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• CommentRowNumber1.
• CommentAuthorZhen Lin
• CommentTimeFeb 10th 2015

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

• CommentRowNumber2.
• CommentAuthorTim_Porter
• CommentTimeFeb 11th 2015

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.)

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeFeb 11th 2015
• (edited Feb 11th 2015)

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

I have also added pointer to Beck 67.

• CommentRowNumber4.
• CommentAuthorTim_Porter
• CommentTimeAug 28th 2018
• (edited Aug 28th 2018)

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.

• CommentRowNumber5.
• CommentAuthorRichard Williamson
• CommentTimeAug 28th 2018
• (edited Aug 28th 2018)

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.

• CommentRowNumber6.
• CommentAuthorn.mertes
• CommentTimeMay 17th 2020

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 $M$?

• CommentRowNumber7.
• CommentAuthorn.mertes
• CommentTimeJun 9th 2020
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.