> **Possible Duplicate:**  
> [Is there an example of a scheme X whose reduction X_red is affine but X is not affine?](http://mathoverflow.net/questions/95/is-there-an-example-ofa-scheme-x-whose-reduction-x-red-is-affine-but-x-is-not-af)  


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 $X$.