A topological space $X$ is said to be spectral if it satisfies all of the following conditions:
1) $X$ is quasicompact and quasiseparated
2) $X$ is sober (over irreducible component has a unique generic point) .
3) $X$ has a basis of quasicompact open subsets
Hochster proved that a spectral space is homeomorphic to the spectrum of a ring, i.e. is affine. So projective space $\mathbb{P}^n_k$ over a field (with the Zariski topology) is not a spectral space. Which one(s) of the above three conditions fails? It seems to me that all three hold... so I am confused about something.
Projective space is a spectral space, so it is homeomorphic to the spectrum of a ring. For example, $\mathbb{P}^1$ over an algebraically closed field $k$ is homeomorphic to $\mathbb{A}^1$, which is the spectrum of $k[x]$. Being homeomorphic is a much weaker condition than being isomorphic as varieties.