Polynomial differential form

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

${\mathrm{\Omega }}_{\text{poly}}^{*}([n])=\mathbb{\mathbb{Q}}[t_0,...,t_n,d{t}_0,...,d{t}_n]/(\sum t_i-1,\sum d{t}_i).$

Varying n, it determines the simplicial commutative dg algebra:

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

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

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.
• 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