Polynomial differential form: Difference between revisions
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:<math>\Omega^*_{\text{poly}}([n])= \mathbb{Q}[t_0, ..., t_n, dt_0, ..., dt_n]/(\sum t_i - 1, \sum dt_i).</math> |
:<math>\Omega^*_{\text{poly}}([n])= \mathbb{Q}[t_0, ..., t_n, dt_0, ..., dt_n]/(\sum t_i - 1, \sum dt_i).</math> |
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Varying ''n'', it determines the [[simplicial commutative ring|simplicial commutative dg algebra]]: |
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:<math>\Omega^*_{\text{poly}}</math> |
:<math>\Omega^*_{\text{poly}}</math> |
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(each <math>u: [n] \to [m]</math> induces the map <math>\Omega^*_{\text{poly}}([m]) \to \Omega^*_{\text{poly}}([n]), t_i \mapsto \sum_{u(j)=i} t_j</math>). |
(each <math>u: [n] \to [m]</math> induces the map <math>\Omega^*_{\text{poly}}([m]) \to \Omega^*_{\text{poly}}([n]), t_i \mapsto \sum_{u(j)=i} t_j</math>). |
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== References == |
== References == |
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* Aldridge Bousfield and V. K. A. M. Gugenheim, §1 and §2 of: On PL De Rham Theory and Rational Homotopy Type, Memoirs of the A. M. S., vol. 179, 1976. |
* Aldridge Bousfield and V. K. A. M. Gugenheim, §1 and §2 of: On PL [[De Rham theorem|De Rham Theory]] and [[Rational homotopy type|Rational Homotopy Type]], Memoirs of the A. M. S., vol. 179, 1976. |
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* {{cite |
* {{cite arXiv|last=Hinich|first=Vladimir|date=1997-02-11|title=Homological algebra of homotopy algebras|arxiv=q-alg/9702015}} |
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== External links == |
== External links == |
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Latest revision as of 05:23, 13 May 2024
In algebra, the ring of polynomial differential forms on the standard n-simplex is the differential graded algebra:[1]
![{\displaystyle \Omega _{\text{poly}}^{*}([n])=\mathbb {Q} [t_{0},...,t_{n},dt_{0},...,dt_{n}]/(\sum t_{i}-1,\sum dt_{i}).}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/8561d29e88a143455f96301cf790bb56fd218cea)
Varying n, it determines the simplicial commutative dg algebra:
(each induces the map ).
![{\displaystyle u:[n]\to [m]}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/a39fd8924b9cb09430e0cc1161803300fe55a2b5)
![{\displaystyle \Omega _{\text{poly}}^{*}([m])\to \Omega _{\text{poly}}^{*}([n]),t_{i}\mapsto \sum _{u(j)=i}t_{j}}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/3d60e3ec6f36c3f7bdc8bec9c08cf236e21bcabe)
References
[edit]- ^ Hinich 1997, § 4.8.1.
- Aldridge Bousfield and V. K. A. M. Gugenheim, §1 and §2 of: On PL De Rham Theory and Rational Homotopy Type, Memoirs of the A. M. S., vol. 179, 1976.
- Hinich, Vladimir (1997-02-11). "Homological algebra of homotopy algebras". arXiv:q-alg/9702015.
External links
[edit]- https://ncatlab.org/nlab/show/differential+forms+on+simplices
- https://mathoverflow.net/questions/220532/polynomial-differential-forms-on-bg
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