Given a projective scheme $X$, say over $\mathbb{C}$, another $\mathbb{C}$-scheme $S$ and a coherent and torsion free (as an $O_X$-module) $M_n(O_X)$-module $F$.

Now we can use Morita equivalence to get a sheaf of $O_X$-modules $G$, which is easier to handle. Given a deformation $\mathcal{G}$ of $G$ over $S$, i.e. a sheaf of coherent $O_{X\times S}$-modules, flat as an $O_S$-module.

Can i get a deformation $\mathcal{F}$ of $F$ over $S$ from this?

That is if $\pi: X\times S \rightarrow X$ is the projection, $\mathcal{F}$ should be a $\pi^{\*}M_n(O_X)=M_n(O_{X\times S})$-module, coherent as on $O_{X\times S}$-module and flat as an $O_S$-module.

Can i then just take $\mathcal{F}:=\mathcal{G}\otimes_{O_{X\times S}} \pi^{\*}(O_X^n)$. This is a $\pi^{\*}M_n(O_X)$-module, and flat over $S$, because $\mathcal{G}$ is. If $s_0$ is the point for which the fiber of $\mathcal{G}$ is $G$, i.e. $\mathcal{G}_{s_0}=G$, the fiber of $\mathcal{F}$ over $s_0$ should be $F$, because $G$ and $F$ are Morita equivalent. So this should give the desired deformation.

But this seems to good to be true. Are there any pitfalls in this construction? Or does Morita equivalence really behave very good in flat families?

I could't find anything in the literature about this subject. Does anybody know a reference for this?


Your answer is correct, no pitfalls on the way.


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