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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 26th 2013
   • (edited Nov 26th 2013)
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   Author: Urs
   Format: MarkdownItexas last week, I have created an entry that collects some of the recent edits scattered over the nLab supposedly in one coherent story, it's _[[basics of étale cohomology]]_ Should be expanded a bit more. But not tonight.

   as last week, I have created an entry that collects some of the recent edits scattered over the nLab supposedly in one coherent story, it’s

   basics of étale cohomology

   Should be expanded a bit more. But not tonight.

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 26th 2013
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   Author: Todd_Trimble
   Format: MarkdownItexI made a slight adjustment in definition 3, since I don't think the nilradical is necessarily nilpotent (it is if the ring is finitely presented). In the material just after that definition, what does $Sh(CRing)$ signify? Sheaves with respect to what topology?

   I made a slight adjustment in definition 3, since I don’t think the nilradical is necessarily nilpotent (it is if the ring is finitely presented).

   In the material just after that definition, what does $Sh(CRing)$ signify? Sheaves with respect to what topology?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeNov 26th 2013
   • (edited Nov 26th 2013)
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   Author: Urs
   Format: MarkdownItexOkay, thanks. That $Sh(CRing)$ should just be $PSh(CRing)$ at this point, I have changed it. There is a corefelctive embedding of reduced rings into all rings and the left/right Kan extensions of the embedding and the coreflection induces adjoints on the presheaves.

   Okay, thanks.

   That $Sh(CRing)$ should just be $PSh(CRing)$ at this point, I have changed it. There is a corefelctive embedding of reduced rings into all rings and the left/right Kan extensions of the embedding and the coreflection induces adjoints on the presheaves.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 26th 2013
   • (edited Nov 26th 2013)
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   Author: Urs
   Format: MarkdownItexedited a bit more at [[basics of etale cohomology]], changed the overall structure a bit by moving the discussion of the Amitsur descent theorem up to before the discussion of etale sheaves. Currently it seems that the proof there that $0 \to N \to N \otimes_A B \to \cdots$ is exact for $A \to B$ faithfully flat only works is $N$ is flat over $A$. I should write out a more general proof, eventually...

   edited a bit more at basics of etale cohomology, changed the overall structure a bit by moving the discussion of the Amitsur descent theorem up to before the discussion of etale sheaves.

   Currently it seems that the proof there that $0 \to N \to N \otimes_A B \to \cdots$ is exact for $A \to B$ faithfully flat only works is $N$ is flat over $A$. I should write out a more general proof, eventually…

5. • CommentRowNumber5.
   • CommentAuthorMarc Hoyois
   • CommentTimeNov 26th 2013
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   Author: Marc Hoyois
   Format: MarkdownItex> I made a slight adjustment in definition 3, since I don't think the nilradical is necessarily nilpotent (it is if the ring is finitely presented). That's a good point, and further, it's not obvious that being $\int_{inf}$-modal is equivalent to being formally étale: I can only see that the latter implies the former for finitely presented morphisms. Is this equivalence proved in Tamme? > Currently it seems that the proof there that $0 \to N \to N \otimes_A B \to \cdots$ is exact for $A \to B$ faithfully flat only works is $N$ is flat over $A$. I should write out a more general proof, eventually... The proof of exactness of the Amitsur complex shows that it is split exact after faithfully flat base change. This implies that it remains exact after tensoring with any $A$-module.

   I made a slight adjustment in definition 3, since I don’t think the nilradical is necessarily nilpotent (it is if the ring is finitely presented).

   That’s a good point, and further, it’s not obvious that being $\int_{inf}$-modal is equivalent to being formally étale: I can only see that the latter implies the former for finitely presented morphisms. Is this equivalence proved in Tamme?

   Currently it seems that the proof there that $0 \to N \to N \otimes_A B \to \cdots$ is exact for $A \to B$ faithfully flat only works is $N$ is flat over $A$. I should write out a more general proof, eventually…

   The proof of exactness of the Amitsur complex shows that it is split exact after faithfully flat base change. This implies that it remains exact after tensoring with any $A$-module.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeNov 26th 2013
   • (edited Nov 26th 2013)
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   Author: Urs
   Format: MarkdownItexAh, right, this needs attention. Thanks. I suppose I should restrict $CRing$ to $CRing^{fin}$ throughout in the discussion of $\int_{inf}$. That is in any case the right "geometry" to consider (e.g. DAG5, 2.5).

   Ah, right, this needs attention. Thanks. I suppose I should restrict $CRing$ to $CRing^{fin}$ throughout in the discussion of $\int_{inf}$. That is in any case the right “geometry” to consider (e.g. DAG5, 2.5).

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeNov 26th 2013
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   Author: Urs
   Format: MarkdownItexOtherwise I should change it to the version where we consider the site of rings equipped with an explict choice of nilpotent extension. This is how Rosenberg-Kontsevich did it ([here](http://ncatlab.org/nlab/show/formally+etale+morphism#KontsevichRosenbergSpaces)) and how Zoran likes to do it. I thought that's overly heavy for the simple purpose of characterizing formally etale morphisms by modalities, but maybe I was making it not only as simple as possible, but even simpler.

   Otherwise I should change it to the version where we consider the site of rings equipped with an explict choice of nilpotent extension. This is how Rosenberg-Kontsevich did it (here) and how Zoran likes to do it. I thought that’s overly heavy for the simple purpose of characterizing formally etale morphisms by modalities, but maybe I was making it not only as simple as possible, but even simpler.

8. • CommentRowNumber8.
   • CommentAuthorMarc Hoyois
   • CommentTimeNov 26th 2013
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   Author: Marc Hoyois
   Format: MarkdownItex> Ah, right, this needs attention. Thanks. I suppose I should restrict $CRing$ to $CRing^{fin}$ throughout in the discussion of $\int_{inf}$. That is in any case the right "geometry" to consider (e.g. DAG5, 2.5). Even then, $\int_{inf}$-modality only provides lifts for nilpotent extensions of reduced rings, not arbitrary rings...

   Ah, right, this needs attention. Thanks. I suppose I should restrict $CRing$ to $CRing^{fin}$ throughout in the discussion of $\int_{inf}$. That is in any case the right “geometry” to consider (e.g. DAG5, 2.5).

   Even then, $\int_{inf}$-modality only provides lifts for nilpotent extensions of reduced rings, not arbitrary rings…

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeNov 26th 2013
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   Author: Urs
   Format: MarkdownItexRight, as I just said, if that's a problem, then I need to change it to the site of rings equipped with a given extension, as Rosenberg-Kontsevich originally did it.

   Right, as I just said, if that’s a problem, then I need to change it to the site of rings equipped with a given extension, as Rosenberg-Kontsevich originally did it.

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2013
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    Author: Urs
    Format: MarkdownItexadded a note that one might better use the category of [[infinitesimal ring extensions]]. Don't have more energy for tonight, will come back to this later.

    added a note that one might better use the category of infinitesimal ring extensions. Don’t have more energy for tonight, will come back to this later.