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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 21st 2014
   • (edited May 21st 2014)
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   Author: Urs
   Format: MarkdownItexwe didn't have any entry defining _[[coherent cohomology]]_, did we? (I notice that we are lacking also an entry _[[coherent object]]_. That really needs to be created.)

   we didn’t have any entry defining coherent cohomology, did we?

   (I notice that we are lacking also an entry coherent object. That really needs to be created.)

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeMay 22nd 2014
   • (edited May 22nd 2014)
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   Author: zskoda
   Format: MarkdownItexMaybe it would be better that the related entry [[triangulated categories of sheaves]] be titled in singular instead. There is nothing fundamentally plural about the concept (like one has [[cohesive topos]]). Now this point is TRICKY: the concept of coherent sheaf in Grothendieck EGA is far more subtle than the concept of coherent object. The two definitions agree for the sheaves above noetherian schemes where the simplified definition a la Hartshorne holds. You see, the thing is also about the passage to the local ! Coherent object is about a global property while the sheaf version is also about behaviour under restrictions to small open subsets. Therefore please do not defined coherent sheaves as coherent objects in some category of sheaves unless we work with very specific setups where this is true.

   Maybe it would be better that the related entry triangulated categories of sheaves be titled in singular instead. There is nothing fundamentally plural about the concept (like one has cohesive topos).

   Now this point is TRICKY: the concept of coherent sheaf in Grothendieck EGA is far more subtle than the concept of coherent object. The two definitions agree for the sheaves above noetherian schemes where the simplified definition a la Hartshorne holds. You see, the thing is also about the passage to the local ! Coherent object is about a global property while the sheaf version is also about behaviour under restrictions to small open subsets.

   Therefore please do not defined coherent sheaves as coherent objects in some category of sheaves unless we work with very specific setups where this is true.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 30th 2014
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   Author: Urs
   Format: MarkdownItexOkay, thanks, I added a "Noetherian" qualifier. Somebody should expand on that, anyway. Also I have added now a pointer to * [[Dmitri Orlov]], _Derived categories of coherent sheaves and equivalences between them_, Russian Math. Surveys, 58 (2003), 3, 89-172 ([English transl. pdf](http://www.mi.ras.ru/~orlov/papers/Uspekhi2003.pdf), [Russian orig. pdf](http://www.mi.ras.ru/~orlov/papers/uspehwin.pdf)) and cited the equivalence to coherent sheaves over Noetherian schemes.

   Okay, thanks, I added a “Noetherian” qualifier. Somebody should expand on that, anyway.

   Also I have added now a pointer to

   • Dmitri Orlov, Derived categories of coherent sheaves and equivalences between them, Russian Math. Surveys, 58 (2003), 3, 89-172 (English transl. pdf, Russian orig. pdf)

   and cited the equivalence to coherent sheaves over Noetherian schemes.

4. • CommentRowNumber4.
   • CommentAuthoradeelkh
   • CommentTimeMay 30th 2014
   • (edited May 30th 2014)
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   Author: adeelkh
   Format: MarkdownItexThat survey is very nice, but I think SGA 6 is a better reference for these fundamental facts, since it contains proofs and has a more thorough discussion of the situation. The coherent version of the statement is Exp. II, Corollaire 2.2.2.1 (and is mentioned on [[triangulated categories of sheaves]]) while the quasi-coherent version is Proposition 3.7, b) of the same Exposé (with more general assumptions). At some point, I would like to rewrite the page [[triangulated categories of sheaves]]. I think it should be split into different pages on the various variants (quasi-coherent, coherent, perfect complexes). I don't have energy for that at the moment, though. Regarding the noetherian hypothesis for the coherent statement, that's not surprising since even the structure sheaf is not coherent without this assumption. What is a _coherent object_?

   That survey is very nice, but I think SGA 6 is a better reference for these fundamental facts, since it contains proofs and has a more thorough discussion of the situation. The coherent version of the statement is Exp. II, Corollaire 2.2.2.1 (and is mentioned on triangulated categories of sheaves) while the quasi-coherent version is Proposition 3.7, b) of the same Exposé (with more general assumptions).

   At some point, I would like to rewrite the page triangulated categories of sheaves. I think it should be split into different pages on the various variants (quasi-coherent, coherent, perfect complexes). I don’t have energy for that at the moment, though.

   Regarding the noetherian hypothesis for the coherent statement, that’s not surprising since even the structure sheaf is not coherent without this assumption.

   What is a coherent object?

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMay 30th 2014
   • (edited May 30th 2014)
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   Author: Urs
   Format: MarkdownItexOkay, thanks, I have now added all these pointers [here](http://ncatlab.org/nlab/show/coherent+cohomology#Properties).

   Okay, thanks, I have now added all these pointers here.

6. • CommentRowNumber6.
   • CommentAuthorZhen Lin
   • CommentTimeMay 31st 2014
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   Author: Zhen Lin
   Format: MarkdownItexFollowing Johnstone, a coherent object in a topos is an object $X$ with these properties: * $X$ is compact, i.e. the top element of $Sub (X)$ is compact. * For all morphisms $U \to X$, if $U$ is compact, then so is (the domain of) the kernel pair. This is more-or-less a direct translation of the usual definition of coherent sheaf on a locally ringed space.

   Following Johnstone, a coherent object in a topos is an object $X$ with these properties:

   • $X$ is compact, i.e. the top element of $Sub (X)$ is compact.
   • For all morphisms $U \to X$, if $U$ is compact, then so is (the domain of) the kernel pair.

   This is more-or-less a direct translation of the usual definition of coherent sheaf on a locally ringed space.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeMay 31st 2014
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   Author: Urs
   Format: MarkdownItexYou are kindly invited to add something to _[[coherent object]]_.

   You are kindly invited to add something to coherent object.

8. • CommentRowNumber8.
   • CommentAuthorzskoda
   • CommentTimeJun 2nd 2014
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   Author: zskoda
   Format: MarkdownItexCoherent objects in abelian categories are defined e,g. in Popescu's book.

   Coherent objects in abelian categories are defined e,g. in Popescu’s book.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeJun 2nd 2014
   • (edited Jun 3rd 2014)
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   Author: Urs
   Format: MarkdownItexJust to clarify: I am not asking for the definition, but am suggesting that somebody writes an entry about it. You know how the joke goes: > A: Excuse me, do you happen to know the directions from here to the train station? > B: Sorry, no, I don't. > A: Okay, so pay attention, it's like this: from here you first go straight this way, then take the second to left,then... :-)

   Just to clarify: I am not asking for the definition, but am suggesting that somebody writes an entry about it.

   You know how the joke goes:

   A: Excuse me, do you happen to know the directions from here to the train station?

   B: Sorry, no, I don’t.

   A: Okay, so pay attention, it’s like this: from here you first go straight this way, then take the second to left,then…

   :-)

10. • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 2nd 2014
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    Author: Todd_Trimble
    Format: MarkdownItexThat joke was recently retold in a Saturday Night Live [sketch](http://www.nbc.com/saturday-night-live/video/tourists/2781088), about 2 minutes in.

    That joke was recently retold in a Saturday Night Live sketch, about 2 minutes in.

11. • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeJun 2nd 2014
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    Author: zskoda
    Format: MarkdownItexUrs, if one recalls something, that is if one invested some time in past some time into it. Sometimes a day, sometimes a month or more. That makes you noticing some traps or inconsistencies. Experience which took lots of past time is in a way precious even if it faded away to be operable. If one is working on some subject CURRENTLY that one is ready to invest time and can effectively navigate in it, and probably has a reason to spend time on it, though one may lack some particular insight or experience. A side observer is NOT currently working on it, nor would be easy for him to change to that topic and operatively write anything about it; he can help only by offering some insight which is a remnant of past spent time.

    Urs, if one recalls something, that is if one invested some time in past some time into it. Sometimes a day, sometimes a month or more. That makes you noticing some traps or inconsistencies. Experience which took lots of past time is in a way precious even if it faded away to be operable. If one is working on some subject CURRENTLY that one is ready to invest time and can effectively navigate in it, and probably has a reason to spend time on it, though one may lack some particular insight or experience. A side observer is NOT currently working on it, nor would be easy for him to change to that topic and operatively write anything about it; he can help only by offering some insight which is a remnant of past spent time.

12. • CommentRowNumber12.
    • CommentAuthoradeelkh
    • CommentTimeJun 2nd 2014
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    Author: adeelkh
    Format: MarkdownItexI put something at [[coherent object]], but I am not very familiar with topos theory, so please feel free to add more. In particular, it would be nice if someone wrote down explicitly the link with the usual definition of coherent sheaf.

    I put something at coherent object, but I am not very familiar with topos theory, so please feel free to add more. In particular, it would be nice if someone wrote down explicitly the link with the usual definition of coherent sheaf.

13. • CommentRowNumber13.
    • CommentAuthorZhen Lin
    • CommentTimeJun 2nd 2014
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    Author: Zhen Lin
    Format: MarkdownItexErr, there should be a huge warning that "compact" here does _not_ mean finitely presented...

    Err, there should be a huge warning that “compact” here does not mean finitely presented…

14. • CommentRowNumber14.
    • CommentAuthoradeelkh
    • CommentTimeJun 3rd 2014
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    Author: adeelkh
    Format: MarkdownItexYeah, I have no idea what compact is supposed to mean here though.

    Yeah, I have no idea what compact is supposed to mean here though.

15. • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJun 3rd 2014
    • (edited Jun 3rd 2014)
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    Author: Urs
    Format: MarkdownItexThanks, Adeel, for starting something! That's the way to go.. I have briefly added pointers to related material in the DAG series [here at "coherent object"](http://ncatlab.org/nlab/show/coherent%20object#InInfinityToposTheory) and [here at "coherent infinity-topos"](http://ncatlab.org/nlab/show/coherent+%28infinity%2C1%29-topos#Examples). Also created a stub for _[[Deligne-Lurie completeness theorem]]_. But the relation to coherent sheaves still needs more discussion.

    Thanks, Adeel, for starting something! That’s the way to go..

    I have briefly added pointers to related material in the DAG series here at “coherent object” and here at “coherent infinity-topos”. Also created a stub for Deligne-Lurie completeness theorem.

    But the relation to coherent sheaves still needs more discussion.

16. • CommentRowNumber16.
    • CommentAuthorzskoda
    • CommentTimeJun 3rd 2014
    • (edited Jun 3rd 2014)
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    Author: zskoda
    Format: MarkdownItexNow I recall that I put the definition of coherent object in AB5-category context within the entry [[finite type]] on April 21 (when I had the access to * [[Nicolae Popescu]], _Abelian categories with applications to rings and modules_, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. [MR0340375](http://www.ams.org/mathscinet-getitem?mr=0340375), which I do not have these days). I will copy it into [[coherent object]]. I still think that finite type is an algebraic condition and that [[rational homotopy theory]] FTOC is not that suitable there. > An object $X$ in an [[AB5-category]] $C$ is __of finite type__ if one of the following equivalent conditions hold: > (i) any complete directed set $\{X_i\}_{i\in I}$ of [[subobject]]s of $X$ is stationary > (ii) for any complete directed set $\{Y_i\}_{i\in I}$ of subobjects of an object $Y$ the natural morphism $colim_i C(X,Y_i) \to C(X,Y)$ is an isomorphism. > An object $X$ is __finitely presented__ if it is of finite type and if for any epimorphism $p:Y\to X$ where $Y$ is of finite type, it follows that $ker\,p$ is also of finite type. An object $X$ in an AB5 category is __coherent__ if it is of finite type and for any morphism $f: Y\to X$ of finite type $ker\,f$ is of finite type. > For an exact sequence $0\to X'\to X\to X''\to 0$ in an AB5 category the following hold: > (a) if $X'$ and $X''$ are finitely presented, then $X$ is finitely presented; > (b) if $X$ is finitely presented and $X'$ of finite type, then $X''$ is finitely presented; > (c) if $X$ is coherent and $X'$ of finite type then $X''$ is also coherent. > For a module $M$ over a ring $R$ this is equivalent to $M$ being finitely generated $R$-module. It is finitely presented if it is finitely presented in the usual sense of existence of short exact sequence of the form $R^I\to R^J\to M\to 0$ where $I$ and $J$ are finite.

    Now I recall that I put the definition of coherent object in AB5-category context within the entry finite type on April 21 (when I had the access to

    • Nicolae Popescu, Abelian categories with applications to rings and modules, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. MR0340375,

    which I do not have these days). I will copy it into coherent object. I still think that finite type is an algebraic condition and that rational homotopy theory FTOC is not that suitable there.

    An object $X$ in an AB5-category $C$ is of finite type if one of the following equivalent conditions hold:

    (i) any complete directed set $\{X_i\}_{i\in I}$ of subobjects of $X$ is stationary

    (ii) for any complete directed set $\{Y_i\}_{i\in I}$ of subobjects of an object $Y$ the natural morphism $colim_i C(X,Y_i) \to C(X,Y)$ is an isomorphism.

    An object $X$ is finitely presented if it is of finite type and if for any epimorphism $p:Y\to X$ where $Y$ is of finite type, it follows that $ker\,p$ is also of finite type. An object $X$ in an AB5 category is coherent if it is of finite type and for any morphism $f: Y\to X$ of finite type $ker\,f$ is of finite type.

    For an exact sequence $0\to X'\to X\to X''\to 0$ in an AB5 category the following hold:

    (a) if $X'$ and $X''$ are finitely presented, then $X$ is finitely presented;

    (b) if $X$ is finitely presented and $X'$ of finite type, then $X''$ is finitely presented;

    (c) if $X$ is coherent and $X'$ of finite type then $X''$ is also coherent.

    For a module $M$ over a ring $R$ this is equivalent to $M$ being finitely generated $R$-module. It is finitely presented if it is finitely presented in the usual sense of existence of short exact sequence of the form $R^I\to R^J\to M\to 0$ where $I$ and $J$ are finite.

17. • CommentRowNumber17.
    • CommentAuthorZhen Lin
    • CommentTimeJun 3rd 2014
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    Author: Zhen Lin
    Format: MarkdownItexI fixed the definition of coherent object in an ordinary topos.

    I fixed the definition of coherent object in an ordinary topos.