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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJul 27th 2010

tried to edit Ext a bit. But this needs to be expanded, eventually.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJul 27th 2010

And a tiny little bit at Tor.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeAug 29th 2012

Added a list of notions of cohomology expressible as Ext-groups to Ext.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeAug 30th 2012

I have added to Ext a section Contravariant Ext on ordinary objects with the explicit definition by resolutions. But mainly I added after this a bit of explicit discussion of how to see that the standard formula $H^n(Hom(P_\bullet,A) )$ computes indeed the homotopy classes $X \to \mathbf{B}^n A$.

More general abstract discussion along these lines is planned there or at derived functors in homological algebra, but is not done yes.

• CommentRowNumber5.
• CommentAuthorZhiyuYuan
• CommentTimeDec 28th 2020

In the subsubsection ‘1-Extensions over single objects’ it reads:

• σ is any choice of lifts of Q→A through P→A, which exists by definition since P is a projective object,

Should it not be:

• σ is any choice of lifts of Q→X through P→X, which exists by definition since Q is a projective object,

?

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeDec 28th 2020
• (edited Dec 28th 2020)

Yes, thanks for the alert. I have fixed it now (here).

(And while I was at it, I also adjusted some of the formatting in the Definition.)

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeJan 14th 2021

for completeness I have added (here) statement of more examples of $Ext^1$-groups of abelian groups (just going along Boardman’s lecture notes)

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJan 17th 2021

added the statement (here), that $Ext^n_R$ sends both $\oplus_i$ in the first variable and $\prod_i$ in the second variable to $\prod_i$-s.

• CommentRowNumber9.
• CommentAuthorBartek
• CommentTimeDec 26th 2021

in the section “Various notions of cohomology expressed by Ext”:

corrected the definition of Lie Algebra cohomology, and slightly modified the group cohomology definition to account for arbitrary commutative rings