An exact sequence in the first variable for torsor cohomology: the 2-dimensional theory of obstructions (Q1072635)

At last, the expected obstruction theory for \textit{P. G. Glenn}'s torsor cohomology [ibid. 25, 33--105 (1982; Zbl 0487.18015)] has been worked out. The authors derive an eight-term exact sequence \[ \begin{multlined} 0\to\Hom(R,A)\to\Hom(B,A)\to\mathrm{Tors}^0(p,A)\to\mathrm{Tors}^1(R,N)\to\\ \to\mathrm{Tors}^1(B,A)\to\mathrm{Tors}^1(p,A)\to\mathrm{Tors}^2(R,A)\to\mathrm{Tors}^2(B,A)\end{multlined} \] associated to a regular epimorphism \(p: B\to R\) and abelian group \(A\) (in any exact category \(\mathcal E)\). The proof of exactness at \(\mathrm{Tors}^1(p,A)\) uses the obstruction theory developed in section 5 of the paper. An interesting side result is Proposition 1.9: \(\mathrm{Tors}^1(-,-): \mathcal E^{op}\times Ab(\mathcal E)\to Ab\) is characterized by two conditions ((a) there is a natural connecting homomorphism \(\mathrm{Tors}^0(p,A)\to\mathrm{Tors}^1(R,A)\) rendering \[ \Hom(B,A)\to\mathrm{Tors}^0(p,A)\to \mathrm{Tors}^1(R,A)\to\mathrm{Tors}^1(B,A) \] exact for any regular epimorphism \(p: B\to R\), and (b) for any \(x\) in \(\mathrm{Tors}^1(R,A)\) there is a regular epimorphism \(p: B\to R\) so that \(p^*(x)=0)\). Section 8 of the paper shows how the general results specialize to a variety of associative, unitary algebras over a commutative ring. For such a variety, 8.7 verifies the equivalence of groupoids and crossed modules. Although technical in nature, the paper is well-written.