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Let $p:\mathbb{G}_m\to \operatorname{Spec} k$ be the structure map, and let $T$ be an algebraic $k$-torus viewed as an étale sheaf over $k$. Why is the cokernel of the canonical map $T\to p_*p^*T$ ...

This question was prompted by the two following: Constructible étale sheaves on X are étale algebraic spaces over X Naive question about constructing constructible sheaves If I have a ...