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Kevin Buzzard
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Contrary to what I guessed initially, I now think the question has a great answer: the functor is representable if and only if $M$ is locally free, and the proof is EGA I, 9.4.10.

Edit: this is an answer to an earlier version of the question.

Kevin Buzzard
  • 41.4k
  • 13
  • 166
  • 245