Quasi-separated Schemes

Defn. A topological space is quasi-separated if the intersection of any two quasi-compact open sets is quasi-compact.  (In a separated space, the intersection of any two affine open sets is affine).

Claim. Affine schemes are quasi-separated.
Proof.  The quasi-compact open subsets of an affine schemes are all finite unions of distinguished open sets. So their intersection will again be a finite union of distinguished open sets.

(Note: quasicompact scheme $\leftrightarrow$ has a finite cover by affine opens).

Lemma. A scheme is quasi-separated iff the intersection of any two affine open subsets is a finite union of affine open subsets.

Proof. Suppose a scheme $X$ is quasi-separated. Take any two affine open subsets of $X$. Since they are quasi-compact, their intersection is also quasi-compact. As their intersection can be covered by affine open sets (e.g. by distinguished open sets), it can be covered by finitely many distinguished open sets.

Conversely, suppose the intersection of any two affine open subsets of $X$ is a finite union of affine open subsets. Take two quasi-compact open sets of $X$. Then each of this open set, being a quasi-compact scheme, is a finite union of affine opens. Their intersection will be the union of intersection of their affine open sets, which by hypothesis is a finite union of affine opens.
------
In practice, we will see the hypothesis "$X$ is quasi-compact and quasi-separated" appear often. Usually, we would transfer such a hypothesis to the following statement, by using the above Lemma.

Proposition. A scheme $X$ is quasi-compact and quasi-separated iff $X$ can be covered by a finite number of affine open subsets, any two of which have intersection also covered by a finite number of affine open subsets.

Proof. $X$ is quasi-compact iff $X$ has a finite cover by affine opens. Then combine with Lemma above.
----------------
Claim. Projective $A$-schemes are quasi-compact and quasi-separated.
Proof.  Let $S$ be a finitely generated graded ring over $A$ (i.e. $S_0 = A$) (by definition of projective $A$-schemes). Suppose $f_1, \ldots, f_n$ are generators of $S$ over $A$. Let $X = \text{Proj} A$. Then $D_{+}(f_i)$ cover $X$.  Notice $D_+{f_i} = \text{Spec} ((S_{f_i})_0)$ where $(S_{f_i})_0$ denotes the $0$-th graded piece of $S_{f_i}$, and $D_+(f_i) \cap D_+(f_j) = D_+(f_i f_j)$.
------------------
Example. (Non-quasi-separated Scheme).

Let $X = \text{Spec}k[x_1, x_2, \ldots]$. Let $\mathfrak{m}$ be the maximal ideal $(x_1, x_2, \ldots).$. Let $U = X - [\mathfrak{m}].$ Let $Y $ be the result of gluing two copies of $X$ along $U$. Then $Y$ is not quasi-separated.
Proof. The two copies of $X$ are both quasi-compact, being affine schemes. However, their intersection is $U$, which is not quasi-compact.

--------------------
Claim. Locally Noetherian schemes are quasi-separated.