Cohomology of Groups at Gregory Berhuy's Book - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T00:37:14Z http://mathoverflow.net/feeds/question/100520 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100520/cohomology-of-groups-at-gregory-berhuys-book Cohomology of Groups at Gregory Berhuy's Book Zachi Evenor 2012-06-24T11:27:39Z 2012-06-25T13:29:25Z <p>Regarding <em>Gregory Berhuy</em>'s book "<em>An Introduction to Galois Cohomology and its Applications</em>":</p> <p>The book defined a cohomology sets for non-abelian $G$-groups. Let $A$ be a $G$-group, we define a 1-cocycle as follows: it is a map $$\alpha : G \to A \quad , \quad \sigma \mapsto \alpha_\sigma$$ such that for all $\alpha \in A$: $\alpha_1 = 1$ (1 is the unit element of each group) and it satisfies the following relation for all $\sigma , \tau \in G$: $$\alpha_{\sigma \tau} = \alpha_\sigma \sigma \cdot \alpha_\tau \ .$$</p> <p>In page 52, Berhuy's uses that fact to deduce that for an exact sequence of groups $$1 \to A \stackrel{f}{\to} B \stackrel{g}{\to} C \to 1$$ where $f(A)$ is a central subgroup of $b$, that $$g( \beta_\sigma (\sigma \cdot \beta_\tau) \beta_{\sigma \tau}^{-1}) = \gamma_\sigma \sigma \cdot \gamma_\tau \gamma_{\sigma \tau}^{-1} = 1$$ however, in page 53, he writes $$f(\alpha_{\sigma, \tau}) = \beta_\sigma (\sigma \cdot \beta_\tau) \beta_{\sigma \tau}^{-1}$$ (notice the brackets on $\sigma \cdot \beta_\tau$) but he doesn't say it is equal to 1, why? If we use the 1-cocycle definition (the equation with the $\alpha$'s) then $$\beta_\sigma (\sigma \cdot \beta_\tau) = \beta_{\sigma \tau}$$ </p> <p>Where is the difference and why?</p> http://mathoverflow.net/questions/100520/cohomology-of-groups-at-gregory-berhuys-book/100593#100593 Answer by Ronnie Brown for Cohomology of Groups at Gregory Berhuy's Book Ronnie Brown 2012-06-25T13:29:25Z 2012-06-25T13:29:25Z <p>Without answering your question in particular, you might like to look at the paper </p> <p>R. Brown, Fibrations of groupoids'', <em>J. Algebra</em> 15 (1970) 103-132.</p> <p>which uses semidirect products to turn derivations (or crossed morphisms, as the term is used there) into sections and also uses exact sequences of a fibration of groupoids to deal with exact sequences: see Theorem 5.7 of that paper. </p>