Talk:Cotangent complex

Notation

What's the $\mathbf {L}$ that appears in the section "Early work on cotangent complexes"? When has it been defined? I think it hould be explained (and I am not able to do it, of course!!!) —Preceding unsigned comment added by 89.0.144.46 (talk) 13:33, 5 April 2010 (UTC)

It stands for "left derived functor of". $\mathbf {L} f^{*}$ is the left derived functor of the pullback functor $f^{*}$. This is standard derived category notation; I've added a link from here to the derived category article. Ozob (talk) 22:05, 5 April 2010 (UTC)

Questions on the Article

1) In the definition of cotangent complex you say "For simplicity, we will consider only the case of commutative rings" but then you go on saying "Suppose that A and B are simplicial rings and that B is an A-algebra". Are these sentences in contrast? 2) In the bibliography there is Lichtenbaum-Schlessinger's paper, but it is not clear (to me) that in the case of morphism of affine schemes the definitions coincide. Thanks 84.103.208.40 (talk) 14:36, 25 May 2010 (UTC)

1. No, I left out the word "simplicial" the first time. The case of simplicial ringed topoi is (for the very limited amount we're doing here) really the same as the case of simplicial commutative rings.
2. I'm not sure quite what your question is here. Do you mean to say that you don't see why the definition given of Lichtenbaum and Schlessinger is equivalent to that given in the article (followin Illusie and Quillen)? I think the best thing I can do is quote Quillen. From his On the (co-)homology of commutative rings:

   In [16], Lichtenbaum and Schlessinger define a satisfactory cohomology theory in dimension q ≤ 2 in the sense that their cohomology extends the exact sequence of 0.1 [of modules of differentials] to nine terms. Their method uses a free differential graded anti-commutative A-algebra resolution of B. It may be shown that their cohomology group T^q(B/A, M) coincides with our D^q(B/A, M) for q ≤ 2 and that free differential graded anti-commutative A-algebra resolutions of B may be used to compute D*(B/A, M) when B is of characteristic zero (§9).

3. It's also worth mentioning that the article is woefully incomplete when it comes to historical definitions of the cotangent complex. (Well, it's also woefully incomplete about applications, about which it says nothing. Not even deformation theory, which was the original motivation. Being the main author, I suppose I have only myself to blame for this. On the other hand I also have myself to thank for writing up as much as I did.) It doesn't describe Lichtenbaum and Schlessinger's construction, nor does it mention Grothendieck's. (Grothedieck's Categories cofibrees et complexe cotangent relatif also contains the strangest remark I've ever seen in a mathematics book. If I understood the French right, he says something like, "For lack of sufficient reading of the works of Chairman Mao, I have failed to find a suitable definition of the cotangent complex.") If you'd like to add to the article, please do so! Ozob (talk) 02:21, 26 May 2010 (UTC)

Ok, thank you for your help. I will go to my University library and take Quillen paper. Even if I am not french I can understand it: I suppose that your translation "I have failed to find" is a translation of "j'ai failli trouver". If it is the case it means "I have nearly found". But I don't know the originl french sentence, so I am just making some hypothesis. I prefer not to touch this wikipedia article since I am new in this domain, so I cannot add interesting remarks!!!. Thank you again. 84.103.208.40 (talk) 07:40, 26 May 2010 (UTC)

Discuss deformation theory

Instead of adding a section for the deformation theory using cotangent complexes, this formalism of deformation theory should be included on this page. In addition, applications such as the deformations of maps should be included, giving an explanation for

• https://mathoverflow.net/questions/114963/tangent-space-of-the-stack-overline-mathcalm-g-nx-beta — Preceding unsigned comment added by Wundzer (talk • contribs) 21:58, 28 April 2020 (UTC)