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    $\begingroup$ What about finite direct sums? (Or is that a basic sheaf theory fact?) $\endgroup$ Commented Jul 19, 2020 at 14:05
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    $\begingroup$ @ChrisGerig Flabbyness is preserved for arbitrary direct products and a finite sum of abelian sheaves is the same as their finite product. $\endgroup$ Commented Jul 19, 2020 at 14:24
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    $\begingroup$ Thank you, Alexander: this is a perfect counterexample. I hope it will find its place in some basic book using sheaves, or maybe become an item in the Stacks Project. $\endgroup$ Commented Jul 19, 2020 at 18:08
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    $\begingroup$ @GeorgesElencwajg It will. :-) I am writing a sequel to ams.org/open-math-notes/omn-view-listing?listingId=110823 that will be about homological methods in commutative algebra. The first half or so is about homological algebra per se, and there sheaves appear as a fundamental recurring example. I am putting Alexander's counterexample as a guided exercise. $\endgroup$ Commented Jul 24, 2020 at 13:48
  • $\begingroup$ Dear@Andrea, this is wonderful news! When I wrote about my hope, I actually thought that its realization was not very probable: I'm happy you are showing me that I was unduly pessimistic! I am delighted that you are writing a follow-up to the book I downloaded a few months ago from the AMS site, and which I much appreciated. Thanks a lot for that great document and my best wishes for the future one. $\endgroup$ Commented Jul 25, 2020 at 7:55