extended by linearity to the direct sum. The cohomology objects $h^i(M \otimes \Omega_R^{\bullet})$, which are $k$-spaces, are called the *de Rham cohomology spaces* of the ${\mathcal{D}}$-module $M$, and are denoted $H^i_{dR}(M)$.

We have defined the de Rham complex of a ${\mathcal{D}}$-module using a chosen regular system of parameters $x_1, \ldots, x_n$ for the formal power series ring $R$. There is an alternate definition from which it is easier to see that this complex does not depend on the chosen parameters, based on the characterization of ${\mathcal{D}}$-modules in terms of *integrable connections*. Let $\Omega_R^1$ be the $R$-module of ($\mathfrak{m}$-adically) *continuous* Kähler differentials of $R$ over $k$ [@EGAIVa 20.7.14], and $d: R \rightarrow \Omega_R^1$ the corresponding universal continuous derivation. In coordinates, if $x_1, \ldots, x_n$ is a regular system of parameters for $R$, we have $\Omega_R^1 \simeq \oplus_i R \, dx_i$ and $d(f) = \sum_i \partial_i(f) \, dx_i$ for all $f \in R$. However, $\Omega_R^1$ and $d$ can also be defined using a universal property, with no reference to coordinates: every $\mathfrak{m}$-adically continuous derivation $\delta: R \rightarrow M$ where $M$ is an $R$-module factors uniquely through $d$. Now recall that a *connection* on an $R$-module $M$ is a $k$-linear map $\nabla: M 