# Flatness of modules over dual numbers

Let $$X$$ be a smooth, affine complex surface, and $$M$$ be a coherent $$\mathcal{O}_X$$-module. Denote by $$D:=\mbox{Spec}(\mathbb{C}[t]/(t^2))$$ and $$X_D:=X \times_{\mathbb{C}} D$$, the trivial deformation of $$X$$. Let $$0 \to M \to M' \to M \to 0$$ be a short exact sequence of $$\mathcal{O}_X$$-modules. Under the natural morphism $$\mathcal{O}_{X_D} \to \mathcal{O}_X,$$ we can consider any $$\mathcal{O}_X$$-module as an $$\mathcal{O}_{X_D}$$-module. My question is: Is $$M'$$ going to be $$D$$-flat, considered as an $$\mathcal{O}_{X_D}$$-module?

• Your formulation is unclear. Is the action of $t$ on $M'$ intended to be the composition of the epimorphism $M'\to M$ and the monomorphism $M\to M'$? – Jason Starr Feb 5 at 13:20
• As worded I think Chen is not asking for that, and so the answer is no. – Phil Tosteson Feb 5 at 14:41
• @PhilTosteson. I agree with you. As formulated, the OP seems to be asking whether $M'$, considered as an $\mathcal{O}_X$-module, is then flat when considered as an $\mathcal{O}_{X_D}$-module, and the answer to that question is "no". However, if the OP changes the $\mathcal{O}_{X_D}$-module structure as in my comment, then $M'$ is flat as an $\mathcal{O}_{X_D}$-module. – Jason Starr Feb 5 at 14:50
• @JasonStarr and Phil: Thank you for the comments. I was considering $M'$ as an $\mathcal{O}_X$-module. – Chen Feb 6 at 20:59