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Lin Chen
Lin Chen
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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?

IfI 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 $\ge 2$)?

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?

If the answer is no, do we have a counterexample when $X$ is a smooth variety and $U$ is the complement of a smooth closed subvariety $Z$ (of codimension $\ge 2$)?

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 $\ge 2$)?

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Lin Chen
Lin Chen
  • 115
  • 4

Direct image of flat module along quasi-affine morphism

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?

If the answer is no, do we have a counterexample when $X$ is a smooth variety and $U$ is the complement of a smooth closed subvariety $Z$ (of codimension $\ge 2$)?