On the basis of Dupont's work, we exhibit a cocycle in the simplicial de Rham complex which represents the Chern character. We also prove the related conjecture due to Brylinski. This gives a way to construct a cocycle in a local truncated complex.
"The Chern Character in the Simplicial De Rham Complex." Nihonkai Math. J. 26 (1) 1 - 13, 2015.