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Here is the definition of closed immersion given on Stacks Project.

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In Hartshorne (II, Section 3), a closed immersion of schemes is only defined by the first two properties. What does "locally generated by sections" mean? And why is it considered to be an essential characteristic of a closed immersion?

D_S
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    Hartshorne defines this in the category of schemes, hence the map in (2) is a map of quasi-coherent sheaves on $X$, hence the kernel is also quasi-coherent and thus automatically locally generated by sections, since it is locally of the form $M^\sim$ for a module $M$. – MooS Jan 06 '17 at 07:23
  • Can you explain what is meant by "locally generated by sections?" – D_S Jan 06 '17 at 23:07
  • Maybe it is too late now. But for the notion of locally generated by sections, one may turn to https://stacks.math.columbia.edu/tag/01B1 . – Hetong Xu Jan 21 '22 at 07:19
  • For the reason why the condition is needed, one may turn to the discussion at the begining of this section: https://stacks.math.columbia.edu/tag/01C1 . The following post https://math.stackexchange.com/questions/4235752/ideal-sheaf-is-quasi-coherent-if-and-only-if-it-is-locally-generated-by-sections may helps when understanding the discussion in the Stacks Project. – Hetong Xu Jan 21 '22 at 11:55

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