By Quillen-Lichtenbaum theorem the weight $i$ mod $l$ motivic complex is quasi-isomorphic to $\tau^{\leq i}R\alpha_{*}\mu_l^{\otimes i}$ where $\alpha$ is the forgetful functor sending etale sheaves to Zariski sheaves. By the purity (excision) for motivic cohmology this implies that for a codimension $c$ closed immersion $f: Z\hookrightarrow X$ of smooth schemes $f^!\tau^{\leq i}R\alpha_{*}\mu_l^{\otimes i}=\tau^{\leq i-c}R\alpha_{*}\mu_l^{\otimes (i-c)}[-2c]$. I was wondering whether it is possible to show the purity directly and without Quillen-Lichtenbaum?
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