Let $\mathcal F$ a coherent sheaf over an affine subset U, then we can consider it as an R module, if $U=Spec(R)$. Let $R$ an algebra over an algebraically closed field $\mathbb K$ and consider the category of Artin local $\mathbb K$-algebras with residue field $\mathbb K$.
An infinitesimal deformation of a R-module M is the data of an $R\otimes A$-module $M_A$, which is flat on $A$, together with an ismorphism $\varphi:M_A\otimes_A\mathbb K\rightarrow M$. If $$\cdots\rightarrow P^{-n}\rightarrow\cdots P^{-1}\rightarrow P^0\rightarrow M\rightarrow 0$$ is a projective resolution of $M$, is a well know fact that the flatness of $M$ allows us to lift the projective resolution to an exact complex of $R\otimes_{\mathbb K}A$-modules flats on $A$:
$$\cdots\rightarrow P^{-n}\otimes_{\mathbb K} A\rightarrow\cdots P^{-1}\otimes_{\mathbb K} A\rightarrow P^0\otimes_{\mathbb K} A\rightarrow M_A\rightarrow 0.$$
I would like to show the following statement: If $M_A$ and $M_A'$ are isomorphic deformations of $M$,then the isomorphism between them lifts to an isomorphism betwen the deformed complex $(P^*\otimes_{\mathbb K} A, d_A)$ and $(P^*\otimes_{\mathbb K} A, d'_A)$, which a priori are two differnt deformation of the projective resolution of $M$.