The Stacks project

37.31 Limit arguments

Some lemmas involving limits of schemes, and Noetherian approximation. We stick mostly to the affine case. Some of these lemmas are special cases of lemmas in the chapter on limits.

Lemma 37.31.1. Let $f : X \to S$ be a morphism of affine schemes, which is of finite presentation. Then there exists a cartesian diagram

\[ \xymatrix{ X_0 \ar[d]_{f_0} & X \ar[l]^ g \ar[d]^ f \\ S_0 & S \ar[l] } \]

such that

  1. $X_0$, $S_0$ are affine schemes,

  2. $S_0$ of finite type over $\mathbf{Z}$,

  3. $f_0$ is finite of finite type.

Proof. Write $S = \mathop{\mathrm{Spec}}(A)$ and $X = \mathop{\mathrm{Spec}}(B)$. As $f$ is of finite presentation we see that $B$ is of finite presentation as an $A$-algebra, see Morphisms, Lemma 29.21.2. Thus the lemma follows from Algebra, Lemma 10.127.18. $\square$

Lemma 37.31.2. Let $f : X \to S$ be a morphism of affine schemes, which is of finite presentation. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite presentation. Then there exists a diagram as in Lemma 37.31.1 such that there exists a coherent $\mathcal{O}_{X_0}$-module $\mathcal{F}_0$ with $g^*\mathcal{F}_0 = \mathcal{F}$.

Proof. Write $S = \mathop{\mathrm{Spec}}(A)$, $X = \mathop{\mathrm{Spec}}(B)$, and $\mathcal{F} = \widetilde{M}$. As $f$ is of finite presentation we see that $B$ is of finite presentation as an $A$-algebra, see Morphisms, Lemma 29.21.2. As $\mathcal{F}$ is of finite presentation over $\mathcal{O}_ X$ we see that $M$ is of finite presentation as a $B$-module, see Properties, Lemma 28.16.2. Thus the lemma follows from Algebra, Lemma 10.127.18. $\square$

Lemma 37.31.3. Let $f : X \to S$ be a morphism of affine schemes, which is of finite presentation. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite presentation and flat over $S$. Then we may choose a diagram as in Lemma 37.31.2 and sheaf $\mathcal{F}_0$ such that in addition $\mathcal{F}_0$ is flat over $S_0$.

Proof. Write $S = \mathop{\mathrm{Spec}}(A)$, $X = \mathop{\mathrm{Spec}}(B)$, and $\mathcal{F} = \widetilde{M}$. As $f$ is of finite presentation we see that $B$ is of finite presentation as an $A$-algebra, see Morphisms, Lemma 29.21.2. As $\mathcal{F}$ is of finite presentation over $\mathcal{O}_ X$ we see that $M$ is of finite presentation as a $B$-module, see Properties, Lemma 28.16.2. As $\mathcal{F}$ is flat over $S$ we see that $M$ is flat over $A$, see Morphisms, Lemma 29.25.2. Thus the lemma follows from Algebra, Lemma 10.168.1. $\square$

Lemma 37.31.4. Let $f : X \to S$ be a morphism of affine schemes, which is of finite presentation and flat. Then there exists a diagram as in Lemma 37.31.1 such that in addition $f_0$ is flat.

Proof. This is a special case of Lemma 37.31.3. $\square$

Lemma 37.31.5. Let $f : X \to S$ be a morphism of affine schemes, which is smooth. Then there exists a diagram as in Lemma 37.31.1 such that in addition $f_0$ is smooth.

Proof. Write $S = \mathop{\mathrm{Spec}}(A)$, $X = \mathop{\mathrm{Spec}}(B)$, and as $f$ is smooth we see that $B$ is smooth as an $A$-algebra, see Morphisms, Lemma 29.34.2. Hence the lemma follows from Algebra, Lemma 10.138.14. $\square$

Lemma 37.31.6. Let $f : X \to S$ be a morphism of affine schemes, which is of finite presentation with geometrically reduced fibres. Then there exists a diagram as in Lemma 37.31.1 such that in addition $f_0$ has geometrically reduced fibres.

Proof. Apply Lemma 37.31.1 to get a cartesian diagram

\[ \xymatrix{ X_0 \ar[d]_{f_0} & X \ar[l]^ g \ar[d]^ f \\ S_0 & S \ar[l]_ h } \]

of affine schemes with $X_0 \to S_0$ a finite type morphism of schemes of finite type over $\mathbf{Z}$. By Lemma 37.24.5 the set $E \subset S_0$ of points where the fibre of $f_0$ is geometrically reduced is a constructible subset. By Lemma 37.24.2 we have $h(S) \subset E$. Write $S_0 = \mathop{\mathrm{Spec}}(A_0)$ and $S = \mathop{\mathrm{Spec}}(A)$. Write $A = \mathop{\mathrm{colim}}\nolimits _ i A_ i$ as a direct colimit of finite type $A_0$-algebras. By Limits, Lemma 32.4.10 we see that $\mathop{\mathrm{Spec}}(A_ i) \to S_0$ has image contained in $E$ for some $i$. After replacing $S_0$ by $\mathop{\mathrm{Spec}}(A_ i)$ and $X_0$ by $X_0 \times _{S_0} \mathop{\mathrm{Spec}}(A_ i)$ we see that all fibres of $f_0$ are geometrically reduced. $\square$

Lemma 37.31.7. Let $f : X \to S$ be a morphism of affine schemes, which is of finite presentation with geometrically irreducible fibres. Then there exists a diagram as in Lemma 37.31.1 such that in addition $f_0$ has geometrically irreducible fibres.

Proof. Apply Lemma 37.31.1 to get a cartesian diagram

\[ \xymatrix{ X_0 \ar[d]_{f_0} & X \ar[l]^ g \ar[d]^ f \\ S_0 & S \ar[l]_ h } \]

of affine schemes with $X_0 \to S_0$ a finite type morphism of schemes of finite type over $\mathbf{Z}$. By Lemma 37.25.7 the set $E \subset S_0$ of points where the fibre of $f_0$ is geometrically irreducible is a constructible subset. By Lemma 37.25.2 we have $h(S) \subset E$. Write $S_0 = \mathop{\mathrm{Spec}}(A_0)$ and $S = \mathop{\mathrm{Spec}}(A)$. Write $A = \mathop{\mathrm{colim}}\nolimits _ i A_ i$ as a direct colimit of finite type $A_0$-algebras. By Limits, Lemma 32.4.10 we see that $\mathop{\mathrm{Spec}}(A_ i) \to S_0$ has image contained in $E$ for some $i$. After replacing $S_0$ by $\mathop{\mathrm{Spec}}(A_ i)$ and $X_0$ by $X_0 \times _{S_0} \mathop{\mathrm{Spec}}(A_ i)$ we see that all fibres of $f_0$ are geometrically irreducible. $\square$

Lemma 37.31.8. Let $f : X \to S$ be a morphism of affine schemes, which is of finite presentation with geometrically connected fibres. Then there exists a diagram as in Lemma 37.31.1 such that in addition $f_0$ has geometrically connected fibres.

Proof. Apply Lemma 37.31.1 to get a cartesian diagram

\[ \xymatrix{ X_0 \ar[d]_{f_0} & X \ar[l]^ g \ar[d]^ f \\ S_0 & S \ar[l]_ h } \]

of affine schemes with $X_0 \to S_0$ a finite type morphism of schemes of finite type over $\mathbf{Z}$. By Lemma 37.26.6 the set $E \subset S_0$ of points where the fibre of $f_0$ is geometrically connected is a constructible subset. By Lemma 37.26.2 we have $h(S) \subset E$. Write $S_0 = \mathop{\mathrm{Spec}}(A_0)$ and $S = \mathop{\mathrm{Spec}}(A)$. Write $A = \mathop{\mathrm{colim}}\nolimits _ i A_ i$ as a direct colimit of finite type $A_0$-algebras. By Limits, Lemma 32.4.10 we see that $\mathop{\mathrm{Spec}}(A_ i) \to S_0$ has image contained in $E$ for some $i$. After replacing $S_0$ by $\mathop{\mathrm{Spec}}(A_ i)$ and $X_0$ by $X_0 \times _{S_0} \mathop{\mathrm{Spec}}(A_ i)$ we see that all fibres of $f_0$ are geometrically connected. $\square$

Lemma 37.31.9. Let $d \geq 0$ be an integer. Let $f : X \to S$ be a morphism of affine schemes, which is of finite presentation all of whose fibres have dimension $d$. Then there exists a diagram as in Lemma 37.31.1 such that in addition all fibres of $f_0$ have dimension $d$.

Proof. Apply Lemma 37.31.1 to get a cartesian diagram

\[ \xymatrix{ X_0 \ar[d]_{f_0} & X \ar[l]^ g \ar[d]^ f \\ S_0 & S \ar[l]_ h } \]

of affine schemes with $X_0 \to S_0$ a finite type morphism of schemes of finite type over $\mathbf{Z}$. By Lemma 37.28.3 the set $E \subset S_0$ of points where the fibre of $f_0$ has dimension $d$ is a constructible subset. By Lemma 37.28.2 we have $h(S) \subset E$. Write $S_0 = \mathop{\mathrm{Spec}}(A_0)$ and $S = \mathop{\mathrm{Spec}}(A)$. Write $A = \mathop{\mathrm{colim}}\nolimits _ i A_ i$ as a direct colimit of finite type $A_0$-algebras. By Limits, Lemma 32.4.10 we see that $\mathop{\mathrm{Spec}}(A_ i) \to S_0$ has image contained in $E$ for some $i$. After replacing $S_0$ by $\mathop{\mathrm{Spec}}(A_ i)$ and $X_0$ by $X_0 \times _{S_0} \mathop{\mathrm{Spec}}(A_ i)$ we see that all fibres of $f_0$ have dimension $d$. $\square$

Lemma 37.31.10. Let $f : X \to S$ be a morphism of affine schemes, which is standard syntomic (see Morphisms, Definition 29.30.1). Then there exists a diagram as in Lemma 37.31.1 such that in addition $f_0$ is standard syntomic.

Proof. This lemma is a copy of Algebra, Lemma 10.136.12. $\square$

Lemma 37.31.11. (Noetherian approximation and combining properties.) Let $P$, $Q$ be properties of morphisms of schemes which are stable under base change. Let $f : X \to S$ be a morphism of finite presentation of affine schemes. Assume we can find cartesian diagrams

\[ \vcenter { \xymatrix{ X_1 \ar[d]_{f_1} & X \ar[l] \ar[d]^ f \\ S_1 & S \ar[l] } } \quad \text{and}\quad \vcenter { \xymatrix{ X_2 \ar[d]_{f_2} & X \ar[l] \ar[d]^ f \\ S_2 & S \ar[l] } } \]

of affine schemes, with $S_1$, $S_2$ of finite type over $\mathbf{Z}$ and $f_1$, $f_2$ of finite type such that $f_1$ has property $P$ and $f_2$ has property $Q$. Then we can find a cartesian diagram

\[ \xymatrix{ X_0 \ar[d]_{f_0} & X \ar[l] \ar[d]^ f \\ S_0 & S \ar[l] } \]

of affine schemes with $S_0$ of finite type over $\mathbf{Z}$ and $f_0$ of finite type such that $f_0$ has both property $P$ and property $Q$.

Proof. The given pair of diagrams correspond to cocartesian diagrams of rings

\[ \vcenter { \xymatrix{ B_1 \ar[r] & B \\ A_1 \ar[u] \ar[r] & A \ar[u] } } \quad \text{and}\quad \vcenter { \xymatrix{ B_2 \ar[r] & B \\ A_2 \ar[u] \ar[r] & A \ar[u] } } \]

Let $A_0 \subset A$ be a finite type $\mathbf{Z}$-subalgebra of $A$ containing the image of both $A_1 \to A$ and $A_2 \to A$. Such a subalgebra exists because by assumption both $A_1$ and $A_2$ are of finite type over $\mathbf{Z}$. Note that the rings $B_{0, 1} = B_1 \otimes _{A_1} A_0$ and $B_{0, 2} = B_2 \otimes _{A_2} A_0$ are finite type $A_0$-algebras with the property that $B_{0, 1} \otimes _{A_0} A \cong B \cong B_{0, 2} \otimes _{A_0} A$ as $A$-algebras. As $A$ is the directed colimit of its finite type $A_0$-subalgebras, by Limits, Lemma 32.10.1 we may assume after enlarging $A_0$ that there exists an isomorphism $B_{0, 1} \cong B_{0, 2}$ as $A_0$-algebras. Since properties $P$ and $Q$ are assumed stable under base change we conclude that setting $S_0 = \mathop{\mathrm{Spec}}(A_0)$ and

\[ X_0 = X_1 \times _{S_1} S_0 = \mathop{\mathrm{Spec}}(B_{0, 1}) \cong \mathop{\mathrm{Spec}}(B_{0, 2}) = X_2 \times _{S_2} S_0 \]

works. $\square$


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