7 Properties of schemes and of morphisms of schemes
We introduce some further notions, giving names to important properties of schemes and scheme morphisms. One of the goals is to obtain a better understanding of the connection between the theory of schemes and the more classical point of view taken in the first chapter. Along the way, we will illustrate how fiber products of schemes are useful to “translate” notions from topology to scheme theory.
Recall the definition of a reduced ring:
For every open \(U \subseteq X\), the ring \(\Gamma (U, {\mathscr O}_X)\) is reduced.
For every affine open \(U \subseteq X\), the ring \(\Gamma (U, {\mathscr O}_X)\) is reduced.
There exists an affine open cover \(X = \bigcup _i U_i\) such that for every \(i\) the ring \(\Gamma (U_i, {\mathscr O}_X)\) is reduced.
For every \(x\in X\), the ring \({\mathscr O}_{X, x}\) is reduced.
See Problem 33.
Every domain is reduced, but not conversely. More precisely, a reduced ring \(R\) is a domain if and only if it has a unique minimal prime ideal (necessarily the zero ideal), if and only if \(\operatorname{Spec}(R)\) is irreducible. This leads to the notion of integral scheme, see below. Before we come to it, we discuss irreducibility and generic points in the context of schemes; cf. Section 2.3.
From the corresponding result for affine schemes, we obtain the existence and uniqueness of generic points.
The scheme \(X\) is reduced and irreducible.
For every non-empty open \(U \subseteq X\), the ring \(\Gamma (U, {\mathscr O}_X)\) is a domain.
For every non-empty affine open \(U \subseteq X\), the ring \(\Gamma (U, {\mathscr O}_X)\) is a domain.
If \(X\) is an integral scheme, then all the local rings \({\mathscr O}_{X, x}\) are domains (note that the converse does not hold, though). Since \(X\) is irreducible, it has a (unique) generic point \(\eta \in X\). The local ring \(K(X) := {\mathscr O}_{X, \eta }\) is a domain with only one prime ideal, and hence a field, called the field of rational functions of \(X\).
Recall that we have defined the notion of open subscheme: If \(X\) is a scheme and \(U \subseteq X\) an open subset of (the topological space) \(X\), restricting the structure sheaf of \(X\) to \(U\) equips \(U\) with the structure of a scheme, and schemes of this form are called open subschemes. The inclusion \(U\to X\) naturally is a scheme morphism. A scheme morphism \(j\colon U\to X\) is called an open immersion, if the continous map \(j\) is a homeomorphism onto an open subset of \(X\), and under this homeomorphism the structure sheaf of \(U\) is identified with the restriction \({\mathscr O}_{X|j(U)}\). Equivalently, \(j\) factors through an isomorphism with an open subscheme of \(X\) and the natural inclusion of that open subscheme into \(X\). (The difference between open subschemes and open immersions is similar to the difference (in the context of sets) between subsets and injective maps.)
There is also a notion of closed subscheme which is, however, slightly more involved. The main reason is that unlike open subschemes, closed subschemes are not determined by the underlying closed subset of \(X\).
For the definition, recall that a morphism of sheaves is called surjective, if the induced maps on stalks are surjective at all points of the underlying topological space. If \(f\colon {\mathscr F}\to {\mathscr G}\) is a morphism of sheaves, we have the notion of the image sheaf \(\mathop{\rm im}(f)\) (the sheafification of the presheaf \(U\mapsto \mathop{\rm im}({\mathscr F}(U)\to {\mathscr G}(U))\)), and \(f\) is surjective, if and only if \(\mathop{\rm im}(f) = {\mathscr G}\). For a surjective homomorphism \(f\colon {\mathscr F}\to {\mathscr G}\) of rings, the kernel sheaf \(\operatorname{Ker}(f)\) (defined by \(U\mapsto \operatorname{Ker}({\mathscr F}(U)\to {\mathscr G}(U))\)) is an ideal sheaf in \({\mathscr F}\) (i.e., for every open \(U\), the subset \(\operatorname{Ker}(f)(U) \subseteq {\mathscr F}(U)\) is an ideal), and there is an isomorphism \({\mathscr G}\cong {\mathscr F}/\operatorname{Ker}(f)\), where the quotient sheaf is defined as the sheafification of the presheaf \(U\mapsto {\mathscr F}(U)/\operatorname{Ker}(f)(U)\).
Note that \(Z\) is defined by \({\mathscr I}\). One can show that for any ideal sheaf \({\mathscr I}\subseteq {\mathscr O}_X\) the set \({\rm supp}({\mathscr O}/{\mathscr I}) := \left\{ x\in X; ({\mathscr O}_X/{\mathscr I})_x\ne 0 \right\} \) (the support of \({\mathscr O}_X/{\mathscr I}\)) is closed in \(X\). Thus a closed subscheme “is” really the same as an ideal sheaf \({\mathscr I}\) such that \(({\rm supp}({\mathscr O}/{\mathscr I}), {\mathscr O}/{\mathscr I})\) is a scheme. (This latter property is not automatic; there are ideal sheaves for which it is not satisfied.)
If \(Z \subseteq X\) is a closed subscheme with corresponding ideal sheaf \({\mathscr I}\), we have a natural scheme morphism \(Z\to X\) which on topological spaces is the inclusion map \(i\) and on sheaves is given by \({\mathscr O}_X\to {\mathscr O}_X/{\mathscr I}= i_* {\mathscr O}_Z\). Thus, if \(Z\) is a closed subscheme, we can recover \({\mathscr I}\) from the structure sheaf of \(Z\) and the natural inclusion \(i\colon Z\to X\) as \({\mathscr I}= \operatorname{Ker}({\mathscr O}_X\to i_*{\mathscr O}_Z)\).
Let \(R\) be a ring, \({\mathfrak a}\subseteq R\) an ideal. Then \(V({\mathfrak a}) \subseteq \operatorname{Spec}(R)\) is a closed subscheme. We will prove below that all closed subschemes of an affine scheme have this form.
Let \(R\) be a ring, and let \(I \subseteq R[X_0, \dots , X_n]\) be a homogeneous ideal. Then \(V_+(I) \subseteq \mathbb {P}^n_R\) is a closed subscheme. (We have seen that \(V_+(I)\) topologically is a closed subset. The sheaf homomorphism \({\mathscr O}_{\mathbb {P}^n_R}\to {\mathscr O}_{V_+(I)}\) is surjective; this can be checked on the standard open charts, thus reducing to the affine case.) One can show that every closed subscheme of \(\mathbb {P}^n_R\) has this form.
Let \(R\) be a ring, \(X = \operatorname{Spec}(R)\), and let \(Z \subseteq X\) be a closed subscheme. Then there exists a unique ideal \({\mathfrak a}\subseteq R\) such that \(Z = V({\mathfrak a})\) (i.e., \(Z\) and \(V({\mathfrak a})\) are defined by the same ideal sheaf in \({\mathscr O}_X\)).
The ideal \({\mathfrak a}\) is given as
In particular, every closed subscheme of an affine scheme is itself affine.
It is easy to check that \(I_{V({\mathfrak a})} = {\mathfrak a}\). Therefore it is enough to show that for every closed subscheme \(Z\), we have \(Z = V(I_Z)\).
The ring homomorphism \(A\to \Gamma (Z, {\mathscr O}_Z)\) factors through \(A/I_Z\), and we want to show that the natural morphism \(Z\to \operatorname{Spec}(A/I_Z)\) is an isomorphism. Replacing \(A\) by \(A/{\mathscr I}\), we may assume that \(\varphi \colon A\to \Gamma (Z, {\mathscr O}_Z)\) is injective (and then want to show that \(Z\cong \operatorname{Spec}(A)\)).
(I) Let us show that the map \(Z\to \operatorname{Spec}(A)\) is a homeomorphism. We know already that it is injective and closed; it remains to show its surjectivity. We will show that \(Z\) is not contained in a proper closed subset of \(\operatorname{Spec}(A)\). Let \(s\in A\) with \(Z \subseteq V(s)\); we will show that then necessarily \(s\) is nilpotent, i.e., \(V(s) = \operatorname{Spec}(A)\).
In view of the injectivity of \(\varphi \), it is enough to check that \(\varphi (s)\) is nilpotent. Let \(Z = \bigcup _i V_i\) be a finite affine open cover (\(Z\) is quasi-compact since it is closed in the quasi-compact affine scheme \(\operatorname{Spec}(A)\)). Then \(\Gamma (Z, {\mathscr O}_Z)\) injects into the product \(\prod _i \Gamma (V_i, {\mathscr O}_Z)\), so it is enough to show that all restrictions \(s_{|V_i}\) are nilpotent. But \(V_i \subseteq Z \subseteq V(s)\) implies \(V_i = V_{V_i}(s_{|V_i})\), so \(s_{|V_i}\in \Gamma (V_i, {\mathscr O}_Z)\) is indeed nilpotent.
(II) To conclude the proof, we show that \(Z=\operatorname{Spec}(R)=X\) as schemes, i.e., the sheaf homomorphism \({\mathscr O}_X\to {\mathscr O}_Z\) is an isomorphism (we identify \(Z=X\) as topological spaces in view of Step (I)). This sheaf homomorphism is surjective by assumption, and it remains to show the injectivity. We check this on the stalks.
So let \({\mathfrak p}\subset R\) be a prime ideal (i.e., a point of \(X\)), and consider the ring homomorphism \(R_{{\mathfrak p}}={\mathscr O}_{X, {\mathfrak p}}\to {\mathscr O}_{Z, {\mathfrak p}}\). It is then enough to show that for all \(g\in A\) with \(\frac{g}{1}\mapsto 0\in {\mathscr O}_{Z, {\mathfrak p}}\) we have \(g=0\).
So fix \(g\) with this property. Let \(V \subseteq Z\) be an affine open neighborhood of \({\mathfrak p}\) such that \(\varphi (g)_{|V} = 0\). Choose an affine open cover \(Z = \bigcup _{i=1}^n V_i\) with \(V_1 = V\).
Let \(s\in A\) with \(D(s) \subseteq V\).
Claim. There exists \(N\ge 0\) such that \(\varphi (s^Ng)=0\in \Gamma (Z, {\mathscr O}_Z)\).
We can check this on each \(V_i\) separately. By assumption \(\varphi (g_{|V}) = 0\); it remains to handle the case \(i {\gt} 1\). Since \(D_{V_i}(\varphi (s)_{|V_i}) = D(s)\cap V_i \subseteq V\cap V_i\), we have \(g_{|D_{V_i}(\varphi (s)_{|V_i})} = 0\), i.e., the image of \(\varphi (g)\) in the localization \(\Gamma (V_i, {\mathscr O}_Z)_{\varphi (s)_{|V_i}}\) is \(=0\). Thus \((\varphi (s)_{|V_i})^{N_i} \varphi (g)_{|V_i} = 0\) for some \(N_i\).
The claim is proved. Since \(\varphi \) is injective, it follows that \(s^N g = 0\), so the image of \(g\) in \(A_s\) is \(=0\). A fortiori, \(g\) maps to \(0\) in \(A_{{\mathfrak p}}\). This is what remained to show.
If \(X=\operatorname{Spec}(R)\) is affine and \(Z = V({\mathfrak a})\) as a set, for an ideal \({\mathfrak a}\subseteq R\), then the scheme \(V(\sqrt{{\mathfrak a}})\) is the unique reduced closed subscheme of \(X\) with underlying set \(Z\) (recall, cf. Proposition 2.9, that \(V({\mathfrak a})\) and \(V({\mathfrak b})\) have the same underlying closed subset, if and only if the radicals of \({\mathfrak a}\) and \({\mathfrak b}\) are equal; the quotient of a ring by an ideal is reduced if and only if the ideal is a radical ideal).
In the general case, in view of the uniqueness statement, we may obtain the desired closed subscheme by using gluing of schemes.
Combining the notions of open and closed subscheme, we obtain the following notion. Recall that a subset of a topological space is called locally closed, if it can be written as the intersection of an open and of a closed subset.
If \(Y \subseteq X\) is a subscheme, then the topological space of \(Y\) is a locally closed subset of \(X\). We have a natural scheme morphism \(Y\to X\) (which may be obtained as the composition \(Y\to U\to X\) with \(U \subseteq X\) open as in the definition), and the sheaf homomorphism \({\mathscr O}_X\to i_*{\mathscr O}_Y\) (where \(i\colon Y\to X\) is the inclusion) induces a surjection \({\mathscr O}_{X, i(y)}\to (i_*{\mathscr O}_Y)_{i(y)}\) for every \(y\in Y\).
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