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1. • CommentRowNumber1.
   • CommentAuthorStephan A Spahn
   • CommentTimeMay 21st 2012
   • PermaLink
   Author: Stephan A Spahn
   Format: MarkdownItexI changed the name of the page [[Frobenius map]] to [[Frobenius morphism]] and added the descriptions à la Demazure to it.

   I changed the name of the page Frobenius map to Frobenius morphism and added the descriptions à la Demazure to it.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeMay 21st 2012
   • PermaLink
   Author: zskoda
   Format: MarkdownItexNow [[Frobenius map]] exists in old version which is a temporary cache bug version. [[Frobenius morphism]] has a redirect Frobenius morphism which is surely superfluous, while it is missing the redirect o the old name Frobenius map and Frobenious map which should exist there (this way it is very often called in algebraic geometry). I will leave it to you to see/learn how to resolve this typical situation in renaming :)

   Now Frobenius map exists in old version which is a temporary cache bug version. Frobenius morphism has a redirect Frobenius morphism which is surely superfluous, while it is missing the redirect o the old name Frobenius map and Frobenious map which should exist there (this way it is very often called in algebraic geometry). I will leave it to you to see/learn how to resolve this typical situation in renaming :)

3. • CommentRowNumber3.
   • CommentAuthorhilbertthm90
   • CommentTimeMay 21st 2012
   • PermaLink
   Author: hilbertthm90
   Format: MarkdownItexCan I suggest as someone that fought a lot with the notation of Demazure that every time we use it to put something on the nLab we update the notation? For example, if I recall correctly $Sp_k A$ just means Spec A (thought of as a k-scheme).

   Can I suggest as someone that fought a lot with the notation of Demazure that every time we use it to put something on the nLab we update the notation? For example, if I recall correctly $Sp_k A$ just means Spec A (thought of as a k-scheme).

4. • CommentRowNumber4.
   • CommentAuthorStephan A Spahn
   • CommentTimeMay 22nd 2012
   • (edited May 22nd 2012)
   • PermaLink
   Author: Stephan A Spahn
   Format: MarkdownItex> I will leave it to you to see/learn how to resolve this typical situation in renaming :) I have an instiki installation of my own and there never such an issue happened. I deleted the auto-redirects. Clicking ''edit'' in ''Frobenius map'' gives the Home Page of the nlab. [Housekeeping](http://golem.ph.utexas.edu/wiki/instiki/show/Housekeeping) says that this page probably will be deleted automatically. > Can I suggest as someone that fought a lot with the notation of Demazure that every time we use it to put something on the nLab we > update the notation? For example, if I recall correctly $Sp_k A$ just means Spec A (thought of as a k-scheme). Where else do you suggest to update notation - or the presentation of the topic in general? In Demazure $Sp_k A$ as well as $Sp A$ are used. Since skalar extension- and restriction plays an important role in some details here, to omit the $k$ in the subscript does not contribute to clarity in these cases - I think.

   I will leave it to you to see/learn how to resolve this typical situation in renaming :)

   I have an instiki installation of my own and there never such an issue happened. I deleted the auto-redirects. Clicking ”edit” in ”Frobenius map” gives the Home Page of the nlab. Housekeeping says that this page probably will be deleted automatically.

   Can I suggest as someone that fought a lot with the notation of Demazure that every time we use it to put something on the nLab we > update the notation? For example, if I recall correctly $Sp_k A$ just means Spec A (thought of as a k-scheme).

   Where else do you suggest to update notation - or the presentation of the topic in general? In Demazure $Sp_k A$ as well as $Sp A$ are used. Since skalar extension- and restriction plays an important role in some details here, to omit the $k$ in the subscript does not contribute to clarity in these cases - I think.

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeMay 22nd 2012
   • (edited May 22nd 2012)
   • PermaLink
   Author: zskoda
   Format: MarkdownItex> I have an instiki installation of my own You should teach me this, I'd be grateful, if you are in ubuntu ? I agree that $Spec_k$ is better than $Spec$ in such situations: but $Sp$ is a bit old fashioned in algebraic geometry ($Sp$ nowdays more often used for some spectra in other areas of mathematics, and occasionally for traces).

   I have an instiki installation of my own

   You should teach me this, I’d be grateful, if you are in ubuntu ?

   I agree that $Spec_k$ is better than $Spec$ in such situations: but $Sp$ is a bit old fashioned in algebraic geometry ($Sp$ nowdays more often used for some spectra in other areas of mathematics, and occasionally for traces).

6. • CommentRowNumber6.
   • CommentAuthorhilbertthm90
   • CommentTimeMay 22nd 2012
   • (edited May 22nd 2012)
   • PermaLink
   Author: hilbertthm90
   Format: MarkdownItexI wasn't so concerned with the Spec issue. I just wanted to pre-emptively prevent things like the category $M_k E$ (k-functors from k-rings to set??) or the notation $\underline{O}_k$ for affine line (just to name a few on the first page of the book).

   I wasn’t so concerned with the Spec issue. I just wanted to pre-emptively prevent things like the category $M_k E$ (k-functors from k-rings to set??) or the notation $\underline{O}_k$ for affine line (just to name a few on the first page of the book).

7. • CommentRowNumber7.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 23rd 2012
   • (edited May 23rd 2012)
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   Author: DavidRoberts
   Format: MarkdownItex$M_k E$ probably stands for something something like Modules for $Ring_k$ in $Ens$, the category of sets, but I agree it's terrible notation.

   $M_k E$ probably stands for something something like Modules for $Ring_k$ in $Ens$, the category of sets, but I agree it’s terrible notation.

8. • CommentRowNumber8.
   • CommentAuthorZhen Lin
   • CommentTimeMay 23rd 2012
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItex$M$, if I remember correctly, stands for "model". (It is the category of all $U$-small rings, where $U$ is a Grothendieck universe contained in the "real" universe $V$.)

   $M$, if I remember correctly, stands for “model”. (It is the category of all $U$-small rings, where $U$ is a Grothendieck universe contained in the “real” universe $V$.)

9. • CommentRowNumber9.
   • CommentAuthorStephan A Spahn
   • CommentTimeMay 23rd 2012
   • (edited May 23rd 2012)
   • PermaLink
   Author: Stephan A Spahn
   Format: MarkdownItex> You should teach me this, I'd be grateful, if you are in ubuntu ? I am not in ubuntu but in mac osx. Following simply the [instruction](http://golem.ph.utexas.edu/wiki/instiki/show/Installation) on the instiki site might be successfull if your device has the right underlying architecture (in my case one problem was that instiki needed to be converted into 64 bit format). If you plan to do an instiki installation on a web host you should assure that this server meets the requirements of instiki: Dreamhost which is recommended by the instiki site does not meet this requirements - at least it did not when I tried it and it was not possible to install it there. In the end a friend of mine who is an it-specialist did the installation for me on his own private server. But I don't know if my information concerning this are still up to date since I just see that the [instiki installation site](http://golem.ph.utexas.edu/wiki/instiki/show/Installation) has been updated on April 30, 2012 - after my trial. And I see that there is a section on ubuntu on this site. > $M_k E$ $M_k E$ stands for the category of the copresheaves on the category of $k$-rings (=rings $R$ with a specified morphism $k\to R$) $M_k$. > $M$, if I remember correctly, stands for "model". (It is the category of all $U$-small rings, where $U$ is a Grothendieck universe contained in the "real" universe $V$.) In ''lectures on $p$-divisible groups'' such size issues are omitted. Nevetheless this might be the reason for the letter $M$. But I think it is not a good notation since it contains no information saying that rings are meant. Also if it stands for ''module'' there is information loss...

   You should teach me this, I’d be grateful, if you are in ubuntu ?

   I am not in ubuntu but in mac osx. Following simply the instruction on the instiki site might be successfull if your device has the right underlying architecture (in my case one problem was that instiki needed to be converted into 64 bit format). If you plan to do an instiki installation on a web host you should assure that this server meets the requirements of instiki: Dreamhost which is recommended by the instiki site does not meet this requirements - at least it did not when I tried it and it was not possible to install it there. In the end a friend of mine who is an it-specialist did the installation for me on his own private server.

   But I don’t know if my information concerning this are still up to date since I just see that the instiki installation site has been updated on April 30, 2012 - after my trial. And I see that there is a section on ubuntu on this site.

   $M_k E$

   $M_k E$ stands for the category of the copresheaves on the category of $k$-rings (=rings $R$ with a specified morphism $k\to R$) $M_k$.

   $M$, if I remember correctly, stands for “model”. (It is the category of all $U$-small rings, where $U$ is a Grothendieck universe contained in the “real” universe $V$.)

   In ”lectures on $p$-divisible groups” such size issues are omitted. Nevetheless this might be the reason for the letter $M$. But I think it is not a good notation since it contains no information saying that rings are meant. Also if it stands for ”module” there is information loss…

10. • CommentRowNumber10.
    • CommentAuthorZhen Lin
    • CommentTimeMay 23rd 2012
    • (edited May 23rd 2012)
    • PermaLink
    Author: Zhen Lin
    Format: MarkdownItexAh. It's explained briefly at the beginning of [Demazure and Gabriel, _Groupes algébriques_]. The reason, I think, is that they want to think of a scheme as a functor $\mathbf{CRing} \to \mathbf{Set}$.

    Ah. It’s explained briefly at the beginning of [Demazure and Gabriel, Groupes algébriques]. The reason, I think, is that they want to think of a scheme as a functor $\mathbf{CRing} \to \mathbf{Set}$.

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMar 16th 2017
    • (edited Mar 16th 2017)
    • PermaLink
    Author: Urs
    Format: MarkdownItexAt _[[Frobenius morphism]]_ there was a typo in the section _[In terms of symmetric products](https://ncatlab.org/nlab/show/Frobenius+morphism#InTermsOfSymmetricProducts)_: where $\alpha_V$ was introduced, its domain was given as $TS^p V$, while from the lines right below the domain is really meant to be $V^{(p)}$. I have fixed that. But what actually is $V^{(p)}$? This section also forgets to say what its definition of th Frobenius morphism $F_X$ actually is.

    At Frobenius morphism there was a typo in the section In terms of symmetric products:

    where $\alpha_V$ was introduced, its domain was given as $TS^p V$, while from the lines right below the domain is really meant to be $V^{(p)}$.

    I have fixed that.

    But what actually is $V^{(p)}$?

    This section also forgets to say what its definition of th Frobenius morphism $F_X$ actually is.

12. • CommentRowNumber12.
    • CommentAuthorSamuel Adrian Antz
    • CommentTimeFeb 3rd 2024
    • PermaLink
    Author: Samuel Adrian Antz
    Format: MarkdownItexAdded the Frobenius homomorphism being a natural transformation. <a href="https://ncatlab.org/nlab/revision/diff/Frobenius+morphism/29">diff</a>, <a href="https://ncatlab.org/nlab/revision/Frobenius+morphism/29">v29</a>, <a href="https://ncatlab.org/nlab/show/Frobenius+morphism">current</a>

    Added the Frobenius homomorphism being a natural transformation.

    diff, v29, current