I think Patrick N. had a point, so (unable to think up a more creative response) I deleted the offending sentence from Remark 2.11.

]]>added pointer to:

- Peter Hilton, Urs Stammbach, p. 129 of:
*A course in homological algebra*, Springer-Verlag, New York, 1971, Graduate Texts in Mathematics, Vol. 4 (doi:10.1007/978-1-4419-8566-8, pdf)

In Remark 2.11, we read:

In other words, a projective resolution of a chain complex in an abelian category \mathcal{A} is a projective resolution of an object in a category of chain complexes Ch_\bullet(\mathcal{A}).

I do not think this quite matches what is written in 2.10, which seems to imply that any exact sequence of chain complexes of projectives would suffice. A chain complex of projectives is not the same thing as a projective object in the category of chain complexes. See Weibel's Homological Algebra book, Exercise 2.2.1 - a projective object in the category of chain complexes is a split exact chain complex of projectives.

I would suggest that some discussion be added that clarifies why we are theoretically justified in using these resolutions by chain complexes of projectives, even though they are not projective. Do chain complexes of projectives play any special role in the category of chain complexes that makes them appropriate here?

The page does not clarify the role of the fully projective or proper resolutions. I actually came here hoping to find out what they were useful for, beyond the proof of existence.

- Patrick N. ]]>

In other words, a projective resolution of a chain complex in an abelian category ]]>

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.

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.

]]>added a section *Definition – F-resolutions of an object*.

Will now add a corresponding Properties-section…

]]>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.

]]>I have added a section *Projective resolutions adapted to general group cohomology* .

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…

]]>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…

add statement of the remark that over a principal ideal domain (such as $\mathbb{Z}$) there is always a projective resolution of length-1.

]]>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.)

]]>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 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.

]]>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 have started adding something in a new section *Examples - Projective resolutions adapted to group cohomology*.

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 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.

]]>Thanks!

]]>I did wind up adding some more words to the proof of proposition 4 at projective object.

]]>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.)

]]>Seems good to me – thanks. I might add another sentence to that later.

]]>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 $Ab(E)$.

]]>