$ and the base is integral.

For an integer $i$, we consider the the l-adic sheaf $R^{i}f_{*}\mathbb{Q_l}$ and for $y\in Y$, let $w(y)=w (H^{i}(X_{\bar{y}},\mathbb{Q_l}))$ the weight of the fiber of the sheaf at $\bar{y}$.

Is it semi-continuous for specialisation, that is to say, if $y'$ is in the closure of {y}, do we have $w(y')\leq w(y)$?

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For weights a-la BBD there is an equality for any proper $f$ (since $f_*=f_!$); see Stabilities 5.1.14. –  Mikhail Bondarko Apr 12 at 15:57