Lemma 115.11.1. Let $(R, \mathfrak m, \kappa )$ be a local ring. Let $X \subset \mathbf{P}^ n_ R$ be a closed subscheme. Assume that $R = \Gamma (X, \mathcal{O}_ X)$. Then the special fibre $X_ k$ is geometrically connected.
115.11 Results on schemes
Lemmas that seem superfluous.
Proof. This is a special case of More on Morphisms, Theorem 37.53.5. $\square$
Lemma 115.11.2. Let $X$ be a Noetherian scheme. Let $Z_0 \subset X$ be an irreducible closed subset with generic point $\xi $. Let $\mathcal{P}$ be a property of coherent sheaves on $X$ such that
For any short exact sequence of coherent sheaves if two out of three of them have property $\mathcal{P}$ then so does the third.
If $\mathcal{P}$ holds for a direct sum of coherent sheaves then it holds for both.
For every integral closed subscheme $Z \subset Z_0 \subset X$, $Z \not= Z_0$ and every quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ Z$ we have $\mathcal{P}$ for $(Z \to X)_*\mathcal{I}$.
There exists some coherent sheaf $\mathcal{G}$ on $X$ such that
$\text{Supp}(\mathcal{G}) = Z_0$,
$\mathcal{G}_\xi $ is annihilated by $\mathfrak m_\xi $, and
property $\mathcal{P}$ holds for $\mathcal{G}$.
Then property $\mathcal{P}$ holds for every coherent sheaf $\mathcal{F}$ on $X$ whose support is contained in $Z_0$.
Proof. The proof is a variant on the proof of Cohomology of Schemes, Lemma 30.12.5. In exactly the same manner as in that proof we see that any coherent sheaf whose support is strictly contained in $Z_0$ has property $\mathcal{P}$.
Consider a coherent sheaf $\mathcal{G}$ as in (3). By Cohomology of Schemes, Lemma 30.12.2 there exists a sheaf of ideals $\mathcal{I}$ on $Z_0$ and a short exact sequence
\[ 0 \to \left((Z_0 \to X)_*\mathcal{I}\right)^{\oplus r} \to \mathcal{G} \to \mathcal{Q} \to 0 \]
where the support of $\mathcal{Q}$ is strictly contained in $Z_0$. In particular $r > 0$ and $\mathcal{I}$ is nonzero because the support of $\mathcal{G}$ is equal to $Z$. Since $\mathcal{Q}$ has property $\mathcal{P}$ we conclude that also $\left((Z_0 \to X)_*\mathcal{I}\right)^{\oplus r}$ has property $\mathcal{P}$. By (2) we deduce property $\mathcal{P}$ for $(Z_0 \to X)_*\mathcal{I}$. Slotting this into the proof of Cohomology of Schemes, Lemma 30.12.5 at the appropriate point gives the lemma. Some details omitted. $\square$
Lemma 115.11.3. Let $X$ be a Noetherian scheme. Let $\mathcal{P}$ be a property of coherent sheaves on $X$ such that
For any short exact sequence of coherent sheaves if two out of three of them have property $\mathcal{P}$ then so does the third.
If $\mathcal{P}$ holds for a direct sum of coherent sheaves then it holds for both.
For every integral closed subscheme $Z \subset X$ with generic point $\xi $ there exists some coherent sheaf $\mathcal{G}$ such that
$\text{Supp}(\mathcal{G}) = Z$,
$\mathcal{G}_\xi $ is annihilated by $\mathfrak m_\xi $, and
property $\mathcal{P}$ holds for $\mathcal{G}$.
Then property $\mathcal{P}$ holds for every coherent sheaf on $X$.
Proof. This follows from Lemma 115.11.2 in exactly the same way that Cohomology of Schemes, Lemma 30.12.6 follows from Cohomology of Schemes, Lemma 30.12.5. $\square$
Lemma 115.11.4. Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $s \in \Gamma (X, \mathcal{L})$ be a section. Let $\mathcal{F}' \subset \mathcal{F}$ be quasi-coherent $\mathcal{O}_ X$-modules. Assume that
$X$ is quasi-compact,
$\mathcal{F}$ is of finite type, and
$\mathcal{F}'|_{X_ s} = \mathcal{F}|_{X_ s}$.
Then there exists an $n \geq 0$ such that multiplication by $s^ n$ on $\mathcal{F}$ factors through $\mathcal{F}'$.
Proof. In other words we claim that $s^ n\mathcal{F} \subset \mathcal{F}' \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}$ for some $n \geq 0$. In other words, we claim that the quotient map $\mathcal{F} \to \mathcal{F}/\mathcal{F}'$ becomes zero after multiplying by a power of $s$. This follows from Properties, Lemma 28.17.3. $\square$
Lemma 115.11.5. Let $f : X \to Y$ be a morphism schemes. Assume
$X$ and $Y$ are integral schemes,
$f$ is locally of finite type and dominant,
$f$ is either quasi-compact or separated,
$f$ is generically finite, i.e., one of (1) – (5) of Morphisms, Lemma 29.51.7 holds.
Then there is a nonempty open $V \subset Y$ such that $f^{-1}(V) \to V$ is finite locally free of degree $\deg (X/Y)$. In particular, the degrees of the fibres of $f^{-1}(V) \to V$ are bounded by $\deg (X/Y)$.
Proof. We may choose $V$ such that $f^{-1}(V) \to V$ is finite. Then we may shrink $V$ and assume that $f^{-1}(V) \to V$ is flat and of finite presentation by generic flatness (Morphisms, Proposition 29.27.1). Then the morphism is finite locally free by Morphisms, Lemma 29.48.2. Since $V$ is irreducible the morphism has a fixed degree. The final statement follows from this and Morphisms, Lemma 29.57.3. $\square$
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