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In Geometric Invariant Theory, by Mumford, Fogarty, and Kirwan, if there is a mention of descent theory, it almost always comes along with a reference to SGA 8, Theorem 5.2 (see the end of the proof of prop 6.9 on page 119, for example). Now I know SGA 8 was never made, but I was wondering:

  1. Does anyone have a good guess as to what this theorem should say?

  2. Does anyone have a good reference for a quick and "hands off" introduction to descent theory? I am really just looking to understand the "gist" of it.

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    $\begingroup$ I wouldn't call this a quick introduction, but look at FGA, or FGA Explained (amazon.com/…) $\endgroup$ – Mahdi Majidi-Zolbanin Jan 7 '12 at 16:27
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    $\begingroup$ You can also look at his Technique de descente et théorèmes d'existence en géométrie algébrique, Parts I,II available on Numdam.org $\endgroup$ – Mahdi Majidi-Zolbanin Jan 7 '12 at 16:30
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    $\begingroup$ Actually, what Mumford cites as SGA 8 is what we nowadays call SGA 1, chapter VIII. $\endgroup$ – user2035 Jan 7 '12 at 16:43
  • $\begingroup$ You should post that as an answer, a-fortiori. $\endgroup$ – David Roberts Jan 7 '12 at 22:15
  • $\begingroup$ SGA8 is mentioned as LNM 407: Cohomologie cristalline des schémas de caractéristique P.Berthelot (See the bibliography on p.41, 43 on "Applications of Sheaves" LNM 753) But LNM 407 isnt about descent theory (only a little, no in the sense that Mumford book ask). I guess that "Le methode de la descente" by J. Giraud , Memoire del SMF , n2, 1964, could be a useful reference, is very hard for a no lover of category theory. This was the old central work about descente theory, and was the J. Giraud thesis did under Grothendieck (as professor). $\endgroup$ – Buschi Sergio Jan 8 '12 at 13:47
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For question 1, see the comment above.

Collecting the answers to question 2:

  • Grothendieck's original FGA, starting with TDTE I
  • Vistoli's chapter in FGA explained, for the connection with stacks
  • What is descent theory? for a very short overview
  • Bosch-Lütkebohmert-Raynaud, Néron Models (recommended by BCnrd in the above-mentioned thread)
  • Waterhouse, Introduction to Affine Group Schemes, containing a 20-page introduction primarily concerned with the affine case

"Community wiki" post, feel free to modify.

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