# Problems in some parts of Monique Hakim thesis?

In "Reminiscences of Grothendieck and his school" Luc Illusie says:

"I heard from Deligne that there were problems in some parts. (of Monique Hakim thesis).

Topos annelés et schémas relatifs, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64, Springer, Berlin, New York (1972).

My question is for a reference for these "problems". What are the problems? Where?

• I noticed there is more and more questions of this type. problems in someones thesis and that this and that said such and such about someones thesis. This seems more like history than mathematics.not many people have enough time to read 100 some page of refereed thesis and find typos. Possibly main ideas are already published.the whole issue is somehow strange and a bit disgraceful. Let say there are mistakes so what? – BigM Mar 31 '16 at 2:56
• This would be a much better question if you clearly stated the specific statements which you care about knowing the validity of. – John Pardon Mar 31 '16 at 3:12
• @BigM I understand all about "disgracefulness" of this sort of request especially if it turns out that there were only typos. But, I understand that MO is also a technical site and false beliefs have be corrected (see related questions) or a caveat (or database of technical errors) is of great value, even if errors should not paralyse creativity. +1 – Duchamp Gérard H. E. Mar 31 '16 at 3:15
• Why not email Deligne (and/or Illusie) directly to ask them about this? Then (in the spirit of John Pardon's suggestion) you could pose an MO question which says "Such-and-such in this reference is unclear to me for this-or-that reason. Does anyone have advice on how to deal with this?" – nfdc23 Mar 31 '16 at 3:30
• I think this may be a decent question, because (now that vague doubts are in the public domain and have been there for decades) it clarifies what parts of Hakim's thesis have passed muster, and what parts we ought to be more careful when citing. – Daniel Moskovich Mar 31 '16 at 10:56

The published version of the paper (it is free), yet apparently not all versions in circulation, has a footnote on that sentence:

(13) Added in April 2010: Deligne doesn’t think there was anything wrong but remembers that the objects she defined over analytic spaces were not the desired ones.

Thus, it seems there is no direct problem.

The bibliographic details are (the relevant part is on page 1110, fifth page of the pdf):

Luc Illusie, Alexander Beilinson, Spencer Bloch, Vladimir Drinfeld, and et al., Reminiscences of Grothendieck and his school, Notices Amer. Math. Soc. 57 (2010), no. 9, 1106--1115.

• Could you please provide a link? The one I found (via googling) does not have this – მამუკა ჯიბლაძე Mar 31 '16 at 18:28
• It's in the published version, which is free ams.org/notices/201009/rtx100901106p.pdf (I'd be surprised if searching for the text of the footnote would not turn it up.) – user9072 Mar 31 '16 at 18:36
• You are right, I should just google again. It is the second one in the search results (the first is this answer of yours). – მამუკა ჯიბლაძე Mar 31 '16 at 18:55
• @მამუკაჯიბლაძე I update the post, to include these thing more clearly. // You are welcome, Matt! – user9072 Mar 31 '16 at 22:21

I have no idea if it's what Illusie and Deligne are referring to, but W. D. Gillam points out some strange behavior of Hakim's Spec functor in Remark 11 of this paper. Gillam says that "there is no meaningful situation in which Hakim's Spec functor agrees with the 'usual' one," and that "if $X$ is a locally ringed space at least one of whose local rings has positive Krull dimension, Hakim's sequence of spectra yields an infinite strictly descending sequence of [morphisms of ringed spaces] $\cdots\to\text{Spec}(\text{Spec}_{\ \!}X)\to\text{Spec}_{\ \!}X\to X$."