Extensions realizing affine datum : the Wells derivation

TL;DR

This paper extends the Wells derivation to arbitrary varieties, establishing an exact sequence linking automorphisms and cohomology groups, with specific refinements for varieties with a difference term and nonabelian extensions.

Contribution

It generalizes the Wells derivation for affine datum extensions across various algebraic varieties, including nonabelian cases and varieties with a difference term.

Findings

Established an exact sequence connecting automorphisms and cohomology groups.

Proved the existence of a homomorphism relating kernel-preserving automorphisms to extensions.

Refined the theorem for varieties with a difference term and nonabelian extensions.

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.