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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.)
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.
Okay, thanks, I added a “Noetherian” qualifier. Somebody should expand on that, anyway.
Also I have added now a pointer to
and cited the equivalence to coherent sheaves over Noetherian schemes.
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?
Okay, thanks, I have now added all these pointers here.
Following Johnstone, a coherent object in a topos is an object $X$ with these properties:
This is more-or-less a direct translation of the usual definition of coherent sheaf on a locally ringed space.
You are kindly invited to add something to coherent object.
Coherent objects in abelian categories are defined e,g. in Popescu’s book.
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…
:-)
That joke was recently retold in a Saturday Night Live sketch, about 2 minutes in.
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.
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.
Err, there should be a huge warning that “compact” here does not mean finitely presented…
Yeah, I have no idea what compact is supposed to mean here though.
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.
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
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.
I fixed the definition of coherent object in an ordinary topos.
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