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In
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
, the ring of polynomial differential forms on the standard ''n''-simplex is the
differential graded algebra In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded associative algebra with an added chain complex structure that respects the algebra structure. __TOC__ Definition A differential graded a ...
: :\Omega^*_( = \mathbb _0, ..., t_n, dt_0, ..., dt_n(\sum t_i - 1, \sum dt_i). Varying ''n'', it determines the simplicial commutative dg algebra: :\Omega^*_ (each u: \to /math> induces the map \Omega^*_( \to \Omega^*_( , t_i \mapsto \sum_ t_j).


References

* Aldridge Bousfield and V. K. A. M. Gugenheim, §1 and §2 of: On PL De Rham Theory and
Rational Homotopy Type In mathematics and specifically in topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored. It was founded by and . This simplification of homo ...
, Memoirs of the A. M. S., vol. 179, 1976. *


External links

* https://ncatlab.org/nlab/show/differential+forms+on+simplices * https://mathoverflow.net/questions/220532/polynomial-differential-forms-on-bg Differential algebra Ring theory {{algebra-stub