Lemma 32.14.1. Let $f : X \to S$ be a quasi-compact morphism of schemes. Let $g : T \to S$ be a morphism of schemes. Let $t \in T$ be a point and $Z \subset X_ T$ be a closed subscheme such that $Z \cap X_ t = \emptyset$. Then there exists an open neighbourhood $V \subset T$ of $t$, a commutative diagram

$\xymatrix{ V \ar[d] \ar[r]_ a & T' \ar[d]^ b \\ T \ar[r]^ g & S, }$

and a closed subscheme $Z' \subset X_{T'}$ such that

1. the morphism $b : T' \to S$ is locally of finite presentation,

2. with $t' = a(t)$ we have $Z' \cap X_{t'} = \emptyset$, and

3. $Z \cap X_ V$ maps into $Z'$ via the morphism $X_ V \to X_{T'}$.

Moreover, we may assume $V$ and $T'$ are affine.

Proof. Let $s = g(t)$. During the proof we may always replace $T$ by an open neighbourhood of $t$. Hence we may also replace $S$ by an open neighbourhood of $s$. Thus we may and do assume that $T$ and $S$ are affine. Say $S = \mathop{\mathrm{Spec}}(A)$, $T = \mathop{\mathrm{Spec}}(B)$, $g$ is given by the ring map $A \to B$, and $t$ correspond to the prime ideal $\mathfrak q \subset B$.

As $X \to S$ is quasi-compact and $S$ is affine we may write $X = \bigcup _{i = 1, \ldots , n} U_ i$ as a finite union of affine opens. Write $U_ i = \mathop{\mathrm{Spec}}(C_ i)$. In particular we have $X_ T = \bigcup _{i = 1, \ldots , n} U_{i, T} = \bigcup _{i = 1, \ldots n} \mathop{\mathrm{Spec}}(C_ i \otimes _ A B)$. Let $I_ i \subset C_ i \otimes _ A B$ be the ideal corresponding to the closed subscheme $Z \cap U_{i, T}$. The condition that $Z \cap X_ t = \emptyset$ signifies that $I_ i$ generates the unit ideal in the ring

$C_ i \otimes _ A \kappa (\mathfrak q) = (B \setminus \mathfrak q)^{-1}\left( C_ i \otimes _ A B/\mathfrak q C_ i \otimes _ A B \right)$

Since $I_ i (B \setminus \mathfrak q)^{-1}(C_ i \otimes _ A B) = (B \setminus \mathfrak q)^{-1} I_ i$ this means that $1 = x_ i/g_ i$ for some $x_ i \in I_ i$ and $g_ i \in B$, $g_ i \not\in \mathfrak q$. Thus, clearing denominators we can find a relation of the form

$x_ i + \sum \nolimits _ j f_{i, j}c_{i, j} = g_ i$

with $x_ i \in I_ i$, $f_{i, j} \in \mathfrak q$, $c_{i, j} \in C_ i \otimes _ A B$, and $g_ i \in B$, $g_ i \not\in \mathfrak q$. After replacing $B$ by $B_{g_1 \ldots g_ n}$, i.e., after replacing $T$ by a smaller affine neighbourhood of $t$, we may assume the equations read

$x_ i + \sum \nolimits _ j f_{i, j}c_{i, j} = 1$

with $x_ i \in I_ i$, $f_{i, j} \in \mathfrak q$, $c_{i, j} \in C_ i \otimes _ A B$.

To finish the argument write $B$ as a colimit of finitely presented $A$-algebras $B_\lambda$ over a directed set $\Lambda$. For each $\lambda$ set $\mathfrak q_\lambda = (B_\lambda \to B)^{-1}(\mathfrak q)$. For sufficiently large $\lambda \in \Lambda$ we can find

1. an element $x_{i, \lambda } \in C_ i \otimes _ A B_\lambda$ which maps to $x_ i$,

2. elements $f_{i, j, \lambda } \in \mathfrak q_{i, \lambda }$ mapping to $f_{i, j}$, and

3. elements $c_{i, j, \lambda } \in C_ i \otimes _ A B_\lambda$ mapping to $c_{i, j}$.

After increasing $\lambda$ a bit more the equation

$x_{i, \lambda } + \sum \nolimits _ j f_{i, j, \lambda }c_{i, j, \lambda } = 1$

will hold. Fix such a $\lambda$ and set $T' = \mathop{\mathrm{Spec}}(B_\lambda )$. Then $t' \in T'$ is the point corresponding to the prime $\mathfrak q_\lambda$. Finally, let $Z' \subset X_{T'}$ be the scheme theoretic image of $Z \to X_ T \to X_{T'}$. As $X_ T \to X_{T'}$ is affine, we can compute $Z'$ on the affine open pieces $U_{i, T'}$ as the closed subscheme associated to $\mathop{\mathrm{Ker}}(C_ i \otimes _ A B_\lambda \to C_ i \otimes _ A B/I_ i)$, see Morphisms, Example 29.6.4. Hence $x_{i, \lambda }$ is in the ideal defining $Z'$. Thus the last displayed equation shows that $Z' \cap X_{t'}$ is empty. $\square$

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