Polynomial differential form: Difference between revisions

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

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

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

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

Hinich, Vladimir (1997-02-11). "Homological algebra of homotopy algebras". arXiv:q-alg/9702015.

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 Varing ''n'', it determines the [[simplicial commutative ring|simplicial commutative dg algebra]]:
 :<math>\Omega^*_{\text{poly}}</math>
-(for each <math>[n] \to [m]</math>, the map <math>\Omega^*_{\text{poly}}([m]) \to \Omega^*_{\text{poly}}([n])</math> is given by )
+(each <math>[n] \to [m]</math> induces the map <math>\Omega^*_{\text{poly}}([m]) \to \Omega^*_{\text{poly}}([n]), t_i \mapsto \sum_{u(j)=i} = t_j</math>).
 == References ==