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Similar questions have been asked on this site, including by myself, but none of these have been given a satisfying answer.

The question is: Why does the Grassmannian scheme represent the Grassmannian functor? I have seen many books and articles about this, and they all treat it as an exercise to the reader. I am willing to admit that I may be too stupid for the exercise, but is there a textbook or survey article that explains this in détail? I mean it is somehow obvious, but I'd like to see it worked out. Please, if possible, no stacks-project or EGA.

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    $\begingroup$ I think there is no explanation that is more detailed than the one in EGA I (2nd edition). $\endgroup$ May 2, 2021 at 13:45
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    $\begingroup$ I suspect that the reason for leaving it as an exercise is that the details are boring. $\endgroup$ May 2, 2021 at 16:03
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    $\begingroup$ The Grassmannian is a special case of a quot scheme. The construction of quot schemes is done in detail in Nitsure's article in the volume 'FGA explained' (but I agree that this might be an overkill...) $\endgroup$ May 2, 2021 at 16:07
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    $\begingroup$ In fact, what is "the" Grassmann scheme? You can construct it as a quotient of a linear group, or as a subscheme of projective space via Plücker equations, or by glueing affine pieces (the latter being essentially the EGA way). The proof will be different in each case. $\endgroup$ May 2, 2021 at 16:09
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    $\begingroup$ ...there is also Görtz-Wedhorn, Algebraic Geometry, p. 209. $\endgroup$ May 2, 2021 at 16:10

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