Separatedness is encoded by the diagonal morphism and by uniqueness in the valuative criterion.
Separated and quasi-separated morphisms (Tag D-T2J3Q)
A morphism \(f : X \to Y\) is separated if the diagonal \[ \Delta_{X/Y} : X \to \fiberprod{X}{Y}{X} \] is a closed immersion. It is quasi-separated if \(\Delta_{X/Y}\) is quasicompact, equivalently if for every affine open \(V \subseteq Y\) the scheme \(f^{-1}(V)\) is quasi-separated in the sense below.
(Quasicompact and quasi-separated)
A scheme \(X\) is quasicompact if every open cover of \(X\) has a finite subcover, and a morphism \(f: X \to Y\) is quasicompact if \(f^{-1}(V)\) is quasicompact for every affine open \(V \subseteq Y\). A scheme \(X\) is quasi-separated if the intersection of any two quasicompact open subsets of \(X\) is quasicompact, equivalently if the intersection of any two affine open subsets is quasicompact; this is quasi-separatedness of \(X \to \Spec \ZZ\).
A topological space \(X\) is Hausdorff if and only if the diagonal is closed in \(X\times X\) with the product topology. An irreducible variety of positive dimension is not Hausdorff in the Zariski topology, since any two nonempty open subsets meet; and the underlying space of \(\fiberprod{X}{Y}{X}\) is in general not the product space. Separatedness asks that the diagonal be closed in the fibre product of schemes: \(\AA^1_k\) is separated over \(k\), while the line \(X\) with a doubled origin is not, since the closure of the diagonal in \(X\times_kX\) contains the point \((0_1,0_2)\) for the two origins \(0_1\ne0_2\).
Every separated morphism is quasi-separated, and every morphism from a Noetherian scheme is quasi-separated. Gluing two copies of \(\Spec k[x_1,x_2,\ldots]\) along the complement of the origin gives a scheme that is not quasi-separated: the intersection of the two affine copies is not quasicompact.
Two morphisms agreeing on a dense open agree (Tag PR-QX0VL)
Let \(f : X \to Y\) be separated, let \(Z\) be a reduced scheme over \(Y\), and let \(g, h : Z \to X\) be \(Y\)-morphisms agreeing on a dense open \(U \subseteq Z\). Then \(g = h\).
This is the good property to name, and the proof shows why both hypotheses are there. The pair \((g,h)\) gives \(Z \to \fiberprod{X}{Y}{X}\), and the locus where \(g\) and \(h\) agree is the preimage of the diagonal. Separatedness makes that preimage closed; it contains the dense \(U\), so it is all of \(Z\) topologically. Reducedness upgrades the topological statement to an equality of morphisms, because a closed subscheme of a reduced scheme with the whole space as support is the whole scheme.
So the analogue for quasi-separated is weaker in exactly the way the diagonal is: quasicompactness of the diagonal gives a retrocompact agreement locus, not a closed one, and the conclusion fails. Quasi-separatedness is what one assumes to make cohomology and direct images behave, not to make morphisms equal.
The line with doubled origin, and dropping each hypothesis (Tag FE-H8DY5)
Glue two copies of \(\AA^1_k\) along \(\AA^1 \sm \ts{0}\) by the identity. The result \(X\) has two origins, and \(X \to \Spec k\) is not separated: the two inclusions \(\AA^1 \to X\) agree on the dense open \(\AA^1 \sm \ts{0}\) and differ at \(0\), so the agreement locus is not closed.
By the proposition this is the failure of separatedness, not merely an illustration of it.
(Separated but not reduced)
Take \(Z = \Spec k[\varepsilon]/(\varepsilon^2)\) and \(X = \AA^1 = \Spec k[x]\), with \(g\) and \(h\) given by \(x \mapsto 0\) and \(x \mapsto \varepsilon\). Both are morphisms to a separated \(X\), and they agree on the unique point of \(Z\) — indeed on every reduced subscheme of \(Z\) — but \(g \neq h\). The dense open here is not proper, and it need not be: on a nonreduced base, agreeing topologically is not agreeing.
The two examples separate the roles of separatedness and reducedness. The doubled line is also a standard example of a scheme that is not a classical variety, showing why separatedness is imposed in the scheme-theoretic definition of a variety.
Properness
Proper morphisms and the valuative criteria (Tag D-8XX95)
A morphism is proper if it is separated, of finite type, and universally closed.
(Valuative criteria for valuation rings)
Let \(f \colon X \to Y\) be a quasiseparated morphism. For every valuation ring \(R\) with fraction field \(K\) and every commutative square \(\Spec K \to X\), \(\Spec R \to Y\) over \(f\), consider the lifts \(\Spec R \to X\) making both triangles commute.
\(f\) is separated if and only if every such square has at most one lift.
If \(f\) is of finite type, \(f\) is proper if and only if every such square has exactly one lift.
(Valuative criteria for discrete valuation rings)
Let \(f: X \to Y\) be of finite type with \(X\) Noetherian. For every discrete valuation ring \(R\) with fraction field \(K\) and every commutative square \[ \Spec K \to X, \qquad \Spec R \to Y , \] consider the lifts \(\Spec R \to X\) making both triangles commute.
\(f\) is separated if and only if every such square has at most one lift.
\(f\) is proper if and only if every such square has exactly one lift.
For a discrete valuation ring \(R\) with fraction field \(K\), \(\Spec R\) has one closed point and one generic point, like the germ of a smooth curve at a point, and \(\Spec K\) is the punctured germ. A lift \(\Spec R\to X\) extends a map of the punctured germ across the puncture. Uniqueness of extensions is the analogue of the Hausdorff property (limits of sequences are unique), and existence is the analogue of compactness (limits exist).
(A closed morphism that is not universally closed)
\(\AA^1_k \to \Spec k\) is separated, of finite type, and closed, but not proper: after base change along \(\AA^1_k\to\Spec k\), the projection \(\AA^1 \times \AA^1 \to \AA^1\) sends the closed hyperbola \(V(xy-1)\) onto \(\AA^1 \sm \ts{0}\), which is not closed.
Let \(f\colon X\to Y\) be of finite type with \(X\) noetherian. Then \(f\) is separated exactly when, for every valuation ring \(R\) with fraction field \(K\), a morphism \(\Spec K\to X\) over a given \(\Spec R\to Y\) has at most one lift \(\Spec R\to X\), and proper exactly when it has exactly one.
A projective morphism is proper. For a connected reduced scheme \(X\) proper over an algebraically closed field \(k\), every regular function \(f\colon X\to\AA^1_k\) is constant: composed with \(\AA^1_k\subset\PP^1_k\), its image is closed, connected, and misses \(\infty\), so it is a point.