A nonconstant morphism of smooth projective curves is finite and flat.
Going up and going down, read on spectra (Tag PR-RJ3P7)
Let \(\phi: A \to B\) be a ring map with \(B\) integral over \(\phi(A)\), and \(f: \Spec B \to \Spec A\) the induced map, \(f(\mfq) = \phi^{-1}(\mfq)\).
(Lying over) If \(\phi\) is injective, then for every prime \(\mfp \subseteq A\) there is a prime \(\mfq \subseteq B\) with \(\phi^{-1}(\mfq) = \mfp\); so \(f\) is surjective.
(Going up) If \(\mfp_1 \subseteq \cdots \subseteq \mfp_n\) is a chain of primes of \(A\) and \(\mfq_1 \subseteq \cdots \subseteq \mfq_m\), \(m < n\), is a chain of primes of \(B\) with \(\phi^{-1}(\mfq_i) = \mfp_i\), then it extends to a chain \(\mfq_1 \subseteq \cdots \subseteq \mfq_n\) with \(\phi^{-1}(\mfq_i) = \mfp_i\) for all \(i\).
In particular \(f\) is a closed map. [@AM18]
(Going down)
Let \(A \subseteq B\) be domains with \(A\) integrally closed and \(B\) integral over \(A\). If \(\mfp_1 \supseteq \cdots \supseteq \mfp_n\) is a chain of primes of \(A\) and \(\mfq_1 \supseteq \cdots \supseteq \mfq_m\), \(m < n\), is a chain of primes of \(B\) with \(\mfq_i \cap A = \mfp_i\), then it extends to a chain \(\mfq_1 \supseteq \cdots \supseteq \mfq_n\) with \(\mfq_i \cap A = \mfp_i\). Going down also holds for every flat ring map \(A \to B\). [@AM18]
The translation is the content of the answer: going up says that a chain of primes in \(A\) ascending from one that is hit can be lifted, which is exactly the statement that the image of a closed set is closed. So going up is the algebraic form of “a finite morphism is closed”, hence of “a finite morphism is proper”.
Going up together with incomparability makes \(\dim B = \dim A\) for an integral extension, which is why a finite surjective morphism cannot drop dimension. Going down is what makes heights match, \(\operatorname{ht} \mfq = \operatorname{ht}(\mfq \cap A)\), under its hypotheses: geometrically, every generization of \(f(\mfq)\) in \(\Spec A\) lifts to a generization of \(\mfq\).
The two are the reason finite morphisms behave like branched covers: closed, surjective, with finite fibres, and preserving dimension.
By going up, an integral ring map \(A\to B\) induces a closed map \(\Spec B\to\Spec A\), so finite morphisms are closed. Going down holds for flat ring maps, and for integral extensions \(A\subseteq B\) of domains with \(A\) integrally closed; for such an integral extension it gives \(\operatorname{ht}\mathfrak q=\operatorname{ht}(\mathfrak q\cap A)\) for every prime \(\mathfrak q\) of \(B\).
Flatness and fibres
Flat morphisms (Tag D-MORFLAT)
\(f : X \to Y\) 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.
For a flat projective morphism to a noetherian scheme, the Hilbert polynomial of the fibres is locally constant on the base.
For a finite flat morphism \(f\colon X\to Y\) with \(Y\) noetherian and connected, \(f_*\OO_X\) is locally free of constant rank \(d\), and every fibre has length \(d\).
A nonconstant morphism of smooth curves is flat, because a torsion-free finitely generated module over a discrete valuation ring is free.
For a nonconstant morphism \(f\colon X\to Y\) of smooth projective curves, every fibre \(f^{-1}(y)\), counted with multiplicity, therefore has degree \(\deg f\).
Where flatness fails
The blowup and the normalisation of a node are not flat (Tag FE-MORNOTFLAT)
Let \(\pi : \Bl_0 \AA^2 \to \AA^2\). Every fibre over \(p \neq 0\) is a single point, and \(\pi^{-1}(0) \cong \PP^1\). Flatness over a reduced connected base forces the fibres to be equidimensional, so \(\pi\) is not flat.
(Normalisation of a node)
Let \(X\) be a nodal cubic with node \(q\) and let \(f : \tilde X \to X\) be its normalisation. If \(f\) were flat then \(f_* \OO_{\tilde X}\) would be a flat coherent sheaf, hence locally free, and of rank \(1\) hence invertible. But \(q\) has two preimages, so \((f_* \OO_{\tilde X})_q\) needs two generators over \(\OO_{X,q}\), and an invertible sheaf needs one.
The two examples are the two ways flatness fails, and they are worth separating. The blowup fails because the fibre dimension jumps; the normalisation fails because the fibre length jumps, from \(1\) to \(2\), while the dimension is constant at \(0\). So “equidimensional fibres” is necessary and not sufficient, and the invariant that flatness really keeps constant is the Hilbert polynomial.
No nontrivial normalisation is flat, for the same rank argument, and a ramified map of curves in the sense of a map that is not finite locally free is not either. The contrast to draw is with a ramified map of smooth curves, which is finite flat of constant degree: there the ramification shows up in the fibre being a fat point, not in the degree dropping.
For a projective morphism to an integral noetherian scheme, constancy of the Hilbert polynomial of the fibres implies flatness (Flatness is constancy of the Hilbert polynomial).
Flatness in examples
Flatness of the affine line over the integers, and faithful flatness as faithfulness (Tag FE-SCHFFLAT)
\(\AA^1_\ZZ = \Spec \ZZ[x] \to \Spec \ZZ\) is flat, because \(\ZZ[x]\) is a free \(\ZZ\)-module with basis \(1, x, x^2, \ldots\), and free modules are flat. More generally \(\AA^n_R \to \Spec R\) is flat for every ring \(R\).
An \(R\)-module \(M\) is faithfully flat if and only if \(M\) is flat and the functor \(- \otimes_R M\) on \(R\)-modules is faithful.
Flatness is needed on both sides: faithfulness of \(- \otimes M\) alone does not give flatness. For \(R = \ZZ\) and \(M = \ZZ \oplus \ZZ/2\), \(- \otimes M\) is faithful because \(M\) has \(\ZZ\) as a direct summand, but \(M\) is not flat, since \(\ZZ \xrightarrow{2} \ZZ\) tensored with \(\ZZ/2\) is the zero map on \(\ZZ/2 \neq 0\).
Finite morphisms of smooth varieties of equal dimension are flat (Tag PR-MIRACLEFLAT)
Let \(f \colon X \to Y\) be a finite morphism of varieties over a field with \(X\) Cohen–Macaulay, \(Y\) smooth, and \(\dim X = \dim Y\) at every point of \(X\). Then \(f\) is flat. In particular a finite surjective morphism between smooth varieties of the same dimension is flat. [@Har10a, Exercise III.10.9]
The hypotheses cannot be dropped. The inclusion of the origin \(\Spec k \to \AA^1_k\) is finite and both schemes are smooth, but it is not flat, since \(k = k[t]/(t)\) is a torsion \(k[t]\)-module; here the dimensions differ. The normalization \(\AA^1 \to V(y^2 - x^3)\), \(t \mapsto (t^2, t^3)\), is finite between varieties of the same dimension with smooth source, but the target is not smooth, and the map is not flat: it is birational of degree one and not an isomorphism.
Criteria for flatness
The local and infinitesimal criteria for flatness (Tag T-FLATCRIT)
Let \(A\) be a Noetherian ring, \(I \subseteq A\) an ideal, \(B\) a Noetherian \(A\)-algebra with \(IB\) contained in the Jacobson radical of \(B\), and \(M\) a finitely generated \(B\)-module. The following are equivalent.
\(M\) is flat over \(A\).
\(M/IM\) is flat over \(A/I\) and \(\Tor_1^A(A/I, M) = 0\).
\(M/I^{n}M\) is flat over \(A/I^{n}\) for every \(n \geq 1\).
In particular, if \(A \to B\) is a local homomorphism of Noetherian local rings with residue field \(k\) of \(A\) and \(M\) is a finitely generated \(B\)-module, then \(M\) is flat over \(A\) if and only if \(\Tor_1^A(k, M) = 0\).
Condition 2 is the local criterion for flatness; with \(I = \mathfrak{m}_A\) the hypothesis on \(M/\mathfrak{m}_A M\) is automatic, because every module over the field \(A/\mathfrak{m}_A\) is flat. Condition 3 is the infinitesimal criterion for flatness: flatness over \(A\) is detected on the infinitesimal thickenings \(\Spec A/I^n\) of \(\Spec A/I\). Geometrically, for a morphism \(f \colon X \to Y\) of locally Noetherian schemes, a point \(x \in X\) and \(y = f(x)\), a coherent sheaf \(\mathcal{F}\) on \(X\) is flat over \(Y\) at \(x\) if and only if \(\Tor_1^{\OO_{Y,y}}(\kappa(y), \mathcal{F}_x) = 0\).
Let \(A = k[t]_{(t)}\) and \(B = (A[x]/(tx))_{(t, x)}\). Then \(\Tor_1^A(k, B)\) is the \(t\)-torsion of \(B\), computed from \(0 \to A \xrightarrow{t} A \to k \to 0\), and it contains \(x \neq 0\). So \(B\) is not flat over \(A\): the family \(V(tx) \to \Spec k[t]\) has fibre the point \(x = 0\) over \(t \neq 0\) and the line \(\AA^1\) over \(t = 0\).