Polynomial differential form explained

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

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

References

• 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. 1997-02-11. Homological algebra of homotopy algebras. q-alg/9702015.

External links

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