Extensions realizing affine datum : central extensions

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 a central extension equipped with affine datum. This is used to develop a Schur-Hopf 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, assuming the existence of an idempotent, 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; for example, any variety of groups with multiple operators in the parlance of P.J. Higgins or algebras of Loday-type - analogous results recently established for these algebras can be recovered by specialization.