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![]() | This article contains a translation of Luc Illusie from fr.wikipedia. |
![]() | This article contains a translation of Luc Illusie from de.wikipedia. |
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)
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.
The « easier variant of the complex » is in fact due to Grothendieck, not Berthelot.
Thank you for all your comments ! Cgolds ( talk) 08:09, 3 September 2016 (UTC)
This article must adhere to the biographies of living persons (BLP) policy, even if it is not a biography, because it contains material about living persons. Contentious material about living persons that is unsourced or poorly sourced must be removed immediately from the article and its talk page, especially if potentially libellous. If such material is repeatedly inserted, or if you have other concerns, please report the issue to this noticeboard.If you are a subject of this article, or acting on behalf of one, and you need help, please see this help page. |
![]() | This article is rated Stub-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||||||||||||
|
![]() | This article contains a translation of Luc Illusie from fr.wikipedia. |
![]() | This article contains a translation of Luc Illusie from de.wikipedia. |
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)
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.
The « easier variant of the complex » is in fact due to Grothendieck, not Berthelot.
Thank you for all your comments ! Cgolds ( talk) 08:09, 3 September 2016 (UTC)