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Does anybody have a good reference on the Grassmanian and its universal property?

I am reading this paper on Quot schemes: https://arxiv.org/abs/math/0504590

Where the Grassmanian is constructed, but its representability and its universal quotient are "exercises". In particular exercise (2) in chapter 1.

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I suggest you:

  1. Eisenbud, Harris - The Geometry of Schemes, (2000) Springer Verlag, paragraph III.2.7;
  2. Eisenbud, Harris - 3264 & All That, Intersection Theory in Algebraic Geometry, chapters 3 and 4 (click);
  3. Görtz, Wedhorn - Algebraic Geometry I, (2010) Vieweg+Teubner Verlag, paragraphs from 8.4 to 8.10;
  4. Kleiman S. L. - Geometry on Grassmannians and Applications to Splitting Bundles and Smoothing Cycles, Pubblications Mathématiques de l'I.H.É.S., 36 (1969) 281-297 (clack);
  5. Vakil - FOAG (December 29 2015 version), paragraph 16.7 (clock).
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    $\begingroup$ And EGA I, $\S$ 9.7 (Springer edition). $\endgroup$ – Laurent Moret-Bailly Dec 20 '16 at 11:12
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    $\begingroup$ A very elegantly written and comprehensive treatment of this and related matters in the context of analytic spaces, but which applies verbatim to schemes (and seems to be the original source upon which the treatment in the revised EGA I from 1971 was based) is given in Exposes 9--14 by Grothendieck in the Seminaire Cartan, tome 13 (1960-61), especially Expose 12 for the case of Grassmannians; see numdam.org/numdam-bin/feuilleter?j=SHC $\endgroup$ – nfdc23 Dec 20 '16 at 13:52
  • $\begingroup$ I found Kleiman's exposition particularly easy to grasp, possibly because its construction of the Grassmanian is quite similar to the one in my original paper. @nfdc23 that sounds very interesting! But I'll need some more time to wrestle with that french. :P $\endgroup$ – foaly Dec 20 '16 at 15:49

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