The residue field at \(x \in X\) is \(\kappa(x) \da \OO_{X,x}/\mfm_x\); for \(x = \mfp \in \Spec A\) this is \(\Frac(A/\mfp) = A_\mfp/\mfp A_\mfp\). To evaluate \(f \in \OO_X(U)\) at \(x\) is to take its image under \(\OO_X(U) \to \OO_{X,x} \to \kappa(x)\).
A closed point is one whose closure is itself, corresponding to a maximal ideal in an affine chart. A generic point of an irreducible closed \(Z\) is \(\eta\) with \(\closure{\ts{\eta}} = Z\). For \(X\) over \(k\), a rational point is an \(x\) with \(\kappa(x) = k\).
Write \(\tilde x \leadsto x\), “\(\tilde x\) specialises to \(x\)”, when \(x \in \closure{\ts{\tilde x}}\); equivalently \(\mfp_{\tilde x} \subseteq \mfp_x\) in an affine chart.
(General points and general fibres)
Let \(X\) be an irreducible scheme. A property holds at a general point of \(X\) if there is a dense open \(U \subseteq X\) such that it holds at every point of \(U\). For a morphism \(f \colon X \to Y\) with \(Y\) irreducible, a property holds for the general fibre of \(f\) if there is a dense open \(V \subseteq Y\) such that the fibre \(X_y\) has it for every \(y \in V\).
Every nonempty open subset of an irreducible scheme contains its generic point. On \(\AA^1_k = \Spec k[t]\) over an algebraically closed field \(k\), the property “\(\kappa(x) = k\)” holds at every closed point and fails at the generic point \(\eta\), where \(\kappa(\eta) = k(t)\); the set of closed points is dense but not open. For the family \(V(xy - t) \to \AA^1_t\), the general fibre is a smooth conic \(xy = c\) with \(c \neq 0\), while the generic fibre is the curve \(xy = t\) over the field \(k(t)\).
A section of the structure sheaf evaluates in the residue field \(\kappa(x)\), which can vary with \(x\); so a section is not in general a function from \(\abs X\) to one fixed field. On \(\Spec \ZZ\) the element \(5\) evaluates to \(0 \in \FF_5\), to \(5 \in \FF_7\), and to \(5 \in \QQ\) at the generic point. On a nonreduced scheme a nonzero section can vanish at every point: \(\eps \in k[\eps]/\eps^2\). So a section of \(\OO_X\) is not determined by its values at the points of \(X\).
Specialisation is a partial order on the points of a scheme; on \(\Spec A\) it is inclusion of primes. For a variety over an algebraically closed field, the points of the classical variety are the closed points of the associated scheme. For a discrete valuation ring \(R\) with fraction field \(K\), \(\Spec R\) consists of a generic point specialising to a closed point, and the valuative criteria (D-8XX95) test lifts of morphisms \(\Spec K\to X\) to \(\Spec R\).