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Extensions realizing affine datum : the Wells derivation

Alexander Wires

TL;DR

This work extends Wells's derivation to extensions realizing affine datum across arbitrary varieties, delivering an exact sequence $1 \to \mathrm{Der}(Q,A^{\alpha,\tau},\ast) \to \mathrm{Aut}_{\alpha} A \to C(Q,A^{\alpha,\tau},\ast) \xrightarrow{W_T} H^{2}_{\mathcal{U}}(Q,A^{\alpha,\tau},\ast)$ and clarifying how kernel-preserving automorphisms decompose as a semidirect product with derivations. The Wells derivation $W_T$ encodes the cohomological obstruction to enlarging compatible automorphisms to full automorphisms of the extension, and the framework is refined for varieties with a weakly-associative difference term and for nonabelian extensions in $R$-modules expanded by multilinear operations. The paper also shows how these results specialize to central and abelian kernels, with explicit representations of compatible automorphisms and their actions via modules, culminating in a unified perspective that extends Wells's theorem beyond groups to broad algebraic settings. Overall, the results provide a cohesive cohomological and automorphism-theoretic decomposition of extension structures across diverse algebraic varieties.

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.

Extensions realizing affine datum : the Wells derivation

TL;DR

This work extends Wells's derivation to extensions realizing affine datum across arbitrary varieties, delivering an exact sequence and clarifying how kernel-preserving automorphisms decompose as a semidirect product with derivations. The Wells derivation encodes the cohomological obstruction to enlarging compatible automorphisms to full automorphisms of the extension, and the framework is refined for varieties with a weakly-associative difference term and for nonabelian extensions in -modules expanded by multilinear operations. The paper also shows how these results specialize to central and abelian kernels, with explicit representations of compatible automorphisms and their actions via modules, culminating in a unified perspective that extends Wells's theorem beyond groups to broad algebraic settings. Overall, the results provide a cohesive cohomological and automorphism-theoretic decomposition of extension structures across diverse algebraic varieties.

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 -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 -modules expanded by multilinear operations.
Paper Structure (4 sections, 8 theorems, 84 equations)

This paper contains 4 sections, 8 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Demonstrations for Theorem \ref{['thm:2']} and Theorem \ref{['thm:3']}
  4. Demonstration for Theorem \ref{['thm:4']}

Key Result

Theorem 1.1

(wells) Let $\pi: G \rightarrow Q$ be a surjective group homomorphism with $K = \ker \pi$. There are homomorphisms and a set map $C(Q,K,\alpha) \rightarrow H_{\bar{\alpha}}^{2}(Q,ZK)$ such that is exact.

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • proof
  • proof
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • ...and 7 more