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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?

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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 $M$. – 2017-01-06
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    Can you explain what is meant by "locally generated by sections?" – 2017-01-06

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