Report on Research in Teams Project: Universal Higher Extensions

In one stream of development, one partner in the project had just achieved an interpretation of group cohomology or Lie-algebra cohomology in terms of higher-dimensional central extensions, as developed by Rodelo–Van der Linden [14, 15]. It extends the classical interpretation of the second group cohomology in terms of equivalence classes of short exact sequences with central kernel. This development builds upon the notion of semi-abelian category as in Janelidze–Márki–Tholen [11], and the concept of higher central extension developed within the framework of categorical Galois theory based on the work of Janelidze et al. (See [2, 5, 9, 10].) In addition, methods from the theory of simplicial groups are used. In a parallel and complementary stream of development, the second partner in the project had just achieved a proof of existence of universal n-step extensions of modules over an arbitrary unitary ring [12]. This development involves certain higher torsion theories, some potentially noncommutative. It immediately has several applications ranging from