Hochschild cohomology of a sheaf of associative algebras

Question

Assume that $X$ is a complex manifold. Let $\delta: X\to X\times X$ be the diagonal map. Assume that $\mathcal{A}_X$ is a $\mathbb C_X$-algebra and $\mathcal{M}_X$ is a left $\mathcal{A}_X\otimes_{\mathbb C_X}\mathcal{A}_X^{op}$-module. I would like to know if there is a well-defined notion of a Hochschild cohomology for $\mathcal{A}_X$ with values in $\mathcal{M}_X$? In particular, is there some intrinsic problem with a naive adaptation of Richard Swan's definition of Hochschild cohomology along the lines of $$H^{\bullet}(\mathcal{A}_X, \mathcal{M}_X):=Ext_{\mathcal{A}_{X\times X}}^{\bullet}(\delta_*\mathcal{A}_X, \delta_*\mathcal{M}_X)=H^{\bullet}(R\Gamma_{X\times X}\circ R\mathcal{Hom}_{\mathcal{A}_{X\times X}}(\delta_*\mathcal{A}_X, \delta_*\mathcal{M}_X))$$ where $\Gamma_{X\times X}$ is the global section functor $\Gamma(X\times X, -)$.