Polynomial differential form: Difference between revisions
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In algebra, the ring of '''polynomial differential forms''' on the standard ''n''-simplex is the [[differential graded algebra]]: |
In algebra, the ring of '''polynomial differential forms''' on the standard ''n''-simplex is the [[differential graded algebra]]:<ref>{{harvnb|Hinich|loc=§ 4.8.1.}}</ref> |
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:<math>\Omega^*_{\text{poly}}([n])= \mathbb{Q}[t_0, ..., t_n, dt_0, ..., dt_n]/(\sum t_i - 1, \sum t_i).</math> |
:<math>\Omega^*_{\text{poly}}([n])= \mathbb{Q}[t_0, ..., t_n, dt_0, ..., dt_n]/(\sum t_i - 1, \sum t_i).</math> |
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Varing ''n'', it determines the [[simplicial commutative ring|simplicial commutative dg algebra]]: |
Varing ''n'', it determines the [[simplicial commutative ring|simplicial commutative dg algebra]]: |
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== References == |
== References == |
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{{reflist}} |
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* {{cite arxiv|last=Hinich|first=Vladimir|date=1997-02-11|title=Homological algebra of homotopy algebras|eprint=q-alg/9702015}} |
* {{cite arxiv|last=Hinich|first=Vladimir|date=1997-02-11|title=Homological algebra of homotopy algebras|eprint=q-alg/9702015}} |
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Revision as of 01:59, 8 October 2019
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 t_{i}).}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/29923d34afd6c004520393d8c038f292f64b2096)
Varing n, it determines the simplicial commutative dg algebra:
(for each , the map is given by )
![{\displaystyle [n]\to [m]}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/e72209b3231888a6e39b555e454a0e87fde5e0a1)
![{\displaystyle \Omega _{\text{poly}}^{*}([m])\to \Omega _{\text{poly}}^{*}([n])}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/e0c4cc181ab0b48a4c2d2f9b544a2b6551fe6019)
References
- Hinich, Vladimir (1997-02-11). "Homological algebra of homotopy algebras". arXiv:q-alg/9702015.
External links
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