Polynomial differential form

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

$\Omega _{\text{poly}}^{*}([n])=\mathbb {Q} [t_{0},...,t_{n},dt_{0},...,dt_{n}]/(\sum t_{i}-1,\sum t_{i}).$

Varing n, it determines the simplicial commutative dg algebra:

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

(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

• 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