Closed Subschemes

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

In §Schemes, ⁋Lemma 2 we saw that for an affine scheme \(\Spec A\), any element \(f\) defines an open affine subscheme \(D(f)\cong \Spec A_f\); in particular, to compare the two structure sheaves we applied [Topology] §Sheaves, ⁋Lemma 11 to

\[(\Spec\epsilon)^\sharp: \mathcal{O}_{\Spec A} \rightarrow (\Spec \epsilon)_\ast \mathcal{O}_{\Spec A_f}\]

obtained from \(\epsilon: A \rightarrow A_f\), yielding

\[(\Spec\epsilon \lvert^{D(f)})^\sharp: \mathcal{O}_{D(f)} \rightarrow (\Spec\epsilon\rvert^{D(f)})_\ast \mathcal{O}_{\Spec A_f}\]

and from the fact that \(\Spec A_f\) is an open subset of \(\Spec A\) we could conclude that this is an isomorphism.

On the other hand, by the second result of §The Spectrum, ⁋Proposition 9, given an affine scheme \(\Spec A\) and an ideal \(\mathfrak{a}\) of \(A\), the \(\Spec\) functor gives

\[\Spec\pi: \Spec A/\mathfrak{a}\rightarrow \Spec A\]

and we know that \(\Spec\pi\) is injective and its image is the closed set \(Z(\mathfrak{a})\). In this case as well, just as above we consider the canonical decomposition

\[\Spec A/\mathfrak{a}\overset{\Spec\pi\vert^{Z(\mathfrak{a})}}{\longrightarrow} Z(\mathfrak{a}) \overset{\iota}{\longrightarrow}\Spec A\]

and from

\[(\Spec\pi)^\sharp: \mathcal{O}_{\Spec A} \rightarrow (\Spec\pi)_\ast \mathcal{O}_{\Spec A/\mathfrak{a}}\]

we can construct a morphism of sheaves on \(Z(\mathfrak{a})\)

\[\iota^{-1} \mathcal{O}_{\Spec A} \rightarrow (\Spec\pi\vert^{Z(\mathfrak{a})})_\ast \mathcal{O}_{\Spec A/\mathfrak{a}}\]

but we have not defined a scheme structure on \(Z(\mathfrak{a})\), and therefore we do not know the relationship between \(\iota^{-1}\mathcal{O}_{\Spec A}\) and \(\mathcal{O}_{Z(\mathfrak{a})}\), nor is there any guarantee that this is an isomorphism. In fact it is far more likely not to be an isomorphism, because \(\iota^{-1}\mathcal{O}_{\Spec A}\) is defined using only the topological data of the closed set \(Z(\mathfrak{a})\) from the structure sheaf of \(\Spec A\), whereas \((\Spec\pi)_\ast\mathcal{O}_{\Spec A/\mathfrak{a}}\) also carries algebraic information about the ring \(A/\mathfrak{a}\).

Example 1 For example, fix a field \(\mathbb{K}\) and consider the affine line \(\mathbb{A}_\mathbb{K}^1=\Spec \mathbb{K}[\x]\). Then there are canonical surjections

\[\pi_1:\mathbb{K}[\x] \rightarrow \mathbb{K}[\x]/(\x)\cong \mathbb{K},\qquad \pi_2:\mathbb{K}[\x] \rightarrow \mathbb{K}[\x]/(\x^2)\]

and concretely \(\pi_1\) and \(\pi_2\) are defined by \(\x\mapsto 0+(\x)\) and \(\x\mapsto \x+(\x^2)\) respectively.

On the other hand, since \(\mathbb{K}[\x]/(\x)\cong \mathbb{K}\), the space \(\Spec \mathbb{K}[\x]/(\x)\) has only a single point \((0)\). Similarly \(\Spec \mathbb{K}[\x]/(\x^2)\) also has only a single point. This is because there is a one-to-one correspondence between prime ideals of \(\mathbb{K}[\x]/(\x^2)\) and prime ideals of \(\mathbb{K}[\x]\) containing \(\x^2\), and since \(\mathbb{K}[\x]\) is a principal ideal domain, writing a prime ideal of \(\mathbb{K}[\x]\) as \((p(\x))\), for this ideal to contain \(\x^2\) we must have \(p(\x)\) dividing \(\x^2\), which forces \(p(\x)=\x\).

Therefore, considering the scheme morphisms

\[\Spec\pi_1:\Spec \mathbb{K}[\x]/(\x) \rightarrow \Spec \mathbb{K}[\x],\qquad \Spec\pi_2:\Spec \mathbb{K}[\x]/(\x^2) \rightarrow \Spec \mathbb{K}[\x]\]

defined by these, as continuous maps we know that \(\Spec\pi_1\) sends the unique point \((0)\) of \(\Spec \mathbb{K}[\x]/(\x)\) to the point \((\x)\) of \(\Spec \mathbb{K}[\x]\), and \(\Spec\pi_2\) sends the unique point \((\x)\) of \(\Spec \mathbb{K}[\x]/(\x^2)\) to the point \((\x)\) of \(\Spec \mathbb{K}[\x]\). That is, as continuous maps they define the same function, but of course \(\Spec \mathbb{K}[\x]/(\x)\) and \(\Spec \mathbb{K}[\x]/(\x^2)\) are not isomorphic as schemes.

Naturally, the structure sheaf we desire is of the form \((\Spec\pi)_\ast \mathcal{O}_{\Spec A/\mathfrak{a}}\), which contains algebraic information, and we will examine at the end of this post how this relates to \(\iota^{-1}\mathcal{O}_{\Spec A}\).

Closed Subschemes

As seen above, our model for a closed subscheme is the canonical projection \(\pi: A \rightarrow A/\mathfrak{a}\) and the scheme morphism

\[(\Spec \pi, (\Spec\pi)^\sharp): \Spec A/\mathfrak{a} \rightarrow\Spec A\]

arising from it. Here \(\Spec\pi\) is an injective continuous map giving a homeomorphism between \(\Spec A/\mathfrak{a}\) and a closed subset of \(\Spec A\), and \(\Spec\pi^\sharp: \mathcal{O}_{\Spec A} \rightarrow (\Spec\pi)_\ast \mathcal{O}_{\Spec A/\mathfrak{a}}\) is obtained from §Affine Scheme, ⁋Proposition 9.

On the other hand, the most important property of the ring homomorphism \(\pi: A \rightarrow A/\mathfrak{a}\) is that \(\pi\) is surjective, and indeed given any surjective ring homomorphism \(\phi: A \rightarrow B\), by the first isomorphism theorem

\[B=\im\phi\cong A/\ker\phi\]

so this property exactly characterizes \(\pi\). On the other hand, recalling [Commutative Algebra] §Properties of Localization, ⁋Proposition 4, the surjectivity of \(\pi\) can be checked by examining whether the localization \(\pi_\mathfrak{p}: A_\mathfrak{p} \rightarrow (A/\mathfrak{a})_{\mathfrak{p}}\) at any prime ideal \(\mathfrak{p}\) is surjective, which geometrically amounts to looking at the stalk at any point \(\mathfrak{p}\) of the affine scheme \(\Spec A\), and therefore by [Topology] §Sheaves, ⁋Proposition 15 this is equivalent to \((\Spec\pi)^\sharp\) being surjective.

Definition 2 A scheme morphism \(\iota: Z \rightarrow X\) is called a closed embedding if \(\iota\) is a homeomorphism between \(Z\) and a closed subset of \(X\), and the sheaf morphism \(\iota^\sharp: \mathcal{O}_X \rightarrow \iota_\ast \mathcal{O}_Z\) is surjective.

For two closed embeddings \(\iota: Z \rightarrow X\) and \(\iota': Z' \rightarrow X\) into \(X\), if there exists an isomorphism \(\theta: Z' \rightarrow Z\) such that \(\iota'=\iota\circ \theta\), we say these are equivalent, and we call this equivalence class a closed subscheme of \(X\).

The condition on the continuous map \(\iota\) is clear, and the intuition for \(\iota^\sharp\) also admits a geometric interpretation: it means that functions on \(Z\), or more precisely on \(\iota(Z)\), should all be obtained by restricting functions on \(X\) to \(Z\). Or, conversely, given any function on \(Z\) it should be possible to extend it locally in a neighborhood of each point to a function on \(X\). Here the surjectivity of sheaves is a condition on stalks, so it does not mean global extension. For example, considering \(X=\mathbb{P}^1\) and the reduced closed subscheme \(Z\) consisting of two points in it, the map \(\Gamma(X,\mathcal{O}_X)=\mathbb{K}\) to \(\Gamma(Z,\mathcal{O}_Z)=\mathbb{K}\times \mathbb{K}\) is not surjective. On the other hand, it is worth contrasting with the case where \(\iota\) is an open embedding. In this case \(\iota^\sharp:\mathcal{O}_X \rightarrow \iota_\ast\mathcal{O}_Z\) itself is not an isomorphism. For example, considering \(X=\mathbb{A}^1_k=\Spec k[t]\) and its open part \(Z=D(t)=\Spec k[t,t^{-1}]\), we have \((\iota_\ast\mathcal{O}_Z)(X)=k[t,t^{-1}]\) so \(k[t] \rightarrow k[t,t^{-1}]\) is not surjective. The correct statement is that since \(\iota\) maps \(Z\) to an open set, we have \(\iota^{-1}\mathcal{O}_X\cong\mathcal{O}_Z\), that is, an isomorphism between stalks at each point of \(\iota(Z)\) is induced.

This definition is natural, but it is somewhat different in flavor from the properties of scheme morphisms we defined in previous posts. Therefore we examine the following equivalent condition.

Proposition 3 For a scheme morphism \(\varphi: X \rightarrow Y\), the following two conditions are equivalent.

1. \(\varphi\) is a closed embedding.
2. \(\varphi\) is an affine morphism, and whenever an affine open subset \(V\cong \Spec B\) of \(Y\) is given, for its preimage \(\varphi^{-1}(V)\cong \Spec A\) the map \(B \rightarrow A\) is surjective.

Proof

First assume the second condition and show that \(\varphi\) is a closed embedding. Cover \(Y\) by affine open subsets \(\{V_i=\Spec B_i\}\); then by assumption \(\varphi^{-1}(V_i)\cong \Spec A_i\) and the corresponding \(\beta_i: B_i \rightarrow A_i\) is surjective. Then by the first isomorphism theorem, setting \(\mathfrak{b}_i=\ker\beta_i\) we have \(A_i\cong B_i/\mathfrak{b}_i\), and therefore the restriction of \(\varphi\) to \(\varphi^{-1}(V_i)\) is \(\Spec\pi\) defined by the canonical projection \(\pi: B_i \rightarrow B_i/\mathfrak{b}_i\).

Now by §The Spectrum, ⁋Proposition 9, \(\Spec\pi\) is injective and its image is the closed set \(Z(\mathfrak{b}_i)\), and \(\Spec\pi\) is a homeomorphism onto this image. From this we first see that \(\varphi\) is injective. Indeed, if \(\varphi(x)=\varphi(x')\) then choosing \(V_i\) containing this point we have \(x,x'\in \varphi^{-1}(V_i)\), and the restriction of \(\varphi\) to \(\varphi^{-1}(V_i)\) is injective. Also for each \(i\), \(\varphi(X)\cap V_i=Z(\mathfrak{b}_i)\) is a closed subset of \(V_i\) and \(\{V_i\}\) is an open cover of \(Y\), so \(\varphi(X)\) is a closed subset of \(Y\). Finally for any open set \(U\) of \(X\), since \(\varphi\) is injective we have

\[\varphi(U)\cap V_i=\varphi(U\cap \varphi^{-1}(V_i))\]

and the right hand side is an open subset of \(\varphi(X)\cap V_i\), so \(\varphi(U)\) is an open subset of \(\varphi(X)\). That is, \(\varphi\) is a homeomorphism between \(X\) and the closed subset \(\varphi(X)\) of \(Y\).

Next we show that \(\varphi^\sharp\) is surjective. By [Topology] §Sheaves, ⁋Proposition 15 it suffices to check stalks at each \(y\in Y\). If \(y\not\in \varphi(X)\), then since \(\varphi(X)\) is closed there exists an open neighborhood \(W\) of \(y\) not meeting \(\varphi(X)\), and then \((\varphi_\ast \mathcal{O}_X)(W)=\mathcal{O}_X(\emptyset)=0\) so \((\varphi_\ast \mathcal{O}_X)_y=0\) and there is nothing to show. Now let \(y=\varphi(x)\). Since \(\varphi\) is a homeomorphism onto its image, for any open set \(U\) of \(X\) containing \(x\) there exists an open set \(W\ni y\) of \(Y\) such that \(\varphi(U)=W\cap \varphi(X)\) and then \(\varphi^{-1}(W)=U\). That is, preimages of open neighborhoods of \(y\) are cofinal among open neighborhoods of \(x\), and therefore

\[(\varphi_\ast \mathcal{O}_X)_y=\varinjlim_{W\ni y}\mathcal{O}_X(\varphi^{-1}(W))\cong \mathcal{O}_{X,x}\]

Now choose \(i\) with \(y\in V_i\) and let \(\mathfrak{q}\) be the prime ideal of \(B_i\) corresponding to \(y\), and \(\mathfrak{p}=\mathfrak{q}/\mathfrak{b}_i\) the prime ideal of \(A_i\) corresponding to \(x\); then by §Affine Scheme, ⁋Lemma 8 the morphism between stalks at \(y\) is the localization of \(\beta_i\)

\[(B_i)_\mathfrak{q} \rightarrow (A_i)_\mathfrak{p}\cong (B_i/\mathfrak{b}_i)_\mathfrak{q}\]

But localization is an exact functor ([Commutative Algebra] §Properties of Localization, ⁋Proposition 2), so this morphism is surjective, and therefore \(\varphi^\sharp\) is surjective. That is, \(\varphi\) is a closed embedding.

The opposite direction is not formal. Assume \(\varphi\) is a closed embedding, fix an affine open subset \(V=\Spec B\) of \(Y\), and write \(W=\varphi^{-1}(V)\). Just as in the previous argument, from the fact that \(\varphi\) is a homeomorphism onto its image, for any \(\mathfrak{q}=\varphi(x)\in \varphi(X)\cap V\) we have \((\varphi_\ast \mathcal{O}_X)_\mathfrak{q}\cong \mathcal{O}_{X,x}\) and at points outside \(\varphi(X)\) we have \((\varphi_\ast \mathcal{O}_X)_\mathfrak{q}=0\). That is, we know the stalks of \(\varphi_\ast \mathcal{O}_X\). However, knowing only this does not tell us what sections \(\varphi_\ast \mathcal{O}_X\) has over open subsets of \(V\), and in particular we cannot tell whether \(W\) is an affine scheme. What we need for this is the fact that for a closed embedding \(\varphi\), both \(\varphi_\ast \mathcal{O}_X\) and the ideal sheaf \(\ker\varphi^\sharp\) are quasi-coherent, that is, for any affine open subset \(\Spec B\) of \(Y\) and any \(f\in B\) the canonical morphism

\[\left((\varphi_\ast \mathcal{O}_X)(\Spec B)\right)_f \rightarrow (\varphi_\ast \mathcal{O}_X)(D(f))\]

is an isomorphism. This is a condition of exactly the same form as the localization condition required for ideals in Proposition 6, but since we do not yet know whether \(\varphi\) is an affine morphism, we cannot obtain this with the tools we have. Therefore we will only assert this fact without proof, and complete the remaining argument with the tools we already have. This fact is proved by §Quasi-coherent Sheaves, ⁋Proposition 18 as the quasi-coherence of \(\varphi_\ast \mathcal{O}_X\).

Set \(C=(\varphi_\ast \mathcal{O}_X)(V)=\Gamma(W, \mathcal{O}_W)\) and \(\beta=\varphi^\sharp(V): B \rightarrow C\). Then since the \(D(f)\) form a base for \(V\), repeating the argument of §Affine Scheme, ⁋Lemma 8 verbatim, from the above fact we obtain for any \(\mathfrak{q}\in V\)

\[(\varphi_\ast \mathcal{O}_X)_\mathfrak{q}\cong C_\mathfrak{q}\]

Here \(C_\mathfrak{q}\) is the localization of the \(B\)-module \(C\) at \(\mathfrak{q}\), and this isomorphism is induced by the restriction maps.

First, \(\beta\) is surjective. Indeed, since \(\varphi^\sharp\) is surjective, by [Topology] §Sheaves, ⁋Proposition 15 the stalk morphism \(B_\mathfrak{q} \rightarrow C_\mathfrak{q}\) at each \(\mathfrak{q}\) is surjective, and this is the localization of the \(B\)-module homomorphism \(\beta\), so by [Commutative Algebra] §Properties of Localization, ⁋Proposition 4 the map \(\beta\) is surjective. Therefore, setting \(\mathfrak{b}=\ker\beta\), we have \(C\cong B/\mathfrak{b}\).

Second, \(W\) is an affine scheme. Topologically, \(\mathfrak{q}\in V\) belongs to \(\varphi(X)\) if and only if the stalk computed above is nonzero. We have already seen that the stalk is zero at points outside \(\varphi(X)\), and when \(\mathfrak{q}=\varphi(x)\) the stalk is the local ring \(\mathcal{O}_{X,x}\) so it is nonzero. But \((\varphi_\ast \mathcal{O}_X)_\mathfrak{q}\cong (B/\mathfrak{b})_\mathfrak{q}\) and this being nonzero is equivalent to \(\mathfrak{b}\subseteq \mathfrak{q}\), so

\[\varphi(X)\cap V=Z(\mathfrak{b})\]

Now applying the adjunction of §Affine Scheme, ⁋Theorem 13 to the identity map \(C \rightarrow \Gamma(W, \mathcal{O}_W)\) we obtain a canonical morphism \(\sigma: W \rightarrow \Spec C\), and by naturality of the adjunction we have \(\Spec\beta\circ \sigma=\varphi\vert_W\). On the other hand, \(\Spec\beta: \Spec B/\mathfrak{b} \rightarrow \Spec B\) is a homeomorphism onto \(Z(\mathfrak{b})\) (§The Spectrum, ⁋Proposition 9), and \(\varphi\vert_W\) is also a homeomorphism onto \(\varphi(X)\cap V=Z(\mathfrak{b})\), so \(\sigma\) is a homeomorphism. Also for any \(x\in W\) and \(\mathfrak{q}=\varphi(x)\), the morphism induced by \(\sigma\) on stalks is \(C_\mathfrak{q} \rightarrow \mathcal{O}_{W,x}\) by §Affine Scheme, ⁋Lemma 8, which is the same as the morphism induced by restriction maps, that is, the isomorphism \(C_\mathfrak{q}\cong (\varphi_\ast \mathcal{O}_X)_\mathfrak{q}\cong \mathcal{O}_{W,x}\) obtained above. Therefore \(\sigma\) is a homeomorphism and an isomorphism on all stalks, hence an isomorphism of locally ringed spaces.

From the above, \(W\cong \Spec C=\Spec B/\mathfrak{b}\) is an affine scheme and \(B \rightarrow C\) is surjective. Since \(V\) was an arbitrary affine open subset of \(Y\), the morphism \(\varphi\) is affine and the second condition holds.

Then any closed embedding can always be thought of locally as coming from a suitable \(\pi: A \rightarrow A/\mathfrak{a}\) as examined above. In particular, if \(Y\) is an affine scheme \(\Spec B\), then by the above equivalence any closed embedding \(\varphi: X \rightarrow Y\) into \(Y\) corresponds exactly to \(B \rightarrow B/\mathfrak{b}\).

Properties of Closed Embeddings

By Proposition 3, any closed embedding is always affine-local on target, and closed embeddings are closed under composition. Moreover the following holds.

Proposition 4 Any closed embedding is always a finite morphism.

Proof

Let a closed embedding \(\varphi: X \rightarrow Y\) be given. By Proposition 3, \(\varphi\) is an affine morphism, and for any affine open subset \(V\cong \Spec B\) of \(Y\) we have \(\varphi^{-1}(V)\cong\Spec A\) and the corresponding ring homomorphism \(\beta: B \rightarrow A\) is surjective. Then for any \(a\in A\) there exists \(b\in B\) such that \(a=\beta(b)=b\cdot 1\), so \(A\) is generated by \(1\) as a \(B\)-module, and therefore \(\beta\) is a finite ring homomorphism. (Fourth condition of [Commutative Algebra] §Integral Extensions, ⁋Definition 3) Now by §Properties of Scheme Morphisms, ⁋Definition 10, the morphism \(\varphi\) is finite.

In light of the geometric intuition for (quasi-)finite morphisms built in §Properties of Scheme Morphisms, ⁋Example 16, it is obvious that at least closed embeddings should always be quasi-finite, and here we have the further geometric interpretation that they are in fact finite.

Definition 5 For any scheme \(Z\), among subsheaves \(\mathcal{I}\) of \(\mathcal{O}_Z\) such that \(\mathcal{I}(U)\) is an ideal of \(\mathcal{O}_Z(U)\) for each open set \(U\), we call those the ideal sheaves of \(Z\). In particular, for a closed embedding \(\iota: Z \rightarrow X\), we call the subsheaf \(\ker\iota^\sharp\) of \(\mathcal{O}_X\) the ideal sheaf defined by \(\iota\), and denote it by \(\mathcal{I}_{Z/X}\).

That is, there exists the following exact sequence

\[0 \rightarrow \mathcal{I}_{Z/X} \rightarrow \mathcal{O}_X \rightarrow \iota_\ast \mathcal{O}_Z \rightarrow 0\]

Therefore for any affine open subset \(U=\Spec A\) of \(X\) we have

\[0 \rightarrow \mathcal{I}_{Z/X}(U) \rightarrow \mathcal{O}_X(U)\cong A \rightarrow \iota_\ast \mathcal{O}_Z(U) \rightarrow 0\]

so \(\mathcal{I}_{Z/X}(U)\) becomes an ideal of \(A\), which justifies this name.

We saw right after Proposition 3 that closed subschemes of an arbitrary affine scheme \(Y=\Spec B\) correspond exactly to ideals of \(B\). On the other hand, since any scheme is built by gluing affine schemes, if ideals are defined on each such affine scheme and they satisfy suitable gluing conditions, then a closed subscheme of the original scheme will be defined through them.

Proposition 6 Suppose that for each affine open subset \(\Spec A\) of a scheme \(X\), an ideal \(\mathcal{I}(A)\subseteq A\) is given. If for each \(f\in A\), the map \(A \rightarrow A_f\) induces an isomorphism \(\mathcal{I}(A_f)\cong \mathcal{I}(A)_f\), then these data induce a unique closed subscheme \(Z\hookrightarrow X\).

Proof

First cover \(X\) by affine open subsets \(\{\Spec A_i\}\). Then what we need to show is that for any \(i,j\), the closed subscheme defined by the ideal \(\mathcal{I}(A_i)\) on \(\Spec A_i\) and the closed subscheme defined by the ideal \(\mathcal{I}(A_j)\) on \(\Spec A_j\) define the same closed subscheme on the intersection of \(\Spec A_i\) and \(\Spec A_j\).

First, from §The Topology of Schemes, ⁋Lemma 11 (Nike) we can cover the intersection of \(\Spec A_i\) and \(\Spec A_j\) by principal open subsets

\[\Spec (A_i)_{f_i}\cong\Spec (A_j)_{f_j}\]

Now restricting the closed subscheme defined by \(\mathcal{I}(A_i)\) on \(\Spec A_i\) to \(D(f_i)\cong\Spec (A_i)_{f_i}\) gives the closed subscheme defined by the ideal \(\mathcal{I}(A_i)_{f_i}\), and by the given assumption \(\mathcal{I}(A_i)_{f_i}\cong \mathcal{I}((A_i)_{f_i})\). Here \(\Spec (A_i)_{f_i}\) and \(\Spec (A_j)_{f_j}\) are the same open subset of \(X\), so through this isomorphism \(\mathcal{I}((A_i)_{f_i})\) and \(\mathcal{I}((A_j)_{f_j})\) denote the same ideal, and therefore the two closed subschemes agree on this open set.

These locally obtained closed subschemes glue well. Indeed two overlapping pieces agree on principal open subsets covering the overlap by the above argument, and the identifications on them all come from restriction maps so they glue to a single morphism on the overlap ([Topology] §Sheaves, ⁋Definition 1), and therefore the cocycle condition also holds automatically. Then by §Schemes, ⁋Lemma 9 they glue to a single scheme \(Z\) and closed embedding \(Z \rightarrow X\). Uniqueness follows because by Proposition 3 a closed subscheme on an affine open subset is completely determined by the ideal on it. That is, two closed subschemes realizing the given data agree on each piece of an affine open covering of \(X\), and therefore are equivalent.

Now let an arbitrary scheme \(X\) and a global section \(s\in \Gamma(X, \mathcal{O}_X)\) be given. Then for each affine cover \(U\cong\Spec A\), the restriction \(s\vert_U\) defines the ideal \(\mathcal{I}(A)=(s\vert_U)\) of \(A\), and the \(\mathcal{I}(A)\) defined in this way trivially satisfy the condition of Proposition 6.

Definition 7 For a scheme \(X\) and a global section \(s\in \Gamma(X, \mathcal{O}_X)\), the scheme \(Z(s)\) defined as above is called the vanishing scheme of \(s\).

More generally, it is also clear how \(Z(S)\) should be defined for a set \(S\) of global sections, and therefore in particular when \(X=\Spec A\) and \(S=\mathfrak{a}\) is an ideal of \(A\), it is clear how \(Z(\mathfrak{a})\) should be defined, and this is obtained by pushing forward the structure sheaf of the affine scheme \(\Spec A/\mathfrak{a}\) to the closed set \(Z(\mathfrak{a})\) via \(\Spec\pi\). Henceforth we always think of \(Z(\mathfrak{a})\) as being equipped with such a scheme structure.

Definition 8 A scheme morphism \(\varphi: X \rightarrow Y\) is called a locally closed embedding if there exists a suitable open subscheme \(\iota:Z\hookrightarrow Y\) of \(Y\) such that through the canonical decomposition

\[X\overset{\varphi\vert^Z}{\longrightarrow}Z\overset{\iota}{\longrightarrow} Y\]

the map \(\varphi\vert^Z\) is a closed embedding.

Then any locally closed embedding is always locally of finite type. To check this, fix an affine open subset \(V=\Spec B\) of \(Y\). First \(Z\cap V\) is covered by principal open subsets \(D(f)\cong \Spec B_f\) of \(\Spec B\) (§The Spectrum, ⁋Lemma 11), and \(B \rightarrow B_f\) is of finite type since it adjoins only \(1/f\). Also since \(\varphi\vert^Z\) is a closed embedding, by Proposition 3 the preimage of each \(D(f)\) is affine and its coordinate ring is a quotient of \(B_f\), so the composition \(B \rightarrow B_f \rightarrow B_f/\mathfrak{b}\) is also of finite type. That is, \(\varphi^{-1}(V)\) is covered by affine open subsets on which \(B \rightarrow \mathcal{O}_X(-)\) is of finite type, and therefore by §Properties of Scheme Morphisms, ⁋Lemma 13 we obtain the same conclusion for any affine open subset of \(\varphi^{-1}(V)\).

Image of a Scheme Morphism

Now we define the image of a scheme morphism. Naturally, when an arbitrary scheme morphism \(\varphi: X \rightarrow Y\) is given, we would want its image \(\im\varphi\) to also be equipped with a scheme structure. However, as a subset of the topological space \(Y\), \(\im\varphi\) may be neither open nor closed, so it seems difficult to define a structure sheaf on \(\im\varphi\) using the structure sheaf of \(Y\).

The solution to this is to define the scheme-theoretic image of \(\varphi\) as the smallest closed subscheme containing the image of \(\varphi\). For this we must first examine what it means for one closed subscheme of \(X\) to be smaller than another.

Lemma 9 Let two closed embeddings \(\iota_1: Z_1 \rightarrow X\), \(\iota_2: Z_2 \rightarrow X\) be given. Then there exists a suitable scheme morphism \(\varphi: Z_1 \rightarrow Z_2\) satisfying \(\iota_1=\iota_2\circ\varphi\) if and only if \(\mathcal{I}_{Z_2/X}\subseteq \mathcal{I}_{Z_1/X}\). In this case \(\varphi\) becomes a closed embedding.

Proof

First suppose there exists \(\varphi\) satisfying \(\iota_1=\iota_2\circ\varphi\). Then \(\iota_1^\sharp\) is the following composition

\[\mathcal{O}_X\overset{\iota_2^\sharp}{\longrightarrow}(\iota_2)_\ast \mathcal{O}_{Z_2}\overset{(\iota_2)_\ast \varphi^\sharp}{\longrightarrow}(\iota_2)_\ast \varphi_\ast \mathcal{O}_{Z_1}=(\iota_1)_\ast \mathcal{O}_{Z_1}\]

so \(\ker\iota_2^\sharp\subseteq \ker\iota_1^\sharp\), and hence by Definition 5 we have \(\mathcal{I}_{Z_2/X}\subseteq \mathcal{I}_{Z_1/X}\).

Conversely assume \(\mathcal{I}_{Z_2/X}\subseteq \mathcal{I}_{Z_1/X}\). Choosing an arbitrary affine open subset \(U=\Spec A\) of \(X\), by Proposition 3 the set \(\iota_k^{-1}(U)\) is an affine open subset, and from the exact sequence right after Definition 5 we have

\[\iota_k^{-1}(U)\cong \Spec A/\mathfrak{a}_k,\qquad \mathfrak{a}_k=\mathcal{I}_{Z_k/X}(U)\]

and the restriction of \(\iota_k\) to \(\iota_k^{-1}(U)\) corresponds to the canonical projection \(A \rightarrow A/\mathfrak{a}_k\). By assumption \(\mathfrak{a}_2\subseteq \mathfrak{a}_1\), so \(A \rightarrow A/\mathfrak{a}_1\) factors uniquely through \(A \rightarrow A/\mathfrak{a}_2\), and the resulting \(\pi_U: A/\mathfrak{a}_2 \rightarrow A/\mathfrak{a}_1\) is surjective. Therefore by the discussion right before Definition 2, \(\varphi_U=\Spec\pi_U: \iota_1^{-1}(U) \rightarrow \iota_2^{-1}(U)\) is a closed embedding, and by construction the restriction of \(\iota_1\) to \(\iota_1^{-1}(U)\) is the composition of the restriction of \(\iota_2\) and \(\varphi_U\).

Now it remains to show that \(\varphi_U\) and \(\varphi_{U'}\) agree on the intersection for two affine open subsets \(U=\Spec A\), \(U'\) of \(X\). By §The Topology of Schemes, ⁋Lemma 11 (Nike) we can cover \(U\cap U'\) by open sets that are principal open in both \(U\) and \(U'\), and since localization is an exact functor ([Commutative Algebra] §Properties of Localization, ⁋Proposition 2), on such \(D(f)\cong \Spec A_f\) we have \(\mathcal{I}_{Z_k/X}(D(f))=\mathfrak{a}_kA_f\). Therefore both \(\varphi_U\) and \(\varphi_{U'}\) correspond on \(D(f)\) to the canonical projection \(A_f/\mathfrak{a}_2A_f \rightarrow A_f/\mathfrak{a}_1A_f\) induced by \(\mathfrak{a}_2A_f\subseteq \mathfrak{a}_1A_f\), so they agree. Then by §Morphisms of Schemes, ⁋Proposition 1 they glue to a scheme morphism \(\varphi: Z_1 \rightarrow Z_2\), and by construction \(\iota_1=\iota_2\circ\varphi\).

Finally we show that any \(\varphi\) satisfying \(\iota_1=\iota_2\circ\varphi\) is a closed embedding. For an affine open subset \(U\) of \(X\) we have \(\varphi^{-1}(\iota_2^{-1}(U))=\iota_1^{-1}(U)\), and the ring homomorphism \(A/\mathfrak{a}_2 \rightarrow A/\mathfrak{a}_1\) corresponding to the restriction of \(\varphi\) to this open set must commute with the two canonical projections from \(A\), so it can only be the above \(\pi_U\). But since \(\iota_2\) is an affine morphism, as \(U\) runs over an affine open cover of \(X\) the sets \(\iota_2^{-1}(U)\) form an affine open cover of \(Z_2\), and since closed embeddings are affine-local on target (Proposition 3), \(\varphi\) is a closed embedding.

For two closed subschemes \(Z_1,Z_2\) of a scheme \(X\), if there exists a closed embedding \(\varphi:Z_1 \rightarrow Z_2\), let us think of \(Z_1\) as a smaller closed subscheme than \(Z_2\).

Definition 10 Let an arbitrary scheme morphism \(\varphi: X \rightarrow Y\) be given. Then we say the image of \(\varphi\) is contained in a closed subscheme \(\iota: Z \rightarrow Y\) if the following composition

\[\mathcal{I}_{Z/Y} \rightarrow \mathcal{O}_Y \rightarrow \varphi_\ast \mathcal{O}_X\]

is zero. In this case, the smallest closed subscheme of \(Y\) containing the image of \(\varphi\) is called the scheme-theoretic image of \(\varphi\).

If in the above \(Y\) is an affine scheme \(\Spec B\), then closed subschemes of \(Y\) are completely determined by ideals \(\mathfrak{b}\) of \(B\). Therefore in this case, the scheme-theoretic image of \(Y\) will be the closed subscheme of \(Y\) defined by the kernel of \(\mathcal{O}_Y \rightarrow \varphi_\ast \mathcal{O}_X\). In the more special case where \(X\) is also an affine scheme, since \(\mathcal{O}_Y \rightarrow \varphi_\ast \mathcal{O}_X\) comes from a ring homomorphism \(\phi\), we can perform explicit computations.

Example 11 Let us examine a slight variation of the example of a closed embedding from Example 1. In this example we write \(\mathbb{K}[\x]/(\x^2)\) as \(\mathbb{K}[\epsilon]/(\epsilon^2)\) for distinction.

We know by [Algebraic Structures] §Algebras, ⁋Proposition 8 that a \(\mathbb{K}\)-algebra homomorphism \(\phi:\mathbb{K}[\x_1,\ldots, \x_n] \rightarrow \mathbb{K}[\epsilon]/(\epsilon^2)\) is completely determined by the values of the \(\x_i\). So let \(\phi(\x_i)=a_i+b_i\epsilon\). If there exists \(b_i\neq 0\), we can show that \(\phi\) is surjective, and therefore \(\Spec\phi\) is a closed embedding and the scheme-theoretic image of \(\Spec\phi\) is the closed subscheme defined by \(\Spec\phi\) itself. Concretely writing this out, \(\Spec\phi\) sends the unique prime ideal \((\epsilon)\) of \(\mathbb{K}[\epsilon]/(\epsilon^2)\) to the maximal ideal of \(\Spec \mathbb{K}[\x_1,\ldots, \x_n]\)

\[(\Spec\phi)((\epsilon))=\phi^{-1}((\epsilon))=(\x_1-a_1,\ldots, \x_n-a_n)\]

Indeed \(\phi(\x_i-a_i)=b_i\epsilon\in(\epsilon)\) so \((\x_1-a_1,\ldots, \x_n-a_n)\subseteq\phi^{-1}((\epsilon))\), and since the left side is a maximal ideal and the right side is a proper ideal, this inclusion is an equality. That is, as a continuous map \(\Spec\phi\) sends the one-point space \(\Spec \mathbb{K}[\epsilon]/(\epsilon^2)\) to the point \((a_1,\ldots, a_n)\) of \(\mathbb{A}^n\).

Geometrically \(\Spec\phi\) corresponds to the tangent vector \((b_1,\ldots, b_n)\) at the point \((a_1,\ldots, a_n)\) of \(\mathbb{A}^n\). This can be checked from the fact that for any \(f\in \mathbb{K}[\x_1,\ldots, \x_n]\)

\[\phi(f)=f(a)+\left(\sum_{i=1}^nb_i\frac{\partial f}{\partial \x_i}(a)\right)\epsilon\]

holds, that is, the \(\epsilon\)-coefficient of \(\phi(f)\) is exactly the directional derivative in the direction of the vector \((b_1,\ldots, b_n)\) at the point \((a_1,\ldots, a_n)\). More generally, thinking of \(\Spec \mathbb{K}[\epsilon]/(\epsilon^k)\) instead of \(\Spec \mathbb{K}[\epsilon]/(\epsilon^2)\), the \(\epsilon^j\)-coefficients of \(\phi(f)\) give the \(j\)-th order Taylor coefficients of \(f\). In characteristic zero this amounts to looking at derivatives up to order \(k-1\), but one must be careful that this is not the case in positive characteristic.

In the above example we did assume that \(X\) is an affine scheme, but \(\varphi^\sharp:\mathcal{O}_Y \rightarrow \varphi_\ast \mathcal{O}_X\) is anyway information contained in the scheme morphism \(\varphi\), so there is nothing new here. The difference appears when we generalize \(Y\) to a general scheme: when an arbitrary affine open subset \(V=\Spec B\) of \(Y\) is given, the ideal

\[\mathcal{I}(V):=\ker(\varphi^\sharp(V))\subseteq B\]

defines a closed subscheme of \(V\), but whether these can be glued to give a single closed subscheme defined on all of \(Y\) is a different problem. Of course we will use Proposition 6 for this, and this assumption is satisfied in particular when \(X\) is a reduced scheme or \(\varphi\) is quasi-compact.

Corollary 12 Let a scheme morphism \(\varphi: X \rightarrow Y\) be given. If \(X\) is reduced, or \(\varphi\) is quasi-compact, then the ideal sheaf \(\mathcal{I}\) defined above satisfies the condition of Proposition 6 and therefore \(\mathcal{I}\) defines a closed subscheme of \(Y\), and this becomes the scheme-theoretic image of \(\varphi\).

Proof

Fix an affine open subset \(V=\Spec B\) of \(Y\) and \(f\in B\), and write \(U=\varphi^{-1}(V)\), \(U'=\varphi^{-1}(D(f))\). Then what Proposition 6 requires is that the canonical map \(\mathcal{I}(V)_f \rightarrow \mathcal{I}(D(f))\) be an isomorphism. For convenience let \(g\) be the image of \(f\) under \(\varphi^\sharp(V): B \rightarrow \mathcal{O}_X(U)\).

Since \(\varphi\) is a morphism of locally ringed spaces, \(\varphi^\sharp_x\) is a local homomorphism at each \(x\in U\), and therefore \(U'\) is exactly the set of points where the stalk of \(g\) does not belong to the maximal ideal of \(\mathcal{O}_{X,x}\). In particular, for any affine open subset \(\Spec A\) of \(U\) we have \(U'\cap \Spec A=D(g\vert_{\Spec A})\), so the restriction of \(g\) to \(U'\) is a unit of \(\mathcal{O}_X(U')\), and therefore by the universal property of [Commutative Algebra] §Localization, ⁋Proposition 6 the restriction map \(\mathcal{O}_X(U) \rightarrow \mathcal{O}_X(U')\) induces a canonical map

\[\alpha: \mathcal{O}_X(U)_g \rightarrow \mathcal{O}_X(U')\]

Also from \(\varphi^\sharp\) being a sheaf morphism, \(\varphi^\sharp(D(f)): B_f \rightarrow \mathcal{O}_X(U')\) is the composition of the localization \(B_f \rightarrow \mathcal{O}_X(U)_g\) of \(\varphi^\sharp(V)\) and \(\alpha\). But localization is an exact functor ([Commutative Algebra] §Properties of Localization, ⁋Proposition 2), so

\[\ker\bigl(B_f \rightarrow \mathcal{O}_X(U)_g\bigr)=\ker\bigl(\varphi^\sharp(V)\bigr)_f=\mathcal{I}(V)_f\]

and therefore if \(\alpha\) is injective then \(\mathcal{I}(D(f))=\ker (\varphi^\sharp(D(f)))=\mathcal{I}(V)_f\) and the condition of Proposition 6 holds. Now we show that \(\alpha\) is injective under each of the two assumptions.

First suppose \(\varphi\) is quasi-compact. Then \(U\) is quasi-compact so it is covered by finitely many affine open subsets \(\Spec A_1,\ldots, \Spec A_n\). If \(s\in \mathcal{O}_X(U)\) satisfies \(s\vert_{U'}=0\), then for each \(l\) the restriction of \(s\vert_{\Spec A_l}\in A_l\) to \(U'\cap \Spec A_l=D(g\vert_{\Spec A_l})\) is \(0\), so for suitable \(n_l\) we have \((g^{n_l}s)\vert_{\Spec A_l}=0\). Since there are finitely many \(l\), choosing a common \(N\) we have \(g^Ns=0\) on all \(\Spec A_l\), and therefore by the first condition of [Topology] §Sheaves, ⁋Definition 1 we have \(g^Ns=0\). That is, \(s/g^m=0\) in \(\mathcal{O}_X(U)_g\), so \(\alpha\) is injective.

Now suppose \(X\) is reduced. (§Algebraic Structure of Schemes, ⁋Definition 1) Let \(s\in \mathcal{O}_X(U)\) satisfy \(s\vert_{U'}=0\) and consider \(gs\); at points of \(U'\) the stalk of \(s\) is \(0\) so the stalk of \(gs\) is \(0\), and at points \(x\) not in \(U'\) the stalk of \(g\) belongs to the maximal ideal of \(\mathcal{O}_{X,x}\). Therefore for any affine open subset \(\Spec A\) of \(U\), \((gs)\vert_{\Spec A}\) belongs to all prime ideals of \(A\), and from [Commutative Algebra] §Properties of Localization, ⁋Corollary 8 and the fact that \(A\) is a reduced ring we get \((gs)\vert_{\Spec A}=0\). Then again by the sheaf condition \(gs=0\), and therefore \(s/g^m=(gs)/g^{m+1}=0\), so \(\alpha\) is injective.

From the above, by Proposition 6 the sheaf \(\mathcal{I}\) uniquely induces a closed subscheme \(\iota: Z \rightarrow Y\). That this contains the image of \(\varphi\) in the sense of Definition 10 follows because for any affine open subset \(V\) of \(Y\) we have \(\mathcal{I}_{Z/Y}(V)=\ker (\varphi^\sharp(V))\), so the composition \(\mathcal{I}_{Z/Y}(V) \rightarrow \mathcal{O}_Y(V) \rightarrow (\varphi_\ast \mathcal{O}_X)(V)\) is zero, and affine open subsets form a base for \(Y\). Conversely for any closed subscheme \(\iota': Z' \rightarrow Y\) of \(Y\) containing the image of \(\varphi\), the same composition being zero gives \(\mathcal{I}_{Z'/Y}(V)\subseteq \ker (\varphi^\sharp(V))=\mathcal{I}_{Z/Y}(V)\), and since both ideal sheaves are subsheaves of \(\mathcal{O}_Y\) we obtain \(\mathcal{I}_{Z'/Y}\subseteq \mathcal{I}_{Z/Y}\) from this. Therefore by Lemma 9 there exists a closed embedding \(Z \rightarrow Z'\), and hence \(Z\) is the smallest closed subscheme containing the image of \(\varphi\), that is, the scheme-theoretic image of \(\varphi\).

Assuming the above condition and checking the image of \(\varphi\) on each affine open subset, we can verify that the scheme-theoretic image of \(\varphi\) is the closure of the image of \(\varphi\) (as a continuous map) equipped with a structure sheaf defined on it.

Without the assumption of Corollary 12, this does not happen.

Example 13 Define a scheme \(X\) by the following formula

\[X=\coprod_{k\geq 0} \Spec \mathbb{K}[\epsilon]/(\epsilon^k)\]

and let \(Y=\Spec \mathbb{K}[\x]\). Now on each component of \(X\) we can define a scheme morphism \(X \rightarrow Y\) via \(\x\mapsto \epsilon\). Then from Example 11 we know that the image of \(X \rightarrow Y\) (as a continuous map) is the single point \(0\in \mathbb{A}^1\).

However, the scheme-theoretic image of the scheme morphism \(\varphi:X \rightarrow Y\) is not \(0\). For this, observe the morphism \(\varphi^\sharp:\mathcal{O}_Y \rightarrow \varphi_\ast \mathcal{O}_X\) between structure sheaves. Then for an element \(f\) of \(\mathcal{O}_Y\) to satisfy \(\varphi^\sharp(f)=0\), the \(k\)-th order approximation of \(f\) must be zero for arbitrary \(k\), so necessarily \(f=0\). That is, \(\mathcal{I}_{Z/Y}\) must be \(0\), and from this we know that the scheme-theoretic image of \(\varphi\) is itself.

Reduced Scheme Structure on a Closed Set

At the beginning of this post we could define two structure sheaves on an arbitrary closed set \(Z(\mathfrak{a})\) of an affine scheme \(\Spec A\): \((\Spec\pi)_\ast \mathcal{O}_{\Spec A/\mathfrak{a}}\) and \(\iota^{-1} \mathcal{O}_{\Spec A}\). Among these we decided to think of \((\Spec\pi)_\ast \mathcal{O}_{\Spec A/ \mathfrak{a}}\) as the correct scheme structure defined on \(Z(\mathfrak{a})\). Now we examine \(\iota^{-1} \mathcal{O}_{\Spec A}\).

More generally, consider an arbitrary scheme \(Y\) and a closed subset \(X\) of \(Y\). Then for any open set \(\Spec B\) of \(Y\), the closed subset \(X\cap \Spec B\) of \(\Spec B\) can be written in the form \(Z(\mathfrak{b})\) for a radical ideal \(\mathfrak{b}\) of \(B\) by §The Spectrum, ⁋Theorem 15. Moreover, since \(\mathfrak{b}\) is by definition the largest among ideals \(\mathfrak{b}'\) of \(B\) such that \(X\cap \Spec B= Z(\mathfrak{b}')\), by Lemma 9 it is the smallest closed subscheme structure that can be put on \(X\cap \Spec B\).

Definition 14 For an arbitrary closed subset \(X\) of a scheme \(Y\), we call the scheme structure defined on \(X\) above the reduced scheme structure and write it as \(X^\red\).

Then in particular when \(X=Y\), for any affine subset \(\Spec B\) writing \(\Spec B=Z(0)\) gives \(\mathfrak{b}=\mathfrak{N}(B)\), and \(B/\mathfrak{N}(B)\) becomes a reduced ring. On the other hand, the sheaf morphism

\[\iota^{-1}\mathcal{O}_{\Spec A} \rightarrow (\Spec\pi\vert^{Z(\mathfrak{a})})_\ast \mathcal{O}_{\Spec A/\mathfrak{a}}\]

examined above is the canonical sheaf morphism induced from restriction, that is, from the adjunction \(\iota^{-1}\dashv \iota_\ast\). However, one must not confuse this with the scheme morphism given by Lemma 9, because \((Z(\mathfrak{a}),\iota^{-1}\mathcal{O}_{\Spec A})\) is generally not a scheme. For example, if \(A=\mathbb{K}[\x]\) and \(\mathfrak{a}=(\x)\), then \(Z(\mathfrak{a})\) is a single point and the stalk on it is \(\mathbb{K}[\x]_{(\x)}\), but the global section ring of an affine scheme consisting of a single point must have only one prime ideal, so this locally ringed space is not isomorphic to any scheme. The canonical morphism actually given by Lemma 9 goes from the reduced structure defined by the radical of \(\mathfrak{a}\) to the given structure, that is, in the direction \(\Spec (A/\sqrt{\mathfrak{a}}) \rightarrow \Spec (A/\mathfrak{a})\).

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References

[Har] R. Hartshorne, Algebraic geometry. Graduate texts in mathematics. Springer, 1977.
[Vak] R. Vakil, The rising sea: Foundation of algebraic geometry. Available online.

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