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Let $i: Z \rightarrow X$ be a closed embedding. Need $i^*$ of an injective sheaf of abelian groups be injective? Need $i^*$ of a flabby sheaf be flabby?

Thanks :D!

Also (maybe should be a separate question) but in general does $f^*$ commute with direct products, and does it admit a left adjoint (for open embeddings yes, namely $j_!$)? Hints would be preferred over solutions for these latter two questions.

Edit: The last questions are both no: the simplest $f^*$ is for $pt \rightarrow X$, i.e. taking stalks, and taking stalks doesn't commute with arbitrary products - I cooked up a counterexample with $X = \mathbb{C}$, and countably many copies of the sheaf of holomorphic functions. In particular, $f^*$ can't have a left adjoint.

uncookedfalcon
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For sheaves with values in the category of abelian groups, the reciprocal image is an exact functor (EGA, 0, 3.7.2).

Further it commutes with inductive limits, direct sums, and tensor products (EGA, 0, 4.3.2 and 4.3.3).

Edit: if $i: U\to X$ is the inclusion of an open subset, then $i^*$ has a left-adjoint $i_!$ which is exact, hence $i^*$ preserves injective objects. But I'm not sure about the closed case.

Edit: (Stacks, 15.6.3) shows that for a closed immersion $i: Z \to X$, the functor $i_*$ has a right adjoint, but it says nothing about a left-adjoint to $i^*$. I would guess it doesn't exist in general.

  • I'm happy with all of this, but I can't see the connection with the question. What did you have in mind? Thanks :D – uncookedfalcon Mar 05 '13 at 07:17
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    Exact functors preserve finite products. – Zhen Lin Mar 05 '13 at 08:09
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    Right. I also thought that exact functors preserve injective objects, but this is not true. But it does imply that the right-adjoint, $f_*$, preserves injective objects; in general whenever F and G are adjoint functors on abelian categories C and D, where D has enough injectives, then F is exact iff G preserves injectives. –  Mar 05 '13 at 09:48
  • @Adeel: I know this is old but I was wondering if the same you wrote in your first Edit holds for any flat map $f$, not just open immersions. That is, if $f$ is a flat map, does $f^\ast$ preserve injective objects? Thanks in advance :) – Brenin Sep 28 '15 at 20:26
  • @Brenin, I don't know, better to ask as a new question! –  Sep 28 '15 at 21:17