Extensions realizing affine datum : central extensions
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866908502497165312 |
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| author | Wires, Alexander |
| author_facet | Wires, Alexander |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_15963 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Extensions realizing affine datum : central extensions Wires, Alexander Rings and Algebras 08A35, 03C05 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. |
| title | Extensions realizing affine datum : central extensions |
| topic | Rings and Algebras 08A35, 03C05 |
| url | https://arxiv.org/abs/2312.15963 |