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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 1st 2016
   • (edited Feb 1st 2016)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI will be compiling something that ought to work as lecture notes for a course that introduces stable homotopy theory for people with background in homotopy theory, and aimed at understanding the Adams-Novikov spectral sequence, together with some extra material on the modern picture via descent down to $Spec(\mathbb{S})$. Just because as an $n$Lab entry that fits well into the existing growing lecture note series titled "geometry of physics", I am putting that now into an entry that is titled * _[[geometry of physics -- stable homotopy types]]_ (as continuation of the previous _[[geometry of physics -- homotopy types]]_) but for the time being there won't be any physics here, except maybe in the guise of some links on further reading as it gets to the meaning of the stratification of $Spec(MU)$. For the moment the entry has mainly just the intended skeleton, I will be adjusting that a little more and then start filling it with serious content.

   I will be compiling something that ought to work as lecture notes for a course that introduces stable homotopy theory for people with background in homotopy theory, and aimed at understanding the Adams-Novikov spectral sequence, together with some extra material on the modern picture via descent down to $Spec(\mathbb{S})$.

   Just because as an $n$Lab entry that fits well into the existing growing lecture note series titled “geometry of physics”, I am putting that now into an entry that is titled

   • geometry of physics – stable homotopy types

   (as continuation of the previous geometry of physics – homotopy types) but for the time being there won’t be any physics here, except maybe in the guise of some links on further reading as it gets to the meaning of the stratification of $Spec(MU)$.

   For the moment the entry has mainly just the intended skeleton, I will be adjusting that a little more and then start filling it with serious content.

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeFeb 2nd 2016
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItex>special importance is carried by those [[E-∞ rings]] such that $Spec(E) \to Spec(\mathbb{S})$ is already a [[covering]] is there another way to characterise these E-∞ rings? In the ordinary case of rings whose specs cover $\mathbb{Z}$ this is having characteristic 0? Does an E-∞ ring have a characteristic?

   special importance is carried by those E-∞ rings such that $Spec(E) \to Spec(\mathbb{S})$ is already a covering

   is there another way to characterise these E-∞ rings?

   In the ordinary case of rings whose specs cover $\mathbb{Z}$ this is having characteristic 0? Does an E-∞ ring have a characteristic?

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeFeb 2nd 2016
   • (edited Feb 2nd 2016)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexNaturally there's work here: * Markus Szymik, _Commutative S-algebras of prime characteristics and applications to unoriented bordism_, [arXiv:1211.3239](http://arxiv.org/abs/1211.3239) * Markus Szymik, _String bordism and chromatic characteristics_, [arXiv:1312.4658](http://arxiv.org/abs/1312.4658) the latter giving a refinement beyond primes as charactistic. And * Andrew Baker, _Characteristics for E∞ ring spectra_, [arXiv:1405.3695](http://arxiv.org/abs/1405.3695)

   Naturally there’s work here:

   • Markus Szymik, Commutative S-algebras of prime characteristics and applications to unoriented bordism, arXiv:1211.3239
   • Markus Szymik, String bordism and chromatic characteristics, arXiv:1312.4658

   the latter giving a refinement beyond primes as charactistic. And

   • Andrew Baker, Characteristics for E∞ ring spectra, arXiv:1405.3695
4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeFeb 2nd 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexDavid, re #2, I am not sure if I see the connection here, could you expand on what you have in mind? For instance $Spec(\mathbb{Z})$ trivially covers itself.

   David, re #2, I am not sure if I see the connection here, could you expand on what you have in mind? For instance $Spec(\mathbb{Z})$ trivially covers itself.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeFeb 2nd 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am splitting off entries for facts that are usefully pointed to in their own right, such as * _[[Thom's theorem]]_ * _[[universal orientation on MU]]_ and I have been adding further literature pointers and cross-links there. To be expanded further.

   I am splitting off entries for facts that are usefully pointed to in their own right, such as

   • Thom’s theorem

   • universal orientation on MU

   and I have been adding further literature pointers and cross-links there. To be expanded further.

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeFeb 2nd 2016
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexUrs re #2, probably a mistaken tired thought. I was wondering for which rings, $R$, is $Spec(R) \to Spec(\mathbb{Z})$ a cover. Is that not when $\mathbb{Z} \to R$ is some kind of mono? And doesn't that connect to the characteristic of $R$. Anyway, the trip took me to see that $E_{\infty}$-rings have characteristics too.

   Urs re #2, probably a mistaken tired thought. I was wondering for which rings, $R$, is $Spec(R) \to Spec(\mathbb{Z})$ a cover. Is that not when $\mathbb{Z} \to R$ is some kind of mono? And doesn’t that connect to the characteristic of $R$.

   Anyway, the trip took me to see that $E_{\infty}$-rings have characteristics too.