Solutions to The Rising Sea (Vakil)

Proof:

For the forward direction, suppose the diagonal \( \Delta \subset X \times X \) is closed and let \( x \neq y \in X \). Then \( (x, y) \in (X \times X) \backslash \Delta \) by definition, which by assumption is an open set. Thus, we can find some neighborhood \( W \subseteq (X \times X) \backslash \Delta \). By definition of the base on the product topology \( X \times X \), we may simply assume that \( W = U \times V \) for some open \( U, V \subseteq X \). To see that \( U \cap V = \emptyset \), suppose to the contrary that there exists some \( a \in U \cap V \). Then \( (a, ) \in (U \times V) \cap \Delta \) — a contradiction.

Conversely, suppose that \( X \) is Hausdorff; it suffices to show that \( (X \times X) \backslash \Delta \) is open. Fix some \( (x, y) \in (X \times X) \backslash \Delta \) so that \( x \neq y \) in \(X\). Then by assumption we can find some disjoint neighborhoods \( U, V \) of \(x, y\), respectively. Then by a similar argument to that above, \( U \times V \) is an open set in \(X \times X\) which does not intersect \( \Delta \).

Proof:

Following the hint, we wish to show that monomorphisms are separated. Suppose \( \pi : X \to Y \) is a monomorphism of schemes, and consider the Cartesian square

To show that \( X \) satisfies the universal property of the fibre product, suppose \( W \) is another scheme together with morphisms \( \alpha, \beta : W \to X \) such that \( \pi \circ \alpha = \pi \circ \beta \). Then as \( \pi \) is a monomorphism, \( \alpha = \beta \) so the unique choice of map \( W \to X \) is obviously \( \alpha = \beta \). Since universal properties are unique up to unique isomorphism, we obtain a unique isomorphism \( X \cong X \times_Y X \), which in this case is simply the diagonal morphism. Since isomorphisms are trivially closed embeddings, the result follows.

Proof:

Following the proof of Proposition 10.1.3, consider a morphism of affine schemes \( \pi : \spec A \to \spec B \) and the induced morphism of rings \( \pi^\sharp : B \to A \). By Exercise 9.1.B we know that \( \spec A \times_{\spec B} \spec A \cong \spec ( A \otimes_B A )\). Then as a morphism of affine schemes, the diagonal morphism \( \Delta : \spec A \to \spec (A \otimes_B A) \) corresponds to the ring morphism \( A \otimes_B A \to A \) given by \( a_1 \otimes a_2 \mapsto a_1a_2 \). As this is clearly a surjective morphism, the diagonal morphism is a closed embedding.

Proof:

Let \( X \) denote the line with double origin, \( U = \spec k[u] = \A^1 \) and \( V = \spec k[t] = \A^1 \). As the two affine lines cover the double origin, we get a cover for \( X \times_k X \) given by \( U \times_k U, V \times_k V, U \times_k V, V \times_k U \) — under the fibre product all of these copies of \( \A^2 \) are glued together except at the origin, giving us \( X \times_k X \) is isomorphic to \( \A^2 \) with four origins. Now the images \( \Delta(U) = U \times_k U \) and \( \Delta(V) = V\times_k V\) tell us that the image of the diagonal is precisely \( \A^2 \) with two copies of the origin (missing the copies in \( U \times V\) and \( V \times U \)). In particular, the preimage of \( \Delta^{-1}( U \times V ) \) is isomorphic to \( \A^1 \backslash \{ 0 \} \); thus our diagonal is locally of the form \( k[u, t] \to k[u]_u \) given by \( u \mapsto u \), \( t \mapsto 0 \) which is not surjective. Thus, the diagonal is not a closed embedding.

Proof:

Let \(X\), \( Y \) be irreducible \( \overline{k} \)-varieties; by definition, this implies that \( X \) and \( Y \) are integral, finite-type, separated \( \overline{k} \)-schemes. By Exercise 9.4.E, we know that \( X \times_{\overline{k}} Y \) is an integral, finite-type \( k \)-scheme. Jumping ahead to Proposition 10.1.10, we see that separatedness is preserved under base-change, so that \( X \times_{\overline{k}} Y \to X \) is separated. By Proposition 10.1.13, we know the composition of separated morphisms is separated, so \( (X \times_{\overline{k}} Y) \to Y \to \spec \overline{k} \) is separated.

Proof:

For the forward direction, suppose that for any \( U, V \subset X\) with \( \pi(U), \pi(V) \subset \spec A \) we have \( U \cap V \) is a finite union of affine opens. Since quasicompactness of a morphism is affine-local by Exercise 7.3.C, we may fix some open affine \( \spec C \subset X \times_Y X \) — by construction of the fibred product in Theorem 9.1.1, we may assume without loss of generality that \( \spec C = U \times_A V \) for some affine opens \( U, V \subset X \) over \( \spec A \). Then by Proposition 10.1.6, the preimage of \( U \times_A V \) under \( \delta_\pi \) is \( U \cap V \), which by assumption is a finite union of open affines and thus quasicompact.

Conversely, as quasiseparatedness is affine-local on the target in the previous sense, we may simply take \( Y = \spec A \). Suppose \( U, V \subset X \) are affine opens and consider \( U \times_A V \subset X \times_A X \), which is again an affine open by Exercise 9.1.B. Then the preimage of \( U \times_A V \) is quasicompact by assumption; again quoting Proposition 10.1.6, we know that the preimage is precisely \( U \cap V \) proving the claim.

Proof:

The hint literally solves the problem here.

Proof:

This should be immediate from Exercise 10.1.G and Exercise 7.3.C(b).

Alternatively, for the forward direction if we fix \( \spec A \subset Y \) and consider \( \{U_i\} \) an open cover of \( \pi^{-1}(\spec A) \) in \(X\), then the image of \( \delta_\pi \vert_{\pi^{-1}(\spec A)} \) is contained in the \( U_i \times_A U_i \). However each \( U_i \times_A U_i \) is affine; since quasicompactness is local on the target as mentioned in the hint, we have that each \( U_i \to U_i \times_A U_i \) is quasicompact (as \( X \to X \times_A X \) is), which implies that \( \pi^{-1}(\spec A) \to \pi^{-1}(\spec A) \times_A \pi^{-1}(\spec A) \) is quasicompact (again using the affine-local property). In the reverse direction, let \( \{\spec B_j\} \) be an affine open cover for \( Y \) and assume each \( \pi^{-1}(\spec B_j) \to \spec B_j \) is quasiseparated (such that \( \pi^{-1}(\spec B_j) \to \pi^{-1}(\spec B_j) \times_{B_j} \pi^{-1}(\spec B_j) \) is quasicompact). If we suppose each \( \pi^{-1}(\spec B_j) \) has an affine open cover \( \{ U_{ij} \} \), then clearly the collection of these across all \( j \) is an affine open cover for \(X\). We know that each \( U_{ij} \times_{B_j} U_{ij} \) is affine, and as quasicompactness is affine-local on the target we have that each \( U_{ij} \to U_{ij} \times_{B_j} U_{ij} \) is quasicompact. But then as these \( U_{ij} \) form a cover for \( X \) we indeed have that \( X \to X \times_Y X \) is quasicompact.

Proof:

Let \( X \) be an \( A \)-scheme and \( \pi : X \to \spec A \) the structure morphism. Since \( \spec \Z \) is final in the category of schemes, \( X \to \spec A \to \spec \Z \) must be the unique structure morphism of \(X\) as a \( \Z\)-scheme. Since morphisms of affine schemes are separated by Exercise 10.1.C and the composition of separated morphisms is separated by Proposition 10.1.13, we have that \( X \) is separated as a \(\Z\)-scheme.

In the reverse direction, consider the class \( P \) of separated morphisms, which as the Propositions have shown is stable under base change and composition. Following the notation of the Cancellation Theorem, let \( \rho : \spec A \to \spec \Z \) denote the unique structure morphism of \( \spec A \) as a \(\Z\)-scheme. Assume \( \tau = \rho \circ \pi \) is separated (i.e. in \(P \)). Since \( \delta_\rho : \spec A \to \spec (A \otimes_\Z A ) \) is again a morphism of affine schemes, we know that it is separated. Then by the Cancellation Theorem, we must have \( \pi \) is in \( P \), i.e. separated.

Proof:

1. By Exercise 8.1.L and Exercise 10.1.B, the classes of locally finite type morphisms and separated morphisms both contain locally closed embeddings. Then the claim simply follows by Theorem 10.1.19(i).

2. Let \( P \) be the class of quasicompact morphisms. By Exercise 7.3.B(b), and morphism from a locally Noetherian scheme is quasiseparated; in particular, \( \rho \) is quasiseparated so the claim follows by Theorem 10.1.19(iii).

3. Take \( \spec A \subset Z \) and suppose \( \{ V_i \} \) is an affine open cover of \( \rho^{-1}(\spec A) \). By letting our affine open \( \spec A \) vary, the collection of our covers \( V_{i} \) will cover \( Y \) — since quasiseparatedness is affine-local on the target, it suffices to show that \( \pi : \pi^{-1}(V_i) \to V_i \) is quasiseparated. If we let \( U_1, U_2 \subset X \) with \( \pi(U_1), \pi(U_2) \subset V_i \), then \( \rho \circ \pi(U_1), \rho \circ \pi(U_2) \subseteq \spec A \) so \( U_1 \cap U_2 \) — by assumption \( \rho \circ \pi \) is quasiseparated, which remains true when restricted to an affine cover, so that \( U_1 \cap U_2 \) is quasicompact.

Proof:

Let \( X \) and \( Y \) be \( k \)-varieties,\( \sigma_X : X \to \spec k \), \( \sigma_Y : Y \to \spec k \) denote the structure morphisms, and \( \pi : X \to Y \) be a morphism of such \(k\)-varieties. Since a morphism of \( k \)-varieties is at the very least a morphism of \( k \)-schemes, we know that \( \sigma_X = \sigma_Y \circ \pi \). To see that \( \pi \) is finite type, notice that the class \( P \) of finite-type morphisms contains closed embeddings — thus, by Theorem 10.1.19(ii), since \( \sigma_Y \) is by definition separated we must have \( \pi \) is of finite type.

To see that \( \pi \) is in fact separated, we use a similar approach. Let \( P \) be the class of separated morphisms; again we have that \( \sigma_X \) is in \( P \). By Proposition 10.1.3 we know that \( \delta_{\sigma_Y} : Y \to Y \times_k Y \) is a locally closed embedding, and by Exercise 10.1.B that locally closed embeddings are separable — in particular, diagonal morphisms are always separable, so by Theorem 10.1.19 we have that \( \pi : X \to Y \) must be separable.

Proof:

This is an immediate consequence of Theorem 10.1.19(i) and (ii) to the following commutative diagram:

Indeed in the first case \( \textrm{id}_X \) is trivially a closed embedding and thus locally closed embedding, and by Proposition 10.1.3 the diagonal is always a locally closed embedding so that \( \mu \) must be a locally closed embedding by (i). In the second case, we simply use the fact that \( \mu \) is assumed to be separable to apply (ii).

To give an easy example of when \( \sigma \) is not a closed embedding, recall from Exercise 10.1.D that the line with double origin \(Z\) has canonical cover by two copies of \( \A^1_k \). Taking \( X = \A^1 \) to be one of the copies, \( \sigma : \A^1_k \hookrightarrow Z\) to be the natural inclusion, and \( \mu : Z \to \A^1_k \) to be the natural projection, it should be clear that \( \mu \) cannot be separated from the following diagram:

Indeed if \( \mu \) were separatedm then as \( \A^1 \) (as an affine variety) is separated, the composition \( Z \to \A^1_k \to \spec k \) would be precisely the structure morphism on \( Z \) implying that \(Z\) is separated — a contradiction. It is also fairly easy to see that \( \sigma : \A^1 \to Z \) is not a closed embedding, as the image is not closed (the compliment is the other copy of the origin cannot be open, as every open set containing that origin must contain the other).

Proof:

Since \( P \) is preserved under base change, we know that \( X \times_Y Y^{red} \to Y^{red} \) is also in \( P\). Recall that the reduction of a scheme \( Z^{red} \to Z \) is a locally closed embedding, as on open affines it is given by \( A \mapsto A / \mathfrak{N}(A) \). Thus, both the morphisms \( X^{red} \to X\) and \( Y^{red} \to Y \) are closed embeddings, which by assumption are themselves in \( P \). Moreover, as we know that closed embeddings are separated ( Exercise 10.1.C) and preserved by base change, this tells us that the morphism \( X \times_Y Y^{red} \) is a separated morphism. Applying Theorem 10.1.19(ii) to the leftmost triangle, this tells us that \( X^{red} \to X \times_Y Y^{red} \) must also be in \( P \). Since \( P \) is closed under compositions, we have that \( X^{red} \to X \times_Y Y^{red} \to Y^{red} \) is in \( P \).

Proof:

By Exercise 9.5.R, consider the class \( P \) consisting of universally injective morphisms — from the discussion at the end of §9.5 we know that locally closed embeddings are in \( P \). It follows from Theorem 10.1.19(i) that \( \pi \) must be universally injective.

Proof:

1. For the forward direction, suppose that \( \delta_\pi : X \to X \times_Y X \) is surjective, let \( f: Y^\prime \to Y \) be arbitrary and denote \( X^\prime = Y^\prime \times_Y X \) with \( \widetilde{\pi} : X^\prime \to Y^\prime \) the pullback. Since surjectivity is stable under base change, we also have \( \delta_{\widetilde{\pi}} \) is surjective. Thus, for any \( x_1, x_2 \in X^\prime \) with \( \widetilde{\pi}(x_1) = \widetilde{\pi}(x_2) \) we necessarily have \( (x_1, x_2) \in X^\prime \times_{Y^\prime} X^\prime \). Since the diagonal is surjective, there exists some \( x \in X^\prime \) that makes some overall diagram commute — in particular, this would imply \( x = x_1 = x_2 \) so that \( \widetilde{\pi} \) is injective.

   Conversely, suppose \( \pi \) is universally injective and take some \( (x_1, x_2) \in X \times_Y X \) such that \( \pi(x_1) = \pi(x_2) \) by construction / commutivity of the fibred product. As \( \pi \) is universally injective, it is injective in the regular sense by considering the trivial base change \( \textrm{id}_Y : Y \to Y \). Thus, \( x = x_1 = x_2\) implying that \(\delta_\pi(x) = (x_1, x_2) \).

2. By the previous part, we know that if \( \pi : X \to Y \) is universally injective then \( \delta_\pi \) is surjective and thus an isomorphism (which is clearly a closed embedding).

3. By possibly passing to \( X(\overline{k}) = \spec \overline{k} \times_Y X \), we may assume \( X \) is reduced. Then from Exercise 7.4.E, a morphism is surjective if and only if it is surjective on closed points (notice we do not make use of the seperatedness of a variety there in the previous proof as it is unnecessary). Then as \( \pi \) is universally injective iff \( \delta_\pi \) is surjective, it suffices to check on closed points.

Proof:

Following the hint quite religiously, we use existence of the fibred product above to define a scheme \( V \) together with a canonical projection \( i : V \to X \), which itself must be a locally closed embedding since \( \delta \) is, and locally closed embeddings are stable under base change. In fact, the argument will not change whatsoever if we assume that \( V \) is a locally closed subscheme of \(X\), though this will make the set-theoretic argument cleaner. In this case, it should be obvious that \( V \) consists of points of \( X \) on which \( \pi \) and \( \pi^\prime \) agree, as the composition \( V \to Y \to Y \times_X Y \) along the top is given by \( x \mapsto \pi(x) \mapsto ( \pi(x), \pi(x) ) \) and the bottom composition is simply \( ( \pi(x), \pi^\prime(x)) \). By construction of the fibred product, \( V \) is the universal object on which these agree, which holds if and only if \( \pi(x) = \pi^\prime(x) \).

Now suppose \( W \) is an arbitrary scheme and \( \mu \) another morphism such that \( \pi \circ \mu = \pi^\prime \circ \mu \) agree — the composition of either clearly gives us a map \( W \to Y \) such that the Cartesian square

Then by the universal property of the fibre product there is a unique morphism \( j : W \to V \) such that \( \mu = i \circ j \).

In the case that \( Y \to Z \) is separated, we have by definition that the diagonal \(\delta : Y \to Y \times_Z Y \) is a closed embedding. Since closed embeddings are stable under base change, we have that \( i : V \hookrightarrow X \) is also a closed embedding.

Proof:

Consider the set \( V \) from the previous exercise as the largest closed subscheme on which \( \pi \) and \( \pi^\prime \) agree, where we are now taking \( Z = \spec \overline{k} \) — by assumption \( X(k) \subset V \). As \( Y \) is a \( \overline{k} \)-variety, \( Y \to \spec \overline{k} \) is separated so that \( i : V \hookrightarrow X \) is a closed \( \overline{k} \)-subvariety. If we assume \( X \backslash V \) is nonempty, then it is an open set — since varieties are by definition finite-type, Exercise 5.3.F tells us that the closed points are dense. Thus, \( X \backslash V \) has a closed point, a contradiction.

This argument applies regardless of whether the field is algebraically closed.

Proof:

Take \( W = \A^1_k \), and \( \pi, \pi^\prime : W \to X \) to be the two natural inclusions (i.e. each mapping to distinct origin). Then the domain of agreement is \( \A^1_k \backslash \{ 0 \} \) which is clearly not closed.

Proof:

The second example was explained in the previous exercise, as the domain of agreement \( D(t) \) obviously cannot be extended to all of \( \A^1 \). For the second example, notice that on \( D(x) \subset \spec k[x, y] / (y^2, xy) \) by making \( x \) a unit we must equivalently have \( xy \) vanishes \( \Rightarrow y \) vanishes — i.e. \( \Big( \spec k[x, y] / (y^2, xy) \Big)_x \cong \spec \spec k[x, y]_x / (y) \). Thus, the map \( t \mapsto x + y \) is the same as \( t \mapsto x \). However, on the complement \( V(x) \) we obtain the first morphism is the zero map while the second is nontrivial (though nilpotent).

Proof:

Following the hint, we let \( \xi^\prime \) and \( \xi \) be two representatives, so that by defintion there exists a dense open set \( Z \subset U \cap V \) on which the two agree. By the reduced-to-separated theorem we may simply take \( Z = U \cap V \). Note that the claim will follow by showing that the graph of each representative \( \xi^\prime \) agrees with the graph on the domain of definition (using transitivity) — thus we may assume without loss of generality that \( V = X \) (implying \( \xi = \pi \)) and \( U \) is a dense open subset.

Notice that for any morphism of schemes \( f : W \to Z \) (with the same properties as \( X, Y \) above), we have a canonical projection \( p : \Gamma_f \to W \) given by \( (w, f(w) ) \mapsto w \) and inclusion \( \gamma_f : W \to \Gamma_f \) given by \( w \mapsto (w, f(w)) \) which are mutually inverse. Thus, \( \Gamma_\pi \cong X \) and \( \Gamma_{\xi^\prime} \cong U \). Since \( U \) is dense in \( X \) it follows that \( \Gamma_{\xi^\prime} \) is dense in \( \Gamma_\pi \) as desired.

Proof:

By the previous exercise, it should be clear that the rational map cannot be extended over the origin since \( \textrm{Bl}_{(0, 0)} \A^2_k \) is not isomorphic to \( \A^2_k \times \P^1_k \) (this will in fact follow once we have shown \( \textrm{Bl}_{(0, 0)} \A^2_k \) is the graph of our rational map). Consider our representative \( \pi : \A^2_k \backslash \{ (0, 0) \} \to \P^1 \) and its corresponding graph \( \Gamma_\pi = \{ \Big( (x, y), [x, y] \Big) \subset \A^2_k \times_k \P^1_k \} \). To see that this agrees with the the previous blow-up construction (namely the subscheme \( \textrm{Bl}_{(0, 0)} \A^2_k \) consisting of tuples \( (x, y), [u, v] \) with \( xv = yu \)), we clearly have \( \Gamma_\pi \subseteq \textrm{Bl}_{(0,0)} \A^2_k \) since \( xy = xy \). Conversely, away from the origin we knew that \( \textrm{Bl}_{(0,0)} \A^2_k \to \A^2_k \) given by the natural projection is an isomorphism thus inducing an isomorphism on our graph. We discount the exceptional divisor as that will correspond to the locus of indeterminacy.

Proof:

Recall that an effective Cartier divisor is a locally principle closed subscheme generated by a non-zerodivisor, and since local principality is an affine-local condition we may assume without loss of generality that \( X = \spec A \). By restricting to a possibly smaller affine open subset, we may assume our Cartier divisor is in fact principal so that \( D = V(t) \) for some non-zerodivisor \( t \in A \). Suppose we let \( \phi : D(t) \to Y \) be some morphism, and let \( \phi_1, \phi_2 : \spec A \to Y \) be two extensions of \( \phi \) to \( \spec A \). Using the same notation as above, let \( V \) be the closed subscheme (as \(Y\) is separated) on which \( \phi_1 = \phi_2 \) — by definition we clearly have \( D(t) \subset V\). Now consider the scheme-theoretic closure of \( D(t) \) — since \( t \) is a non-zero divisor, \( A \hookrightarrow A_t \) is an injection. Thus, for any closed subscheme \( Z \hookrightarrow \spec A \) and function \( f \) vanishing on \( Z \), we have \( f \) always pulls back to \( 0 \) on \(D(t)\). In particular, this tells us that the scheme-theoretic closure of \( D(t) \) is \( \spec A \) and therefore \( V = \spec A \) so that \( \pi_1 = \pi_2 \) everywhere.

Proof:

By §5.5.4 since \( U \) is locally Noetherian we know that a set is dense if and only if it contains all the associated points. Thus, we may use the same argument as above by considering \( V \) to be the closed subscheme where \( \pi = \pi^\prime \) and showing that since this contains a dense open subset we must have \( V = U \).

Proof:

Consider the base change by the same morphism \( \A^1_\C \to \spec \C \) — we obtain a map \( \A^1_\C \times_\C \A^1_\C \to \A^1_\C\) given by projection onto the first coordinate. However this is not a close map in the Zariski topology as \( V(xy - 1) \) maps to \( \A^1_\C \backslash \{ 0 \} \).

Proof:

By the previous proposition, we know that \( \pi \) must be proper as well. Since finite type = quasicompact + locally of finite type, this necessarily tells us that \( X \) is quasicompact and thus by §8.3.6 we have that the scheme theoretic image of \( \pi \) is a closed subscheme \( \iota : \pi(X) \hookrightarrow Y \). Since separatedness and finite-type are preserved under base change, we may assume without loss of generality that \( \pi \) is surjective. Thus, the problem reduces to showing that \( Y \) is universally closed. Consider the base change

Since \( X^\prime = X \times_Z Z^\prime \to Z^\prime \) and \( X^\prime \to X \to Y \), we get an induced morphism \( \pi^\prime : X^\prime \to Y^\prime \) which is necessarily surjective by Exercise 9.4.D. Additionally, as a morphism of \( Z^\prime \)-schemes we should have \( \tau^\prime = \rho^\prime \circ \pi^\prime \) — since \( \tau \) is proper, \( \tau^\prime \) is closed. Then for any closed subset \( W \subset Y \), we have by continuity and surjectivity that \( (\pi^\prime)^{-1}(W) \) is closed in \( X \). Since \( \tau^\prime \) is a closed map we get \( \tau^\prime( \pi^{-1}(W) ) = \rho^\prime(W) \) is closed, so that \( \rho \) is a universally closed morphism as desired.