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A theorem of Tate states that for any affinoid space $X=\mathrm{Sp}\:A$ and a coherent sheaf $F$ we have $H^i(X,F)=0$ for all $i>0$.

Is this theorem true for merely quasi-coherent sheaves? The naive analogue in the algebraic case is true so I am not sure what to expect.

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  • $\begingroup$ How do you define quasi-coherent sheaves in this context? $\endgroup$
    – Simon Wadsley
    Commented Aug 4, 2019 at 14:13
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    $\begingroup$ There is no known satisfactory definition of a q-c sheaf in an analytic setting, either complex or rigid. The broadest definition is a sheaf that is locally a filtered direct limit of coherent sheaves. The theorem is false in that generality (which is usually taken as a refutation of that defn). Cohomology commutes with filtered direct limits, so a global filtered direct limit has no cohomology, but Gabber gives examples, both complex and rigid, of a sheaf with $H^1$ that is loc filt direct limit. complex case $\endgroup$
    – Ben Wieland
    Commented Aug 4, 2019 at 14:35
  • $\begingroup$ @BenWieland looks like that solves it, many thanks! $\endgroup$
    – user143954
    Commented Aug 4, 2019 at 17:02

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