Dimension
Fibre dimension and upper semicontinuity (Tag T-MORFIBDIM)
Let \(k\) be an infinite field and \(X \subseteq \AA^{n}_k\) an affine variety of dimension \(d\). Then there exists a finite morphism \(X \to \AA^{d}\) which is the restriction to \(X\) of a linear map \(\AA^{n} \to \AA^{d}\).
(Noether normalization (projective))
Let \(k\) be an infinite field and \(X \subseteq \PP^{n}_k\) a projective variety of dimension \(d\). Then there exists a finite morphism \(X \to \PP^{d}\) which is the restriction to \(X\) of a linear projection \(\PP^{n} \dashrightarrow \PP^{d}\) with centre a linear subspace \(\PP^{k} \subseteq \PP^{n}\) disjoint from \(X\), where \(k + d = n - 1\).
The linear forms are chosen from a nonempty Zariski open subset of a space of linear maps, which has \(k\)-points because \(k\) is infinite. Over a finite field a linear map need not exist: over \(\FF_2\), no linear map \(\AA^2 \to \AA^1\) restricts to a finite morphism on the irreducible cubic \(V(xy(x+y) + 1)\). Noether normalization still holds over every field if nonlinear maps such as \((x_1 - x_n^{e_1}, \ldots, x_{n-1} - x_n^{e_{n-1}})\) are allowed.
(Fibre dimension)
Let \(f : X \to Y\) be a dominant morphism of integral schemes of finite type over a field, and set \(e = \dim X - \dim Y\). Then every irreducible component of every nonempty fibre \(X_y\) has dimension at least \(e\), and there is a nonempty open \(V \subseteq Y\) with \(\dim X_y = e\) for every \(y \in V\).
(Semicontinuity)
The function \(x \mapsto \dim_x X_{f(x)}\) is upper semicontinuous on \(X\): the locus where the fibre dimension is at least \(n\) is closed. If \(f\) is flat then \(\dim_x X_{f(x)} = \dim_x X - \dim_{f(x)} Y\) at every point, so the fibre dimension is locally constant.
Upper semicontinuity means fibre dimension can jump up on special loci, never down. The blowup is the model: fibre dimension \(0\) generically, \(1\) over the origin, and the jump locus \(\ts{0}\) is closed. It cannot go the other way, because a component of a fibre is cut out by \(\dim Y\) equations locally and Krull’s height theorem bounds the drop.
The two halves are used differently. The lower bound proves positive-dimensional fibres in dimension-comparison arguments. The generic equality gives the dimension formula \(\dim X = \dim Y + \dim(\text{general fibre})\).
Flatness removes this jumping by forcing the expected local fibre dimension everywhere.
For the blowup \(\Bl_0\AA^2\to\AA^2\), the fibre over \(0\) is \(\PP^1\) and every other fibre is a point. For a dominant morphism \(f\colon X\to Y\) of varieties, \(\dim X=\dim Y+\dim X_y\) for \(y\) in a dense open subset of \(Y\).
Miracle flatness (Tag T-MORMIRACLE)
Let \(f : X \to Y\) be a local homomorphism of Noetherian local rings, or a morphism of schemes locally of finite type, with \(X\) Cohen–Macaulay and \(Y\) regular. If \[ \dim_x X_{f(x)} = \dim_x X - \dim_{f(x)} Y \] at every point, that is, the fibres have the expected dimension, then \(f\) is flat.
This is the theorem that makes flatness checkable, since the hypotheses are about dimension and the conclusion is about modules. In the form one uses it: a morphism between smooth varieties with equidimensional fibres of the expected dimension is flat.
It also explains the blowup counterexample rather than merely contradicting it: \(\Bl_0 \AA^2 \to \AA^2\) has a fibre of dimension \(1\) where the expected dimension is \(0\), so the hypothesis fails at exactly the point where flatness does.
The displayed dimension formula is itself a consequence of flatness in general, so the theorem says that for a Cohen–Macaulay source over a regular base the necessary condition is sufficient. The Cohen–Macaulay hypothesis rules out embedded depth defects that would otherwise allow the expected dimension formula without flatness.
If \(X\) is Cohen–Macaulay, \(Y\) is regular, and \(\dim_xX_{f(x)}=\dim\OO_{X,x}-\dim\OO_{Y,f(x)}\) at every point \(x\), then \(f\) is flat.
Images
Zariski's main theorem (Tag T-MORZMT)
A separated quasi-finite morphism \(f : X \to Y\) of Noetherian schemes factors as \[ X \injects \overline{X} \to Y \] with the first an open immersion and the second finite.
(Zariski's form)
Let \(f : X \to Y\) be a birational projective morphism of Noetherian integral schemes with \(Y\) normal. Then \(f_* \OO_X = \OO_Y\) and every fibre of \(f\) is connected. In particular a birational projective morphism to a normal \(Y\) that is quasi-finite is an isomorphism. Equivalently, \(\dim f^{-1}(y) > 0\) unless \(f^{-1}(y)\) is a singleton.
(Purity of the exceptional locus)
Let \(f : X \to Y\) be as above, with \(X\) and \(Y\) normal projective varieties over \(\CC\) \((\)the statement holds for Noetherian integral schemes with \(Y\) normal\()\). The exceptional locus \[ E = E(f) := \bigcup_{\dim f^{-1}(y) > 0} f^{-1}(y) \] is Zariski-closed in \(X\). If \(y_{0} \in f(E)\) is a smooth point of \(Y\), then \(E\) has codimension \(1\) in \(X\) near the points of the fibre \(f^{-1}(y_{0})\). In particular if \(Y\) is smooth then \(E\) is a projective hypersurface (pure of codimension \(1\)).
The first form is the structural statement: quasi-finite is finite minus a compactification, which is exactly what the hyperbola projection showed, and the theorem says that gap is the only one. Read alongside “finite equals proper plus quasi-finite”, it says a quasi-finite morphism fails to be finite only by missing points of a finite one.
The second form is the one used in birational geometry, and normality is where all the weight sits. Without it the theorem fails: the normalisation of a nodal cubic is a birational projective quasi-finite morphism that is not an isomorphism, and the two preimages of the node are exactly the disconnected fibre that normality would forbid. So the theorem is the precise sense in which a normal variety cannot be improved by a birational morphism without contracting something positive-dimensional.
Stein factorisation (Tag T-MORSTEIN)
Let \(f : X \to Y\) be a proper morphism of Noetherian schemes. Then \(f\) factors as \[ X \mapsvia{g} Y' \da \Spec_Y f_* \OO_X \mapsvia{h} Y \] with \(g\) proper with connected fibres and \(g_* \OO_X = \OO_{Y'}\), and \(h\) finite.
The factorisation separates the two ways a proper morphism can fail to be an isomorphism: \(g\) contracts connected fibres, while \(h\) is finite. Thus a proper morphism factors into a contraction followed by a finite cover.
\(f_* \OO_X\) is a coherent sheaf of \(\OO_Y\)-algebras because \(f\) is proper, and it is finite because coherent, which is where properness enters; this is why there is no Stein factorisation for a general morphism. Since \(g\) is surjective with connected fibres, the connected components of the fibre \(X_y\) are the fibres of \(g\) over the points of the finite set \(h^{-1}(y)\), so they correspond bijectively to the points of \(h^{-1}(y)\). This count is at most the degree of \(h\) over \(y\), the length \(\dim_{\kappa(y)} (f_* \OO_X \otimes \kappa(y))\), and can be smaller: for \(f = h \colon \AA^1_\CC \to \AA^1_\CC\), \(t \mapsto t^2\), which is finite with \(g = \id\), the fibre over \(0\) is the single point \(\Spec \CC[t]/(t^2)\) of length \(2\).
The immediate corollary is the connectedness statement in Zariski’s main theorem: if \(f_* \OO_X = \OO_Y\) then \(Y' = Y\), so \(h\) is the identity and the fibres of \(f\) are connected. In the other direction, for a proper morphism of varieties with \(Y\) normal and \(f\) birational, the Stein factorisation is what proves the fibres connected without a separate argument.
A proper morphism \(f\colon X\to Y\) of noetherian schemes factors as \(X\xrightarrow{f'}Y'\xrightarrow{g}Y\) with \(Y'=\underline{\Spec}_Y f_*\OO_X\), \(g\) finite, and \(f'_*\OO_X=\OO_{Y'}\), so every fibre of \(f'\) is connected. For \(X\) integral, \(f\) birational, and \(Y\) normal, \(g\) is finite and birational onto a normal scheme, hence an isomorphism, and every fibre of \(f\) is connected. The normalization of the nodal cubic is finite and birational, and its fibre over the node is two points. The same map is not flat: its fibre over the node has length \(2\) and every other fibre has length \(1\).
Connected fibres and integral closure
$\OO_Y \to \pi_* \OO_X$ is an isomorphism exactly when $Y$ is integrally closed in $K(X)$ (Tag P-AGXMISCOCONNECTED)
Show that given \(\pi: X \rightarrow Y\), \(Y\) is integrally closed in \(K(X)\) over \(K(Y)\) if and only if \(\OO_{Y} \rightarrow \pi_{*} \OO_{X}\) is an isomorphism (i.e., \(\pi\) is \(\OO\)-connected).
Solution
Assume that \(X\) and \(Y\) are integral Noetherian schemes, \(\pi:X\to Y\) is proper and dominant, and \(X\) is normal. Say that \(Y\) is integrally closed in \(K(X)\) if \(\OO_Y(V)\) is integrally closed in \(K(X)\) for every affine open \(V\subseteq Y\). Dominance gives an inclusion \[ K(Y)\subseteq K(X). \]
<1>1. Let \[ V=\Spec A\subseteq Y \] be a nonempty affine open and put \[ U=\pi^{-1}(V), \qquad B=\Gamma(U,\OO_X). \] Then \(B\) is a finite \(A\)-algebra contained in \(K(X)\).
The restricted morphism \[ \pi_U:U\longrightarrow V \] is proper. By proper pushforward of coherent sheaves, or equivalently the finite part of Stein factorisation T-MORSTEIN, \[ (\pi_U)_*\OO_U \] is a coherent \(\OO_V\)-algebra. Since \(V\) is affine, this means \[ B=\Gamma(U,\OO_U) \] is a finite \(A\)-module.
Because \(\pi\) is dominant and \(V\) is nonempty, \(U\) is a nonempty open subset of the integral scheme \(X\). Every regular function on \(U\) determines a rational function on \(X\), giving an injection \[ B\injects K(X). \]
<1>2. Every element of \(B\) is integral over \(A\).
By step <1>1, \(B\) is finite as an \(A\)-module. Every element of a finite \(A\)-algebra is integral over \(A\). Therefore \[ B \subseteq \overline A^{\,K(X)}, \] where \(\overline A^{\,K(X)}\) denotes the integral closure of \(A\) in \(K(X)\).
<1>3. Since \(X\) is normal, every element of \(K(X)\) integral over \(A\) belongs to \(B\).
Let \[ \alpha\in K(X) \] be integral over \(A\). Thus \(\alpha\) satisfies a monic equation \[ \alpha^n+a_{n-1}\alpha^{n-1}+\cdots+a_0=0, \qquad a_i\in A. \]
Fix \(x\in U\). The map \[ A\longrightarrow\OO_{X,x} \] induced by \(\pi\) carries the same equation into \(\OO_{X,x}\). Hence \(\alpha\) is integral over \(\OO_{X,x}\). Since \(X\) is normal, \(\OO_{X,x}\) is integrally closed in \(K(X)\), so \[ \alpha\in\OO_{X,x}. \]
This holds for every \(x\in U\). Thus \(\alpha\) is regular in a neighborhood of every point of \(U\). These local representatives are all the same rational function and therefore agree on overlaps, so they glue to a section of \(\OO_X(U)\). Hence \[ \alpha\in B. \] Therefore \[ \overline A^{\,K(X)} \subseteq B. \]
<1>4. Consequently, \[ \boxed{ \Gamma(\pi^{-1}V,\OO_X) = \overline{\Gamma(V,\OO_Y)}^{\,K(X)} } \] for every nonempty affine open \(V\subseteq Y\).
Step <1>2 gives \[ B\subseteq\overline A^{\,K(X)}, \] and step <1>3 gives the reverse inclusion.
<1>5. If \(Y\) is integrally closed in \(K(X)\), then \[ \OO_Y\xrightarrow{\sim}\pi_*\OO_X. \]
By hypothesis, for every nonempty affine open \[ V=\Spec A\subseteq Y, \] the ring \(A\) is integrally closed in \(K(X)\): \[ A=\overline A^{\,K(X)}. \] Step <1>4 therefore gives \[ \Gamma(V,\OO_Y) = A = \Gamma(\pi^{-1}V,\OO_X) = \Gamma(V,\pi_*\OO_X). \] These identifications are induced by the natural sheaf map and are compatible with restriction. Hence that map is an isomorphism.
<1>6. Conversely, if \[ \OO_Y\xrightarrow{\sim}\pi_*\OO_X, \] then \(Y\) is integrally closed in \(K(X)\).
Let \[ V=\Spec A\subseteq Y \] be nonempty affine. The sheaf isomorphism gives \[ A = \Gamma(V,\OO_Y) \cong \Gamma(\pi^{-1}V,\OO_X) =B. \] By step <1>4, \[ B=\overline A^{\,K(X)}. \] Therefore \[ A=\overline A^{\,K(X)}. \] Since this holds on every affine open, \(Y\) is integrally closed in \(K(X)\).
<1>7. Under these hypotheses, \[ \boxed{ Y\text{ is integrally closed in }K(X) \iff \OO_Y\xrightarrow{\sim}\pi_*\OO_X. } \]
Step <1>5 proves the forward implication and step <1>6 proves the reverse implication.
<1>8. Q.E.D.
Step <1>7 is the asserted equivalence.
The problem states no hypotheses on \(X\), \(Y\), or \(\pi\). The solution assumes \(X\) and \(Y\) integral Noetherian, \(\pi\) proper and dominant, so that \(K(Y) \subseteq K(X)\), and \(X\) normal. The implication from \(\OO_Y \cong \pi_* \OO_X\) to integral closedness fails without normality of \(X\). Let \(Y\) be the cuspidal cubic \(V(y^2 - x^3)\) and \(\pi = \id_Y\). Then \(\OO_Y \to \pi_* \OO_X\) is an isomorphism, but \(t = y/x \in K(Y)\) satisfies \(t^2 = x\), so it is integral over \(\OO_Y\) and does not lie in \(\OO_Y\).