Quillen (co)homology of divided power algebras over an operad
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913643223842816 |
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| author | Dokas, Ioannis Frankland, Martin Ikonicoff, Sacha |
| author_facet | Dokas, Ioannis Frankland, Martin Ikonicoff, Sacha |
| contents | Barr--Beck cohomology, put into the framework of model categories by Quillen, provides a cohomology theory for any algebraic structure, for example André--Quillen cohomology of commutative rings. Quillen cohomology has been studied notably for divided power algebras and restricted Lie algebras, both of which are instances of divided power algebras over an operad $P$: the commutative and Lie operad respectively. In this paper, we investigate the Quillen cohomology of divided power algebras over an operad $P$, identifying Beck modules, derivations, and Kähler differentials in that setup. We also compare the cohomology of divided power algebras over $P$ with that of $P$-algebras, and work out some examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18049 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quillen (co)homology of divided power algebras over an operad Dokas, Ioannis Frankland, Martin Ikonicoff, Sacha Rings and Algebras Algebraic Topology Primary 18M70, Secondary 17B50, 17B56, 13D03 Barr--Beck cohomology, put into the framework of model categories by Quillen, provides a cohomology theory for any algebraic structure, for example André--Quillen cohomology of commutative rings. Quillen cohomology has been studied notably for divided power algebras and restricted Lie algebras, both of which are instances of divided power algebras over an operad $P$: the commutative and Lie operad respectively. In this paper, we investigate the Quillen cohomology of divided power algebras over an operad $P$, identifying Beck modules, derivations, and Kähler differentials in that setup. We also compare the cohomology of divided power algebras over $P$ with that of $P$-algebras, and work out some examples. |
| title | Quillen (co)homology of divided power algebras over an operad |
| topic | Rings and Algebras Algebraic Topology Primary 18M70, Secondary 17B50, 17B56, 13D03 |
| url | https://arxiv.org/abs/2403.18049 |