Talk:Luc Illusie

Problem with a paragraph of the current text
I have a problem (several problems, in fact) with the paragraph : »Results in Illusie's dissertation have been called impressing for their "overwhelming generality", but the need for highly elaborate techniques is seen as a disadvantage. The mere definition of Illusie's cotangent complex has been called complicated, and therefor a variant of the cotangent complex by Pierre Berthelot has been praised ».

I do not think that it accurately reflects the situation, nor the source. In agreement with Vincent Verheyen who wrote it and found my explanation in the "edit summary" unclear, I shall try here to explain in more detail my reasons.

The source is Langholf, Fabian (27 November 2011). Atiyah classes with values in the truncated cotangent complex (PDF) (Diploma). University of Bonn. p. 1.

[Note that the reference is given to the arxiv version, but there is a published version, Mathematische Nachrichten 286, 1305-1325 (2013), DOI: 10.1002/mana.201200069]

F. Langholf writes : « Illusie’s results impress by their overwhelming generality, but they require highly elaborate techniques. Already the definition of his cotangent complex is complicated. Fortunately, there is an easier variant of this complex, introduced by Berthelot in [SGA6, Sect. VIII.2]. It is obtained from Illusie’s cotangent complex by truncation (see [Ill, Cor. III.1.2.9.1]), thus we call it the truncated cotangent complex LX|S of a morphism X → S. Recently, using the easier complex, Huybrechts and Thomas managed to prove results similar to Illusie’s with more elementary methods that also cover the deformation theory of complexes as objects in the derived category. »

This is a comment on the definition and variants of the cotangent complex in various situations. Obviously, depending on the applications one has in mind, a certain degree of generality is needed and for his own work, Langholf only needed a less general version, and thus was happy to be able to use a more elementary construction.

My first problem, thus, is with relevancy. This discussion of some variants may be relevant in the article on the complex cotangent, but in my opinion, not in the article on Illusie himself. Or we should include all variants of every result he proved or concept he introduced (for instance there is another, even more complicated, version elaborated by Olson and Gabber in 2005, based in particular on Illusie’s work, see M. Olsson, The logarithmic cotangent complex. Math. Ann. 333 (2005), no. 4, 859-931] !

I have two other, even more serious, problems with the use of this reference (which is a Diplomarbeit, that is a work between a Master’s thesis and a Ph D).

1) The paragraph in Wikipedia as it stands now does not for me reflect the meaning of the source and in particular it sounds more critical of Illusie’s construction than the source. It may be a problem of English, but in any case, this should be corrected.

2) The « easier variant of the complex » is in fact due to Grothendieck (Categories cofibrées additives et complexe cotangent relatif. Lecture Notes in Mathematics, No. 79 Springer-Verlag, Berlin-New York 1968), not Berthelot. Berthelot presented it in SGA6, VIII.2. This was a point of departure of Illusie’s more general construction, as it is explained in "Reminiscences of Grothendieck and His School", Notices of the AMS, 2010, http://www.ams.org/notices/201009/rtx100901106p.pdf, p. 1109. The formulation of the source is a bit ambiguous on this point (and I do not know what the author exactly means with the word « introduces »), but the Wikipedia formulation attributes the origin of the concept itself to Berthelot, which is not the case. It may also suggest that the easier variant is a later simplification of Illusie’s construction, while it was a first step (and not sufficient for some applications).

For these reasons, this paragraph did not seem appropriate to me. This is why I replaced it (keeping the issue of generality) by a paragraph explaining briefly why Illusie (and others) needed his generalization of Grothendieck’s construction.

I hope that this explains more clearly why I changed the paragraph. Thank you. Cgolds (talk) 17:37, 30 August 2016 (UTC)


 * Dear Cgolds. Thank you for your elaborate clarifications. Above, you wrote:

"This is a comment on the definition and variants of the cotangent complex in various situations. Obviously, depending on the applications one has in mind, a certain degree of generality is needed and for his own work, Langholf only needed a less general version, and thus was happy to be able to use a more elementary construction."


 * Further more, you addressed a problem of relevancy, regarding any criticism or praise of Luc Illusie's results related to the cotangent complex. Would you agree that Illusie's results regarding the 'cotangent complex' are perhaps of his most notable mathematical results? If so, I would suggest a minor discussion of those results is not redundant on his page at the English Wikipedia, as he is mostly notable through his mathematical works. At the same time (coming back to the passage which I mentioned above), I understand that your criticism claims that the work by Fabian Langholf only refers to a specific use of the cotangent complex. Nevertheless, I fail to understand which specific use that would be. But further more, I would fail to understand why that specific use (and the accompanying respective praise and criticism of the 'cotangent complex') would be merely marginal? Mathematicians can come along and claim this or that use (and such or such praise/criticism) is marginal or not. However, if we do not get a published source for such claims, then where to draw the line, you see? Those are my remarks on the first things you wrote.  You also wrote:

"The « easier variant of the complex » is in fact due to Grothendieck, not Berthelot."


 * I cannot find such clear words in the source ("Reminiscences of Grothendieck and His School") you stated. Perhaps you could help me a bit on that?


 * I hope third party editors could come along and voice their opinions, on those matters for which attention has been asked.
 * — Preceding unsigned comment added by Verheyen Vincent (talk • contribs) 14:23, 1 September 2016 (UTC)


 * Dear Vincent Verheysen


 * Illusie's thesis is certainly not his most important result, he has many others ! And I explained this result briefly in the paragraph intended to replace yours (now : just under your paragraph as you put yours back), while your paragraph did not explain Illusie's result or its use : it just says that Langholf does not use it at all in his Diplomarbeit (and add criticisms which are not in Langholf).  Langholf uses a previous version of the cotangent complex, due to Grothendieck and simpler (and the resultats developed later with this version), and sufficient for Langholf's uses.  You can for instance read the review of his paper in Zentralblatt. As for Grothendieck and Berthelot, I gave the references above. I refered to the paper on "Reminiscences"  not for this, but for the development of Illusie's idea and thesis. Yours Cgolds (talk) 09:09, 2 September 2016 (UTC)


 * (indented above par.) I agree with Cgolds that the paragraph he finds problematic does not belong in this article. One possibility to include the relevant content would be to have a discussion, in a more general context, of Illusie's very abstract and general style of mathematics (which I guess is an avatar of Grothendieck's vision) versus a more "pragmatist" approach to mathematics which eschews large technical baggage in favour of immediate readability. Of course such a discussion would have to be correctly sourced (with more relevant sources than an informal remark in a thesis manuscript). jraimbau (talk) 09:52, 2 September 2016 (UTC)
 * It is clear from this discussion that Illusie's definition of the cotangent complex and more generally the results in Illusie's dissertation appeared at a period where this subject were rapidly evolving, and that most variants of the definition of the cotangent complex use Illusie's results and refer to it. Therefore any criticism of his results is unfair, if this criticism is not placed in an accurate historical analysis of the subject. Thus, such a criticism is misplaced here; if it would be relevant, it should be placed in the historical section of Cotangent complex.
 * Moreover, this criticism is original research, as it appears in a master thesis, which is not a reliable source. Thus Langholf quotation does not belongs to Wikipedia.
 * Therefore, I completely agree with . D.Lazard (talk) 10:48, 2 September 2016 (UTC)
 * The paragraph is simply unacceptable. It reveals a deep misunderstanding of the nature of the cotangent complex and of the importance of Illusie's work. The cotangent complex is a homotopical construction, and the general framework for making homotopical constructions in algebraic geometry was poorly understood at the time.  To call the two-term complex introduced by Grothendieck a "simpler variant" is to miss the point. Grothendieck gave the best definition he could and proved that in some cases it behaved correctly; Illusie gave a definition that worked unconditionally.  Moreover, the cited source does not even "praise" Grothendieck's construction to the exclusion of Illusie's; rather, it simply remarks that Illusie's definition is by its nature difficult to compute with.  Rightly understood, this is not so much a criticism as a remark on the difficulty of studying highly general situations.
 * I was going to remove the paragraph myself, but D. Lazard beat me to it. Ozob (talk) 12:56, 2 September 2016 (UTC)

Thank you for all your comments  ! Cgolds (talk) 08:09, 3 September 2016 (UTC)