Example 1

Take $A$ to be the ring $k[X_1, ..., X_n]$. Then $\Omega_{A/k}^1$ is a free $A$-module with basis $dX_1, ..., dX_n$ (III, p. 134, example). Consequently, if, for every subset $I = \{ i_1, ..., i_p \}$ of $\{ 1, n \}$, one sets $dX_I = dX_{i_1} \wedge ... \wedge dX_{i_p}$ (with $i_1 < ... < i_p$), the $A$-module $\Omega_{A/k}^p$ admits as basis the elements $dX_I$, where $I$ runs through the set of subsets of $\{ 1, n \}$ of cardinal $p$. One has

where $n(I, i)$ denotes the number of elements of $I$ strictly less than $i$.

The cycles of $Z^p(\Omega_{A/k})$ are therefore the elements $\omega = \sum_{\operatorname{Card}(I) = p} P_I dX_I$ such that, for every subset $J$ with $(p + 1)$ elements of $\{ 1, n \}$, one has:

The element $\omega$ is a boundary if one can choose, for every subset $J \subset \{1, n\}$ with $(p - 1)$ elements, a polynomial $Q_J \in A$ in such a way that:

We shall see in § 9 that the de Rham complex of $A$ over $k$ is acyclic in degrees $> 0$ if $k$ is a $\mathbf{Q}$-algebra (X, p. 159, remark 4).