polynomial differential form

In algebra, the ring of polynomial differential forms on the standard n-simplex is the differential graded algebra:{{harvnb|Hinich|1997|loc=§ 4.8.1.}}

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

: $\Omega^*_{\text{poly}}$

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

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.
{{cite arXiv|last=Hinich|first=Vladimir|date=1997-02-11|title=Homological algebra of homotopy algebras|arxiv=q-alg/9702015}}