The $\bar{\mathrm{\partial }}$-equation on a non-reduced analytic space

Let $X$ be a, possibly non-reduced, analytic space of pure dimension. We introduce a notion of $\bar{\mathrm{\partial }}$-equation on $X$ and prove a Dolbeault-Grothendieck lemma. We obtain fine sheaves ${\mathcal{A}}_X^q$ of $(0,q)$-currents, so that the associated Dolbeault complex yields a resolution of the structure sheaf ${\mathcal{O}}_X$. Our construction is based on intrinsic semi-global Koppelman formulas on $X$.