Polynomial differential form

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

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

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

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

• 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年02月11日). "Homological algebra of homotopy algebras". arXiv:q-alg/9702015 .

• https://ncatlab.org/nlab/show/differential+forms+on+simplices
• https://mathoverflow.net/questions/220532/polynomial-differential-forms-on-bg

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