Chain complexes of symmetric categorical groups (Q1770537)

The authors revisit previous results using recent techniques developed in higher-dimensional homological algebra. The Hattori-Villamayor-Zelinsky long exact sequence connecting Amitsur cohomology groups of a commutative algebra with coefficients \(\mathcal U\) (the group of units) and \({\mathcal Pic}\) (the Picard Group) and the Ulbrich cohomology are derived as special instances of general results on the symmetric categorical groups. The kernel and the cokernel of a morphism between symmetric categorical groups previously introduced are refined here by the notions of kernel and cokernel relative to a natural transformation. Using these, the authors define the cohomology categorical groups of a complex of symmetric categorical groups. Two definitions are investigated as in the case of abelian groups, giving equivalent categorical groups. Extensions of categorical groups are introduced and a long 2-exact sequence of cohomology categorical groups is associated to any extension of complexes of symmetric categorical groups. As a special case of this sequence, the authors obtain the Ulbrich cohomology, the Hattori-Villamayor-Zelinsky exact sequences and a special case of the kernel-cokernel lemma for symmetric categorical groups. Simplicial cohomology with coefficients in a symmetric categorical group is also discussed.