\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

The Stacks project

36.55 Reduced fibre theorem

In this section we discuss the simplest kind of theorem of the kind advertised by the title. Although the proof of the result is kind of laborious, in essence it follows in a straightforward manner from Epp's result on eliminating ramification, see More on Algebra, Theorem 15.98.23.

Let $A$ be a Dedekind domain with fraction field $K$. Let $X$ be a scheme flat and of finite type over $A$. Let $L$ be a finite extension of $K$. Let $B$ be the integral closure of $A$ in $L$. Then $B$ is a Dedekind domain (Algebra, Lemma 10.119.18). Let $X_ B = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(B)$ be the base change. Then $X_ B \to \mathop{\mathrm{Spec}}(B)$ is of finite type (Morphisms, Lemma 28.14.4). Hence $X_ B$ is Noetherian (Morphisms, Lemma 28.14.6). Thus the normalization $\nu : Y \to X_ B$ exists (see Morphisms, Definition 28.51.1 and the discussion following). Picture

36.55.0.1
\begin{equation} \label{more-morphisms-equation-normalized-base-change} \xymatrix{ Y \ar[rd] \ar[r]_\nu & X_ B \ar[r] \ar[d] & X \ar[d] \\ & \mathop{\mathrm{Spec}}(B) \ar[r] & \mathop{\mathrm{Spec}}(A) } \end{equation}

We sometimes call $Y$ the normalized base change of $X$. In general the morphism $\nu $ may not be finite. But if $A$ is a Nagata ring (a condition that is virtually always satisfied in practice) then $\nu $ is finite and $Y$ is of finite type over $B$, see Morphisms, Lemmas 28.51.10 and 28.17.1.

Taking the normalized base change commutes with composition. More precisely, if $K \subset L \subset M$ are finite extensions of fields with integral closures $A \subset B \subset C$ then the normalized base change $Z$ of $Y \to \mathop{\mathrm{Spec}}(B)$ relative to $L \subset M$ is equal to the normalized base change of $X \to \mathop{\mathrm{Spec}}(A)$ relative to $K \subset M$.

Theorem 36.55.1. Let $A$ be a Dedekind ring with fraction field $K$. Let $X$ be a scheme flat and of finite type over $A$. Assume $A$ is a Nagata ring. There exists a finite extension $K \subset L$ such that the normalized base change $Y$ is smooth over $\mathop{\mathrm{Spec}}(B)$ at all generic points of all fibres.

Proof. During the proof we will repeatedly use that formation of the set of points where a (flat, finitely presented) morphism like $X \to \mathop{\mathrm{Spec}}(A)$ is smooth commutes with base change, see Morphisms, Lemma 28.32.15.

We first choose a finite extension $K \subset L$ such that $(X_ L)_{red}$ is geometrically reduced over $L$, see Varieties, Lemma 32.6.11. Since $Y \to (X_ B)_{red}$ is birational we see applying Varieties, Lemma 32.6.8 that $Y_ L$ is geometrically reduced over $L$ as well. Hence $Y_ L \to \mathop{\mathrm{Spec}}(L)$ is smooth on a dense open $V \subset Y_ L$ by Varieties, Lemma 32.25.7. Thus the smooth locus $U \subset Y$ of the morphism $Y \to \mathop{\mathrm{Spec}}(B)$ is open (by Morphisms, Definition 28.32.1) and is dense in the generic fibre. Replacing $A$ by $B$ and $X$ by $Y$ we reduce to the case treated in the next paragraph.

Assume $X$ is normal and the smooth locus $U \subset X$ of $X \to \mathop{\mathrm{Spec}}(A)$ is dense in the generic fibre. This implies that $U$ is dense in all but finitely many fibres, see Lemma 36.22.3. Let $x_1, \ldots , x_ r \in X \setminus U$ be the finitely many generic points of irreducible components of $X \setminus U$ which are moreover generic points of irreducible components of fibres of $X \to \mathop{\mathrm{Spec}}(A)$. Set $\mathcal{O}_ i = \mathcal{O}_{X, x_ i}$. Let $A_ i$ be the localization of $A$ at the maximal ideal corresponding to the image of $x_ i$ in $\mathop{\mathrm{Spec}}(A)$. By More on Algebra, Proposition 15.98.25 there exist finite extensions $K \subset K_ i$ which are solutions for the extension of discrete valuation rings $A_ i \to \mathcal{O}_ i$. Let $K \subset L$ be a finite extension dominating all of the extensions $K \subset K_ i$. Then $K \subset L$ is still a solution for $A_ i \to \mathcal{O}_ i$ by More on Algebra, Lemma 15.98.4.

Consider the diagram (36.55.0.1) with the extension $L/K$ we just produced. Note that $U_ B \subset X_ B$ is smooth over $B$, hence normal (for example use Algebra, Lemma 10.157.9). Thus $Y \to X_ B$ is an isomorphism over $U_ B$. Let $y \in Y$ be a generic point of an irreducible component of a fibre of $Y \to \mathop{\mathrm{Spec}}(B)$ lying over the maximal ideal $\mathfrak m \subset B$. Assume that $y \not\in U_ B$. Then $y$ maps to one of the points $x_ i$. It follows that $\mathcal{O}_{Y, y}$ is a local ring of the integral closure of $\mathcal{O}_ i$ in $R(X) \otimes _ K L$ (details omitted). Hence because $K \subset L$ is a solution for $A_ i \to \mathcal{O}_ i$ we see that $B_\mathfrak m \to \mathcal{O}_{Y, y}$ is formally smooth (this is the definition of being a "solution"). In other words, $\mathfrak m\mathcal{O}_{Y, y} = \mathfrak m_ y$ and the residue field extension is separable. Hence the local ring of the fibre at $y$ is $\kappa (y)$. This implies the fibre is smooth over $\kappa (\mathfrak m)$ at $y$ for example by Algebra, Lemma 10.138.5. This finishes the proof. $\square$

Lemma 36.55.2 (Variant over curves). Let $f : X \to S$ be a flat, finite type morphism of schemes. Assume $S$ is Nagata, integral with function field $K$, and regular of dimension $1$. Then there exists a finite extension $L/K$ such that in the diagram

\[ \xymatrix{ Y \ar[rd]_ g \ar[r]_-\nu & X \times _ S T \ar[d] \ar[r] & X \ar[d]_ f \\ & T \ar[r] & S } \]

the morphism $g$ is smooth at all generic points of fibres. Here $T$ is the normalization of $S$ in $\mathop{\mathrm{Spec}}(L)$ and $\nu : Y \to X \times _ S T$ is the normalization.

Proof. Choose a finite affine open covering $S = \bigcup \mathop{\mathrm{Spec}}(A_ i)$. Then $K$ is equal to the fraction field of $A_ i$ for all $i$. Let $X_ i = X \times _ S \mathop{\mathrm{Spec}}(A_ i)$. Choose $L_ i/K$ as in Theorem 36.55.1 for the morphism $X_ i \to \mathop{\mathrm{Spec}}(A_ i)$. Let $B_ i \subset L_ i$ be the integral closure of $A_ i$ and let $Y_ i$ be the normalized base change of $X$ to $B_ i$. Let $L/K$ be a finite extension dominating each $L_ i$. Let $T_ i \subset T$ be the inverse image of $\mathop{\mathrm{Spec}}(A_ i)$. For each $i$ we get a commutative diagram

\[ \xymatrix{ g^{-1}(T_ i) \ar[r] \ar[d] & Y_ i \ar[r] \ar[d] & X \times _ S \mathop{\mathrm{Spec}}(A_ i) \ar[d] \\ T_ i \ar[r] & \mathop{\mathrm{Spec}}(B_ i) \ar[r] & \mathop{\mathrm{Spec}}(A_ i) } \]

and in fact the left hand square is a normalized base change as discussed at the beginning of the section. In the proof of Theorem 36.55.1 we have seen that the smooth locus of $Y \to T$ contains the inverse image in $g^{-1}(T_ i)$ of the set of points where $Y_ i$ is smooth over $B_ i$. This proves the lemma. $\square$

Lemma 36.55.3 (Variant with separable extension). Let $A$ be a Dedekind ring with fraction field $K$. Let $X$ be a scheme flat and of finite type over $A$. Assume $A$ is a Nagata ring and that for every generic point $\eta $ of an irreducible component of $X$ the field extension $K \subset \kappa (\eta )$ is separable. Then there exists a finite separable extension $K \subset L$ such that the normalized base change $Y$ is smooth over $\mathop{\mathrm{Spec}}(B)$ at all generic points of all fibres.

Proof. This is proved in exactly the same manner as Theorem 36.55.1 with a few minor modifications. The most important change is to use More on Algebra, Lemma 15.98.26 instead of More on Algebra, Proposition 15.98.25. During the proof we will repeatedly use that formation of the set of points where a (flat, finitely presented) morphism like $X \to \mathop{\mathrm{Spec}}(A)$ is smooth commutes with base change, see Morphisms, Lemma 28.32.15.

Since $X$ is flat over $A$ every generic point $\eta $ of $X$ maps to the generic point of $\mathop{\mathrm{Spec}}(A)$. After replacing $X$ by its reduction we may assume $X$ is reduced. In this case $X_ K$ is geometrically reduced over $K$ by Varieties, Lemma 32.6.8. Hence $X_ K \to \mathop{\mathrm{Spec}}(K)$ is smooth on a dense open by Varieties, Lemma 32.25.7. Thus the smooth locus $U \subset X$ of the morphism $X \to \mathop{\mathrm{Spec}}(A)$ is open (by Morphisms, Definition 28.32.1) and is dense in the generic fibre. This reduces us to the situation of the following paragraph.

Assume $X$ is normal and the smooth locus $U \subset X$ of $X \to \mathop{\mathrm{Spec}}(A)$ is dense in the generic fibre. This implies that $U$ is dense in all but finitely many fibres, see Lemma 36.22.3. Let $x_1, \ldots , x_ r \in X \setminus U$ be the finitely many generic points of irreducible components of $X \setminus U$ which are moreover generic points of irreducible components of fibres of $X \to \mathop{\mathrm{Spec}}(A)$. Set $\mathcal{O}_ i = \mathcal{O}_{X, x_ i}$. Observe that the fraction field of $\mathcal{O}_ i$ is the residue field of a generic point of $X$. Let $A_ i$ be the localization of $A$ at the maximal ideal corresponding to the image of $x_ i$ in $\mathop{\mathrm{Spec}}(A)$. We may apply More on Algebra, Lemma 15.98.26 and we find finite separable extensions $K \subset K_ i$ which are solutions for $A_ i \to \mathcal{O}_ i$. Let $K \subset L$ be a finite separable extension dominating all of the extensions $K \subset K_ i$. Then $K \subset L$ is still a solution for $A_ i \to \mathcal{O}_ i$ by More on Algebra, Lemma 15.98.4.

Consider the diagram (36.55.0.1) with the extension $L/K$ we just produced. Note that $U_ B \subset X_ B$ is smooth over $B$, hence normal (for example use Algebra, Lemma 10.157.9). Thus $Y \to X_ B$ is an isomorphism over $U_ B$. Let $y \in Y$ be a generic point of an irreducible component of a fibre of $Y \to \mathop{\mathrm{Spec}}(B)$ lying over the maximal ideal $\mathfrak m \subset B$. Assume that $y \not\in U_ B$. Then $y$ maps to one of the points $x_ i$. It follows that $\mathcal{O}_{Y, y}$ is a local ring of the integral closure of $\mathcal{O}_ i$ in $R(X) \otimes _ K L$ (details omitted). Hence because $K \subset L$ is a solution for $A_ i \to \mathcal{O}_ i$ we see that $B_\mathfrak m \to \mathcal{O}_{Y, y}$ is formally smooth (this is the definition of being a "solution"). In other words, $\mathfrak m\mathcal{O}_{Y, y} = \mathfrak m_ y$ and the residue field extension is separable. Hence the local ring of the fibre at $y$ is $\kappa (y)$. This implies the fibre is smooth over $\kappa (\mathfrak m)$ at $y$ for example by Algebra, Lemma 10.138.5. This finishes the proof. $\square$

Lemma 36.55.4 (Variant with separable extensions over curves). Let $f : X \to S$ be a flat, finite type morphism of schemes. Assume $S$ is Nagata, integral with function field $K$, and regular of dimension $1$. Assume the field extensions $K \subset \kappa (\eta )$ are separable for every generic point $\eta $ of an irreducible component of $X$. Then there exists a finite separable extension $L/K$ such that in the diagram

\[ \xymatrix{ Y \ar[rd]_ g \ar[r]_-\nu & X \times _ S T \ar[d] \ar[r] & X \ar[d]_ f \\ & T \ar[r] & S } \]

the morphism $g$ is smooth at all generic points of fibres. Here $T$ is the normalization of $S$ in $\mathop{\mathrm{Spec}}(L)$ and $\nu : Y \to X \times _ S T$ is the normalization.

Proof. This follows from Lemma 36.55.3 in exactly the same manner that Lemma 36.55.2 follows from Theorem 36.55.1. $\square$


Comments (2)

Comment #3286 by Ariyan Javanpeykar on

Typo in end of second paragraph. " is of finite" should probably be " is finite"


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