Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory object of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJul 27th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItextried to edit [[Ext]] a bit. But this needs to be expanded, eventually.

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

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJul 27th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexAnd a tiny little bit at [[Tor]].

   And a tiny little bit at Tor.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeAug 29th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexAdded a list of notions of cohomology expressible as Ext-groups to [[Ext]].

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

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeAug 30th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added to _[[Ext]]_ a section _[Contravariant Ext on ordinary objects](http://ncatlab.org/nlab/show/Ext#ContravariantExtOnObject)_ 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.

   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.

5. • CommentRowNumber5.
   • CommentAuthorZhiyuYuan
   • CommentTimeDec 28th 2020
   • PermaLink
   Author: ZhiyuYuan
   Format: MarkdownItexIn 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, ?

   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,

   ?

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeDec 28th 2020
   • (edited Dec 28th 2020)
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes, thanks for the alert. I have fixed it now ([here](https://ncatlab.org/nlab/show/Ext#ExtractCocycleFromExtension)). (And while I was at it, I also adjusted some of the formatting in the Definition.) <a href="https://ncatlab.org/nlab/revision/diff/Ext/24">diff</a>, <a href="https://ncatlab.org/nlab/revision/Ext/24">v24</a>, <a href="https://ncatlab.org/nlab/show/Ext">current</a>

   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.)

   diff, v24, current

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJan 14th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexfor completeness I have added ([here](https://ncatlab.org/nlab/show/Ext#Examples)) statement of more examples of $Ext^1$-groups of abelian groups (just going along Boardman's lecture notes) <a href="https://ncatlab.org/nlab/revision/diff/Ext/25">diff</a>, <a href="https://ncatlab.org/nlab/revision/Ext/25">v25</a>, <a href="https://ncatlab.org/nlab/show/Ext">current</a>

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

   diff, v25, current

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJan 17th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded the statement ([here](https://ncatlab.org/nlab/show/Ext#BasicProperties)), that $Ext^n_R$ sends both $\oplus_i$ in the first variable and $\prod_i$ in the second variable to $\prod_i$-s. <a href="https://ncatlab.org/nlab/revision/diff/Ext/26">diff</a>, <a href="https://ncatlab.org/nlab/revision/Ext/26">v26</a>, <a href="https://ncatlab.org/nlab/show/Ext">current</a>

   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.

   diff, v26, current

9. • CommentRowNumber9.
   • CommentAuthorBartek
   • CommentTimeDec 26th 2021
   • PermaLink
   Author: Bartek
   Format: MarkdownItexin 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 <a href="https://ncatlab.org/nlab/revision/diff/Ext/30">diff</a>, <a href="https://ncatlab.org/nlab/revision/Ext/30">v30</a>, <a href="https://ncatlab.org/nlab/show/Ext">current</a>

   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

   diff, v30, current