Polynomial differential form

In algebra, the ring of polynomial differential forms on the standard n-simplex is the differential graded algebra:

[math]\displaystyle{ \Omega^*_{\text{poly}}([n])= \mathbb{Q}[t_0, ..., t_n, dt_0, ..., dt_n]/(\sum t_i - 1, \sum dt_i). }[/math]

Varying n, it determines the simplicial commutative dg algebra:

[math]\displaystyle{ \Omega^*_{\text{poly}} }[/math]

(each [math]\displaystyle{ u: [n] \to [m] }[/math] induces the map [math]\displaystyle{ \Omega^*_{\text{poly}}([m]) \to \Omega^*_{\text{poly}}([n]), t_i \mapsto \sum_{u(j)=i} t_j }[/math]).

References

• 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

• https://ncatlab.org/nlab/show/differential+forms+on+simplices
• https://mathoverflow.net/questions/220532/polynomial-differential-forms-on-bg