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70.5 Morphisms between integral algebraic spaces

The following lemma characterizes dominant morphisms of finite degree between integral algebraic spaces.

Lemma 70.5.1. Let $S$ be a scheme. Let $X$, $Y$ be integral algebraic spaces over $S$ Let $x \in |X|$ and $y \in |Y|$ be the generic points. Let $f : X \to Y$ be locally of finite type. Assume $f$ is dominant (Morphisms of Spaces, Definition 65.18.1). The following are equivalent:

  1. the transcendence degree of $x/y$ is $0$,

  2. the extension $\kappa (x) \supset \kappa (y)$ (see proof) is finite,

  3. there exist nonempty affine opens $U \subset X$ and $V \subset Y$ such that $f(U) \subset V$ and $f|_ U : U \to V$ is finite,

  4. $f$ is quasi-finite at $x$, and

  5. $x$ is the only point of $|X|$ mapping to $y$.

If $f$ is separated or if $f$ is quasi-compact, then these are also equivalent to

  1. there exists a nonempty affine open $V \subset Y$ such that $f^{-1}(V) \to V$ is finite.

Proof. By elementary topology, we see that $f(x) = y$ as $f$ is dominant. Let $Y' \subset Y$ be the schematic locus of $Y$ and let $X' \subset f^{-1}(Y')$ be the schematic locus of $f^{-1}(Y')$. By the discussion above, using Decent Spaces, Proposition 66.12.4 and Theorem 66.10.2, we see that $x \in |X'|$ and $y \in |Y'|$. Then $f|_{X'} : X' \to Y'$ is a morphism of integral schemes which is locally of finite type. Thus we see that (1), (2), (3) are equivalent by Morphisms, Lemma 29.50.7.

Condition (4) implies condition (1) by Morphisms of Spaces, Lemma 65.33.3 applied to $X \to Y \to Y$. On the other hand, condition (3) implies condition (4) as a finite morphism is quasi-finite and as $x \in U$ because $x$ is the generic point. Thus (1) – (4) are equivalent.

Assume the equivalent conditions (1) – (4). Suppose that $x' \mapsto y$. Then $x \leadsto x'$ is a specialization in the fibre of $|X| \to |Y|$ over $y$. If $x' \not= x$, then $f$ is not quasi-finite at $x$ by Decent Spaces, Lemma 66.18.9. Hence $x = x'$ and (5) holds. Conversely, if (5) holds, then (5) holds for the morphism of schemes $X' \to Y'$ (see above) and we can use Morphisms, Lemma 29.50.7 to see that (1) holds.

Observe that (6) implies the equivalent conditions (1) – (5) without any further assumptions on $f$. To finish the proof we have to show the equivalent conditions (1) – (5) imply (6). This follows from Decent Spaces, Lemma 66.21.4. $\square$

Definition 70.5.2. Let $S$ be a scheme. Let $X$ and $Y$ be integral algebraic spaces over $S$. Let $f : X \to Y$ be locally of finite type and dominant. Assume any of the equivalent conditions (1) – (5) of Lemma 70.5.1. Let $x \in |X|$ and $y \in |Y|$ be the generic points. Then the positive integer

\[ \text{deg}(X/Y) = [\kappa (x) : \kappa (y)] \]

is called the degree of $X$ over $Y$.

Lemma 70.5.3. Let $S$ be a scheme. Let $X$, $Y$, $Z$ be integral algebraic spaces over $S$. Let $f : X \to Y$ and $g : Y \to Z$ be dominant morphisms locally of finite type. Assume any of the equivalent conditions (1) – (5) of Lemma 70.5.1 hold for $f$ and $g$. Then

\[ \deg (X/Z) = \deg (X/Y) \deg (Y/Z). \]

Proof. This comes from the multiplicativity of degrees in towers of finite extensions of fields, see Fields, Lemma 9.7.7. $\square$


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