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II.1.14

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This problem has Hartshorne Height 1.

HAPPY

  • Show the complement of \mbox{Supp} s is open. If  s \in F(U) then since the stalk is a direct limit, for s to map to 0 in the stalk means it maps to 0 in some small enough open neighborhood of the point. So the any point in the complement is an interior point.
  • One example for  \mbox{Supp} F not being closed. Take  X = \mathbb{R} with the usual topology. The for any open set U consider the set of real valued functions which are 0 if  0 \in U. Then  \mbox{Supp} F is  \mathbb{R} \backslash 0

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