De Rham complex

As a partucular example, let us consider , . In this case we get:


This is the de Rham complex for the flat space .


Our is an -dimensional vector space. Let us introduce a basis and call the basis elements :

(this is just a notation, just use letter instead of ). What is ? It is the space of polynomials of , i.e.:


And what is the dual space ? It is the space of “Taylor series”:


The dual space has the following basis:


Finally, what is ? Here it is:


The De Rham differential acts as usual:

This operator is nilpotent. Moreover, it provides the resolution of in the sense that the following complex:


is exact.

Mathematically speaking, the de Rham complex provides:

An injective resolution of the module