My question is a sub-problem I came up while trying to solve the question I proposed here (Definition of the pullback of the sheaf of differentials). But I resume it here anyway for completness:
Consider the morphism of schemes
$f: X \longrightarrow Y$
How can $f^*\Omega_{Y/Z}$ be explicited in terms of a tensor product ?
I hope the question makes sense even without the context given by the link, but I'm so far from understanding the question linked that I cannot even say if it does.
Forgive me if I misunderstood your question. This tensor product is coming from the definition of the pullback of a sheaf of modules.
If $f: X \to Y$ is a morphism of ringed spaces and $\mathcal{F}$ is an $\mathcal{O}_Y$-module, then $f^* \mathcal{F}$ is defined to be $f^{-1}\mathcal{F} \otimes_{f^{-1}\mathcal{O}_Y} \mathcal{O}_X$, where $f^{-1}\mathcal{O}_Y \to \mathcal{O}_X$ is part of the data of $f$, and $f^{-1}$ is the usual pullback of sheaves.
In particular, if $f: \operatorname{Spec} B \to \operatorname{Spec} A$ is a morphism of schemes, and $\mathcal{F} = \tilde{M}$ then $f^*\mathcal{F} = \widetilde{M \otimes_AB}$ is the extension of scalars of $M$.
Edit: For completeness, let me link to a proof in the stacks project of the above fact.