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I am interested in a certain result which says that if we have an open cover $F_i$ of a sheaf $F$ with each $F_i$ representable, then $F$ is representable. The reason I am interested in this is because I am learning about the construction of the fibered product of two schemes (over a fixed base scheme) using representable functors.

Now the only reference I have been able to find is supposed to be proposition 0.4.5.4 in EGA mentioned by Akhil Mathew here. But how do I find this result in EGA? There are so many volumes/chapters that I get confused! If I understand correctly, the 0 in the beginning followed by 4 means chapter 0 of EGA IV yes? But I can't find it there. Can someone familiar with EGA help me please?

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    $\begingroup$ I'd say it's part of chapter 0, that's divided between the volumes. The point might be: If it's in chapter 0 of volume I, it might be in the '71 Springer edition only. $\endgroup$ – Ben Aug 9 '13 at 11:43
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    $\begingroup$ Indeed, the cited result is in the 1971 edition. $\endgroup$ – Zhen Lin Aug 9 '13 at 12:41
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This refers to EGA I (Springer edition from 1971), Chapter 0 (which contains almost 200 pages of "Préliminaires", sic!), Paragraph 4, Proposition 5.4. In a paper I would cite this as [EGA I, Chap. 0, Prop. 4.5.4]. These functorial aspects are not covered in the original EGA I from 1960. I have to admit that it takes getting used to citations to Chapter 0 of EGA, because this chapter is extended in each volume of EGA. For example, EGA IV (Première partie) begins with §§14-23 of Chapter 0, and the actual Chapter 4 starts at page 222. For citations outside of Chapter 0, one doesn't have to add the chapter because this is clear from the volume. For example, [EGA II, Thm. 5.2.1] is Serre's cohomological criterion for affineness.

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  • $\begingroup$ Dear Martin, thanks for your answer. I am looking at my EGA I, chapter 0 but the problem is that there is only upto 4.4.8. Do I have the wrong edition? $\endgroup$ – user38268 Aug 9 '13 at 12:56
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    $\begingroup$ Yes, you have the original edition. You can find the publisher and the date of publication of a book by looking at its first pages. The original EGA volumes are free available at nundam, but the Springer edition of EGA I (which is twice as long as the original edition) can be found at the usual other sources ... $\endgroup$ – Martin Brandenburg Aug 9 '13 at 12:59

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