38.11 Extending properties from an open
In this section we collect a number of results of the form: If f : X \to S is a flat morphism of schemes and f satisfies some property over a dense open of S, then f satisfies the same property over all of S.
Lemma 38.11.1.slogan Let f : X \to S be a morphism of schemes. Let \mathcal{F} be a quasi-coherent \mathcal{O}_ X-module. Let U \subset S be open. Assume
f is locally of finite presentation,
\mathcal{F} is of finite type and flat over S,
U \subset S is retrocompact and scheme theoretically dense,
\mathcal{F}|_{f^{-1}U} is of finite presentation.
Then \mathcal{F} is of finite presentation.
Proof.
The problem is local on X and S, hence we may assume X and S affine. Write S = \mathop{\mathrm{Spec}}(A) and X = \mathop{\mathrm{Spec}}(B). Let N be a finite B-module such that \mathcal{F} is the quasi-coherent sheaf associated to N. We have U = D(f_1) \cup \ldots \cup D(f_ n) for some f_ i \in A, see Algebra, Lemma 10.29.1. As U is schematically dense the map A \to A_{f_1} \times \ldots \times A_{f_ n} is injective. Pick a prime \mathfrak q \subset B lying over \mathfrak p \subset A corresponding to x \in X mapping to s \in S. By Lemma 38.10.9 the module N_\mathfrak q is of finite presentation over B_\mathfrak q. Choose a surjection \varphi : B^{\oplus m} \to N of B-modules. Choose k_1, \ldots , k_ t \in \mathop{\mathrm{Ker}}(\varphi ) and set N' = B^{\oplus m}/\sum Bk_ j. There is a canonical surjection N' \to N and N is the filtered colimit of the B-modules N' constructed in this manner. Thus we see that we can choose k_1, \ldots , k_ t such that (a) N'_{f_ i} \cong N_{f_ i}, i = 1, \ldots , n and (b) N'_\mathfrak q \cong N_\mathfrak q. This in particular implies that N'_\mathfrak q is flat over A. By openness of flatness, see Algebra, Theorem 10.129.4 we conclude that there exists a g \in B, g \not\in \mathfrak q such that N'_ g is flat over A. Consider the commutative diagram
\xymatrix{ N'_ g \ar[r] \ar[d] & N_ g \ar[d] \\ \prod N'_{gf_ i} \ar[r] & \prod N_{gf_ i} }
The bottom arrow is an isomorphism by choice of k_1, \ldots , k_ t. The left vertical arrow is an injective map as A \to \prod A_{f_ i} is injective and N'_ g is flat over A. Hence the top horizontal arrow is injective, hence an isomorphism. This proves that N_ g is of finite presentation over B_ g. We conclude by applying Algebra, Lemma 10.23.2.
\square
Lemma 38.11.2. Let f : X \to S be a morphism of schemes. Let U \subset S be open. Assume
f is locally of finite type and flat,
U \subset S is retrocompact and scheme theoretically dense,
f|_{f^{-1}U} : f^{-1}U \to U is locally of finite presentation.
Then f is of locally of finite presentation.
Proof.
The question is local on X and S, hence we may assume X and S affine. Choose a closed immersion i : X \to \mathbf{A}^ n_ S and apply Lemma 38.11.1 to i_*\mathcal{O}_ X. Some details omitted.
\square
Lemma 38.11.3. Let f : X \to S be a morphism of schemes which is flat and locally of finite type. Let U \subset S be a dense open such that X_ U \to U has relative dimension \leq e, see Morphisms, Definition 29.29.1. If also either
f is locally of finite presentation, or
U \subset S is retrocompact,
then f has relative dimension \leq e.
Proof.
Proof in case (1). Let W \subset X be the open subscheme constructed and studied in More on Morphisms, Lemmas 37.22.7 and 37.22.9. Note that every generic point of every fibre is contained in W, hence it suffices to prove the result for W. Since W = \bigcup _{d \geq 0} U_ d, it suffices to prove that U_ d = \emptyset for d > e. Since f is flat and locally of finite presentation it is open hence f(U_ d) is open (Morphisms, Lemma 29.25.10). Thus if U_ d is not empty, then f(U_ d) \cap U \not= \emptyset as desired.
Proof in case (2). We may replace S by its reduction. Then U is scheme theoretically dense. Hence f is locally of finite presentation by Lemma 38.11.2. In this way we reduce to case (1).
\square
Lemma 38.11.4. Let f : X \to S be a morphism of schemes which is flat and proper. Let U \subset S be a dense open such that X_ U \to U is finite. If also either f is locally of finite presentation or U \subset S is retrocompact, then f is finite.
Proof.
By Lemma 38.11.3 the fibres of f have dimension zero. Hence f is quasi-finite (Morphisms, Lemma 29.29.5) whence has finite fibres (Morphisms, Lemma 29.20.10). Hence f is finite by More on Morphisms, Lemma 37.44.1.
\square
Lemma 38.11.5. Let f : X \to S be a morphism of schemes and U \subset S an open. If
f is separated, locally of finite type, and flat,
f^{-1}(U) \to U is an isomorphism, and
U \subset S is retrocompact and scheme theoretically dense,
then f is an open immersion.
Proof.
By Lemma 38.11.2 the morphism f is locally of finite presentation. The image f(X) \subset S is open (Morphisms, Lemma 29.25.10) hence we may replace S by f(X). Thus we have to prove that f is an isomorphism. We may assume S is affine. We can reduce to the case that X is quasi-compact because it suffices to show that any quasi-compact open X' \subset X whose image is S maps isomorphically to S. Thus we may assume f is quasi-compact. All the fibers of f have dimension 0, see Lemma 38.11.3. Hence f is quasi-finite, see Morphisms, Lemma 29.29.5. Let s \in S. Choose an elementary étale neighbourhood g : (T, t) \to (S, s) such that X \times _ S T = V \amalg W with V \to T finite and W_ t = \emptyset , see More on Morphisms, Lemma 37.41.6. Denote \pi : V \amalg W \to T the given morphism. Since \pi is flat and locally of finite presentation, we see that \pi (V) is open in T (Morphisms, Lemma 29.25.10). After shrinking T we may assume that T = \pi (V). Since f is an isomorphism over U we see that \pi is an isomorphism over g^{-1}U. Since \pi (V) = T this implies that \pi ^{-1}g^{-1}U is contained in V. By Morphisms, Lemma 29.25.15 we see that \pi ^{-1}g^{-1}U \subset V \amalg W is scheme theoretically dense. Hence we deduce that W = \emptyset . Thus X \times _ S T = V is finite over T. This implies that f is finite (after replacing S by an open neighbourhood of s), for example by Descent, Lemma 35.23.23. Then f is finite locally free (Morphisms, Lemma 29.48.2) and after shrinking S to a smaller open neighbourhood of s we see that f is finite locally free of some degree d (Morphisms, Lemma 29.48.5). But d = 1 as is clear from the fact that the degree is 1 over the dense open U. Hence f is an isomorphism.
\square
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