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Let $X$ be $k$-scheme of finite type, $k$ being a (perfect) field, and assume $X\otimes\overline{k}$ is affine. Is $X$ necessarily an affine scheme? What about if $X$ is a $k$-group scheme?

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    $\begingroup$ The property of a morphism (such as structural morphism to an affine base) being affine descends through any fpqc base change. This follows from Serre's cohomological criterion for affineness (which is valid for qcqs schemes) coupled with the compatibility of formation of quasi-coherent sheaf cohomology with respect to flat base change (for qcqs schemes). $\endgroup$
    – BCnrd
    Nov 22, 2010 at 17:54
  • $\begingroup$ Again I wonder why BCnrd does not write this as an answer ... it's perfect! $\endgroup$ Nov 22, 2010 at 17:57
  • $\begingroup$ Once he explained to me his reason for doing so...but anyway, this answers my question. Thanks, Brian! $\endgroup$
    – shenghao
    Nov 22, 2010 at 18:08

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