1
$\begingroup$

It is a known fact from Algebraic Geometry that the complement of an affine open subset of a variety is of pure codimension one. Does the same hold for the complement of a quasi-affine open subset of a variety? I don't know much about codimension, so I hope that this question is not trivial.

$\endgroup$
  • $\begingroup$ Is it Ok for affine variety ? If yes then this is enough since the dimension if a local notion. $\endgroup$ – Lierre Jul 20 '12 at 12:26
1
$\begingroup$

If $V$ is affine, and $Z$ is closed in $V$, then $V\setminus Z$ is quasi-affine (it is open in the affine variety $V$), and its complement in $V$ is $Z$, which has whatever codimension it has. (Anything up to the dimension of $V$.) So there is no real constraint on the codimension of the complement of a quasi-affine open.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.