Extensions realizing affine datum : central extensions (2312.15963v1)

Abstract: The study of extensions realizing affine datum is specialized to central extensions in varieties with a difference term which leads to generalizations of several classical theorems on central extensions from group theory. We establish a 1-dimensional Hochschild-Serre sequence for central extension and affine datum. This is used to develop a Schur-formula which characterizes the $2^{\mathrm{nd}}$-cohomology group of regular datum in terms of the transgression map and commutators in free presentations. We prove sufficient conditions for the existence of covers and provide a cohomological characterization of perfect algebras. The class of varieties with a difference term contain all varieties of algebras with modular congruence lattices and so the present work generalizes the classical results for groups and many analogous results recently established for algebras of Loday-type which can be recovered by specialization.