Not signed in

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

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorZhen Lin
   • CommentTimeFeb 10th 2015
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexCreated [[Beck module]], mentioned it (once) on the [[tangent category]] page.

   Created Beck module, mentioned it (once) on the tangent category page.

2. • CommentRowNumber2.
   • CommentAuthorTim_Porter
   • CommentTimeFeb 11th 2015
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexThere was work done on the `derived module' of a group homomorphism in R. H. Crowell, The derived module of a homomorphism, Advances in Math., 5, (1971), 210 – 238. This is related to both Beck modules and to Fox derivatives. (i discussed this in the Menagerie, and it relates to the linearisation of crossed complexes in Ronnie Brown's work.)

   There was work done on the ‘derived module’ of a group homomorphism in

   R. H. Crowell, The derived module of a homomorphism, Advances in Math., 5, (1971), 210 – 238.

   This is related to both Beck modules and to Fox derivatives. (i discussed this in the Menagerie, and it relates to the linearisation of crossed complexes in Ronnie Brown’s work.)

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeFeb 11th 2015
   • (edited Feb 11th 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItexVery nice! I have added a few more pointers to this from other relevant entries. I have also added pointer to [Beck 67](http://ncatlab.org/nlab/show/Beck+module#Beck67).

   Very nice! I have added a few more pointers to this from other relevant entries.

   I have also added pointer to Beck 67.

4. • CommentRowNumber4.
   • CommentAuthorTim_Porter
   • CommentTimeAug 28th 2018
   • (edited Aug 28th 2018)
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexAdded a reference to a preprint by [[Markus Szymik]] relating Beck modules of quandles to the detection of the unknot. NB: Richard, he is at Trondheim. <a href="https://ncatlab.org/nlab/revision/diff/Beck+module/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/Beck+module/3">v3</a>, <a href="https://ncatlab.org/nlab/show/Beck+module">current</a>

   Added a reference to a preprint by Markus Szymik relating Beck modules of quandles to the detection of the unknot. NB: Richard, he is at Trondheim.

   diff, v3, current

5. • CommentRowNumber5.
   • CommentAuthorRichard Williamson
   • CommentTimeAug 28th 2018
   • (edited Aug 28th 2018)
   • PermaLink
   Author: Richard Williamson
   Format: MarkdownItexHehe, yes, Markus and I were colleagues for a while, and meet occasionally :-). He is a very nice chap! I believe is on sabbatical at the moment, in Cambridge I think.

   Hehe, yes, Markus and I were colleagues for a while, and meet occasionally :-). He is a very nice chap! I believe is on sabbatical at the moment, in Cambridge I think.

6. • CommentRowNumber6.
   • CommentAuthorn.mertes
   • CommentTimeMay 17th 2020
   • PermaLink
   Author: n.mertes
   Format: MarkdownItexLet $A$ be a commutative ring, and let $M$ be an $A$-module. Then any square-zero extension $B\to A$ with kernel $M$ can be viewed as a torsor in $\text{CRing}/A$ for the abelian group object $A\oplus M\to A$ corresponding to $M$. The group action $(A\oplus M)\times_A B\to B$ is such that $((a, m), b)$ is mapped to $b+m$. Is every torsor for $A\oplus M\to A$ necessarily a square-zero extension? More specifically, is the category of torsors for $A\oplus M\to A$ equivalent to the category of square-zero extensions of $A$ by $M$?

   Let $A$ be a commutative ring, and let $M$ be an $A$-module. Then any square-zero extension $B\to A$ with kernel $M$ can be viewed as a torsor in $\text{CRing}/A$ for the abelian group object $A\oplus M\to A$ corresponding to $M$. The group action $(A\oplus M)\times_A B\to B$ is such that $((a, m), b)$ is mapped to $b+m$.

   Is every torsor for $A\oplus M\to A$ necessarily a square-zero extension? More specifically, is the category of torsors for $A\oplus M\to A$ equivalent to the category of square-zero extensions of $A$ by $M$?

7. • CommentRowNumber7.
   • CommentAuthorn.mertes
   • CommentTimeJun 9th 2020
   • PermaLink
   Author: n.mertes
   Format: TextI answered my question listed above in this article https://arxiv.org/abs/2006.04230. I also attempted to generalize this situation to the setting of schemes.

   I answered my question listed above in this article https://arxiv.org/abs/2006.04230. I also attempted to generalize this situation to the setting of schemes.