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At projective resolution I have
spelled out the Definition in lots of detail;
spelled out statement and proof of the existence of resolutions in full detail.
there is a typo: traditiona I would correct it but you are editing it anyway.
It looks nice.
Thanks! Fixed now. And fixed the next typo two lines below, too. :-)
Yes, I’ll be further editing the entry for just a bit more. Am adding more basic facts…
I have added statement and proof of a proposition that spells out how morphism lift to injective resolutions of their codomain. And a few other things.
Have added to projective resolution a detailed proof of the previously omitted statement that the resolution of a morphism along an injective resolution of its codomain is unique up to chain homotopy – currently that’s prop 4 there.
added a section Functorial resolutions and derived functors
There has long been a section Relation to axioms of choice at projective resolution. Maybe somebody who feels responsible for that section could react to the following two points:
Where it says “projective cover” in that section, it seems to me that what is really meant is “epimorphism from a projective”. This is in general different from projective cover as far as I am aware.
Would the whole section not maybe better be moved to Projective object – Existence of enough projectives?
@Urs 7: I wrote that section not long ago, with your earlier approval. Yes, epimorphism from a projective is what’s meant, and I believe that terminology is in fact used in the context of e.g. topos theory, but if I understand what you are saying, it’s that this conflicts with a homological algebra meaning of projective cover that is dual to injective hull. Is that what you meant? There is no problem in rephrasing if that’s what you want.
As for number 2: please feel free to move it as you see fit.
Thanks for the reaction. Maybe I forget about our previous exchange on that section. I’ll move it to the section on enough-projectives, then.
I keep myself wanting to type “projective cover” but, yes, it has a more restrictive meaning among some people, see for instance
So maybe we’ll better change to “epimorphism from a projective”, even though that’s somewhat awkward. Otherwise we could try to enforce on the $n$Lab that the other notion should go by some other name.
It seem that injective hull is more widespread and “projective cover” is an attempt at formally dual terminology. Maybe “cover” was a bad choice for the duall of “hull”, though.
Okay, I have moved that statement about enough projectives in the presence of COSHEP to here.
Just to double-check: I have changed the wording to make it read as follows:
+– {: .num_prop}
Let $E$ be a W-pretopos that satisfies COSHEP. Then $Ab(E)$ has enough projectives.
=–
The idea of proof is that the underlying object of an abelian group $A$ in $E$ admits an epimorphism from a projective object $X \to U(A)$ in $E$, and then the corresponding $F(X) \to A$ is an epimorphism out of a projective in $Ab(E)$.
Seems good to me – thanks. I might add another sentence to that later.
I have added statment and proof of the long exact sequence of right derived functors induced from a short exact sequence…
… based on the lemma that one can always find injective resolutions of short exact sequences by short exact sequences of cochain complexes…
…of which I haven’t fully written out the proof yet.
(Eventually I’ll copy part of this to the entry on derived functors, but for the moment it is convenient to have it all here in one place in order to have easy pointers to the relevant lemmas and pre-propositions.)
I did wind up adding some more words to the proof of proposition 4 at projective object.
Thanks!
I have now written up more details of that proof that in the presence of enough injectives/projectives every short exact sequences has an injective/projective resolution by a short exact sequence.
I have started in a section Derived hom-functor / Ext-functor to spell out some first details. But I need to quit now for the moment.
I have started adding something in a new section Examples - Projective resolutions adapted to group cohomology.
There is now in the entry a detailed proof spelled out of
$Ext^1(G,A) \simeq H^2(G,A) \simeq Ext(G,A) \,.$I ended up putting it mostly into Examples - Projective resolutions adapted to group cocycles with further comments in Derived Hom-functor / Ext functor.
Not that this can’t still be expanded in lots of directions. But I am beginning to think that for the purpose of HAI (schreiber) that’s maybe enough.
I added a subsection on cohomology of cyclic groups to projective resolution, including a discussion of Hilbert’s theorem 90. I’m a little tired now and didn’t insert all the links I might have.
Very nice, indeed. Thanks.
I have added a brief pointer to this from the entry cyclic group, where the reader might be more likely to look for this. Maybe eventually we should copy much of this over to there (and similarly other pieces currently at projective resolution might eventually need to be copied elsewhere).
And all this reminds me that we are badly in need of bossting the entry group cohomology to a status where it contains some actual definitions.
I’ll start editing there now. But I won’t get far. I was about to call it quits already.
I have been expanding a bit more on the basic properties of derived functors in the section Functorial resolutions and derived functors
(As I mentioned above, eventually I will copy this over to the entry on derived functors proper, but for the moment it is very convenient to develop it on that page which has all the relevant lemmas.)
add statement of the remark that over a principal ideal domain (such as $\mathbb{Z}$) there is always a projective resolution of length-1.
There is something wrong in the section Resolutions adapted to abelian group extensions: the maps $F_1 \coloneqq F(U(G)^{2}) \to A$ there give general group 2-cocycles $c : G \times G \to A$. But they need to give symmetric 2-cocycles, $c(g_1, g_2) = c(g_2, g_1)$, in order to classify actual abelian extensions…
To be frank, I got that idea from the end of these lecture notes and adopted it with too little scrutinizing, as it now turns out.
So that resolution there is not correct. The differential $\partial_0 : F_1 \to F_0$ given by $(g_1,g_2) \mapsto g_1 + g_2 - (g_1 +_G g_2)$ has a larger kernel than given by the image of $\partial_1$: the linear combinations of the form $(g_1, g_2) - (g_2, g_1)$ are in the kernel, but not in the image of $\partial_1$. So that has to be quotiented out. Then maps out of $F_1/\sim$ are indeed symmetric. But now the projectivity-property needs attention…
I have added a section Projective resolutions adapted to general group cohomology .
I have worked on the section
meaning to spell out in some detail the proof that in any abelian category $Ext^1(G,A)$ is the isomorphism classes of extensions of $G$ by $A$.
I finished a first go at at now. But will need to go through it again and smoothen out corners.
added a section Definition – F-resolutions of an object.
Will now add a corresponding Properties-section…
okay, now I have spelled out the detailed proof that for the right/left derived functors of some $F$ the $F$-injective/$F$-projective resolutions (hence in particular the $F$-acyclic resolutions) are sufficient: here.
I have finally added subsections Definition - Projective resolution of a chain complex and Properties - Existence and construction of resolutions of chain complexes with the statement and proof that every chain complex has a “fully projective” or “proper” resolution.
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