Extension realizing affine datum: low-dimensional cohomology
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866909343624986624 |
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| author | Wires, Alexander |
| author_facet | Wires, Alexander |
| contents | For arbitrary varieties of universal algebras, we develop the theory around the first and second-cohomology groups characterizing extensions realizing affine datum. Restricted to varieties with a weak-difference term, extensions realizing affine datum are exactly extensions with abelian kernels. This recovers many classic examples of extensions with abelian coefficients since varieties with a weak-difference term give a far-reaching generalization of algebras like groups with multiple operators; indeed, any variety of algebras whose congruences form modular lattices. We introduce a notion of action and its model relation with a set of equations. In varieties with a difference term, central extensions are characterized by a property of their actions. Restricting further to a subclass of varieties with a difference term which still includes groups with multiple operators, we recover a special case of the representation of extensions with abelian kernels. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_16989 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Extension realizing affine datum: low-dimensional cohomology Wires, Alexander Rings and Algebras Logic 08A05, 08A35, 03C05 For arbitrary varieties of universal algebras, we develop the theory around the first and second-cohomology groups characterizing extensions realizing affine datum. Restricted to varieties with a weak-difference term, extensions realizing affine datum are exactly extensions with abelian kernels. This recovers many classic examples of extensions with abelian coefficients since varieties with a weak-difference term give a far-reaching generalization of algebras like groups with multiple operators; indeed, any variety of algebras whose congruences form modular lattices. We introduce a notion of action and its model relation with a set of equations. In varieties with a difference term, central extensions are characterized by a property of their actions. Restricting further to a subclass of varieties with a difference term which still includes groups with multiple operators, we recover a special case of the representation of extensions with abelian kernels. |
| title | Extension realizing affine datum: low-dimensional cohomology |
| topic | Rings and Algebras Logic 08A05, 08A35, 03C05 |
| url | https://arxiv.org/abs/2309.16989 |