The Stacks project

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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05BD. Beware of the difference between the letter 'O' and the digit '0'.