title: Polynomial differential form
text: # Polynomial differential form
  
  In algebra, the ring of polynomial differential forms on the standard n-simplex is the differential graded algebra:
  
      $\Omega _{\text{poly}}^{*}([n])=\mathbb {Q} [t_{0},...,t_{n},dt_{0},...,dt_{n}]/(\sum t_{i}-1,\sum dt_{i}).$
  
  Varying n, it determines the simplicial commutative dg algebra:
  
      $\Omega _{\text{poly}}^{*}$
  
  (each $u:[n]\to [m]$ induces the map $\Omega _{\text{poly}}^{*}([m])\to \Omega _{\text{poly}}^{*}([n]),t_{i}\mapsto \sum _{u(j)=i}t_{j}$).
  
  