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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 1st 2015
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added the pair of references * [[John Francis]], _Derived algebraic geometry over $\mathcal{E}_n$-Rings_ ([pdf](http://math.northwestern.edu/~jnkf/writ/thezrev.pdf)) * [[John Francis]], _The tangent complex and Hochschild cohomology of $\mathcal{E}_n$-rings_ ([pdf](http://math.northwestern.edu/~jnkf/writ/cotangentcomplex.pdf)) to the References-section at various related entries, such as at _[[derived noncommutative geometry]]_. (Thanks to Adeel in the MO-comments [here](http://mathoverflow.net/q/201643/381). He watched me ask the question there on three different forums, before then giving a reply on the fourth ;-)

   I have added the pair of references

   • John Francis, Derived algebraic geometry over $\mathcal{E}_n$-Rings (pdf)

   • John Francis, The tangent complex and Hochschild cohomology of $\mathcal{E}_n$-rings (pdf)

   to the References-section at various related entries, such as at derived noncommutative geometry.

   (Thanks to Adeel in the MO-comments here. He watched me ask the question there on three different forums, before then giving a reply on the fourth ;-)

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 1st 2015
   • PermaLink
   Author: Urs
   Format: MarkdownItexAlso added there and at _[[noncommutative algebraic geometry]]_ pointer to * [[Mikhail Kapranov]], _Noncommutative geometry based on commutator expansions_ ([arXiv:9802041](http://arxiv.org/abs/math/9802041)) which discusses formal NCG perturbing around ordinary abelian schemes. This is pretty much exactly what I had in mind when I asked the question, though of course its intriguing that in $E_{n \geq 2}$-algebraic geoemtry a derived version of this comes out by itself, without being put in by hand.

   Also added there and at noncommutative algebraic geometry pointer to

   • Mikhail Kapranov, Noncommutative geometry based on commutator expansions (arXiv:9802041)

   which discusses formal NCG perturbing around ordinary abelian schemes. This is pretty much exactly what I had in mind when I asked the question, though of course its intriguing that in $E_{n \geq 2}$-algebraic geoemtry a derived version of this comes out by itself, without being put in by hand.

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeApr 1st 2015
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   Author: zskoda
   Format: MarkdownItexI do not understand. Francis' higher rings are commutative...

   I do not understand. Francis’ higher rings are commutative…

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeApr 1st 2015
   • (edited Apr 1st 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThey are in between associative and commutative. For $n = 1$ they are associative, for $n = \infty$ they are commutative. But for $n \geq 2$ their $\pi_0$ is always commutative, in this way they encode, for $n \geq 2$, a kind of formal noncommutative thickening of an abelian affine scheme.

   They are in between associative and commutative. For $n = 1$ they are associative, for $n = \infty$ they are commutative. But for $n \geq 2$ their $\pi_0$ is always commutative, in this way they encode, for $n \geq 2$, a kind of formal noncommutative thickening of an abelian affine scheme.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeApr 1st 2015
   • PermaLink
   Author: Urs
   Format: MarkdownItexOh, wow, now I see that you, Zoran, had hidden an entry on _[[Kapranov's noncommutative geometry]]_ all along.

   Oh, wow, now I see that you, Zoran, had hidden an entry on Kapranov’s noncommutative geometry all along.

6. • CommentRowNumber6.
   • CommentAuthorzskoda
   • CommentTimeApr 1st 2015
   • (edited Apr 1st 2015)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexPlease do not use word "abelian" in this context. Abelian in *commutative* algebraic geometry is about the class of abelian varieties, which is a subclass of group schemes, which are abelian as groups. In the same time they are never affine. So for an algebraic geometer words "abelian affine scheme" do not parse. As far as thickenings coming naturally or not, Rosenberg and Lunts had general framework in 1996 a la Grothendieck of noncommutative thickenings of (generally noncommutative) subschemes within ambient noncommutative in context of abelian categories of quasicoherent sheaves. It is internal viewpoint similar to working in topos theory and nothing is added by hand. Part I <http://www.mpim-bonn.mpg.de/preblob/3894>, Part II <http://www.mpim-bonn.mpg.de/preblob/3916> Third comment: taking derived to noncommutative world is adding some level of commutativity in the picture. You see, in rough terms, abelian categories of sheaves do not reduce to homotopy theory, while triangulated categories do. And homotopy theory is essentially spaces, commutative. As late Rosenberg used to say, the notions are usually much easier as many things from abelian context coalesce once you pass to derived category. Specially in repesentation theory, where the functors induced on the derived categories of representations behave better and are easier to study but loose some information living on abelian level.

   Please do not use word “abelian” in this context. Abelian in commutative algebraic geometry is about the class of abelian varieties, which is a subclass of group schemes, which are abelian as groups. In the same time they are never affine. So for an algebraic geometer words “abelian affine scheme” do not parse.

   As far as thickenings coming naturally or not, Rosenberg and Lunts had general framework in 1996 a la Grothendieck of noncommutative thickenings of (generally noncommutative) subschemes within ambient noncommutative in context of abelian categories of quasicoherent sheaves. It is internal viewpoint similar to working in topos theory and nothing is added by hand. Part I http://www.mpim-bonn.mpg.de/preblob/3894, Part II http://www.mpim-bonn.mpg.de/preblob/3916

   Third comment: taking derived to noncommutative world is adding some level of commutativity in the picture. You see, in rough terms, abelian categories of sheaves do not reduce to homotopy theory, while triangulated categories do. And homotopy theory is essentially spaces, commutative. As late Rosenberg used to say, the notions are usually much easier as many things from abelian context coalesce once you pass to derived category. Specially in repesentation theory, where the functors induced on the derived categories of representations behave better and are easier to study but loose some information living on abelian level.

7. • CommentRowNumber7.
   • CommentAuthorzskoda
   • CommentTimeApr 1st 2015
   • PermaLink
   Author: zskoda
   Format: MarkdownItex5 hidden ? Not only created and reported but even put it in a title of a nforum [thread](http://nforum.mathforge.org/discussion/981/kapranovs-ncg) :)

   5 hidden ? Not only created and reported but even put it in a title of a nforum thread :)