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These days I found a mysterious page on Google books describing a book entitled On the De Rham cohomology of schemes by Grothendieck, Coates, and Jussila.

At once I thought this was an error and Google books had miss-indexed Exposé IX of Dix Exposés sur la Cohomologie de Schémas. However, the references diverge both in their titles:

Crystals and the De Rham Cohomology of Schemes

vs.

On the De Rham cohomology of schemes,

as well as their numbers of pages (53 in Dix Exposés vs. 106 in this reference).

The book is even cited in Cisinski 's Habilitation thesis as

[Gro66] A. Grothendieck « On the de Rham cohomology of schemes » Publ. Math. IHES 29 (1966), p. 93--103.

Maybe this is "On the de Rham cohomology of algebraic varieties", but again, the references diverge in page number and title.

So, does this book really exists, and, if so, is it available anywhere?

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    $\begingroup$ (My reasons for not promptly deciding that this is just an indexing eror is that Google Books usually does a great job of indexing unpublished/unavailable/lost references, such as the second volume of Görtz--Wedhorn's textbook on schemes (see here) or abandoned book projects) $\endgroup$
    – Emily
    Mar 26, 2020 at 18:34
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    $\begingroup$ Past issues of Publ. IHES (like most French journals) are freely available at Numdam: Issue 29 (1966) is here: Grothendieck "On the de Rham cohomology of algebraic varieties" 95-103. $\endgroup$
    – YCor
    Mar 26, 2020 at 18:38
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    $\begingroup$ It might be generated by a Google bug: frequently for a thesis, PhD or Habilitation, Google refers to names of the thesis committee as authors. Also "I. Coates" doesn't seem to exist and it's probably John Coates. Coates was certainly too young in 1965 to be in such a committee. Anyway, it might be, for any reason, that this "google books" page was automatically generated and messes up several facts (it would sound unlikely that a then published 100-page paper of Grothendieck would be now beyond radars). $\endgroup$
    – YCor
    Mar 26, 2020 at 18:48
  • $\begingroup$ Tlön, Uqbar, Orbis Tertius $\endgroup$
    – David Tonhofer
    Mar 27, 2020 at 12:21

2 Answers 2

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I think your hunch is correct that Google is in error, and that "Crystals and the de Rham cohomology of schemes" refers to the article in Dix Exposés. I doubt there is a book by the same name and author. There is also "On the de Rham cohomology of algebraic varieties" but that's different. The first is a long proposal for crystalline cohomology (which hadn't been developed at that point), and the second is an extract from a letter to Atiyah giving a proof of the algebraic de Rham theorem. Does that help?

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  • $\begingroup$ Yes; thanks! My hopes on reading such an article are now shattered, but oh well :P $\endgroup$
    – Emily
    Mar 28, 2020 at 3:21
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As an addendum to Donu's answer, I quote from the Table des matieres of Dix Exposés:

IX GROTHENDIECK (Alexander), Crystals and the De Rham Cohomology of schemes (notes by I. Coates and O. Jussila), IHES Decembre 1966, 54 p.

So it is certainly Dix Exposés. And "I. Coates" is most likely John Coates. According to Wikipedia Coates was born in '45 (so he would've been 21 at the time), and he moved to Paris to study at the ENS after obtaining a Bachelor's degree from ANU.

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  • $\begingroup$ The notes can be found here, page 306 (=p314 of the pdf); it's written "I. Coates" as well at that page. $\endgroup$
    – YCor
    Mar 26, 2020 at 23:46
  • $\begingroup$ Thanks for the information! :) $\endgroup$
    – Emily
    Mar 28, 2020 at 3:24
  • $\begingroup$ PS: I got the confirmation from Pierre Berthelot (who's quoted in this chapter) that it's John coates, and "I. Coates" is a typo. $\endgroup$
    – YCor
    Mar 30, 2020 at 17:25

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