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I found the answer to this question while typing it up, but since I've already written it, it is probably worthwhile to post-and-answer in case someone finds it useful. Let $S=\operatorname{Spec}\...

Which of the following operations on étale sheaves $A$ commute with tensor powers? (eg. for instance $i^*(A^{\otimes n})=(i^*(A))^{\otimes n}$?) $i^*(A)$, $i$ closed immersion. $i_*(A)$ $i^!(A)$ $i_!(...