2_4x Separated Proper Morphisms

II.4: Separated and Proper Morphisms

II.4.1 #to_work

Show that a finite morphism is proper.

II.4.2 #to_work

Let \(S\) be a scheme, let \(X\) be a reduced scheme over \(S\), and let \(Y\) be a separated scheme over \(S\). Let \(f\) and \(g\) be two \(S\)-morphisms of \(X\) to \(Y\) which agree on an open dense subset of \(X\). Show that \(f=g\). Give examples to show that this result fails if either

• \(X\) is nonreduced, or

• \(Y\) is nonseparated. ¹

II.4.3 #to_work

Let \(X\) be a separated scheme over an affine scheme \(S\). Let \(U\) and \(V\) be open affine subsets of \(X\). Then \(U \cap V\) is also affine. Give an example to show that this fails if \(X\) is not separated.

II.4.4. The image of a proper scheme is proper. #to_work

Let \(f: X \rightarrow Y\) be a morphism of separated schemes of finite type over a noetherian scheme \(S\). Let \(Z\) be a closed subscheme of \(X\) which is proper over \(S\). Show that \(f(Z)\) is closed in \(Y\), and that \(f(Z)\) with its image subscheme structure (Ex. 3.11d) is proper over \(S\). ²

II.4.5 #to_work

Let \(X\) be an integral scheme of finite type over a field \(k\), having function field \(K\). We say that a valuation of \(K / k\) (see \(\mathrm{I}, \S 6\) ) has center \(x\) on \(X\) if its valuation ring \(R\) dominates the local ring \(\mathcal{O}_{x, X}\).

• If \(X\) is separated over \(k\), then the center of any valuation of \(K / k\) on \(X\) (if it exists) is unique.

• If \(X\) is proper over \(k\), then every valuation of \(K / k\) has a unique center on \(X\). ³

• * Prove the converses of (a) and (b). ⁴

• If \(X\) is proper over \(k\), and if \(k\) is algebraically closed, show that \(\Gamma\left(X, \mathcal{O}_X\right)=k\). This result generalizes (I, 3.4a). ⁵

II.4.6 #to_work

Let \(f: X \rightarrow Y\) be a proper morphism of affine varieties over \(k\). Then \(f\) is a finite morphism. ⁶

II.4.7. Schemes Over \(\mathbf{R}\). #to_work

For any scheme \(X_0\) over \(\mathbf{R}\), let \(X=X_0 \times_{\mathbf{R}} \mathbf{C}\). Let \(\alpha: \mathbf{C} \rightarrow \mathbf{C}\) be complex conjugation, and let \(\sigma: X \rightarrow X\) be the automorphism obtained by keeping \(X_0\) fixed and applying \(\alpha\) to \(\mathbf{C}\). Then \(X\) is a scheme over \(\mathbf{C}\), and \(\sigma\) is a semi-linear automorphism, in the sense that we have a commutative diagram:

Link to Diagram

Since \(\sigma^2=\mathrm{id}\), we call \(\sigma\) an involution.

• Now let \(X\) be a separated scheme of finite type over \(\mathbf{C}\), let \(\sigma\) be a semilinear involution on \(X\), and assume that for any two points \(x_1, x_2 \in X\), there is an open affine subset containing both of them. (This last condition is satisfied for example if \(X\) is quasi-projective.) Show that there is a unique separated scheme \(X_0\) of finite type over \(\mathbf{R}\), such that \(X_0 \times_{\mathbf{R}} \mathbf{C} \cong X\), and such that this isomorphism identifies the given involution of \(X\) with the one on \(X_0 \times_{\mathbf{R}} \mathbf{C}\) described above.

  For the following statements, \(X_0\) will denote a separated scheme of finite type over \(\mathbf{R}\), and \(X, \sigma\) will denote the corresponding scheme with involution over \(\mathbf{C}\).

• Show that \(X_0\) is affine if and only if \(X\) is.

• If \(X_0, Y_0\) are two such schemes over \(\mathbf{R}\), then to give a morphism \(f_0: X_0 \rightarrow Y_0\) is equivalent to giving a morphism \(f: X \rightarrow Y\) which commutes with the involutions, i.e., \(f \circ \sigma_X=\sigma_Y \circ f\).

• If \(X \cong \mathbf{A}_{\mathbf{C}}^1\), then \(X_0 \cong \mathbf{A}_{\mathbf{R}}^1\).

• If \(X \cong \mathbf{P}_{\mathbf{C}}^1\), then either \(X_0 \cong \mathbf{P}_{\mathbf{R}}^1\), or \(X_0\) is isomorphic to the conic in \(\mathbf{P}_{\mathbf{R}}^2\) given by the homogeneous equation \(x_0^2+x_1^2+x_2^2=0\).

II.4.8 #to_work

Let \(\mathscr{P}\) be a property of morphisms of schemes such that:

• a closed immersion has \(\mathscr{P}\);

• a composition of two morphisms having \(\mathscr{P}\) has \(\mathscr{P}\);

• \(\mathscr{P}\) is stable under base extension.

  Then show that:

• a product of morphisms having \(\mathscr{P}\) has \(\mathscr{P}\);

• if \(f: X \rightarrow Y\) and \(g: Y \rightarrow Z\) are two morphisms, and if \(g \circ f\) has \(\mathscr{P}\) and \(g\) is separated, then \(f\) has \(\mathscr{P}\); ⁷

• If \(f: X \rightarrow Y\) has \(\mathscr{P}\), then \(f_{\text {red }}: X_{\text {red }} \rightarrow Y_{\text {red }}\) has \(\mathscr{P}\).

II.4.9 #to_work

Show that a composition of projective morphisms is projective. ⁸ Conclude that projective morphisms have properties (a)-(f) of (Ex. 4.8) above.

II.4.10 * Chow’s Lemma. #to_work

This result says that proper morphisms are fairly close to projective morphisms.

Let \(X\) be proper over a noetherian scheme \(S\). Then there is a scheme \(X^{\prime}\) and a morphism \(g: X^{\prime} \rightarrow X\) such that \(X^{\prime}\) is projective over \(S\), and there is an open dense subset \(U \subseteq X\) such that \(g\) induces an isomorphism of \(g^{-1}(U)\) to \(U\). Prove this result in the following steps.

• Reduce to the case \(X\) irreducible.

• Show that \(X\) can be covered by a finite number of open subsets \(U_i, i=1, \ldots, n\), each of which is quasi-projective over \(S\). Let \(U_i \rightarrow P_i\) be an open immersion of \(U_i\) into a scheme \(P_i\) which is projective over \(S\).

• Let \(U=\bigcap U_i\), and consider the map \begin{align*} f: U \to X \underset{\scriptscriptstyle {S} }{\times} P_1 \underset{\scriptscriptstyle {S} }{\times} \cdots \underset{\scriptscriptstyle {S} }{\times} P_n \end{align*} deduced from the given maps \(U \rightarrow X\) and \(U \rightarrow P_i\). Let \(X^{\prime}\) be the closed image subscheme structure ⁹ \(f(U)^{-}\). Let \(g: X^{\prime} \rightarrow X\) be the projection onto the first factor, and let \begin{align*} h: X' \to P = P_1 \underset{\scriptscriptstyle {k} }{\times} \cdots \underset{\scriptscriptstyle {k} }{\times} P_n \end{align*} be the projection onto the product of the remaining factors. Show that \(h\) is a closed immersion, hence \(X^{\prime}\) is projective over \(S\).

• Show that \(g^{-1}(U) \rightarrow U\) is an isomorphism, thus completing the proof.

II.4.11 #to_work

If you are willing to do some harder commutative algebra, and stick to noetherian schemes, then we can express the valuative criteria of separatedness and properness using only discrete valuation rings.

• If \(\mathcal{O}, {\mathfrak{m}}\) is a noetherian local domain with quotient field \(K\), and if \(L\) is a finitely generated field extension of \(K\), then there exists a discrete valuation ring \(R\) of \(L\) dominating \(\mathcal{O}\).

  Prove this in the following steps.

  • By taking a polynomial ring over \(\mathcal{O}\), reduce to the case where \(L\) is a finite extension field of \(K\).
  • Then show that for a suitable choice of generators \(x_1, \ldots, x_n\) of \(m\), the ideal \(\mathfrak{a}=\left(x_1\right)\) in \(\mathcal{O}^{\prime}=\mathbb{C}\left[x_2 / x_1, \ldots, x_n / x_1\right]\) is not equal to the unit ideal.
  • Then let \(\mathfrak{p}\) be a minimal prime ideal of \(a\), and let \(\mathcal{O}_p^{\prime}\) be the localization of \(\mathcal{O}^{\prime}\) at \(\mathfrak{p}\). This is a noetherian local domain of dimension 1 dominating \(\mathcal{O}\).
  • Let \(\tilde{\mathscr{O}}_p^{\prime}\) be the integral closure of \(\mathcal{O}_p^{\prime}\) in \(L\). Use the theorem of Krull-Akizuki ¹⁰ to show that \(\tilde{U}_{\mathrm{p}}^{\prime}\) is noetherian of dimension 1.
  • Finally, take \(R\) to be a localization of \(\tilde{{\mathcal{O}}}_p^{\prime}\) at one of its maximal ideals.

• Let \(f: X \rightarrow Y\) be a morphism of finite type of noetherian schemes. Show that \(f\) is separated (respectively, proper) if and only if the criterion of \((4.3)\) (respectively, (4.7)) holds for all discrete valuation rings.

II.4.12 Examples of Valuation Rings. #to_work

Let \(k\) be an algebraically closed field.

• If \(K\) is a function field of dimension 1 over \(k\) then every valuation ring of \(K / k\) (except for \(K\) itself) is discrete. Thus the set of all of them is just the abstract nonsingular curve \(C_K\) of \((I, \S 6)\).

• If \(K / k\) is a function field of dimension two, there are several different kinds of valuations. Suppose that \(X\) is a complete nonsingular surface with function field \(K\).

• If \(Y\) is an irreducible curve on \(X\), with generic point \(x_1\), then the local ring \(R=\mathcal{O}_{x_1, X}\) is a discrete valuation ring of \(K / k\) with center at the (nonclosed) point \(x_1\) on \(X\).

• If \(f: X^{\prime} \rightarrow X\) is a birational morphism, and if \(Y^{\prime}\) is an irreducible curve in \(X^{\prime}\) whose image in \(X\) is a single closed point \(x_0\), then the local ring \(R\) of the generic point of \(Y^{\prime}\) on \(X^{\prime}\) is a discrete valuation ring of \(K / k\) with center at the closed point \(x_0\) on \(X\).

• Let \(x_0 \in X\) be a closed point. Let \(f: X_1 \rightarrow X\) be the blowing-up of \(x_0\) (I, §4) and let \(E_1=f^{-1}\left(x_0\right)\) be the exceptional curve. Choose a closed point \(x_1 \in E_1\), let \(f_2: X_2 \rightarrow X_1\) be the blowing-up of \(x_1\), and let \(E_2=\) \(f_2^{-1}\left(x_1\right)\) be the exceptional curve. Repeat.

  In this manner we obtain a sequence of varieties \(X_i\) with closed points \(x_i\) chosen on them, and for each \(i\), the local ring \(\mathcal{O}_{x_{i+1}, X_{i+1}}\) dominates \(\mathcal{O}_{x_i, X_i}\). Let \(R_0=\bigcup_{i=0}^{\infty} \mathcal{O}_{x_i, X_i}\). Then \(R_0\) is a local ring, so it is dominated by some valuation ring \(R\) of \(K / k\) by (I, 6.1A).

  Show that \(R\) is a valuation ring of \(K / k\), and that it has center \(x_0\) on \(X\). When is \(R\) a discrete valuation ring? ¹¹