Extensions realizing affine datum : the Wells derivation

Abstract

We develop the Wells derivation for extensions realizing affine datum in arbitrary varieties; in particular, we show there is an exact sequence connecting the group of compatible automorphisms determined by the datum and the subgroup of automorphisms of an extension which preserves the extension's kernel. This implies a homomorphism between $2^{\mathrm{nd}}$-cohomology groups which realizes a group of kernel-preserving automorphisms of an extension as itself an extension of a subgroup of compatible automorphisms by the group of derivations of the datum. A refinement of this general Wells's-type theorem is given for a restricted class of varieties with a difference term which include any variety of groups with multiple operators in the sense of Higgins. The same results are obtained for nonabelian extensions in any variety of $R$-modules expanded by multilinear operations.