Polynomial differential form

In algebra, the ring of polynomial differential forms on the standard n-simplex is the differential graded algebra:¹

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

Varying n, it determines the simplicial commutative dg algebra:

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

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