We define the complexification $X_\mathbb{C}\rightarrow \operatorname{Spec} \mathbb{C}$ of a real scheme $X\rightarrow\operatorname{Spec}\mathbb{R}$ to be the fiber product $X_\mathbb{C} = X \times_{\operatorname{Spec}\mathbb{R}} \operatorname{Spec}\mathbb{C}$.
In general, does complexification of real schemes preserve dimension? Smoothness? Would anyone know a reference that discusses properties of this complexification in detail?