(Flat)
\(f : X \to Y\) is flat at \(x\) if \(\OO_{X,x}\) is a flat module over \(\OO_{Y,f(x)}\), and flat if it is flat at every point. More generally \(\mcf \in \mods{\OO_X}\) is flat over \(Y\) at \(x\) if \(\mcf_x\) is a flat \(\OO_{Y,f(x)}\)-module, and \(f\) is flat exactly when \(\OO_X\) is. \(f\) is faithfully flat if it is flat and surjective.
Flatness is a condition on stalks and therefore checkable on affines, where it is the ordinary flatness of \(B \to A\). Over a Dedekind domain, a module is flat if and only if it is torsion-free; hence every dominant morphism from an integral scheme to a regular integral curve is flat. Over a regular base of dimension two this fails: the blowup \(\operatorname{Bl}_0\AA^2\to\AA^2\) is dominant with integral source, and it is not flat, since its fibre over \(0\) has dimension \(1\) while its other fibres are points, whereas a flat morphism \(f\) of schemes of finite type over a field satisfies \(\dim_x X_{f(x)}=\dim_x X-\dim_{f(x)} Y\) at every point \(x\) [@Har10a, Proposition III.9.5].
For \(A\to B\) faithfully flat, an \(A\)-module \(M\) is zero if and only if \(M\tensor_AB=0\), and a complex of \(A\)-modules is exact if and only if it is exact after \(\wait\tensor_AB\) (D-DEFFFLAT).
A flat morphism \(f \colon X \to T\) is regarded as the family of schemes \(X_t = X \times_T \Spec \kappa(t)\) parametrized by the points \(t \in T\) (D-MORFIB). For \(f\) flat and projective over a connected Noetherian base, the Hilbert polynomials of the fibres are constant (T-COHFLATCHI).
(Associated points)
A point \(x\) of a Noetherian scheme \(X\) is an associated point if \(\mfm_x\) is an associated prime of \(0\) in \(\OO_{X,x}\), equivalently if every element of \(\mfm_x\) is a zero divisor in \(\OO_{X,x}\). The generic points of the irreducible components are associated points, and when \(X\) is reduced they are the only ones.
(Flatness over a regular curve)
Let \(f \colon X \to Y\) be a morphism with \(X\) Noetherian and \(Y\) integral, regular and of dimension one. Then \(f\) is flat if and only if every associated point of \(X\) maps to the generic point of \(Y\). In particular, if \(X\) is reduced, \(f\) is flat if and only if every irreducible component of \(X\) dominates \(Y\). [@Har10a]
(Flat limits)
Let \(Y\) be integral, regular and of dimension one, \(P \in Y\) a closed point, and \(X^\circ \subseteq \PP^n_{Y \sm P}\) a closed subscheme flat over \(Y \sm P\). Then there is a unique closed subscheme \(X \subseteq \PP^n_Y\), flat over \(Y\), whose restriction to \(\PP^n_{Y \sm P}\) is \(X^\circ\): the scheme-theoretic closure of \(X^\circ\). [@Har10a]
(What a flat family does not preserve)
The family \(V(xy - t) \subseteq \AA^2 \times \AA^1_t \to \AA^1_t\) is flat, since its total space is integral and dominates the line; its fibres are smooth irreducible hyperbolas for \(t \neq 0\) and the reducible pair of axes at \(t = 0\). The family \(V(y^2 - tx) \to \AA^1_t\) is flat for the same reason; its fibres are reduced parabolas for \(t \neq 0\) and the nonreduced double line \(y^2 = 0\) at \(t = 0\). So irreducibility, reducedness and smoothness can all fail in the special fibre, while the Hilbert polynomial of a flat projective family cannot change.
Dependencies
D-MORFIBThe fibre of a morphism as a base changeuses
Backlinks
T-MORMIRACLEMiracle flatnessusesFE-MORNOTFLATThe blowup and the normalisation of a node are not flatusesD-ETFPPFThe étale and fppf topologies, and the small étale siteuses