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> [Is there an example of a scheme X whose reduction X\_red is affine but X is not affine?](https://mathoverflow.net/questions/95/is-there-an-example-ofa-scheme-x-whose-reduction-x-red-is-affine-but-x-is-not-af) 
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I got a question, which may be very easy, but I didn't figure out it.
Let $X$ be a scheme such that $X\_{\mathrm{red}}$ is an affine scheme. Could we conclude that $X$ itself must be affine also? This question came into my mind because I am thinking if the natural morpihsm $ i: X\_{\mathrm{red}} \rightarrow X$ has such a property that for any open affine subset $U$ of $X\_{\mathrm{red}}$, $i(U)$ is an open affine subset of $X$. Notice that $i$ is an homeomorphism, hence $i(U)$ is an open subset of $X$.



a question about affiness 
If $X$ is noetherian the answer is *yes*.


Indeed, in this case $X$ is affine if and only if $X\_{\textrm{red}}$ is affine.


See [Hartshorne, Algebraic Geometry], Exercise 3.1 p. 216.


