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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 23rd 2010
   • PermaLink
   Author: Urs
   Format: Markdownadded more details to [[cotangent complex]]

   added more details to cotangent complex

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 27th 2017
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexSeeing the discussion of cotangents over [here](https://nforum.ncatlab.org/discussion/7951/microlocal-analysis/), I was wondering what the connection to cotangent complex is. Connections are made in both directions from [[cotangent complex]] to [[Kähler differential]], but there's no link either way to [[cotangent bundle]], which is the target for cotangent vector/space. So what could be added? I see from this [MO answer](https://mathoverflow.net/a/2804/447) >The first example of a cotangent complex is that of a smooth $S$-scheme $X$. Then the cotangent complex in this case is just the cotangent bundle $\Omega_{X/S}$.

   Seeing the discussion of cotangents over here, I was wondering what the connection to cotangent complex is. Connections are made in both directions from cotangent complex to Kähler differential, but there’s no link either way to cotangent bundle, which is the target for cotangent vector/space.

   So what could be added? I see from this MO answer

   The first example of a cotangent complex is that of a smooth $S$-scheme $X$. Then the cotangent complex in this case is just the cotangent bundle $\Omega_{X/S}$.

3. • CommentRowNumber3.
   • CommentAuthorHarry Gindi
   • CommentTimeAug 27th 2017
   • (edited Aug 27th 2017)
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   Author: Harry Gindi
   Format: MarkdownItexThe cotangent sheaf/bundle is the geometric version of the module of Kahler differentials. The Cotangent complex is the derived version. The statement about smooth relative schemes having the cotangent complex concentrated in degree 0 arises from the lifting property, and smoothness means there are no higher-dimensional obstructions if you unravel the definitions. Unramifiedness means that the Kahler differentials vanish, and together (etale), they imply a unique lifting, so the cotangent complex (homotopically) vanishes.

   The cotangent sheaf/bundle is the geometric version of the module of Kahler differentials. The Cotangent complex is the derived version. The statement about smooth relative schemes having the cotangent complex concentrated in degree 0 arises from the lifting property, and smoothness means there are no higher-dimensional obstructions if you unravel the definitions. Unramifiedness means that the Kahler differentials vanish, and together (etale), they imply a unique lifting, so the cotangent complex (homotopically) vanishes.

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 30th 2017
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   Author: David_Corfield
   Format: MarkdownItexThanks. This would be more useful to the world placed in the relevant entry, if it could be made to fit in.

   Thanks. This would be more useful to the world placed in the relevant entry, if it could be made to fit in.

5. • CommentRowNumber5.
   • CommentAuthorHarry Gindi
   • CommentTimeAug 31st 2017
   • (edited Aug 31st 2017)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexHere's an interesting question that I haven't seen answered but surely has a well-known answer: Can you glue "the" (i.e. representatives of) cotangent complexes of open affines together to form a representative of the global cotangent complex on a general scheme? Because the cotangent complex is a homotopical object, it isn't obvious to me that it needs to satisfy the glueing condition strictly. In particular, for a map $S\to T$ of schemes and an atlas of open affines $\{\operatorname{Spec}(U_i) \to S\}$, if you take the cotangent $U_i$ module to be the simplicial $U_i$-module corepresenting $$\operatorname{Der}_{\mathcal{O}_S} (\bar{U_i}, -)$$ where $\bar{U_i}$ is the bar construction on $U_i$ (see André), do the associated quasicoherent simplicial sheaves $\mathbb{L} \Omega_{U_i/T}$ on $\operatorname{Spec}(U_i)$ glue together to the cotangent complex $\mathbb{L}\Omega_{S/T}$ Also, another question: Is the construction strictly local on the base, or only after passing to derived categories? If not the bar construction, are there any constructions that _can_ be glued strictly and/or are local on the base?

   Here’s an interesting question that I haven’t seen answered but surely has a well-known answer: Can you glue “the” (i.e. representatives of) cotangent complexes of open affines together to form a representative of the global cotangent complex on a general scheme? Because the cotangent complex is a homotopical object, it isn’t obvious to me that it needs to satisfy the glueing condition strictly.

   In particular, for a map $S\to T$ of schemes and an atlas of open affines $\{\operatorname{Spec}(U_i) \to S\}$, if you take the cotangent $U_i$ module to be the simplicial $U_i$-module corepresenting

   $$\operatorname{Der}_{\mathcal{O}_S} (\bar{U_i}, -)$$

   where $\bar{U_i}$ is the bar construction on $U_i$ (see André), do the associated quasicoherent simplicial sheaves $\mathbb{L} \Omega_{U_i/T}$ on $\operatorname{Spec}(U_i)$ glue together to the cotangent complex $\mathbb{L}\Omega_{S/T}$

   Also, another question: Is the construction strictly local on the base, or only after passing to derived categories? If not the bar construction, are there any constructions that can be glued strictly and/or are local on the base?

6. • CommentRowNumber6.
   • CommentAuthorjbian
   • CommentTimeFeb 27th 2020
   • PermaLink
   Author: jbian
   Format: Text@Harry Gindi - obviously late answer, but the answer to your question is Example 17.2.4.4 in SAG

   @Harry Gindi - obviously late answer, but the answer to your question is Example 17.2.4.4 in SAG