Let $j: U \to X$ be a quasi-affine open embedding between schemes and $M$ be a flat quasi-coherent $O_U$-module. Is $j_*M$ flat as an $O_X$-module?
I think the answer is no in general, but: do we have a counterexample when $X$ is a smooth variety and $U$ is the complement of a smooth closed subvariety $Z$ (of codimension $\ge 2$)?
Best Answer
Take $X=\mathbb{A}^n$ with $n\geq 3$, $Z=\{0\} $, and for $M$ the kernel of the homomorphism $\mathscr{O}_U^n\rightarrow\mathscr{O}_U$ defined by $(x_1,\ldots ,x_n)$. Then $M$ is locally free, but $(j_*M)_{0}=\operatorname{Ker} (\mathscr{O}_{0}^n\rightarrow \mathfrak{m}_0)$ has projective dimension $n-2$, hence $j_*M$ is not flat.