Example 2

Suppose that $k = \mathbf{C}$ and $A = \mathbf{C} [X_1, ..., X_n]/(P_1, ..., P_r)$, where the $P_i$ are polynomials in $X_1, ..., X_n$, such that the set of points of $\mathbf{C}^n$ where all the $P_i$ vanish is an analytic subvariety $V$ of $\mathbf{C}^n$. One can show that the de Rham cohomology of $A$ over $\mathbf{C}$ is isomorphic to the singular cohomology $H(V, \mathbf{C})$.

Let now $M$ be an $A$-module and $\nabla^0$ a $k$-linear mapping from $M$ into $M \otimes_A \Omega^1_{A/k}$ such that

(one sometimes says that $\nabla^0$ is a connection on the $A$-module $M$).