Properties of Schemes

Introduction

In this chapter we introduce some absolute properties of schemes. A foundational reference is [EGA].

Constructible sets

Constructible and locally constructible sets are introduced in Topology, Section 04ZC. We may characterize locally constructible subsets of schemes as follows.

Lemma

Let \(X\) be a scheme. A subset \(E\) of \(X\) is locally constructible in \(X\) if and only if \(E \cap U\) is constructible in \(U\) for every affine open \(U\) of \(X\).

Proof

Assume \(E\) is locally constructible. Then there exists an open covering \(X = \bigcup U_i\) such that \(E \cap U_i\) is constructible in \(U_i\) for each \(i\). Let \(V \subset X\) be any affine open. We can find a finite open affine covering \(V = V_1 \cup \ldots \cup V_m\) such that for each \(j\) we have \(V_j \subset U_i\) for some \(i = i(j)\). By Topology, Lemma 005J we see that each \(E \cap V_j\) is constructible in \(V_j\). Since the inclusions \(V_j \to V\) are quasi-compact (see Schemes, Lemma 01K4) we conclude that \(E \cap V\) is constructible in \(V\) by Topology, Lemma 053W. The converse implication is immediate.

Lemma

Let \(X\) be a scheme and let \(E \subset X\) be a locally constructible subset. Let \(\xi \in X\) be a generic point of an irreducible component of \(X\).

If \(\xi \in E\), then an open neighbourhood of \(\xi\) is contained in \(E\).

If \(\xi \not \in E\), then an open neighbourhood of \(\xi\) is disjoint from \(E\).

Proof

As the complement of a locally constructible subset is locally constructible it suffices to show (2). We may assume \(X\) is affine and hence \(E\) constructible (Lemma 054C). In this case \(X\) is a spectral space (Algebra, Lemma 090M). Then \(\xi \not \in E\) implies \(\xi \not \in \overline{E}\) by Topology, Lemma 0903 and the fact that there are no points of \(X\) different from \(\xi\) which specialize to \(\xi\).

Lemma

Let \(X\) be a quasi-separated scheme. The intersection of any two quasi-compact opens of \(X\) is a quasi-compact open of \(X\). Every quasi-compact open of \(X\) is retrocompact in \(X\).

Proof

If \(U\) and \(V\) are quasi-compact open then \(U \cap V = \Delta^{-1}(U \times V)\), where \(\Delta : X \to X \times X\) is the diagonal. As \(X\) is quasi-separated we see that \(\Delta\) is quasi-compact. Hence we see that \(U \cap V\) is quasi-compact as \(U \times V\) is quasi-compact (details omitted; use Schemes, Lemma 01JS to see \(U \times V\) is a finite union of affines). The other assertions follow from the first and Topology, Lemma 0069.

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. Then the underlying topological space of \(X\) is a spectral space.

Proof

By Topology, Definition 08YG we have to check that \(X\) is sober, quasi-compact, has a basis of quasi-compact opens, and the intersection of any two quasi-compact opens is quasi-compact. This follows from Schemes, Lemma 01IS and 01IT and Lemma 054D above.

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. Any locally constructible subset of \(X\) is constructible.

Proof

As \(X\) is quasi-compact we can choose a finite affine open covering \(X = V_1 \cup \ldots \cup V_m\). As \(X\) is quasi-separated each \(V_i\) is retrocompact in \(X\) by Lemma 054D. Hence by Topology, Lemma 053W we see that \(E \subset X\) is constructible in \(X\) if and only if \(E \cap V_j\) is constructible in \(V_j\). Thus we win by Lemma 054C.

Lemma

Let \(X\) be a scheme. A subset \(E\) of \(X\) is retrocompact in \(X\) if and only if \(E \cap U\) is quasi-compact for every affine open \(U\) of \(X\).

Proof

Immediate from the fact that every quasi-compact open of \(X\) is a finite union of affine opens.

Lemma

A partition \(X = \coprod_{i \in I} X_i\) of a scheme \(X\) with retrocompact parts is locally finite if and only if the parts are locally constructible.

Proof

See Topology, Definitions 005A, 09XZ, and 0BDS for the definitions of retrocompact, partition, and locally finite.

If the partition is locally finite and \(U \subset X\) is an affine open, then we see that \(U = \coprod_{i \in I} U \cap X_i\) is a finite partition (more precisely, all but a finite number of its parts are empty). Hence \(U \cap X_i\) is quasi-compact and its complement is retrocompact in \(U\) as a finite union of retrocompact parts. Thus \(U \cap X_i\) is constructible by Topology, Lemma 0F2K. It follows that \(X_i\) is locally constructible by Lemma 054C.

Assume the parts are locally constructible. Then for any affine open \(U \subset X\) we obtain a covering \(U = \coprod X_i \cap U\) by constructible subsets. Since the constructible topology is quasi-compact, see Topology, Lemma 0901, this covering has a finite refinement, i.e., the partition is locally finite.

Integral, irreducible, and reduced schemes

Definition

Let \(X\) be a scheme. We say \(X\) is integral if it is nonempty and for every nonempty affine open \(\Spec(R) = U \subset X\) the ring \(R\) is an integral domain.

Lemma

Let \(X\) be a scheme. The following are equivalent.

The scheme \(X\) is reduced, see Schemes, Definition 01J0.

There exists an affine open covering \(X = \bigcup U_i\) such that each \(\Gamma(U_i, \mathcal{O}_X)\) is reduced.

For every affine open \(U \subset X\) the ring \(\mathcal{O}_X(U)\) is reduced.

For every open \(U \subset X\) the ring \(\mathcal{O}_X(U)\) is reduced.

Proof

See Schemes, Lemmas 01J1 and 01J2.

Lemma

Let \(X\) be a scheme. The following are equivalent.

The scheme \(X\) is irreducible.

There exists an affine open covering \(X = \bigcup_{i \in I} U_i\) such that \(I\) is not empty, \(U_i\) is irreducible for all \(i \in I\), and \(U_i \cap U_j \not = \emptyset\) for all \(i, j \in I\).

The scheme \(X\) is nonempty and every nonempty affine open \(U \subset X\) is irreducible.

Proof

Assume (1). By Schemes, Lemma 01IS we see that \(X\) has a unique generic point \(\eta\). Then \(X = \overline{\{\eta\}}\). Hence \(\eta\) is an element of every nonempty affine open \(U \subset X\). This implies that \(\eta \in U\) is dense hence \(U\) is irreducible. It also implies any two nonempty affines meet. Thus (1) implies both (2) and (3).

Assume (2). Suppose \(X = Z_1 \cup Z_2\) is a union of two closed subsets. For every \(i\) we see that either \(U_i \subset Z_1\) or \(U_i \subset Z_2\). Pick some \(i \in I\) and assume \(U_i \subset Z_1\) (possibly after renumbering \(Z_1\), \(Z_2\)). For any \(j \in I\) the open subset \(U_i \cap U_j\) is dense in \(U_j\) and contained in the closed subset \(Z_1 \cap U_j\). We conclude that also \(U_j \subset Z_1\). Thus \(X = Z_1\) as desired.

Assume (3). Choose an affine open covering \(X = \bigcup_{i \in I} U_i\). We may assume that each \(U_i\) is nonempty. Since \(X\) is nonempty we see that \(I\) is not empty. By assumption each \(U_i\) is irreducible. Suppose \(U_i \cap U_j = \emptyset\) for some pair \(i, j \in I\). Then the open \(U_i \amalg U_j = U_i \cup U_j\) is affine, see Schemes, Lemma 01I5. Hence it is irreducible by assumption which is absurd. We conclude that (3) implies (2). The lemma is proved.

Lemma

A scheme \(X\) is integral if and only if it is reduced and irreducible.

Proof

If \(X\) is irreducible, then every affine open \(\Spec(R) = U \subset X\) is irreducible. If \(X\) is reduced, then \(R\) is reduced, by Lemma 01OL above. Hence \(R\) is reduced and \((0)\) is a prime ideal, i.e., \(R\) is an integral domain.

If \(X\) is integral, then for every nonempty affine open \(\Spec(R) = U \subset X\) the ring \(R\) is reduced and hence \(X\) is reduced by Lemma 01OL. Moreover, every nonempty affine open is irreducible. Hence \(X\) is irreducible, see Lemma 01OM.

In Examples, Section 0568 we construct a connected affine scheme all of whose local rings are domains, but which is not integral.

Types of schemes defined by properties of rings

In this section we study what properties of rings allow one to define local properties of schemes.

Definition

Let \(P\) be a property of rings. We say that \(P\) is local if the following hold:

For any ring \(R\), and any \(f \in R\) we have \(P(R) \Rightarrow P(R_f)\).

For any ring \(R\), and \(f_i \in R\) such that \((f_1, \ldots, f_n) = R\) then \(\forall i, P(R_{f_i}) \Rightarrow P(R)\).

Definition

Let \(P\) be a property of rings. Let \(X\) be a scheme. We say \(X\) is locally \(P\) if for any \(x \in X\) there exists an affine open neighbourhood \(U\) of \(x\) in \(X\) such that \(\mathcal{O}_X(U)\) has property \(P\).

This is only a good notion if the property is local. Even if \(P\) is a local property we will not automatically use this definition to say that a scheme is “locally \(P\)” unless we also explicitly state the definition elsewhere.

Lemma

Let \(X\) be a scheme. Let \(P\) be a local property of rings. The following are equivalent:

The scheme \(X\) is locally \(P\).

For every affine open \(U \subset X\) the property \(P(\mathcal{O}_X(U))\) holds.

There exists an affine open covering \(X = \bigcup U_i\) such that each \(\mathcal{O}_X(U_i)\) satisfies \(P\).

There exists an open covering \(X = \bigcup X_j\) such that each open subscheme \(X_j\) is locally \(P\).

Moreover, if \(X\) is locally \(P\) then every open subscheme is locally \(P\).

Proof

Of course (1) \(\Leftrightarrow\) (3) and (2) \(\Rightarrow\) (1). If (3) \(\Rightarrow\) (2), then the final statement of the lemma holds and it follows easily that (4) is also equivalent to (1). Thus we show (3) \(\Rightarrow\) (2).

Let \(X = \bigcup U_i\) be an affine open covering, say \(U_i = \Spec(R_i)\). Assume \(P(R_i)\). Let \(\Spec(R) = U \subset X\) be an arbitrary affine open. By Schemes, Lemma 01IX there exists a standard covering of \(U = \Spec(R)\) by standard opens \(D(f_j)\) such that each ring \(R_{f_j}\) is a principal localization of one of the rings \(R_i\). By Definition 01OP (1) we get \(P(R_{f_j})\). Whereupon \(P(R)\) by Definition 01OP (2).

Here is a sample application.

Lemma

Let \(X\) be a scheme. Then \(X\) is reduced if and only if \(X\) is “locally reduced” in the sense of Definition 01OQ.

Proof

This is clear from Lemma 01OL.

Lemma

The following properties of a ring \(R\) are local.

(Cohen-Macaulay.) The ring \(R\) is Noetherian and CM, see Algebra, Definition 00NC.

(Regular.) The ring \(R\) is Noetherian and regular, see Algebra, Definition 00OD.

(Absolutely Noetherian.) The ring \(R\) is of finite type over \(Z\).

Add more here as needed.¹

Proof

Omitted.

Noetherian schemes

Recall that a ring \(R\) is Noetherian if it satisfies the ascending chain condition of ideals. Equivalently every ideal of \(R\) is finitely generated.

Definition

Let \(X\) be a scheme.

We say \(X\) is locally Noetherian if every \(x \in X\) has an affine open neighbourhood \(\Spec(R) = U \subset X\) such that the ring \(R\) is Noetherian.

We say \(X\) is Noetherian if \(X\) is locally Noetherian and quasi-compact.

Here is the standard result characterizing locally Noetherian schemes.

Lemma

Let \(X\) be a scheme. The following are equivalent:

The scheme \(X\) is locally Noetherian.

For every affine open \(U \subset X\) the ring \(\mathcal{O}_X(U)\) is Noetherian.

There exists an affine open covering \(X = \bigcup U_i\) such that each \(\mathcal{O}_X(U_i)\) is Noetherian.

There exists an open covering \(X = \bigcup X_j\) such that each open subscheme \(X_j\) is locally Noetherian.

Moreover, if \(X\) is locally Noetherian then every open subscheme is locally Noetherian.

Proof

To show this it suffices to show that being Noetherian is a local property of rings, see Lemma 01OR. Any localization of a Noetherian ring is Noetherian, see Algebra, Lemma 00FN. By Algebra, Lemma 00EO we see the second property to Definition 01OP.

Lemma

Any immersion \(Z \to X\) with \(X\) locally Noetherian is quasi-compact.

Proof

A closed immersion is clearly quasi-compact. A composition of quasi-compact morphisms is quasi-compact, see Topology, Lemma 005B. Hence it suffices to show that an open immersion into a locally Noetherian scheme is quasi-compact. Using Schemes, Lemma 01K4 we reduce to the case where \(X\) is affine. Any open subset of the spectrum of a Noetherian ring is quasi-compact (for example combine Algebra, Lemma 00FQ and Topology, Lemmas 0052 and 04ZA).

Lemma

A locally Noetherian scheme is quasi-separated.

Proof

By Schemes, Lemma 01KO we have to show that the intersection \(U \cap V\) of two affine opens of \(X\) is quasi-compact. This follows from Lemma 01OX above on considering the open immersion \(U \cap V \to U\) for example. (But really it is just because any open of the spectrum of a Noetherian ring is quasi-compact.)

Lemma

A (locally) Noetherian scheme has a (locally) Noetherian underlying topological space, see Topology, Definition 0051.

Proof

This is because a Noetherian scheme is a finite union of spectra of Noetherian rings and Algebra, Lemma 00FQ and Topology, Lemma 0053.

Lemma

Any locally closed subscheme of a (locally) Noetherian scheme is (locally) Noetherian.

Proof

Omitted. Hint: Any quotient, and any localization of a Noetherian ring is Noetherian. For the Noetherian case use again that any subset of a Noetherian space is a Noetherian space (with induced topology).

Lemma

A Noetherian scheme has a finite number of irreducible components.

Proof

The underlying topological space of a Noetherian scheme is Noetherian (Lemma 01OZ) and we conclude because a Noetherian topological space has only finitely many irreducible components (Topology, Lemma 0052).

Lemma

Any morphism of schemes \(f : X \to Y\) with \(X\) Noetherian is quasi-compact.

Proof

Use Lemma 01OZ and use that any subset of a Noetherian topological space is quasi-compact (see Topology, Lemmas 0052 and 04ZA).

Here is a fun lemma. It says that every locally Noetherian scheme has plenty of closed points (at least one in every closed subset).

Lemma

Any nonempty locally Noetherian scheme has a closed point. Any nonempty closed subset of a locally Noetherian scheme has a closed point. Equivalently, any point of a locally Noetherian scheme specializes to a closed point.

Proof

The second assertion follows from the first (using Schemes, Lemma 01J3 and Lemma 02IK). Consider any nonempty affine open \(U \subset X\). Let \(x \in U\) be a closed point. If \(x\) is a closed point of \(X\) then we are done. If not, let \(X_0 \subset X\) be the reduced induced closed subscheme structure on \(\overline{\{x\}}\). Then \(U_0 = U \cap X_0\) is an affine open of \(X_0\) by Schemes, Lemma 01IN and \(U_0 = \{x\}\). Let \(y \in X_0\), \(y \not = x\) be a specialization of \(x\). Consider the local ring \(R = \mathcal{O}_{X_0, y}\). This is a Noetherian local ring as \(X_0\) is Noetherian by Lemma 02IK. Denote \(V \subset \Spec(R)\) the inverse image of \(U_0\) in \(\Spec(R)\) by the canonical morphism \(\Spec(R) \to X_0\) (see Schemes, Section 01J5.) By construction \(V\) is a singleton with unique point corresponding to \(x\) (use Schemes, Lemma 01J7). By Algebra, Lemma 02IG we see that \(\dim(R) = 1\). In other words, we see that \(y\) is an immediate specialization of \(x\) (see Topology, Definition 02I9). In other words, any point \(y \not = x\) such that \(x \leadsto y\) is an immediate specialization of \(x\). Clearly each of these points is a closed point as desired.

Lemma

Let \(X\) be a locally Noetherian scheme. Let \(x' \leadsto x\) be a specialization of points of \(X\). Then

there exists a discrete valuation ring \(R\) and a morphism \(f : \Spec(R) \to X\) such that the generic point \(\eta\) of \(\Spec(R)\) maps to \(x'\) and the special point maps to \(x\), and

provided \(x \not = x'\) given a finitely generated field extension \(K/\kappa(x')\), we may arrange it so that the extension \(\kappa(\eta)/\kappa(x')\) induced by \(f\) is isomorphic to the given one.

Proof

First assume \(x' \leadsto x\) is a specialization in \(X\) with \(x \not = x'\) and let \(K/\kappa(x')\) be a finitely generated extension of fields. By Schemes, Lemma 01J7 and the discussion following Schemes, Lemma 01J9 this leads to ring maps \(\mathcal{O}_{X, x} \to \kappa(x') \to K\). Since \(x \not = x'\) the image of \(\mathcal{O}_{X, x}\) in \(K\) is not a field (details omitted). Let \(R \subset K\) be any discrete valuation ring whose field of fractions is \(K\) and which dominates the image of \(\mathcal{O}_{X, x} \to K\), see Algebra, Lemma 00PH. The ring map \(\mathcal{O}_{X, x} \to R\) induces the morphism \(f : \Spec(R) \to X\), see Schemes, Lemma 01J6. This morphism has all the desired properties by construction. If \(x = x'\) then we can set \(R = \kappa(x)[t]_{(t)}\).

Lemma

Let \(S\) be a Noetherian scheme. Let \(T \subset S\) be an infinite subset. Then there exists an infinite subset \(T' \subset T\) such that there are no nontrivial specializations among the points \(T'\).

Proof

Let \(T_0 \subset T\) be the set of \(t \in T\) which do not specialize to another point of \(T\). If \(T_0\) is infinite, then \(T' = T_0\) works. Hence we may and do assume \(T_0\) is finite. Inductively, for \(i > 0\), consider the set \(T_i \subset T\) of \(t \in T\) such that

\(t \not \in T_{i - 1} \cup T_{i - 2} \cup \ldots \cup T_0\),

there exist a nontrivial specialization \(t \leadsto t'\) with \(t' \in T_{i - 1}\), and

for any nontrivial specialization \(t \leadsto t'\) with \(t' \in T\) we have \(t' \in T_{i - 1} \cup T_{i - 2} \cup \ldots \cup T_0\).

Again, if \(T_i\) is infinite, then \(T' = T_i\) works. Let \(d\) be the maximum of the dimensions of the local rings \(\mathcal{O}_{S, t}\) for \(t \in T_0\); then \(d\) is an integer because \(T_0\) is finite and the dimensions of the local rings are finite by Algebra, Proposition 00KQ. Then \(T_i = \emptyset\) for \(i > d\). Namely, if \(t \in T_i\) then we can find a sequence of nontrivial specializations \(t = t_i \leadsto t_{i - 1} \leadsto \ldots \leadsto t_0\) with \(t_0 \in T_0\). As the points \(t = t_i, t_{i - 1}, \ldots, t_0\) are in \(\Spec(\mathcal{O}_{S, t_0})\) (Schemes, Lemma 01J7), we see that \(i \leq d\). Thus \(\bigcup T_i = T_d \cup \ldots \cup T_0\) is a finite subset of \(T\).

Suppose \(t \in T\) is not in \(\bigcup T_i\). Then there must be a specialization \(t \leadsto t'\) with \(t' \in T\) and \(t' \not \in \bigcup T_i\). (Namely, if every specialization of \(t\) is in the finite set \(T_d \cup \ldots \cup T_0\), then there is a maximum \(i\) such that there is some specialization \(t \leadsto t'\) with \(t' \in T_i\) and then \(t \in T_{i + 1}\) by construction.) Hence we get an infinite sequence \[t \leadsto t' \leadsto t'' \leadsto \ldots\] of nontrivial specializations between points of \(T \setminus \bigcup T_i\). This is impossible because the underlying topological space of \(S\) is Noetherian by Lemma 01OY.

Lemma

Let \(S\) be a Noetherian scheme. Let \(T \subset S\) be a subset. Let \(T_0 \subset T\) be the set of \(t \in T\) such that there is no nontrivial specialization \(t' \leadsto t\) with \(t' \in T'\). Then (a) there are no specializations among the points of \(T_0\), (b) every point of \(T\) is a specialization of a point of \(T_0\), and (c) the closures of \(T\) and \(T_0\) are the same.

Proof

Recall that \(\dim(\mathcal{O}_{S, s}) < \infty\) for any \(s \in S\), see Algebra, Proposition 00KQ. Let \(t \in T\). If \(t' \leadsto t\), then by dimension theory \(\dim(\mathcal{O}_{S, t'}) \leq \dim(\mathcal{O}_{S, t})\) with equality if and only if \(t' = t\). Thus if we pick \(t' \leadsto t\) with \(\dim(\mathcal{O}_{T, t'})\) minimal, then \(t' \in T_0\). In other words, every \(t \in T\) is the specialization of an element of \(T_0\).

Lemma

Let \(S\) be a Noetherian scheme. Let \(T \subset S\) be an infinite dense subset. Then there exist a countable subset \(E \subset T\) which is dense in \(S\).

Proof

Let \(T'\) be the set of points \(s \in S\) such that \(\overline{\{s\}} \cap T\) contains a countable subset whose closure is \(\overline{\{s\}}\). Since a finite set is countable we have \(T \subset T'\). For \(s \in T'\) choose such a countable subset \(E_s \subset \overline{\{s\}} \cap T\). Let \(E' = \{s_1, s_2, s_3, \ldots\} \subset T'\) be a countable subset. Then the closure of \(E'\) in \(S\) is the closure of the countable subset \(\bigcup_n E_{s_n}\) of \(T\). It follows that if \(Z\) is an irreducible component of the closure of \(E'\), then the generic point of \(Z\) is in \(T'\).

Denote \(T'_0 \subset T'\) the subset of \(t \in T'\) such that there is no nontrivial specialization \(t' \leadsto t\) with \(t' \in T'\) as in Lemma 0G2R whose results we will use without further mention. If \(T'_0\) is infinite, then we choose a countable subset \(E' \subset T'_0\). By the argument in the first paragraph, the generic points of the irreducible components of the closure of \(E'\) are in \(T'\). However, since one of these points specializes to infinitely many distinct elements of \(E' \subset T'_0\) this is a contradiction. Thus \(T'_0\) is finite, say \(T'_0 = \{s_1, \ldots, s_m\}\). Then it follows that \(S\), which is the closure of \(T\), is contained in the closure of \(\{s_1, \ldots, s_m\}\), which in turn is contained in the closure of the countable subset \(E_{s_1} \cup \ldots \cup E_{s_m} \subset T\) as desired.

Jacobson schemes

Recall that a space is said to be Jacobson if the closed points are dense in every closed subset, see Topology, Section 005T.

Definition

A scheme \(S\) is said to be Jacobson if its underlying topological space is Jacobson.

Recall that a ring \(R\) is Jacobson if every radical ideal of \(R\) is the intersection of maximal ideals, see Algebra, Definition 00G0.

Lemma

An affine scheme \(\Spec(R)\) is Jacobson if and only if the ring \(R\) is Jacobson.

Proof

This is Algebra, Lemma 00G3.

Here is the standard result characterizing Jacobson schemes. Intuitively it claims that Jacobson \(\Leftrightarrow\) locally Jacobson.

Lemma

Let \(X\) be a scheme. The following are equivalent:

The scheme \(X\) is Jacobson.

The scheme \(X\) is “locally Jacobson” in the sense of Definition 01OQ.

For every affine open \(U \subset X\) the ring \(\mathcal{O}_X(U)\) is Jacobson.

There exists an affine open covering \(X = \bigcup U_i\) such that each \(\mathcal{O}_X(U_i)\) is Jacobson.

There exists an open covering \(X = \bigcup X_j\) such that each open subscheme \(X_j\) is Jacobson.

Moreover, if \(X\) is Jacobson then every open subscheme is Jacobson.

Proof

The final assertion of the lemma holds by Topology, Lemma 005X. The equivalence of (5) and (1) is Topology, Lemma 005W. Hence, using Lemma 01P3, we see that (1) \(\Leftrightarrow\) (2). To finish proving the lemma it suffices to show that “Jacobson” is a local property of rings, see Lemma 01OR. Any localization of a Jacobson ring at an element is Jacobson, see Algebra, Lemma 00G6. Suppose \(R\) is a ring, \(f_1, \ldots, f_n \in R\) generate the unit ideal and each \(R_{f_i}\) is Jacobson. Then we see that \(\Spec(R) = \bigcup D(f_i)\) is a union of open subsets which are all Jacobson, and hence \(\Spec(R)\) is Jacobson by Topology, Lemma 005W again. This proves the second property of Definition 01OP.

Many schemes used commonly in algebraic geometry are Jacobson, see Morphisms, Lemma 02J6. We mention here the following interesting case.

Lemma

Examples of Noetherian Jacobson schemes.

If \((R, \mathfrak m)\) is a Noetherian local ring, then the punctured spectrum \(\Spec(R) \setminus \{\mathfrak m\}\) is a Jacobson scheme.

If \(R\) is a Noetherian ring with Jacobson radical \(\text{rad}(R)\) then \(\Spec(R) \setminus V(\text{rad}(R))\) is a Jacobson scheme.

If \((R, I)\) is a Zariski pair (More on Algebra, Definition 0ELY) with \(R\) Noetherian, then \(\Spec(R) \setminus V(I)\) is a Jacobson scheme.

Proof

Proof of (3). Observe that \(\Spec(R) - V(I)\) has a covering by the affine opens \(\Spec(R_f)\) for \(f \in I\). The rings \(R_f\) are Jacobson by More on Algebra, Lemma 0GED. Hence \(\Spec(R) \setminus V(I)\) is Jacobson by Lemma 01P4. Parts (1) and (2) are special cases of (3).

Direct proof of case (1). Since \(\Spec(R)\) is a Noetherian scheme, \(S\) is a Noetherian scheme (Lemma 02IK). Hence \(S\) is a sober, Noetherian topological space (use Schemes, Lemma 01IS). Assume \(S\) is not Jacobson to get a contradiction. By Topology, Lemma 02I7 there exists some non-closed point \(\xi \in S\) such that \(\{\xi\}\) is locally closed. This corresponds to a prime \(\mathfrak p \subset R\) such that (1) there exists a prime \(\mathfrak q\), \(\mathfrak p \subset \mathfrak q \subset \mathfrak m\) with both inclusions strict, and (2) \(\{\mathfrak p\}\) is open in \(\Spec(R/\mathfrak p)\). This is impossible by Algebra, Lemma 02IG.

Normal schemes

Recall that a ring \(R\) is said to be normal if all its local rings are normal domains, see Algebra, Definition 00GV. A normal domain is a domain which is integrally closed in its field of fractions, see Algebra, Definition 0309. Thus it makes sense to define a normal scheme as follows.

Definition

A scheme \(X\) is normal if and only if for all \(x \in X\) the local ring \(\mathcal{O}_{X, x}\) is a normal domain.

This seems to be the definition used in EGA, see [EGA, 0, 4.1.4]. Suppose \(X = \Spec(A)\), and \(A\) is reduced. Then saying that \(X\) is normal is not equivalent to saying that \(A\) is integrally closed in its total ring of fractions. However, if \(A\) is Noetherian then this is the case (see Algebra, Lemma 030C).

Lemma

Let \(X\) be a scheme. The following are equivalent:

The scheme \(X\) is normal.

For every affine open \(U \subset X\) the ring \(\mathcal{O}_X(U)\) is normal.

There exists an affine open covering \(X = \bigcup U_i\) such that each \(\mathcal{O}_X(U_i)\) is normal.

There exists an open covering \(X = \bigcup X_j\) such that each open subscheme \(X_j\) is normal.

Moreover, if \(X\) is normal then every open subscheme is normal.

Proof

This is clear from the definitions.

Lemma

A normal scheme is reduced.

Proof

Immediate from the definitions.

Lemma

Let \(X\) be an integral scheme. Then \(X\) is normal if and only if for every nonempty affine open \(U \subset X\) the ring \(\mathcal{O}_X(U)\) is a normal domain.

Proof

This follows from Algebra, Lemma 030B.

Lemma

Let \(X\) be a scheme such that any quasi-compact open has a finite number of irreducible components. The following are equivalent:

\(X\) is normal, and

\(X\) is a disjoint union of normal integral schemes.

Proof

It is immediate from the definitions that (2) implies (1). Let \(X\) be a normal scheme such that every quasi-compact open has a finite number of irreducible components. If \(X\) is affine then \(X\) satisfies (2) by Algebra, Lemma 030C. For a general \(X\), let \(X = \bigcup X_i\) be an affine open covering. Note that also each \(X_i\) has but a finite number of irreducible components, and the lemma holds for each \(X_i\). Let \(T \subset X\) be an irreducible component. By the affine case each intersection \(T \cap X_i\) is open in \(X_i\) and an integral normal scheme. Hence \(T \subset X\) is open, and an integral normal scheme. This proves that \(X\) is the disjoint union of its irreducible components, which are integral normal schemes.

Lemma

Let \(X\) be a Noetherian scheme. The following are equivalent:

\(X\) is normal, and

\(X\) is a finite disjoint union of normal integral schemes.

Proof

This is a special case of Lemma 0357 because a Noetherian scheme has a Noetherian underlying topological space (Lemma 01OZ and Topology, Lemma 0052).

Lemma

Let \(X\) be a locally Noetherian scheme. The following are equivalent:

\(X\) is normal, and

\(X\) is a disjoint union of integral normal schemes.

Proof

Omitted. Hint: This is purely topological from Lemma 033M.

Remark

Let \(X\) be a normal scheme. If \(X\) is locally Noetherian then we see that \(X\) is integral if and only if \(X\) is connected, see Lemma 033N. But there exists a connected affine scheme \(X\) such that \(\mathcal{O}_{X, x}\) is a domain for all \(x \in X\), but \(X\) is not irreducible, see Examples, Section 0568. This example is even a normal scheme (proof omitted), so beware!

Lemma

Let \(X\) be an integral normal scheme. Then \(\Gamma(X, \mathcal{O}_X)\) is a normal domain.

Proof

Set \(R = \Gamma(X, \mathcal{O}_X)\). It is clear that \(R\) is a domain. Suppose \(f = a/b\) is an element of its fraction field which is integral over \(R\). Say we have \(f^d + \sum_{i = 0, \ldots, d - 1} a_i f^i = 0\) with \(a_i \in R\). Let \(U \subset X\) be a nonempty affine open. Since \(b \in R\) is not zero and since \(X\) is integral we see that also \(b|_U \in \mathcal{O}_X(U)\) is not zero. Hence \(a/b\) is an element of the fraction field of \(\mathcal{O}_X(U)\) which is integral over \(\mathcal{O}_X(U)\) (because we can use the same polynomial \(f^d + \sum_{i = 0, \ldots, d - 1} a_i|_U f^i = 0\) on \(U\)). Since \(\mathcal{O}_X(U)\) is a normal domain (Lemma 033L), we see that \(f_U = (a|_U)/(b|_U) \in \mathcal{O}_X(U)\). It is clear that \(f_U|_V = f_V\) whenever \(V \subset U \subset X\) are nonempty affine open. Hence the local sections \(f_U\) glue to an element \(g \in R = \Gamma(X, \mathcal{O}_X)\). Then \(bg\) and \(a\) restrict to the same element of \(\mathcal{O}_X(U)\) for all \(U\) as above, hence \(bg = a\), in other words, \(g\) maps to \(f\) in the fraction field of \(R\).

Cohen-Macaulay schemes

Recall, see Algebra, Definition 00N8, that a local Noetherian ring \((R, \mathfrak m)\) is said to be Cohen-Macaulay if \(\text{depth}_{\mathfrak m}(R) = \dim(R)\). Recall that a Noetherian ring \(R\) is said to be Cohen-Macaulay if every local ring \(R_{\mathfrak p}\) of \(R\) is Cohen-Macaulay, see Algebra, Definition 00NC.

Definition

Let \(X\) be a scheme. We say \(X\) is Cohen-Macaulay if for every \(x \in X\) there exists an affine open neighbourhood \(U \subset X\) of \(x\) such that the ring \(\mathcal{O}_X(U)\) is Noetherian and Cohen-Macaulay.

Lemma

Let \(X\) be a scheme. The following are equivalent:

\(X\) is Cohen-Macaulay,

\(X\) is locally Noetherian and all of its local rings are Cohen-Macaulay, and

\(X\) is locally Noetherian and for any closed point \(x \in X\) the local ring \(\mathcal{O}_{X, x}\) is Cohen-Macaulay.

Proof

Algebra, Lemma 00NB says that the localization of a Cohen-Macaulay local ring is Cohen-Macaulay. The lemma follows by combining this with Lemma 01OW, with the existence of closed points on locally Noetherian schemes (Lemma 02IL), and the definitions.

Lemma

Let \(X\) be a scheme. The following are equivalent:

The scheme \(X\) is Cohen-Macaulay.

For every affine open \(U \subset X\) the ring \(\mathcal{O}_X(U)\) is Noetherian and Cohen-Macaulay.

There exists an affine open covering \(X = \bigcup U_i\) such that each \(\mathcal{O}_X(U_i)\) is Noetherian and Cohen-Macaulay.

There exists an open covering \(X = \bigcup X_j\) such that each open subscheme \(X_j\) is Cohen-Macaulay.

Moreover, if \(X\) is Cohen-Macaulay then every open subscheme is Cohen-Macaulay.

Proof

Combine Lemmas 01OW and 02IP.

More information on Cohen-Macaulay schemes and depth can be found in Cohomology of Schemes, Section 0340.

Regular schemes

Recall, see Algebra, Definition 00KU, that a local Noetherian ring \((R, \mathfrak m)\) is said to be regular if \(\mathfrak m\) can be generated by \(\dim(R)\) elements. Recall that a Noetherian ring \(R\) is said to be regular if every local ring \(R_{\mathfrak p}\) of \(R\) is regular, see Algebra, Definition 00OD.

Definition

Let \(X\) be a scheme. We say \(X\) is regular, or nonsingular if for every \(x \in X\) there exists an affine open neighbourhood \(U \subset X\) of \(x\) such that the ring \(\mathcal{O}_X(U)\) is Noetherian and regular.

Lemma

Let \(X\) be a scheme. The following are equivalent:

\(X\) is regular,

\(X\) is locally Noetherian and all of its local rings are regular, and

\(X\) is locally Noetherian and for any closed point \(x \in X\) the local ring \(\mathcal{O}_{X, x}\) is regular.

Proof

By the discussion in Algebra preceding Algebra, Definition 00OD we know that the localization of a regular local ring is regular. The lemma follows by combining this with Lemma 01OW, with the existence of closed points on locally Noetherian schemes (Lemma 02IL), and the definitions.

Lemma

Let \(X\) be a scheme. The following are equivalent:

The scheme \(X\) is regular.

For every affine open \(U \subset X\) the ring \(\mathcal{O}_X(U)\) is Noetherian and regular.

There exists an affine open covering \(X = \bigcup U_i\) such that each \(\mathcal{O}_X(U_i)\) is Noetherian and regular.

There exists an open covering \(X = \bigcup X_j\) such that each open subscheme \(X_j\) is regular.

Moreover, if \(X\) is regular then every open subscheme is regular.

Proof

Combine Lemmas 01OW and 02IT.

Lemma

A regular scheme is normal.

Proof

See Algebra, Lemma 0567.

Dimension

The dimension of a scheme is just the dimension of its underlying topological space.

Definition

Let \(X\) be a scheme.

The dimension of \(X\) is just the dimension of \(X\) as a topological spaces, see Topology, Definition 0055.

For \(x \in X\) we denote \(\dim_x(X)\) the dimension of the underlying topological space of \(X\) at \(x\) as in Topology, Definition 0055. We say \(\dim_x(X)\) is the dimension of \(X\) at \(x\).

As a scheme has a sober underlying topological space (Schemes, Lemma 01IS) we may compute the dimension of \(X\) as the supremum of the lengths \(n\) of chains \[T_0 \subset T_1 \subset \ldots \subset T_n\] of irreducible closed subsets of \(X\), or as the supremum of the lengths \(n\) of chains of specializations \[\xi_n \leadsto \xi_{n - 1} \leadsto \ldots \leadsto \xi_0\] of points of \(X\).

Lemma

Let \(X\) be a scheme. The following are equal

The dimension of \(X\).

The supremum of the dimensions of the local rings of \(X\).

The supremum of \(\dim_x(X)\) for \(x \in X\).

Proof

Note that given a chain of specializations \[\xi_n \leadsto \xi_{n - 1} \leadsto \ldots \leadsto \xi_0\] of points of \(X\) all of the points \(\xi_i\) correspond to prime ideals of the local ring of \(X\) at \(\xi_0\) by Schemes, Lemma 01J7. Hence we see that the dimension of \(X\) is the supremum of the dimensions of its local rings. In particular \(\dim_x(X) \geq \dim(\mathcal{O}_{X, x})\) as \(\dim_x(X)\) is the minimum of the dimensions of open neighbourhoods of \(x\). Thus \(\sup_{x \in X} \dim_x(X) \geq \dim(X)\). On the other hand, it is clear that \(\sup_{x \in X} \dim_x(X) \leq \dim(X)\) as \(\dim(U) \leq \dim(X)\) for any open subset of \(X\).

Lemma

Let \(X\) be a scheme. Let \(Y \subset X\) be an irreducible closed subset. Let \(\xi \in Y\) be the generic point. Then \[\text{codim}(Y, X) = \dim(\mathcal{O}_{X, \xi})\] where the codimension is as defined in Topology, Definition 02I3.

Proof

By Topology, Lemma 02I4 we may replace \(X\) by an affine open neighbourhood of \(\xi\). In this case the result follows easily from Algebra, Lemma 00ET.

Lemma

Let \(X\) be a scheme. Let \(x \in X\). Then \(x\) is a generic point of an irreducible component of \(X\) if and only if \(\dim(\mathcal{O}_{X, x}) = 0\).

Proof

This follows from Lemma 02IZ for example.

Lemma

A locally Noetherian scheme of dimension \(0\) is a disjoint union of spectra of Artinian local rings.

Proof

A Noetherian ring of dimension \(0\) is a finite product of Artinian local rings, see Algebra, Proposition 00KJ. Hence an affine open of a locally Noetherian scheme \(X\) of dimension \(0\) has discrete underlying topological space. This implies that the topology on \(X\) is discrete. The lemma follows easily from these remarks.

Lemma

Let \(X\) be a scheme of dimension zero. The following are equivalent

\(X\) is quasi-separated,

\(X\) is separated,

\(X\) is Hausdorff,

every affine open is closed.

In this case the connected components of \(X\) are points and every quasi-compact open of \(X\) is affine. In particular, if \(X\) is quasi-compact, then \(X\) is affine.

Proof

As the dimension of \(X\) is zero, we see that for any affine open \(U \subset X\) the space \(U\) is profinite and satisfies a bunch of other properties which we will use freely below, see Algebra, Lemma 04MG. We choose an affine open covering \(X = \bigcup U_i\).

If (4) holds, then \(U_i \cap U_j\) is a closed subset of \(U_i\), hence quasi-compact, hence \(X\) is quasi-separated, by Schemes, Lemma 01KO, hence (1) holds.

If (1) holds, then \(U_i \cap U_j\) is a quasi-compact open of \(U_i\) hence closed in \(U_i\). Then \(U_i \cap U_j \to U_i\) is an open immersion whose image is closed, hence it is a closed immersion. In particular \(U_i \cap U_j\) is affine and \(\mathcal{O}(U_i) \to \mathcal{O}_X(U_i \cap U_j)\) is surjective. Thus \(X\) is separated by Schemes, Lemma 01KP, hence (2) holds.

Assume (2) and let \(x, y \in X\). Say \(x \in U_i\). If \(y \in U_i\) too, then we can find disjoint open neighbourhoods of \(x\) and \(y\) because \(U_i\) is Hausdorff. Say \(y \not \in U_i\) and \(y \in U_j\). Then \(y \not \in U_i \cap U_j\) which is an affine open of \(U_j\) and hence closed in \(U_j\). Thus we can find an open neighbourhood of \(y\) not meeting \(U_i\) and we conclude that \(X\) is Hausdorff, hence (3) holds.

Assume (3). Let \(U \subset X\) be affine open. Then \(U\) is closed in \(X\) by Topology, Lemma 08YB. This proves (4) holds.

Assume \(X\) satisfies the equivalent conditions (1) – (4). We prove the final statements of the lemma. Say \(x, y \in X\) with \(x \not = y\). Since \(y\) does not specialize to \(x\) we can choose \(U \subset X\) affine open with \(x \in U\) and \(y \not \in U\). Then we see that \(X = U \amalg (X \setminus U)\) is a decomposistion into open and closed subsets which shows that \(x\) and \(y\) do not belong to the same connected component of \(X\). Next, assume \(U \subset X\) is a quasi-compact open. Write \(U = U_1 \cup \ldots \cup U_n\) as a union of affine opens. We will prove by induction on \(n\) that \(U\) is affine. This immediately reduces us to the case \(n = 2\). In this case we have \(U = (U_1 \setminus U_2) \amalg (U_1 \cap U_2) \amalg (U_2 \setminus U_1)\) and the arguments above show that each of the pieces is affine.

Lemma

Let \(x\) be a point of a locally Noetherian scheme \(X\). Then \(\dim_x(X) = 0\) if and only if \(x\) is an isolated point of \(X\).

Proof

If \(x\) is an isolated point, then \(\{x\}\) is an open subset of \(X\) (Topology, Definition 06RM) and hence \(\dim_x(X) = \dim(\{x\}) = 0\) by definition. Conversely, if \(\dim_x(X) = 0\), then there exists an open neighbourhood \(U \subset X\) of \(x\) such that \(\dim(U) = 0\) (Topology, Definition 0055). By Lemma 0AAX we see that the topology on \(U\) is discrete and hence \(x\) is an isolated point.

Catenary schemes

Recall that a topological space \(X\) is called catenary if for every pair of irreducible closed subsets \(T \subset T'\) there exist a maximal chain of irreducible closed subsets \[T = T_0 \subset T_1 \subset \ldots \subset T_e = T'\] and every such chain has the same length. See Topology, Definition 02I1.

Definition

Let \(S\) be a scheme. We say \(S\) is catenary if the underlying topological space of \(S\) is catenary.

Recall that a ring \(A\) is called catenary if for any pair of prime ideals \(\mathfrak p \subset \mathfrak q\) there exists a maximal chain of primes \[\mathfrak p = \mathfrak p_0 \subset \ldots \subset \mathfrak p_e = \mathfrak q\] and all of these have the same length. See Algebra, Definition 00NI.

Lemma

Let \(S\) be a scheme. The following are equivalent

\(S\) is catenary,

there exists an open covering of \(S\) all of whose members are catenary schemes,

for every affine open \(\Spec(R) = U \subset S\) the ring \(R\) is catenary, and

there exists an affine open covering \(S = \bigcup U_i\) such that each \(U_i\) is the spectrum of a catenary ring.

Moreover, in this case any locally closed subscheme of \(S\) is catenary as well.

Proof

Combine Topology, Lemma 02I2, and Algebra, Lemma 02IH.

Lemma

Let \(S\) be a locally Noetherian scheme. The following are equivalent:

\(S\) is catenary, and

locally in the Zariski topology there exists a dimension function on \(S\) (see Topology, Definition 02I9).

Proof

This follows from Topology, Lemmas 02I2, 02IA, and 02IC, Schemes, Lemma 01IS and finally Lemma 01OZ.

It turns out that a scheme is catenary if and only if its local rings are catenary.

Lemma

Let \(X\) be a scheme. The following are equivalent

\(X\) is catenary, and

for any \(x \in X\) the local ring \(\mathcal{O}_{X, x}\) is catenary.

Proof

Assume \(X\) is catenary. Let \(x \in X\). By Lemma 02IX we may replace \(X\) by an affine open neighbourhood of \(x\), and then \(\Gamma(X, \mathcal{O}_X)\) is a catenary ring. By Algebra, Lemma 00NJ any localization of a catenary ring is catenary. Whence \(\mathcal{O}_{X, x}\) is catenary.

Conversely assume all local rings of \(X\) are catenary. Let \(Y \subset Y'\) be an inclusion of irreducible closed subsets of \(X\). Let \(\xi \in Y\) be the generic point. Let \(\mathfrak p \subset \mathcal{O}_{X, \xi}\) be the prime corresponding to the generic point of \(Y'\), see Schemes, Lemma 01J7. By that same lemma the irreducible closed subsets of \(X\) in between \(Y\) and \(Y'\) correspond to primes \(\mathfrak q \subset \mathcal{O}_{X, \xi}\) with \(\mathfrak p \subset \mathfrak q \subset \mathfrak m_{\xi}\). Hence we see all maximal chains of these are finite and have the same length as \(\mathcal{O}_{X, \xi}\) is a catenary ring.

Serre’s conditions

Here are two technical notions that are often useful. See also Cohomology of Schemes, Section 0340.

Definition

Let \(X\) be a locally Noetherian scheme. Let \(k \geq 0\).

We say \(X\) is regular in codimension \(k\), or we say \(X\) has property \((R_k)\) if for every \(x \in X\) we have \[\dim(\mathcal{O}_{X, x}) \leq k \Rightarrow \mathcal{O}_{X, x}\text{ is regular}\]

We say \(X\) has property \((S_k)\) if for every \(x \in X\) we have \(\text{depth}(\mathcal{O}_{X, x}) \geq \min(k, \dim(\mathcal{O}_{X, x}))\).

The phrase “regular in codimension \(k\)” makes sense since we have seen in Section 02IV that if \(Y \subset X\) is irreducible closed with generic point \(x\), then \(\dim(\mathcal{O}_{X, x}) = \text{codim}(Y, X)\). For example condition \((R_0)\) means that for every generic point \(\eta \in X\) of an irreducible component of \(X\) the local ring \(\mathcal{O}_{X, \eta}\) is a field. But for general Noetherian schemes it can happen that the regular locus of \(X\) is badly behaved, so care has to be taken.

Lemma

Let \(X\) be a locally Noetherian scheme. Then \(X\) is regular if and only if \(X\) has \((R_k)\) for all \(k \geq 0\).

Proof

Follows from Lemma 02IT and the definitions.

Lemma

Let \(X\) be a locally Noetherian scheme. Then \(X\) is Cohen-Macaulay if and only if \(X\) has \((S_k)\) for all \(k \geq 0\).

Proof

By Lemma 02IP we reduce to looking at local rings. Hence the lemma is true because a Noetherian local ring is Cohen-Macaulay if and only if it has depth equal to its dimension.

Lemma

Let \(X\) be a locally Noetherian scheme. Then \(X\) is reduced if and only if \(X\) has properties \((S_1)\) and \((R_0)\).

Proof

This is Algebra, Lemma 031R.

Lemma

Let \(X\) be a locally Noetherian scheme. Then \(X\) is normal if and only if \(X\) has properties \((S_2)\) and \((R_1)\).

Proof

This is Algebra, Lemma 031S.

Lemma

Let \(X\) be a locally Noetherian scheme which is normal and has dimension \(\leq 1\). Then \(X\) is regular.

Proof

This follows from Lemma 0345 and the definitions.

Lemma

Let \(X\) be a locally Noetherian scheme which is normal and has dimension \(\leq 2\). Then \(X\) is Cohen-Macaulay.

Proof

This follows from Lemma 0345 and the definitions.

Japanese and Nagata schemes

The notions considered in this section are not prominently defined in EGA. A “universally Japanese scheme” is mentioned and defined in [EGA, IV Corollary 5.11.4]. A “Japanese scheme” is mentioned in [EGA, IV Remark 10.4.14 (ii)] but no definition is given. A Nagata scheme (as given below) occurs in a few places in the literature (see for example [Liu, Definition 8.2.30] and [Greco, Page 142]).

We briefly recall that a domain \(R\) is called Japanese if the integral closure of \(R\) in any finite extension of its fraction field is finite over \(R\). A ring \(R\) is called universally Japanese if for any finite type ring map \(R \to S\) with \(S\) a domain \(S\) is Japanese. A ring \(R\) is called Nagata if it is Noetherian and \(R/\mathfrak p\) is Japanese for every prime \(\mathfrak p\) of \(R\).

Definition

Let \(X\) be a scheme.

Assume \(X\) integral. We say \(X\) is Japanese if for every \(x \in X\) there exists an affine open neighbourhood \(x \in U \subset X\) such that the ring \(\mathcal{O}_X(U)\) is Japanese (see Algebra, Definition 032F).

We say \(X\) is universally Japanese if for every \(x \in X\) there exists an affine open neighbourhood \(x \in U \subset X\) such that the ring \(\mathcal{O}_X(U)\) is universally Japanese (see Algebra, Definition 032R).

We say \(X\) is Nagata if for every \(x \in X\) there exists an affine open neighbourhood \(x \in U \subset X\) such that the ring \(\mathcal{O}_X(U)\) is Nagata (see Algebra, Definition 032R).

Being Nagata is the same thing as being locally Noetherian and universally Japanese, see Lemma 033Z.

Remark

In [Hoobler-finite] a (locally Noetherian) scheme \(X\) is called Japanese if for every \(x \in X\) and every associated prime \(\mathfrak p\) of \(\mathcal{O}_{X, x}\) the ring \(\mathcal{O}_{X, x}/\mathfrak p\) is Japanese. We do not use this definition since there exists a one dimensional Noetherian domain with excellent (in particular Japanese) local rings whose normalization is not finite. See [Hochster-loci, Example 1] or [Heinzer-Levy] or [Traveaux, Exposé XIX]. On the other hand, we could circumvent this problem by calling a scheme \(X\) Japanese if for every affine open \(\Spec(A) \subset X\) the ring \(A/\mathfrak p\) is Japanese for every associated prime \(\mathfrak p\) of \(A\).

Lemma

A Nagata scheme is locally Noetherian.

Proof

This is true because a Nagata ring is Noetherian by definition.

Lemma

Let \(X\) be an integral scheme. The following are equivalent:

The scheme \(X\) is Japanese.

For every affine open \(U \subset X\) the domain \(\mathcal{O}_X(U)\) is Japanese.

There exists an affine open covering \(X = \bigcup U_i\) such that each \(\mathcal{O}_X(U_i)\) is Japanese.

There exists an open covering \(X = \bigcup X_j\) such that each open subscheme \(X_j\) is Japanese.

Moreover, if \(X\) is Japanese then every open subscheme is Japanese.

Proof

This follows from Lemma 01OR and Algebra, Lemmas 032G and 032H.

Lemma

Let \(X\) be a scheme. The following are equivalent:

The scheme \(X\) is universally Japanese.

For every affine open \(U \subset X\) the ring \(\mathcal{O}_X(U)\) is universally Japanese.

There exists an affine open covering \(X = \bigcup U_i\) such that each \(\mathcal{O}_X(U_i)\) is universally Japanese.

There exists an open covering \(X = \bigcup X_j\) such that each open subscheme \(X_j\) is universally Japanese.

Moreover, if \(X\) is universally Japanese then every open subscheme is universally Japanese.

Proof

This follows from Lemma 01OR and Algebra, Lemmas 032S and 032V.

Lemma

Let \(X\) be a scheme. The following are equivalent:

The scheme \(X\) is Nagata.

For every affine open \(U \subset X\) the ring \(\mathcal{O}_X(U)\) is Nagata.

There exists an affine open covering \(X = \bigcup U_i\) such that each \(\mathcal{O}_X(U_i)\) is Nagata.

There exists an open covering \(X = \bigcup X_j\) such that each open subscheme \(X_j\) is Nagata.

Moreover, if \(X\) is Nagata then every open subscheme is Nagata.

Proof

This follows from Lemma 01OR and Algebra, Lemmas 032U and 032V.

Lemma

Let \(X\) be a locally Noetherian scheme. Then \(X\) is Nagata if and only if every integral closed subscheme \(Z \subset X\) is Japanese.

Proof

Assume \(X\) is Nagata. Let \(Z \subset X\) be an integral closed subscheme. Let \(z \in Z\). Let \(\Spec(A) = U \subset X\) be an affine open containing \(z\) such that \(A\) is Nagata. Then \(Z \cap U \cong \Spec(A/\mathfrak p)\) for some prime \(\mathfrak p\), see Schemes, Lemma 01IN (and Definition 01OK). By Algebra, Definition 032R we see that \(A/\mathfrak p\) is Japanese. Hence \(Z\) is Japanese by definition.

Assume every integral closed subscheme of \(X\) is Japanese. Let \(\Spec(A) = U \subset X\) be any affine open. As \(X\) is locally Noetherian we see that \(A\) is Noetherian (Lemma 01OW). Let \(\mathfrak p \subset A\) be a prime ideal. We have to show that \(A/\mathfrak p\) is Japanese. Let \(T \subset U\) be the closed subset \(V(\mathfrak p) \subset \Spec(A)\). Let \(\overline{T} \subset X\) be the closure. Then \(\overline{T}\) is irreducible as the closure of an irreducible subset. Hence the reduced closed subscheme defined by \(\overline{T}\) is an integral closed subscheme (called \(\overline{T}\) again), see Schemes, Lemma 01J3. In other words, \(\Spec(A/\mathfrak p)\) is an affine open of an integral closed subscheme of \(X\). This subscheme is Japanese by assumption and by Lemma 033V we see that \(A/\mathfrak p\) is Japanese.

Lemma

Let \(X\) be a scheme. The following are equivalent:

\(X\) is Nagata, and

\(X\) is locally Noetherian and universally Japanese.

Proof

This is Algebra, Proposition 0334.

This discussion will be continued in Morphisms, Section 0359.

G-schemes

We remark that a ring \(R\) is a G-ring if \(R\) is Noetherian and for every prime ideal \(\mathfrak p\) of \(R\) the ring map \(R_\mathfrak p \to (R_\mathfrak p)^\wedge\) is regular.

Definition

Let \(X\) be a scheme. We say \(X\) is a G-scheme² if for every \(x \in X\) there exists an affine open neighbourhood \(x \in U \subset X\) such that the ring \(\mathcal{O}_X(U)\) is a G-ring (see More on Algebra, Definition 07GH).

Lemma

Let \(X\) be a scheme. The following are equivalent:

\(X\) is a G-scheme,

\(X\) is locally Noetherian and for all \(x \in X\) the ring map \(\mathcal{O}_{X, x} \to \mathcal{O}_{X, x}^\wedge\) is regular, and

\(X\) is locally Noetherian and for any closed point \(x \in X\) the ring map \(\mathcal{O}_{X, x} \to \mathcal{O}_{X, x}^\wedge\) is regular.

Proof

It is clear that (1) implies (2) and (2) implies (3). Assume (3). Let \(x \in X\) be any point. By Lemma 02IL there exists a specialization \(x \leadsto x'\) with \(x'\) closed in \(X\). Then \(\mathcal{O}_{X, x'}\) is a G-ring by More on Algebra, Lemma 07PT. Since \(\mathcal{O}_{X, x}\) is the localization of \(\mathcal{O}_{X, x'}\) at a prime ideal (Schemes, Lemma 01J7), we see that \(\mathcal{O}_{X, x} \to \mathcal{O}_{X, x}^\wedge\) is a regular ring map. Since \(x \in X\) was arbitrary we conclude that for any affine open \(U \subset X\) the Noetherian (Lemma 01OW) ring \(\mathcal{O}_X(U)\) is a G-ring. This proves that (1) holds.

Lemma

Let \(X\) be a scheme. The following are equivalent:

The scheme \(X\) is a G-scheme.

For every affine open \(U \subset X\) the ring \(\mathcal{O}_X(U)\) is a G-ring.

There exists an affine open covering \(X = \bigcup U_i\) such that each \(\mathcal{O}_X(U_i)\) is a G-ring.

There exists an open covering \(X = \bigcup X_j\) such that each open subscheme \(X_j\) is a G-scheme.

Moreover, if \(X\) is a G-scheme then every open subscheme is a G-scheme.

Proof

Combine Lemmas 01OW and 0HA4.

The singular locus

Here is the definition.

Definition

Let \(X\) be a locally Noetherian scheme. The regular locus \(\text{Reg}(X)\) of \(X\) is the set of \(x \in X\) such that \(\mathcal{O}_{X, x}\) is a regular local ring. The singular locus \(\text{Sing}(X)\) is the complement \(X \setminus \text{Reg}(X)\), i.e., the set of points \(x \in X\) such that \(\mathcal{O}_{X, x}\) is not a regular local ring.

The regular locus of a locally Noetherian scheme is stable under generalizations, see the discussion preceding Algebra, Definition 00OD. However, for general locally Noetherian schemes the regular locus need not be open. In More on Algebra, Section 07P6 the reader can find some criteria for when this is the case. We will discuss this further in Morphisms, Section 07R2.

Local irreducibility

Recall that in More on Algebra, Section 06DT we introduced the notion of a (geometrically) unibranch local ring.

Definition

Let \(X\) be a scheme. Let \(x \in X\). We say \(X\) is unibranch at \(x\) if the local ring \(\mathcal{O}_{X, x}\) is unibranch. We say \(X\) is geometrically unibranch at \(x\) if the local ring \(\mathcal{O}_{X, x}\) is geometrically unibranch. We say \(X\) is unibranch if \(X\) is unibranch at all of its points. We say \(X\) is geometrically unibranch if \(X\) is geometrically unibranch at all of its points.

To be sure, it can happen that a local ring \(A\) is geometrically unibranch (in the sense of More on Algebra, Definition 0BPZ) but the scheme \(\Spec(A)\) is not geometrically unibranch in the sense of Definition 0BQ2. For example this happens if \(A\) is the local ring at the vertex of the cone over an irreducible plane curve which has ordinary double point singularity (a node).

Lemma

A normal scheme is geometrically unibranch.

Proof

This follows from the definitions. Namely, a scheme is normal if the local rings are normal domains. It is immediate from the More on Algebra, Definition 0BPZ that a local normal domain is geometrically unibranch.

Lemma

Let \(X\) be a Noetherian scheme. The following are equivalent

\(X\) is geometrically unibranch (Definition 0BQ2),

for every point \(x \in X\) which is not the generic point of an irreducible component of \(X\), the punctured spectrum of the strict henselization \(\mathcal{O}_{X, x}^{sh}\) is connected.

Proof

More on Algebra, Lemma 06DM shows that (1) implies that the punctured spectra in (2) are irreducible and in particular connected.

Assume (2). Let \(x \in X\). We have to show that \(\mathcal{O}_{X, x}\) is geometrically unibranch. By induction on \(\dim(\mathcal{O}_{X, x})\) we may assume that the result holds for every nontrivial generalization of \(x\). We may replace \(X\) by \(\Spec(\mathcal{O}_{X, x})\). In other words, we may assume that \(X = \Spec(A)\) with \(A\) local and that \(A_\mathfrak p\) is geometrically unibranch for each nonmaximal prime \(\mathfrak p \subset A\).

Let \(A^{sh}\) be the strict henselization of \(A\). If \(\mathfrak q \subset A^{sh}\) is a prime lying over \(\mathfrak p \subset A\), then \(A_\mathfrak p \to A^{sh}_\mathfrak q\) is a filtered colimit of étale algebras. Hence the strict henselizations of \(A_\mathfrak p\) and \(A^{sh}_\mathfrak q\) are isomorphic. Thus by More on Algebra, Lemma 06DM we conclude that \(A^{sh}_\mathfrak q\) has a unique minimal prime ideal for every nonmaximal prime \(\mathfrak q\) of \(A^{sh}\).

Let \(\mathfrak q_1, \ldots, \mathfrak q_r\) be the minimal primes of \(A^{sh}\). We have to show that \(r = 1\). By the above we see that \(V(\mathfrak q_1) \cap V(\mathfrak q_j) = \{\mathfrak m^{sh}\}\) for \(j = 2, \ldots, r\). Hence \(V(\mathfrak q_1) \setminus \{\mathfrak m^{sh}\}\) is an open and closed subset of the punctured spectrum of \(A^{sh}\) which is a contradiction with the assumption that this punctured spectrum is connected unless \(r = 1\).

Definition

Let \(X\) be a scheme. Let \(x \in X\). The number of branches of \(X\) at \(x\) is the number of branches of the local ring \(\mathcal{O}_{X, x}\) as defined in More on Algebra, Definition 0C26. The number of geometric branches of \(X\) at \(x\) is the number of geometric branches of the local ring \(\mathcal{O}_{X, x}\) as defined in More on Algebra, Definition 0C26.

Often we want to compare this with the branches of the complete local ring, but the comparison is not straightforward in general; some information on this topic can be found in More on Algebra, Section 0C27.

Lemma

Let \(X\) be a scheme and \(x \in X\). Let \(X_i\), \(i \in I\) be the irreducible components of \(X\) passing through \(x\). Then the number of (geometric) branches of \(X\) at \(x\) is the sum over \(i \in I\) of the number of (geometric) branches of \(X_i\) at \(x\).

Proof

We view the \(X_i\) as integral closed subschemes of \(X\), see Schemes, Definition 01J4 and Lemma 01ON. Observe that the number of (geometric) branches of \(X_i\) at \(x\) is at least \(1\) for all \(i\) (essentially by definition). Recall that the \(X_i\) correspond \(1\)-to-\(1\) with the minimal prime ideals \(\mathfrak p_i \subset \mathcal{O}_{X, x}\), see Algebra, Lemma 00ET. Thus, if \(I\) is infinite, then \(\mathcal{O}_{X, x}\) has infinitely many minimal primes, whence both \(\mathcal{O}_{X, x}^h\) and \(\mathcal{O}_{X, x}^{sh}\) have infinitely many minimal primes (combine Algebra, Lemmas 00FK and 0CAN and the injectivity of the maps \(\mathcal{O}_{X, x} \to \mathcal{O}_{X, x}^h \to \mathcal{O}_{X, x}^{sh}\)). In this case the number of (geometric) branches of \(X\) at \(x\) is defined to be \(\infty\) which is also true for the sum. Thus we may assume \(I\) is finite. Let \(A'\) be the integral closure of \(\mathcal{O}_{X, x}\) in the total ring of fractions \(Q\) of \((\mathcal{O}_{X, x})_{red}\). Let \(A'_i\) be the integral closure of \(\mathcal{O}_{X, x}/\mathfrak p_i\) in the total ring of fractions \(Q_i\) of \(\mathcal{O}_{X, x}/\mathfrak p_i\). By Algebra, Lemma 02LX we have \(Q = \prod_{i \in I} Q_i\). Thus \(A' = \prod A'_i\). Then the equality of the lemma follows from More on Algebra, Lemma 0C37 which expresses the number of (geometric) branches in terms of the maximal ideals of \(A'\).

Lemma

Let \(X\) be a scheme. Let \(x \in X\).

The number of branches of \(X\) at \(x\) is \(1\) if and only if \(X\) is unibranch at \(x\).

The number of geometric branches of \(X\) at \(x\) is \(1\) if and only if \(X\) is geometrically unibranch at \(x\).

Proof

This lemma follows immediately from the definitions and the corresponding result for rings, see More on Algebra, Lemma 0C37.

Characterizing modules of finite type and finite presentation

Let \(X\) be a scheme. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. The following lemma implies that \(\mathcal{F}\) is of finite type (see Modules, Definition 01B5) if and only if \(\mathcal{F}\) is on each open affine \(\Spec(A) = U \subset X\) of the form \(\widetilde M\) for some finite \(A\)-module \(M\). Similarly, \(\mathcal{F}\) is of finite presentation (see Modules, Definition 01BN) if and only if \(\mathcal{F}\) is on each open affine \(\Spec(A) = U \subset X\) of the form \(\widetilde M\) for some finitely presented \(A\)-module \(M\).

Lemma

Let \(X = \Spec(R)\) be an affine scheme. The quasi-coherent sheaf of \(\mathcal{O}_X\)-modules \(\widetilde M\) is a finite type \(\mathcal{O}_X\)-module if and only if \(M\) is a finite \(R\)-module.

Proof

Assume \(\widetilde M\) is a finite type \(\mathcal{O}_X\)-module. This means there exists an open covering of \(X\) such that \(\widetilde M\) restricted to the members of this covering is globally generated by finitely many sections. Thus there also exists a standard open covering \(X = \bigcup_{i = 1, \ldots, n} D(f_i)\) such that \(\widetilde M|_{D(f_i)}\) is generated by finitely many sections. Thus \(M_{f_i}\) is finitely generated for each \(i\). Hence we conclude by Algebra, Lemma 00EO.

Lemma

Let \(X = \Spec(R)\) be an affine scheme. The quasi-coherent sheaf of \(\mathcal{O}_X\)-modules \(\widetilde M\) is an \(\mathcal{O}_X\)-module of finite presentation if and only if \(M\) is an \(R\)-module of finite presentation.

Proof

Assume \(\widetilde M\) is an \(\mathcal{O}_X\)-module of finite presentation. By Lemma 01PB we see that \(M\) is a finite \(R\)-module. Choose a surjection \(R^n \to M\) with kernel \(K\). By Schemes, Lemma 01HV there is a short exact sequence \[0 \to \widetilde{K} \to \bigoplus \mathcal{O}_X^{\oplus n} \to \widetilde{M} \to 0\] By Modules, Lemma 01BP we see that \(\widetilde{K}\) is a finite type \(\mathcal{O}_X\)-module. Hence by Lemma 01PB again we see that \(K\) is a finite \(R\)-module. Hence \(M\) is an \(R\)-module of finite presentation.

Sections over principal opens

Here is a typical result of this kind. We will use a more naive but more direct method of proof in later lemmas.

Lemma

Let \(X\) be a scheme. Let \(f \in \Gamma(X, \mathcal{O}_X)\). Denote \(X_f \subset X\) the open where \(f\) is invertible, see Schemes, Lemma 01HZ. If \(X\) is quasi-compact and quasi-separated, the canonical map \[\Gamma(X, \mathcal{O}_X)_f \longrightarrow \Gamma(X_f, \mathcal{O}_X)\] is an isomorphism. Moreover, if \(\mathcal{F}\) is a quasi-coherent sheaf of \(\mathcal{O}_X\)-modules the map \[\Gamma(X, \mathcal{F})_f \longrightarrow \Gamma(X_f, \mathcal{F})\] is an isomorphism.

Proof

Write \(R = \Gamma(X, \mathcal{O}_X)\). Consider the canonical morphism \[\varphi : X \longrightarrow \Spec(R)\] of schemes, see Schemes, Lemma 01I1. Then the inverse image of the standard open \(D(f)\) on the right hand side is \(X_f\) on the left hand side. Moreover, since \(X\) is assumed quasi-compact and quasi-separated the morphism \(\varphi\) is quasi-compact and quasi-separated, see Schemes, Lemma 01K4 and 01KV. Hence by Schemes, Lemma 01LC we see that \(\varphi_*\mathcal{F}\) is quasi-coherent. Hence we see that \(\varphi_*\mathcal{F} = \widetilde M\) with \(M = \Gamma(X, \mathcal{F})\) as an \(R\)-module. Thus we see that \[\Gamma(X_f, \mathcal{F}) = \Gamma(D(f), \varphi_*\mathcal{F}) = \Gamma(D(f), \widetilde M) = M_f\] which is exactly the content of the lemma. The first displayed isomorphism of the lemma follows by taking \(\mathcal{F} = \mathcal{O}_X\).

Recall that given a scheme \(X\), an invertible sheaf \(\mathcal{L}\) on \(X\), and a sheaf of \(\mathcal{O}_X\)-modules \(\mathcal{F}\) we get a graded ring \(\Gamma_*(X, \mathcal{L}) = \bigoplus\nolimits_{n \geq 0} \Gamma(X, \mathcal{L}^{\otimes n})\) and a graded \(\Gamma_*(X, \mathcal{L})\)-module \(\Gamma_*(X, \mathcal{L}, \mathcal{F}) = \bigoplus\nolimits_{n \in \mathbf{Z}} \Gamma(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n})\) see Modules, Definition 01CV. If we have moreover a section \(s \in \Gamma(X, \mathcal{L})\), then we obtain a map [0B5L]\[\begin{equation} \Gamma_*(X, \mathcal{L}, \mathcal{F})_{(s)} \longrightarrow \Gamma(X_s, \mathcal{F}|_{X_s}) \end{equation}\] which sends \(t/s^n\) where \(t \in \Gamma(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n})\) to \(t|_{X_s} \otimes s|_{X_s}^{-n}\). This makes sense because \(X_s \subset X\) is by definition the open over which \(s\) has an inverse, see Modules, Lemma 01CY.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{L}\) be an invertible sheaf on \(X\). Let \(s \in \Gamma(X, \mathcal{L})\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module.

If \(X\) is quasi-compact, then (0B5L) is injective, and

if \(X\) is quasi-compact and quasi-separated, then (0B5L) is an isomorphism.

In particular, the canonical map \[\Gamma_*(X, \mathcal{L})_{(s)} \longrightarrow \Gamma(X_s, \mathcal{O}_X),\quad a/s^n \longmapsto a \otimes s^{-n}\] is an isomorphism if \(X\) is quasi-compact and quasi-separated.

Proof

Assume \(X\) is quasi-compact. Choose a finite affine open covering \(X = U_1 \cup \ldots \cup U_m\) with \(U_j\) affine and \(\mathcal{L}|_{U_j} \cong \mathcal{O}_{U_j}\). Via this isomorphism, the image \(s|_{U_j}\) corresponds to some \(f_j \in \Gamma(U_j, \mathcal{O}_{U_j})\). Then \(X_s \cap U_j = D(f_j)\).

Proof of (1). Let \(t/s^n\) be an element in the kernel of (0B5L). Then \(t|_{X_s} = 0\). Hence \((t|_{U_j})|_{D(f_j)} = 0\). By Lemma 01P7 we conclude that \(f_j^{e_j} t|_{U_j} = 0\) for some \(e_j \geq 0\). Let \(e = \max(e_j)\). Then we see that \(t \otimes s^e\) restricts to zero on \(U_j\) for all \(j\), hence is zero. Since \(t/s^n\) is equal to \(t \otimes s^e/s^{n + e}\) in \(\Gamma_*(X, \mathcal{L}, \mathcal{F})_{(s)}\) we conclude that \(t/s^n = 0\) as desired.

Proof of (2). Assume \(X\) is quasi-compact and quasi-separated. Then \(U_j \cap U_{j'}\) is quasi-compact for all pairs \(j, j'\), see Schemes, Lemma 01KO. By part (1) we know (0B5L) is injective. Let \(t' \in \Gamma(X_s, \mathcal{F}|_{X_s})\). For every \(j\), there exist an integer \(e_j \geq 0\) and \(t'_j \in \Gamma(U_j, \mathcal{F}|_{U_j})\) such that \(t'|_{D(f_j)}\) corresponds to \(t'_j/f_j^{e_j}\) via the isomorphism of Lemma 01P7. Set \(e = \max(e_j)\) and \[t_j = f_j^{e - e_j} t'_j \otimes q_j^e \in \Gamma(U_j, (\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes e})|_{U_j})\] where \(q_j \in \Gamma(U_j, \mathcal{L}|_{U_j})\) is the trivializing section coming from the isomorphism \(\mathcal{L}|_{U_j} \cong \mathcal{O}_{U_j}\). In particular we have \(s|_{U_j} = f_j q_j\). Using this a calculation shows that \(t_j|_{U_j \cap U_{j'}}\) and \(t_{j'}|_{U_j \cap U_{j'}}\) map to the same section of \(\mathcal{F}\) over \(U_j \cap U_{j'} \cap X_s\). By quasi-compactness of \(U_j \cap U_{j'}\) and part (1) there exists an integer \(e' \geq 0\) such that \[t_j|_{U_j \cap U_{j'}} \otimes s^{e'}|_{U_j \cap U_{j'}} = t_{j'}|_{U_j \cap U_{j'}} \otimes s^{e'}|_{U_j \cap U_{j'}}\] as sections of \(\mathcal{F} \otimes \mathcal{L}^{\otimes e + e'}\) over \(U_j \cap U_{j'}\). We may choose the same \(e'\) to work for all pairs \(j, j'\). Then the sheaf conditions implies there is a section \(t \in \Gamma(X, \mathcal{F} \otimes \mathcal{L}^{\otimes e + e'})\) whose restriction to \(U_j\) is \(t_j \otimes s^{e'}|_{U_j}\). A simple computation shows that \(t/s^{e + e'}\) maps to \(t'\) as desired.

Let \(X\) be a scheme. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Let \(\mathcal{F}\) and \(\mathcal{G}\) be quasi-coherent \(\mathcal{O}_X\)-modules. Consider the graded \(\Gamma_*(X, \mathcal{L})\)-module \[M = \bigoplus\nolimits_{n \in \mathbf{Z}} \Hom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n})\] Next, let \(s \in \Gamma(X, \mathcal{L})\) be a section. Then there is a canonical map [0B5M]\[\begin{equation} M_{(s)} \longrightarrow \Hom_{\mathcal{O}_{X_s}}(\mathcal{F}|_{X_s}, \mathcal{G}|_{X_s}) \end{equation}\] which sends \(\alpha/s^n\) to the map \(\alpha|_{X_s} \otimes s|_{X_s}^{-n}\). The following lemma, combined with Lemma 01PI, says roughly that, if \(X\) is quasi-compact and quasi-separated, the category of finitely presented \(\mathcal{O}_{X_s}\)-modules is the category of finitely presented \(\mathcal{O}_X\)-modules with the multiplicative system of maps \(s^n: \mathcal{F} \to \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}\) inverted.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Let \(s \in \Gamma(X, \mathcal{L})\) be a section. Let \(\mathcal{F}\), \(\mathcal{G}\) be quasi-coherent \(\mathcal{O}_X\)-modules.

If \(X\) is quasi-compact and \(\mathcal{F}\) is of finite type, then (0B5M) is injective, and

if \(X\) is quasi-compact and quasi-separated and \(\mathcal{F}\) is of finite presentation, then (0B5M) is bijective.

Proof

We first prove the lemma in case \(X = \Spec(A)\) is affine and \(\mathcal{L} = \mathcal{O}_X\). In this case \(s\) corresponds to an element \(f \in A\). Say \(\mathcal{F} = \widetilde{M}\) and \(\mathcal{G} = \widetilde{N}\) for some \(A\)-modules \(M\) and \(N\). Then the lemma translates (via Lemmas 01PB and 01PC) into the following algebra statements

If \(M\) is a finite \(A\)-module and \(\varphi : M \to N\) is an \(A\)-module map such that the induced map \(M_f \to N_f\) is zero, then \(f^n\varphi = 0\) for some \(n\).

If \(M\) is a finitely presented \(A\)-module, then \(\Hom_A(M, N)_f = \Hom_{A_f}(M_f, N_f)\).

The second statement is Algebra, Lemma 0583 and we omit the proof of the first statement.

Next, we prove (1) for general \(X\). Assume \(X\) is quasi-compact and hoose a finite affine open covering \(X = U_1 \cup \ldots \cup U_m\) with \(U_j\) affine and \(\mathcal{L}|_{U_j} \cong \mathcal{O}_{U_j}\). Via this isomorphism, the image \(s|_{U_j}\) corresponds to some \(f_j \in \Gamma(U_j, \mathcal{O}_{U_j})\). Then \(X_s \cap U_j = D(f_j)\). Let \(\alpha/s^n\) be an element in the kernel of (0B5M). Then \(\alpha|_{X_s} = 0\). Hence \((\alpha|_{U_j})|_{D(f_j)} = 0\). By the affine case treated above we conclude that \(f_j^{e_j} \alpha|_{U_j} = 0\) for some \(e_j \geq 0\). Let \(e = \max(e_j)\). Then we see that \(\alpha \otimes s^e\) restricts to zero on \(U_j\) for all \(j\), hence is zero. Since \(\alpha/s^n\) is equal to \(\alpha \otimes s^e/s^{n + e}\) in \(M_{(s)}\) we conclude that \(\alpha/s^n = 0\) as desired.

Proof of (2). Since \(\mathcal{F}\) is of finite presentation, the sheaf \(\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})\) is quasi-coherent, see Schemes, Section 01LA. Moreover, it is clear that \[\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) = \SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G}) \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}\] for all \(n\). Hence in this case the statement follows from Lemma 01PW applied to \(\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})\).

Quasi-affine schemes

Definition

A scheme \(X\) is called quasi-affine if it is quasi-compact and isomorphic to an open subscheme of an affine scheme.

Lemma

Let \(A\) be a ring and let \(U \subset \Spec(A)\) be a quasi-compact open subscheme. For \(\mathcal{F}\) quasi-coherent on \(U\) the canonical map \[\widetilde{H^0(U, \mathcal{F})}|_U \to \mathcal{F}\] is an isomorphism.

Proof

Denote \(j : U \to \Spec(A)\) the inclusion morphism. Then \(H^0(U, \mathcal{F}) = H^0(\Spec(A), j_*\mathcal{F})\) and \(j_*\mathcal{F}\) is quasi-coherent by Schemes, Lemma 01LC. Hence \(j_*\mathcal{F} = \widetilde{H^0(U, \mathcal{F})}\) by Schemes, Lemma 01IB. Restricting back to \(U\) we get the lemma.

Lemma

Let \(A\) be a ring and let \(U \subset \Spec(A)\) be a quasi-compact open subscheme. Let \(\mathcal{F}\) be an \(\mathcal{O}_U\)-module and set \(M = \Gamma(U, \mathcal{F})\). The following are equivalent:

\(\mathcal{F}\) is quasi-coherent,

there are \(f_1, \ldots, f_n \in A\) such that \(U = \bigcup D(f_i)\) and \(\mathcal{F}|_{D(f_i)} \cong \widetilde{M_i}\) for some \(A_{f_i}\)-module \(M_i\),

there is an \(A\)-module \(N\) such that \(\mathcal{F} \cong \widetilde N|_U\), and

for every \(f \in A\) such that \(D(f) \subset U\) the canonical map \[M_f \longrightarrow \Gamma(D(f), \mathcal{F})\] is an isomorphism.

If these conditions hold, then in (3) one can take \(N = M\), and the resulting isomorphism is canonical.

Proof

The standard opens contained in \(U\) form a basis for its topology and, as \(U\) is quasi-compact, a finite subcollection covers \(U\). Thus (3) implies (2) by Schemes, Lemma 01I9, and (2) implies (1) because being quasi-coherent is local. Lemma 0EHM proves that (1) implies (3), with \(N = M\) and with the canonical isomorphism asserted above. The same lemma, followed by taking sections over \(D(f)\), proves that (1) implies (4).

Assume (4), and write \(j : U \to \Spec(A)\) for the inclusion. By Schemes, Lemma 01I7, the identity map of \(M\) gives a map \[\widetilde M \longrightarrow j_*\mathcal{F}.\] After restriction to \(U\), this gives a canonical map \(\widetilde M|_U \to \mathcal{F}\). On a standard open \(D(f) \subset U\), the induced map on sections is the map in (4). Consequently this map is an isomorphism on a basis for the topology of \(U\), and hence it is an isomorphism. This proves (3) and the final assertion.

Lemma

Let \(X\) be a scheme. Let \(f \in \Gamma(X, \mathcal{O}_X)\). Assume \(X\) is quasi-compact and quasi-separated and assume that \(X_f\) is affine. Then the canonical morphism \[j : X \longrightarrow \Spec(\Gamma(X, \mathcal{O}_X))\] from Schemes, Lemma 01I1 induces an isomorphism of \(X_f = j^{-1}(D(f))\) onto the standard affine open \(D(f) \subset \Spec(\Gamma(X, \mathcal{O}_X))\).

Proof

This is clear as \(j\) induces an isomorphism of rings \(\Gamma(X, \mathcal{O}_X)_f \to \mathcal{O}_X(X_f)\) by Lemma 01P7 above.

Lemma

Let \(X\) be a scheme. Then \(X\) is quasi-affine if and only if the canonical morphism \[X \longrightarrow \Spec(\Gamma(X, \mathcal{O}_X))\] from Schemes, Lemma 01I1 is a quasi-compact open immersion.

Proof

If the displayed morphism is a quasi-compact open immersion then \(X\) is isomorphic to a quasi-compact open subscheme of \(\Spec(\Gamma(X, \mathcal{O}_X))\) and clearly \(X\) is quasi-affine.

Assume \(X\) is quasi-affine, say \(X \subset \Spec(R)\) is quasi-compact open. This in particular implies that \(X\) is separated, see Schemes, Lemma 01L8. Let \(A = \Gamma(X, \mathcal{O}_X)\). Consider the ring map \(R \to A\) coming from \(R = \Gamma(\Spec(R), \mathcal{O}_{\Spec(R)})\) and the restriction mapping of the sheaf \(\mathcal{O}_{\Spec(R)}\). By Schemes, Lemma 01I1 we obtain a factorization: \[X \longrightarrow \Spec(A) \longrightarrow \Spec(R)\] of the inclusion morphism. Let \(x \in X\). Choose \(r \in R\) such that \(x \in D(r)\) and \(D(r) \subset X\). Denote \(f \in A\) the image of \(r\) in \(A\). The open \(X_f\) of Lemma 01P7 above is equal to \(D(r) \subset X\) and hence \(A_f \cong R_r\) by the conclusion of that lemma. Hence \(D(r) \to \Spec(A)\) is an isomorphism onto the standard affine open \(D(f)\) of \(\Spec(A)\). Since \(X\) can be covered by such affine opens \(D(f)\) we win.

Lemma

Let \(U \to V\) be an open immersion of quasi-affine schemes. Then \[\xymatrix{ U \ar[d] \ar[rr]_-j & & \Spec(\Gamma(U, \mathcal{O}_U)) \ar[d] \\ U \ar[r] & V \ar[r]^-{j'} & \Spec(\Gamma(V, \mathcal{O}_V)) }\] is cartesian.

Proof

The diagram is commutative by Schemes, Lemma 01I1. Write \(A = \Gamma(U, \mathcal{O}_U)\) and \(B = \Gamma(V, \mathcal{O}_V)\). Let \(g \in B\) be such that \(V_g\) is affine and contained in \(U\). This means that if \(f\) is the image of \(g\) in \(A\), then \(U_f = V_g\). By Lemma 01P8 we see that \(j'\) induces an isomorphism of \(V_g\) with the standard open \(D(g)\) of \(\Spec(B)\). Thus \(V_g \times_{\Spec(B)} \Spec(A) \to \Spec(A)\) is an isomorphism onto \(D(f) \subset \Spec(A)\). By Lemma 01P8 again \(j\) maps \(U_f\) isomorphically to \(D(f)\). Thus we see that \(U_f = U_f \times_{\Spec(B)} \Spec(A)\). Since by Lemma 01P9 we can cover \(U\) by \(V_g = U_f\) as above, we see that \(U \to U \times_{\Spec(B)} \Spec(A)\) is an isomorphism.

Lemma

Let \(X\) be a quasi-affine scheme. There exists an integer \(n \geq 0\), an affine scheme \(T\), and a morphism \(T \to X\) such that for every morphism \(X' \to X\) with \(X'\) affine the fibre product \(X' \times_X T\) is isomorphic to \(\mathbf{A}^n_{X'}\) over \(X'\).

Proof

By definition, there exists a ring \(A\) such that \(X\) is isomorphic to a quasi-compact open subscheme \(U \subset \Spec(A)\). Recall that the standard opens \(D(f) \subset \Spec(A)\) form a basis for the topology, see Algebra, Section 00DY. Since \(U\) is quasi-compact we can choose \(f_1, \ldots, f_n \in A\) such that \(U = D(f_1) \cup \ldots \cup D(f_n)\). Thus we may assume \(X = \Spec(A) \setminus V(I)\) where \(I = (f_1, \ldots, f_n)\). We set \[T = \Spec(A[t, x_1, \ldots, x_n]/(f_1 x_1 + \ldots + f_n x_n - 1))\] The structure morphism \(T \to \Spec(A)\) factors through the open \(X\) to give the morphism \(T \to X\). If \(X' = \Spec(A')\) and the morphism \(X' \to X\) corresponds to the ring map \(A \to A'\), then the images \(f'_1, \ldots, f'_n \in A'\) of \(f_1, \ldots, f_n\) generate the unit ideal in \(A'\). Say \(1 = f'_1 a'_1 + \ldots + f'_n a'_n\). The base change \(X' \times_X T\) is the spectrum of \(A'[t, x_1, \ldots, x_n]/(f'_1 x_1 + \ldots + f'_n x_n - 1)\). We claim the \(A'\)-algebra homomorphism \[\varphi : A'[y_1, \ldots, y_n] \longrightarrow A'[t, x_1, \ldots, x_n, x_{n + 1}]/(f'_1 x_1 + \ldots + f'_n x_n - 1)\] sending \(y_i\) to \(a'_i t + x_i\) is an isomorphism. The claim finishes the proof of the lemma. The inverse of \(\varphi\) is given by the \(A'\)-algebra homomorphism \[\psi : A'[t, x_1, \ldots, x_n, x_{n + 1}]/(f'_1 x_1 + \ldots + f'_n x_n - 1) \longrightarrow A'[y_1, \ldots, y_n]\] sending \(t\) to \(-1 + f'_1 y_1 + \ldots + f'_n y_n\) and \(x_i\) to \(y_i + a'_i - a'_i(f'_1 y_1 + \ldots + f'_n y_n)\) for \(i = 1, \ldots, n\). This makes sense because \(\sum f'_ix_i\) is mapped to \[\begin{matrix} \sum f'_i(y_i + a'_i - a'_i(\sum f'_j y_j)) = (\sum f'_iy_i) + 1 - (\sum f'_j y_j) = 1 \end{matrix}\] To see the maps are mutually inverse one computes as follows: \[\begin{matrix} \varphi(\psi(t) = \varphi(-1 + \sum f'_i y_i) = -1 + \sum f'_i (a'_i t + x_i) = t \\ \varphi(\psi(x_i)) = \varphi(y_i + a'_i - a'_i(\sum f'_j y_j)) = a'_i t + x_i + a'_i - a'_i(\sum f'_ja'_jt + f'_jx_j) = x_i \\ \psi(\varphi(y_i)) = \psi(a'_i t + x_i) = a'_i(-1 + \sum f'_j y_j) + y_i + a'_i - a'_i(\sum f'_j y_j) = y_i \end{matrix}\] This finishes the proof.

Flat modules

On any ringed space \((X, \mathcal{O}_X)\) we know what it means for an \(\mathcal{O}_X\)-module to be flat (at a point), see Modules, Definition 05ND (Definition 05NF). For quasi-coherent sheaves on an affine scheme this matches the notion defined in the algebra chapter.

Lemma

Let \(X = \Spec(R)\) be an affine scheme. Let \(\mathcal{F} = \widetilde{M}\) for some \(R\)-module \(M\). The quasi-coherent sheaf \(\mathcal{F}\) is a flat \(\mathcal{O}_X\)-module if and only if \(M\) is a flat \(R\)-module.

Proof

Flatness of \(\mathcal{F}\) may be checked on the stalks, see Modules, Lemma 05NE. The same is true in the case of modules over a ring, see Algebra, Lemma 00HT. And since \(\mathcal{F}_x = M_{\mathfrak p}\) if \(x\) corresponds to \(\mathfrak p\) the lemma is true.

Locally free modules

On any ringed space we know what it means for an \(\mathcal{O}_X\)-module to be (finite) locally free. On an affine scheme this matches the notion defined in the algebra chapter.

Lemma

Let \(X = \Spec(R)\) be an affine scheme. Let \(\mathcal{F} = \widetilde{M}\) for some \(R\)-module \(M\). The quasi-coherent sheaf \(\mathcal{F}\) is a (finite) locally free \(\mathcal{O}_X\)-module of if and only if \(M\) is a (finite) locally free \(R\)-module.

Proof

Follows from the definitions, see Modules, Definition 01C6 and Algebra, Definition 00NW.

We can characterize finite locally free modules in many different ways.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. The following are equivalent:

\(\mathcal{F}\) is a flat \(\mathcal{O}_X\)-module of finite presentation,

\(\mathcal{F}\) is \(\mathcal{O}_X\)-module of finite presentation and for all \(x \in X\) the stalk \(\mathcal{F}_x\) is a free \(\mathcal{O}_{X, x}\)-module,

\(\mathcal{F}\) is a locally free, finite type \(\mathcal{O}_X\)-module,

\(\mathcal{F}\) is a finite locally free \(\mathcal{O}_X\)-module, and

\(\mathcal{F}\) is an \(\mathcal{O}_X\)-module of finite type, for every \(x \in X\) the stalk \(\mathcal{F}_x\) is a free \(\mathcal{O}_{X, x}\)-module, and the function \[\rho_\mathcal{F} : X \to \mathbf{Z}, \quad x \longmapsto \dim_{\kappa(x)} \mathcal{F}_x \otimes_{\mathcal{O}_{X, x}} \kappa(x)\] is locally constant in the Zariski topology on \(X\).

Proof

This lemma immediately reduces to the affine case. In this case the lemma is a reformulation of Algebra, Lemma 00NX. The translation uses Lemmas 01PB, 01PC, 05P0, and 05JM.

Lemma

Let \(X\) be a reduced scheme. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Then the equivalent conditions of Lemma 05P2 are also equivalent to

\(\mathcal{F}\) is an \(\mathcal{O}_X\)-module of finite type and the function \[\rho_\mathcal{F} : X \to \mathbf{Z}, \quad x \longmapsto \dim_{\kappa(x)} \mathcal{F}_x \otimes_{\mathcal{O}_{X, x}} \kappa(x)\] is locally constant in the Zariski topology on \(X\).

Proof

This lemma immediately reduces to the affine case. In this case the lemma is a reformulation of Algebra, Lemma 0FWG.

Locally projective modules

A consequence of the work done in the algebra chapter is that it makes sense to define a locally projective module as follows.

Definition

Let \(X\) be a scheme. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. We say \(\mathcal{F}\) is locally projective if for every affine open \(U \subset X\) the \(\mathcal{O}_X(U)\)-module \(\mathcal{F}(U)\) is projective.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. The following are equivalent

\(\mathcal{F}\) is locally projective, and

there exists an affine open covering \(X = \bigcup U_i\) such that the \(\mathcal{O}_X(U_i)\)-module \(\mathcal{F}(U_i)\) is projective for every \(i\).

In particular, if \(X = \Spec(A)\) and \(\mathcal{F} = \widetilde{M}\) then \(\mathcal{F}\) is locally projective if and only if \(M\) is a projective \(A\)-module.

Proof

First, note that if \(M\) is a projective \(A\)-module and \(A \to B\) is a ring map, then \(M \otimes_A B\) is a projective \(B\)-module, see Algebra, Lemma 05A3. Hence if \(U\) is an affine open such that \(\mathcal{F}(U)\) is a projective \(\mathcal{O}_X(U)\)-module, then the standard open \(D(f)\) is an affine open such that \(\mathcal{F}(D(f))\) is a projective \(\mathcal{O}_X(D(f))\)-module for all \(f \in \mathcal{O}_X(U)\). Assume (2) holds. Let \(U \subset X\) be an arbitrary affine open. We can find an open covering \(U = \bigcup_{j = 1, \ldots, m} D(f_j)\) by finitely many standard opens \(D(f_j)\) such that for each \(j\) the open \(D(f_j)\) is a standard open of some \(U_i\), see Schemes, Lemma 01IW. Hence, if we set \(A = \mathcal{O}_X(U)\) and if \(M\) is an \(A\)-module such that \(\mathcal{F}|_U\) corresponds to \(M\), then we see that \(M_{f_j}\) is a projective \(A_{f_j}\)-module. It follows that \(A \to B = \prod A_{f_j}\) is a faithfully flat ring map such that \(M \otimes_A B\) is a projective \(B\)-module. Hence \(M\) is projective by Algebra, Theorem 05A9.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \(\mathcal{G}\) be a quasi-coherent \(\mathcal{O}_Y\)-module. If \(\mathcal{G}\) is locally projective on \(Y\), then \(f^*\mathcal{G}\) is locally projective on \(X\).

Proof

Follows from Algebra, Lemma 05A3 and Lemma 05JQ.

Extending quasi-coherent sheaves

It is sometimes useful to be able to show that a given quasi-coherent sheaf on an open subscheme extends to the whole scheme.

Lemma

Let \(j : U \to X\) be a quasi-compact open immersion of schemes.

Any quasi-coherent sheaf on \(U\) extends to a quasi-coherent sheaf on \(X\).

Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(X\). Let \(\mathcal{G} \subset \mathcal{F}|_U\) be a quasi-coherent subsheaf. There exists a quasi-coherent subsheaf \(\mathcal{H}\) of \(\mathcal{F}\) such that \(\mathcal{H}|_U = \mathcal{G}\) as subsheaves of \(\mathcal{F}|_U\).

Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(X\). Let \(\mathcal{G}\) be a quasi-coherent sheaf on \(U\). Let \(\varphi : \mathcal{G} \to \mathcal{F}|_U\) be a morphism of \(\mathcal{O}_U\)-modules. There exists a quasi-coherent sheaf \(\mathcal{H}\) of \(\mathcal{O}_X\)-modules and a map \(\psi : \mathcal{H} \to \mathcal{F}\) such that \(\mathcal{H}|_U = \mathcal{G}\) and that \(\psi|_U = \varphi\).

Proof

An immersion is separated (see Schemes, Lemma 01L7) and \(j\) is quasi-compact by assumption. Hence for any quasi-coherent sheaf \(\mathcal{G}\) on \(U\) the sheaf \(j_*\mathcal{G}\) is an extension to \(X\). See Schemes, Lemma 01LC and Sheaves, Section 009Z.

Assume \(\mathcal{F}\), \(\mathcal{G}\) are as in (2). Then \(j_*\mathcal{G}\) is a quasi-coherent sheaf on \(X\) (see above). It is a subsheaf of \(j_*j^*\mathcal{F}\). Hence the kernel \[\mathcal{H} = \Ker(\mathcal{F} \oplus j_* \mathcal{G} \longrightarrow j_*j^*\mathcal{F})\] is quasi-coherent as well, see Schemes, Section 01LA. It is formal to check that \(\mathcal{H} \subset \mathcal{F}\) and that \(\mathcal{H}|_U = \mathcal{G}\) (using the material in Sheaves, Section 009Z again).

Part (3) is proved in the same manner as (2). Just take \(\mathcal{H} = \Ker(\mathcal{F} \oplus j_* \mathcal{G} \to j_*j^*\mathcal{F})\) with its obvious map to \(\mathcal{F}\) and its obvious identification with \(\mathcal{G}\) over \(U\).

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. Let \(U \subset X\) be a quasi-compact open. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(\mathcal{G} \subset \mathcal{F}|_U\) be a quasi-coherent \(\mathcal{O}_U\)-submodule which is of finite type. Then there exists a quasi-coherent submodule \(\mathcal{G}' \subset \mathcal{F}\) which is of finite type such that \(\mathcal{G}'|_U = \mathcal{G}\).

Proof

Let \(n\) be the minimal number of affine opens \(U_i \subset X\), \(i = 1, \ldots , n\) such that \(X = U \cup \bigcup U_i\). (Here we use that \(X\) is quasi-compact.) Suppose we can prove the lemma for the case \(n = 1\). Then we can successively extend \(\mathcal{G}\) to a \(\mathcal{G}_1\) over \(U \cup U_1\) to a \(\mathcal{G}_2\) over \(U \cup U_1 \cup U_2\) to a \(\mathcal{G}_3\) over \(U \cup U_1 \cup U_2 \cup U_3\), and so on. Thus we reduce to the case \(n = 1\).

Thus we may assume that \(X = U \cup V\) with \(V\) affine. Since \(X\) is quasi-separated and \(U\), \(V\) are quasi-compact open, we see that \(U \cap V\) is a quasi-compact open. It suffices to prove the lemma for the system \((V, U \cap V, \mathcal{F}|_V, \mathcal{G}|_{U \cap V})\) since we can glue the resulting sheaf \(\mathcal{G}'\) over \(V\) to the given sheaf \(\mathcal{G}\) over \(U\) along the common value over \(U \cap V\). Thus we reduce to the case where \(X\) is affine.

Assume \(X = \Spec(R)\). Write \(\mathcal{F} = \widetilde M\) for some \(R\)-module \(M\). By Lemma 01PE above we may find a quasi-coherent subsheaf \(\mathcal{H} \subset \mathcal{F}\) which restricts to \(\mathcal{G}\) over \(U\). Write \(\mathcal{H} = \widetilde N\) for some \(R\)-module \(N\). For every \(u \in U\) there exists an \(f \in R\) such that \(u \in D(f) \subset U\) and such that \(N_f\) is finitely generated, see Lemma 01PB. Since \(U\) is quasi-compact we can cover it by finitely many \(D(f_i)\) such that \(N_{f_i}\) is generated by finitely many elements, say \(x_{i, 1}/f_i^N, \ldots, x_{i, r_i}/f_i^N\). Let \(N' \subset N\) be the submodule generated by the elements \(x_{i, j}\). Then the subsheaf \(\mathcal{G}' = \widetilde{N'} \subset \mathcal{H} \subset \mathcal{F}\) works.

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. Any quasi-coherent sheaf of \(\mathcal{O}_X\)-modules is the directed colimit of its quasi-coherent \(\mathcal{O}_X\)-submodules which are of finite type.

Proof

The colimit is directed because if \(\mathcal{G}_1\), \(\mathcal{G}_2\) are quasi-coherent subsheaves of finite type, then the image of \(\mathcal{G}_1 \oplus \mathcal{G}_2 \to \mathcal{F}\) is a quasi-coherent submodule of finite type. Let \(U \subset X\) be any affine open, and let \(s \in \Gamma(U, \mathcal{F})\) be any section. Let \(\mathcal{G} \subset \mathcal{F}|_U\) be the subsheaf generated by \(s\). Then clearly \(\mathcal{G}\) is quasi-coherent and has finite type as an \(\mathcal{O}_U\)-module. By Lemma 01PF we see that \(\mathcal{G}\) is the restriction of a quasi-coherent subsheaf \(\mathcal{G}' \subset \mathcal{F}\) which has finite type. Since \(X\) has a basis for the topology consisting of affine opens we conclude that every local section of \(\mathcal{F}\) is locally contained in a quasi-coherent submodule of finite type. Thus we win.

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(U \subset X\) be a quasi-compact open. Let \(\mathcal{G}\) be an \(\mathcal{O}_U\)-module which is of finite presentation. Let \(\varphi : \mathcal{G} \to \mathcal{F}|_U\) be a morphism of \(\mathcal{O}_U\)-modules. Then there exists an \(\mathcal{O}_X\)-module \(\mathcal{G}'\) of finite presentation, and a morphism of \(\mathcal{O}_X\)-modules \(\varphi' : \mathcal{G}' \to \mathcal{F}\) such that \(\mathcal{G}'|_U = \mathcal{G}\) and such that \(\varphi'|_U = \varphi\).

Proof

The beginning of the proof is a repeat of the beginning of the proof of Lemma 01PF. We write it out carefully anyway.

Let \(n\) be the minimal number of affine opens \(U_i \subset X\), \(i = 1, \ldots , n\) such that \(X = U \cup \bigcup U_i\). (Here we use that \(X\) is quasi-compact.) Suppose we can prove the lemma for the case \(n = 1\). Then we can successively extend the pair \((\mathcal{G}, \varphi)\) to a pair \((\mathcal{G}_1, \varphi_1)\) over \(U \cup U_1\) to a pair \((\mathcal{G}_2, \varphi_2)\) over \(U \cup U_1 \cup U_2\) to a pair \((\mathcal{G}_3, \varphi_3)\) over \(U \cup U_1 \cup U_2 \cup U_3\), and so on. Thus we reduce to the case \(n = 1\).

Thus we may assume that \(X = U \cup V\) with \(V\) affine. Since \(X\) is quasi-separated and \(U\) quasi-compact, we see that \(U \cap V \subset V\) is quasi-compact. Suppose we prove the lemma for the system \((V, U \cap V, \mathcal{F}|_V, \mathcal{G}|_{U \cap V}, \varphi|_{U \cap V})\) thereby producing \((\mathcal{G}', \varphi')\) over \(V\). Then we can glue \(\mathcal{G}'\) over \(V\) to the given sheaf \(\mathcal{G}\) over \(U\) along the common value over \(U \cap V\), and similarly we can glue the map \(\varphi'\) to the map \(\varphi\) along the common value over \(U \cap V\). Thus we reduce to the case where \(X\) is affine.

Assume \(X = \Spec(R)\). By Lemma 01PE above we may find a quasi-coherent sheaf \(\mathcal{H}\) with a map \(\psi : \mathcal{H} \to \mathcal{F}\) over \(X\) which restricts to \(\mathcal{G}\) and \(\varphi\) over \(U\). By Lemma 01PF we can find a finite type quasi-coherent \(\mathcal{O}_X\)-submodule \(\mathcal{H}' \subset \mathcal{H}\) such that \(\mathcal{H}'|_U = \mathcal{G}\). Thus after replacing \(\mathcal{H}\) by \(\mathcal{H}'\) and \(\psi\) by the restriction of \(\psi\) to \(\mathcal{H}'\) we may assume that \(\mathcal{H}\) is of finite type. By Lemma 01PB we conclude that \(\mathcal{H} = \widetilde{N}\) with \(N\) a finitely generated \(R\)-module. Hence there exists a surjection as in the following short exact sequence of quasi-coherent \(\mathcal{O}_X\)-modules \[0 \to \mathcal{K} \to \mathcal{O}_X^{\oplus n} \to \mathcal{H} \to 0\] where \(\mathcal{K}\) is defined as the kernel. Since \(\mathcal{G}\) is of finite presentation and \(\mathcal{H}|_U = \mathcal{G}\) by Modules, Lemma 01BP the restriction \(\mathcal{K}|_U\) is an \(\mathcal{O}_U\)-module of finite type. Hence by Lemma 01PF again we see that there exists a finite type quasi-coherent \(\mathcal{O}_X\)-submodule \(\mathcal{K}' \subset \mathcal{K}\) such that \(\mathcal{K}'|_U = \mathcal{K}|_U\). The solution to the problem posed in the lemma is to set \[\mathcal{G}' = \mathcal{O}_X^{\oplus n}/\mathcal{K}'\] which is clearly of finite presentation and restricts to give \(\mathcal{G}\) on \(U\) with \(\varphi'\) equal to the composition \[\mathcal{G}' = \mathcal{O}_X^{\oplus n}/\mathcal{K}' \to \mathcal{O}_X^{\oplus n}/\mathcal{K} = \mathcal{H} \xrightarrow{\psi} \mathcal{F}.\] This finishes the proof of the lemma.

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. Let \(U \subset X\) be a quasi-compact open. Let \(\mathcal{G}\) be an \(\mathcal{O}_U\)-module.

If \(\mathcal{G}\) is quasi-coherent and of finite type, then there exists a quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{G}'\) of finite type such that \(\mathcal{G}'|_U = \mathcal{G}\).

If \(\mathcal{G}\) is of finite presentation, then there exists an \(\mathcal{O}_X\)-module \(\mathcal{G}'\) of finite presentation such that \(\mathcal{G}'|_U = \mathcal{G}\).

Proof

Part (2) is the special case of Lemma 01PI where \(\mathcal{F} = 0\). For part (1) we first write \(\mathcal{G} = \mathcal{F}|_U\) for some quasi-coherent \(\mathcal{O}_X\)-module by Lemma 01PE and then we apply Lemma 01PF with \(\mathcal{G} = \mathcal{F}|_U\).

The following lemma says that every quasi-coherent sheaf on a quasi-compact and quasi-separated scheme is a filtered colimit of \(\mathcal{O}\)-modules of finite presentation. Actually, we reformulate this in (perhaps more familiar) terms of directed colimits over directed sets in the next lemma.

Lemma

Let \(X\) be a scheme. Assume \(X\) is quasi-compact and quasi-separated. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. There exist

a filtered index category \(\mathcal{I}\) (see Categories, Definition 002V),

a diagram \(\mathcal{I} \to \textit{Mod}(\mathcal{O}_X)\) (see Categories, Section 002D), \(i \mapsto \mathcal{F}_i\),

morphisms of \(\mathcal{O}_X\)-modules \(\varphi_i : \mathcal{F}_i \to \mathcal{F}\)

such that each \(\mathcal{F}_i\) is of finite presentation and such that the morphisms \(\varphi_i\) induce an isomorphism \[\colim_i \mathcal{F}_i = \mathcal{F}.\]

Proof

Choose a set \(I\) and for each \(i \in I\) an \(\mathcal{O}_X\)-module of finite presentation and a homomorphism of \(\mathcal{O}_X\)-modules \(\varphi_i : \mathcal{F}_i \to \mathcal{F}\) with the following property: For any \(\psi : \mathcal{G} \to \mathcal{F}\) with \(\mathcal{G}\) of finite presentation there is an \(i \in I\) such that there exists an isomorphism \(\alpha : \mathcal{F}_i \to \mathcal{G}\) with \(\varphi_i = \psi \circ \alpha\). It is clear from Modules, Lemma 01BC that such a set exists (see also its proof). We denote \(\mathcal{I}\) the category with \(\Ob(\mathcal{I}) = I\) and given \(i, i' \in I\) we set \[\Mor_\mathcal{I}(i, i') = \{\alpha : \mathcal{F}_i \to \mathcal{F}_{i'} \mid \alpha \circ \varphi_{i'} = \varphi_i \}.\] We claim that \(\mathcal{I}\) is a filtered category and that \(\mathcal{F} = \colim_i \mathcal{F}_i\).

Let \(i, i' \in I\). Then we can consider the morphism \[\mathcal{F}_i \oplus \mathcal{F}_{i'} \longrightarrow \mathcal{F}\] which is the direct sum of \(\varphi_i\) and \(\varphi_{i'}\). Since a direct sum of finitely presented \(\mathcal{O}_X\)-modules is finitely presented we see that there exists some \(i'' \in I\) such that \(\varphi_{i''} : \mathcal{F}_{i''} \to \mathcal{F}\) is isomorphic to the displayed arrow towards \(\mathcal{F}\) above. Since there are commutative diagrams \[\xymatrix{ \mathcal{F}_i \ar[r] \ar[d] & \mathcal{F} \ar@{=}[d] \\ \mathcal{F}_i \oplus \mathcal{F}_{i'} \ar[r] & \mathcal{F} } \quad \text{and} \quad \xymatrix{ \mathcal{F}_{i'} \ar[r] \ar[d] & \mathcal{F} \ar@{=}[d] \\ \mathcal{F}_i \oplus \mathcal{F}_{i'} \ar[r] & \mathcal{F} }\] we see that there are morphisms \(i \to i''\) and \(i' \to i''\) in \(\mathcal{I}\). Next, suppose that we have \(i, i' \in I\) and morphisms \(\alpha, \beta : i \to i'\) (corresponding to \(\mathcal{O}_X\)-module maps \(\alpha, \beta : \mathcal{F}_i \to \mathcal{F}_{i'}\)). In this case consider the coequalizer \[\mathcal{G} = \Coker( \mathcal{F}_i \xrightarrow{\alpha - \beta} \mathcal{F}_{i'} )\] Note that \(\mathcal{G}\) is an \(\mathcal{O}_X\)-module of finite presentation. Since by definition of morphisms in the category \(\mathcal{I}\) we have \(\varphi_{i'} \circ \alpha = \varphi_{i'} \circ \beta\) we see that we get an induced map \(\psi : \mathcal{G} \to \mathcal{F}\). Hence again the pair \((\mathcal{G}, \psi)\) is isomorphic to the pair \((\mathcal{F}_{i''}, \varphi_{i''})\) for some \(i''\). Hence we see that there exists a morphism \(i' \to i''\) in \(\mathcal{I}\) which equalizes \(\alpha\) and \(\beta\). Thus we have shown that the category \(\mathcal{I}\) is filtered.

We still have to show that the colimit of the diagram is \(\mathcal{F}\). By definition of the colimit, and by our definition of the category \(\mathcal{I}\) there is a canonical map \[\varphi : \colim_i \mathcal{F}_i \longrightarrow \mathcal{F}.\] Pick \(x \in X\). Let us show that \(\varphi_x\) is an isomorphism. Recall that \[(\colim_i \mathcal{F}_i)_x = \colim_i \mathcal{F}_{i, x},\] see Sheaves, Section 009E. First we show that the map \(\varphi_x\) is injective. Suppose that \(s \in \mathcal{F}_{i, x}\) is an element such that \(s\) maps to zero in \(\mathcal{F}_x\). Then there exists a quasi-compact open \(U\) such that \(s\) comes from \(s \in \mathcal{F}_i(U)\) and such that \(\varphi_i(s) = 0\) in \(\mathcal{F}(U)\). By Lemma 01PF we can find a finite type quasi-coherent subsheaf \(\mathcal{K} \subset \Ker(\varphi_i)\) which restricts to the quasi-coherent \(\mathcal{O}_U\)-submodule of \(\mathcal{F}_i\) generated by \(s\): \(\mathcal{K}|_U = \mathcal{O}_U\cdot s \subset \mathcal{F}_i|_U\). Clearly, \(\mathcal{F}_i/\mathcal{K}\) is of finite presentation and the map \(\varphi_i\) factors through the quotient map \(\mathcal{F}_i \to \mathcal{F}_i/\mathcal{K}\). Hence we can find an \(i' \in I\) and a morphism \(\alpha : \mathcal{F}_i \to \mathcal{F}_{i'}\) in \(\mathcal{I}\) which can be identified with the quotient map \(\mathcal{F}_i \to \mathcal{F}_i/\mathcal{K}\). Then it follows that the section \(s\) maps to zero in \(\mathcal{F}_{i'}(U)\) and in particular in \((\colim_i \mathcal{F}_i)_x = \colim_i \mathcal{F}_{i, x}\). The injectivity follows. Finally, we show that the map \(\varphi_x\) is surjective. Pick \(s \in \mathcal{F}_x\). Choose a quasi-compact open neighbourhood \(U \subset X\) of \(x\) such that \(s\) corresponds to a section \(s \in \mathcal{F}(U)\). Consider the map \(s : \mathcal{O}_U \to \mathcal{F}\) (multiplication by \(s\)). By Lemma 01PI there exists an \(\mathcal{O}_X\)-module \(\mathcal{G}\) of finite presentation and an \(\mathcal{O}_X\)-module map \(\mathcal{G} \to \mathcal{F}\) such that \(\mathcal{G}|_U \to \mathcal{F}|_U\) is identified with \(s : \mathcal{O}_U \to \mathcal{F}\). Again by definition of \(\mathcal{I}\) there exists an \(i \in I\) such that \(\mathcal{G} \to \mathcal{F}\) is isomorphic to \(\varphi_i : \mathcal{F}_i \to \mathcal{F}\). Clearly there exists a section \(s' \in \mathcal{F}_i(U)\) mapping to \(s \in \mathcal{F}(U)\). This proves surjectivity and the proof of the lemma is complete.

Lemma

Let \(X\) be a scheme. Assume \(X\) is quasi-compact and quasi-separated. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. There exist

a directed set \(I\) (see Categories, Definition 00D3),

a system \((\mathcal{F}_i, \varphi_{ii'})\) over \(I\) in \(\textit{Mod}(\mathcal{O}_X)\) (see Categories, Definition 0030)

morphisms of \(\mathcal{O}_X\)-modules \(\varphi_i : \mathcal{F}_i \to \mathcal{F}\)

such that each \(\mathcal{F}_i\) is of finite presentation and such that the morphisms \(\varphi_i\) induce an isomorphism \[\colim_i \mathcal{F}_i = \mathcal{F}.\]

Proof

This is a direct consequence of Lemma 01PJ and Categories, Lemma 0032 (combined with the fact that colimits exist in the category of sheaves of \(\mathcal{O}_X\)-modules, see Sheaves, Section 009E).

Lemma

Let \(X\) be a scheme. Assume \(X\) is quasi-compact and quasi-separated. Let \(\mathcal{F}\) be a finite type quasi-coherent \(\mathcal{O}_X\)-module. Then we can write \(\mathcal{F} = \colim \mathcal{F}_i\) with \(\mathcal{F}_i\) of finite presentation and all transition maps \(\mathcal{F}_i \to \mathcal{F}_{i'}\) surjective.

Proof

Write \(\mathcal{F} = \colim \mathcal{G}_i\) as a filtered colimit of finitely presented \(\mathcal{O}_X\)-modules (Lemma 01PK). We claim that \(\mathcal{G}_i \to \mathcal{F}\) is surjective for some \(i\). Namely, choose a finite affine open covering \(X = U_1 \cup \ldots \cup U_m\). Choose sections \(s_{jl} \in \mathcal{F}(U_j)\) generating \(\mathcal{F}|_{U_j}\), see Lemma 01PB. By Sheaves, Lemma 009F we see that \(s_{jl}\) is in the image of \(\mathcal{G}_i \to \mathcal{F}\) for \(i\) large enough. Hence \(\mathcal{G}_i \to \mathcal{F}\) is surjective for \(i\) large enough. Choose such an \(i\) and let \(\mathcal{K} \subset \mathcal{G}_i\) be the kernel of the map \(\mathcal{G}_i \to \mathcal{F}\). Write \(\mathcal{K} = \colim \mathcal{K}_a\) as the filtered colimit of its finite type quasi-coherent submodules (Lemma 01PG). Then \(\mathcal{F} = \colim \mathcal{G}_i/\mathcal{K}_a\) is a solution to the problem posed by the lemma.

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. Let \(\mathcal{F}\) be a finite type quasi-coherent \(\mathcal{O}_X\)-module. Let \(U \subset X\) be a quasi-compact open such that \(\mathcal{F}|_U\) is of finite presentation. Then there exists a map of \(\mathcal{O}_X\)-modules \(\varphi : \mathcal{G} \to \mathcal{F}\) with (a) \(\mathcal{G}\) of finite presentation, (b) \(\varphi\) is surjective, and (c) \(\varphi|_U\) is an isomorphism.

Proof

Write \(\mathcal{F} = \colim \mathcal{F}_i\) as a directed colimit with each \(\mathcal{F}_i\) of finite presentation, see Lemma 01PK. Choose a finite affine open covering \(X = \bigcup V_j\) and choose finitely many sections \(s_{jl} \in \mathcal{F}(V_j)\) generating \(\mathcal{F}|_{V_j}\), see Lemma 01PB. By Sheaves, Lemma 009F we see that \(s_{jl}\) is in the image of \(\mathcal{F}_i \to \mathcal{F}\) for \(i\) large enough. Hence \(\mathcal{F}_i \to \mathcal{F}\) is surjective for \(i\) large enough. Choose such an \(i\) and let \(\mathcal{K} \subset \mathcal{F}_i\) be the kernel of the map \(\mathcal{F}_i \to \mathcal{F}\). Since \(\mathcal{F}_U\) is of finite presentation, we see that \(\mathcal{K}|_U\) is of finite type, see Modules, Lemma 01BP. Hence we can find a finite type quasi-coherent submodule \(\mathcal{K}' \subset \mathcal{K}\) with \(\mathcal{K}'|_U = \mathcal{K}|_U\), see Lemma 01PF. Then \(\mathcal{G} = \mathcal{F}_i/\mathcal{K}'\) with the given map \(\mathcal{G} \to \mathcal{F}\) is a solution.

Let \(X\) be a scheme. In the following lemma we use the notion of a quasi-coherent \(\mathcal{O}_X\)-algebra \(\mathcal{A}\) of finite presentation. This means that for every affine open \(\Spec(R) \subset X\) we have \(\mathcal{A} = \widetilde{A}\) where \(A\) is a (commutative) \(R\)-algebra which is of finite presentation as an \(R\)-algebra.

Lemma

Let \(X\) be a scheme. Assume \(X\) is quasi-compact and quasi-separated. Let \(\mathcal{A}\) be a quasi-coherent \(\mathcal{O}_X\)-algebra. There exist

a directed set \(I\) (see Categories, Definition 00D3),

a system \((\mathcal{A}_i, \varphi_{ii'})\) over \(I\) in the category of \(\mathcal{O}_X\)-algebras,

morphisms of \(\mathcal{O}_X\)-algebras \(\varphi_i : \mathcal{A}_i \to \mathcal{A}\)

such that each \(\mathcal{A}_i\) is a quasi-coherent \(\mathcal{O}_X\)-algebra of finite presentation and such that the morphisms \(\varphi_i\) induce an isomorphism \[\colim_i \mathcal{A}_i = \mathcal{A}.\]

Proof

First we write \(\mathcal{A} = \colim_i \mathcal{F}_i\) as a directed colimit of finitely presented quasi-coherent sheaves as in Lemma 01PK. For each \(i\) let \(\mathcal{B}_i = \text{Sym}(\mathcal{F}_i)\) be the symmetric algebra on \(\mathcal{F}_i\) over \(\mathcal{O}_X\). Write \(\mathcal{I}_i = \Ker(\mathcal{B}_i \to \mathcal{A})\). Write \(\mathcal{I}_i = \colim_j \mathcal{F}_{i, j}\) where \(\mathcal{F}_{i, j}\) is a finite type quasi-coherent submodule of \(\mathcal{I}_i\), see Lemma 01PG. Set \(\mathcal{I}_{i, j} \subset \mathcal{I}_i\) equal to the \(\mathcal{B}_i\)-ideal generated by \(\mathcal{F}_{i, j}\). Set \(\mathcal{A}_{i, j} = \mathcal{B}_i/\mathcal{I}_{i, j}\). Then \(\mathcal{A}_{i, j}\) is a quasi-coherent finitely presented \(\mathcal{O}_X\)-algebra. Define \((i, j) \leq (i', j')\) if \(i \leq i'\) and the map \(\mathcal{B}_i \to \mathcal{B}_{i'}\) maps the ideal \(\mathcal{I}_{i, j}\) into the ideal \(\mathcal{I}_{i', j'}\). Then it is clear that \(\mathcal{A} = \colim_{i, j} \mathcal{A}_{i, j}\).

Let \(X\) be a scheme. In the following lemma we use the notion of a quasi-coherent \(\mathcal{O}_X\)-algebra \(\mathcal{A}\) of finite type. This means that for every affine open \(\Spec(R) \subset X\) we have \(\mathcal{A} = \widetilde{A}\) where \(A\) is a (commutative) \(R\)-algebra which is of finite type as an \(R\)-algebra.

Lemma

Let \(X\) be a scheme. Assume \(X\) is quasi-compact and quasi-separated. Let \(\mathcal{A}\) be a quasi-coherent \(\mathcal{O}_X\)-algebra. Then \(\mathcal{A}\) is the directed colimit of its finite type quasi-coherent \(\mathcal{O}_X\)-subalgebras.

Proof

If \(\mathcal{A}_1, \mathcal{A}_2 \subset \mathcal{A}\) are quasi-coherent \(\mathcal{O}_X\)-subalgebras of finite type, then the image of \(\mathcal{A}_1 \otimes_{\mathcal{O}_X} \mathcal{A}_2 \to \mathcal{A}\) is also a quasi-coherent \(\mathcal{O}_X\)-subalgebra of finite type (some details omitted) which contains both \(\mathcal{A}_1\) and \(\mathcal{A}_2\). In this way we see that the system is directed. To show that \(\mathcal{A}\) is the colimit of this system, write \(\mathcal{A} = \colim_i \mathcal{A}_i\) as a directed colimit of finitely presented quasi-coherent \(\mathcal{O}_X\)-algebras as in Lemma 05JS. Then the images \(\mathcal{A}'_i = \Im(\mathcal{A}_i \to \mathcal{A})\) are quasi-coherent subalgebras of \(\mathcal{A}\) of finite type. Since \(\mathcal{A}\) is the colimit of these the result follows.

Let \(X\) be a scheme. In the following lemma we use the notion of a finite (resp. integral) quasi-coherent \(\mathcal{O}_X\)-algebra \(\mathcal{A}\). This means that for every affine open \(\Spec(R) \subset X\) we have \(\mathcal{A} = \widetilde{A}\) where \(A\) is a (commutative) \(R\)-algebra which is finite (resp. integral) as an \(R\)-algebra.

Lemma

Let \(X\) be a scheme. Assume \(X\) is quasi-compact and quasi-separated. Let \(\mathcal{A}\) be a finite quasi-coherent \(\mathcal{O}_X\)-algebra. Then \(\mathcal{A} = \colim \mathcal{A}_i\) is a directed colimit of finite and finitely presented quasi-coherent \(\mathcal{O}_X\)-algebras such that all transition maps \(\mathcal{A}_{i'} \to \mathcal{A}_i\) are surjective.

Proof

By Lemma 086M there exists a finitely presented \(\mathcal{O}_X\)-module \(\mathcal{F}\) and a surjection \(\mathcal{F} \to \mathcal{A}\). Using the algebra structure we obtain a surjection \[\text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}) \longrightarrow \mathcal{A}\] Denote \(\mathcal{J}\) the kernel. Write \(\mathcal{J} = \colim \mathcal{E}_i\) as a filtered colimit of finite type \(\mathcal{O}_X\)-submodules \(\mathcal{E}_i\) (Lemma 01PG). Set \[\mathcal{A}_i = \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F})/(\mathcal{E}_i)\] where \((\mathcal{E}_i)\) indicates the ideal sheaf generated by the image of \(\mathcal{E}_i \to \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F})\). Then each \(\mathcal{A}_i\) is a finitely presented \(\mathcal{O}_X\)-algebra, the transition maps are surjections, and \(\mathcal{A} = \colim \mathcal{A}_i\). To finish the proof we still have to show that \(\mathcal{A}_i\) is a finite \(\mathcal{O}_X\)-algebra for \(i\) sufficiently large. To do this we choose an affine open covering \(X = U_1 \cup \ldots \cup U_m\). Take generators \(f_{j, 1}, \ldots, f_{j, N_j} \in \Gamma(U_i, \mathcal{F})\). As \(\mathcal{A}(U_j)\) is a finite \(\mathcal{O}_X(U_j)\)-algebra we see that for each \(k\) there exists a monic polynomial \(P_{j, k} \in \mathcal{O}(U_j)[T]\) such that \(P_{j, k}(f_{j, k})\) is zero in \(\mathcal{A}(U_j)\). Since \(\mathcal{A} = \colim \mathcal{A}_i\) by construction, we have \(P_{j, k}(f_{j, k}) = 0\) in \(\mathcal{A}_i(U_j)\) for all sufficiently large \(i\). For such \(i\) the algebras \(\mathcal{A}_i\) are finite.

Lemma

Let \(X\) be a scheme. Assume \(X\) is quasi-compact and quasi-separated. Let \(\mathcal{A}\) be an integral quasi-coherent \(\mathcal{O}_X\)-algebra. Then

\(\mathcal{A}\) is the directed colimit of its finite quasi-coherent \(\mathcal{O}_X\)-subalgebras, and

\(\mathcal{A}\) is a direct colimit of finite and finitely presented quasi-coherent \(\mathcal{O}_X\)-algebras.

Proof

By Lemma 05JT we have \(\mathcal{A} = \colim \mathcal{A}_i\) where \(\mathcal{A}_i \subset \mathcal{A}\) runs through the quasi-coherent \(\mathcal{O}_X\)-algebras of finite type. Any finite type quasi-coherent \(\mathcal{O}_X\)-subalgebra of \(\mathcal{A}\) is finite (apply Algebra, Lemma 02JJ to \(\mathcal{A}_i(U) \subset \mathcal{A}(U)\) for affine opens \(U\) in \(X\)). This proves (1).

To prove (2), write \(\mathcal{A} = \colim \mathcal{F}_i\) as a colimit of finitely presented \(\mathcal{O}_X\)-modules using Lemma 01PK. For each \(i\), let \(\mathcal{J}_i\) be the kernel of the map \[\text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i) \longrightarrow \mathcal{A}\] For \(i' \geq i\) there is an induced map \(\mathcal{J}_i \to \mathcal{J}_{i'}\) and we have \(\mathcal{A} = \colim \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i)/\mathcal{J}_i\). Moreover, the quasi-coherent \(\mathcal{O}_X\)-algebras \(\text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i)/\mathcal{J}_i\) are finite (see above). Write \(\mathcal{J}_i = \colim \mathcal{E}_{ik}\) as a colimit of finitely presented \(\mathcal{O}_X\)-modules. Given \(i' \geq i\) and \(k\) there exists a \(k'\) such that we have a map \(\mathcal{E}_{ik} \to \mathcal{E}_{i'k'}\) making \[\xymatrix{ \mathcal{J}_i \ar[r] & \mathcal{J}_{i'} \\ \mathcal{E}_{ik} \ar[u] \ar[r] & \mathcal{E}_{i'k'} \ar[u] }\] commute. This follows from Modules, Lemma 01BS. This induces a map \[\mathcal{A}_{ik} = \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i)/(\mathcal{E}_{ik}) \longrightarrow \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_{i'})/(\mathcal{E}_{i'k'}) = \mathcal{A}_{i'k'}\] where \((\mathcal{E}_{ik})\) denotes the ideal generated by \(\mathcal{E}_{ik}\). The quasi-coherent \(\mathcal{O}_X\)-algebras \(\mathcal{A}_{ki}\) are of finite presentation and finite for \(k\) large enough (see proof of Lemma 086N). Finally, we have \[\colim \mathcal{A}_{ik} = \colim \mathcal{A}_i = \mathcal{A}\] Namely, the first equality was shown in the proof of Lemma 086N and the second equality because \(\mathcal{A}\) is the colimit of the modules \(\mathcal{F}_i\).

Gabber’s result

In this section we prove a result of Gabber which guarantees that on every scheme there exists a cardinal \(\kappa\) such that every quasi-coherent module \(\mathcal{F}\) is the union of its quasi-coherent \(\kappa\)-generated subsheaves. It follows that the category of quasi-coherent sheaves on a scheme is a Grothendieck abelian category having limits and enough injectives³.

Definition

Let \((X, \mathcal{O}_X)\) be a ringed space. Let \(\kappa\) be an infinite cardinal. We say a sheaf of \(\mathcal{O}_X\)-modules \(\mathcal{F}\) is \(\kappa\)-generated if there exists an open covering \(X = \bigcup U_i\) such that \(\mathcal{F}|_{U_i}\) is generated by a subset \(R_i \subset \mathcal{F}(U_i)\) whose cardinality is at most \(\kappa\).

Note that a direct sum of at most \(\kappa\) \(\kappa\)-generated modules is again \(\kappa\)-generated because \(\kappa \otimes \kappa = \kappa\), see Sets, Section 000D. In particular this holds for the direct sum of two \(\kappa\)-generated modules. Moreover, a quotient of a \(\kappa\)-generated sheaf is \(\kappa\)-generated. (But the same needn’t be true for submodules.)

Lemma

Let \((X, \mathcal{O}_X)\) be a ringed space. Let \(\kappa\) be a cardinal. There exists a set \(T\) and a family \((\mathcal{F}_t)_{t \in T}\) of \(\kappa\)-generated \(\mathcal{O}_X\)-modules such that every \(\kappa\)-generated \(\mathcal{O}_X\)-module is isomorphic to one of the \(\mathcal{F}_t\).

Proof

There is a set of coverings of \(X\) (provided we disallow repeats). Suppose \(X = \bigcup U_i\) is a covering and suppose \(\mathcal{F}_i\) is an \(\mathcal{O}_{U_i}\)-module. Then there is a set of isomorphism classes of \(\mathcal{O}_X\)-modules \(\mathcal{F}\) with the property that \(\mathcal{F}|_{U_i} \cong \mathcal{F}_i\) since there is a set of glueing maps. This reduces us to proving there is a set of (isomorphism classes of) quotients \(\oplus_{k \in \kappa} \mathcal{O}_X \to \mathcal{F}\) for any ringed space \(X\). This is clear.

Here is the result the title of this section refers to.

Lemma

Let \(X\) be a scheme. There exists a cardinal \(\kappa\) such that every quasi-coherent module \(\mathcal{F}\) is the directed colimit of its quasi-coherent \(\kappa\)-generated submodules.

Proof

Choose an affine open covering \(X = \bigcup_{i \in I} U_i\). For each pair \(i, j\) choose an affine open covering \(U_i \cap U_j = \bigcup_{k \in I_{ij}} U_{ijk}\). Write \(U_i = \Spec(A_i)\) and \(U_{ijk} = \Spec(A_{ijk})\). Let \(\kappa\) be any infinite cardinal \(\geq\) than the cardinality of any of the sets \(I\), \(I_{ij}\).

Let \(\mathcal{F}\) be a quasi-coherent sheaf. Set \(M_i = \mathcal{F}(U_i)\) and \(M_{ijk} = \mathcal{F}(U_{ijk})\). Note that \[M_i \otimes_{A_i} A_{ijk} = M_{ijk} = M_j \otimes_{A_j} A_{ijk}.\] see Schemes, Lemma 01I9. Using the axiom of choice we choose a map \[(i, j, k, m) \mapsto S(i, j, k, m)\] which associates to every \(i, j \in I\), \(k \in I_{ij}\) and \(m \in M_i\) a finite subset \(S(i, j, k, m) \subset M_j\) such that we have \[m \otimes 1 = \sum\nolimits_{m' \in S(i, j, k, m)} m' \otimes a_{m'}\] in \(M_{ijk}\) for some \(a_{m'} \in A_{ijk}\). Moreover, let’s agree that \(S(i, i, k, m) = \{m\}\) for all \(i, j = i, k, m\) as above. Fix such a map.

Given a family \(\mathcal{S} = (S_i)_{i \in I}\) of subsets \(S_i \subset M_i\) of cardinality at most \(\kappa\) we set \(\mathcal{S}' = (S'_i)\) where \[S'_j = \bigcup\nolimits_{(i, k, m)\text{ such that }m \in S_i} S(i, j, k, m)\] Note that \(S_i \subset S'_i\). Note that \(S'_i\) has cardinality at most \(\kappa\) because it is a union over a set of cardinality at most \(\kappa\) of finite sets. Set \(\mathcal{S}^{(0)} = \mathcal{S}\), \(\mathcal{S}^{(1)} = \mathcal{S}'\) and by induction \(\mathcal{S}^{(n + 1)} = (\mathcal{S}^{(n)})'\). Then set \(\mathcal{S}^{(\infty)} = \bigcup_{n \geq 0} \mathcal{S}^{(n)}\). Writing \(\mathcal{S}^{(\infty)} = (S^{(\infty)}_i)\) we see that for any element \(m \in S^{(\infty)}_i\) the image of \(m\) in \(M_{ijk}\) can be written as a finite sum \(\sum m' \otimes a_{m'}\) with \(m' \in S_j^{(\infty)}\). In this way we see that setting \[N_i = A_i\text{-submodule of }M_i\text{ generated by }S^{(\infty)}_i\] we have \[N_i \otimes_{A_i} A_{ijk} = N_j \otimes_{A_j} A_{ijk}.\] as submodules of \(M_{ijk}\). Thus there exists a quasi-coherent subsheaf \(\mathcal{G} \subset \mathcal{F}\) with \(\mathcal{G}(U_i) = N_i\). Moreover, by construction the sheaf \(\mathcal{G}\) is \(\kappa\)-generated.

Let \(\{\mathcal{G}_t\}_{t \in T}\) be the set of \(\kappa\)-generated quasi-coherent subsheaves. If \(t, t' \in T\) then \(\mathcal{G}_t + \mathcal{G}_{t'}\) is also a \(\kappa\)-generated quasi-coherent subsheaf as it is the image of the map \(\mathcal{G}_t \oplus \mathcal{G}_{t'} \to \mathcal{F}\). Hence the system (ordered by inclusion) is directed. The arguments above show that every section of \(\mathcal{F}\) over \(U_i\) is in one of the \(\mathcal{G}_t\) (because we can start with \(\mathcal{S}\) such that the given section is an element of \(S_i\)). Hence \(\colim_t \mathcal{G}_t \to \mathcal{F}\) is both injective and surjective as desired.

Proposition

Let \(X\) be a scheme.

The category \(\QCoh(\mathcal{O}_X)\) is a Grothendieck abelian category. Consequently, \(\QCoh(\mathcal{O}_X)\) has enough injectives and all limits.

The inclusion functor \(\QCoh(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X)\) has a right adjoint⁴ \[Q : \textit{Mod}(\mathcal{O}_X) \longrightarrow \QCoh(\mathcal{O}_X)\] such that for every quasi-coherent sheaf \(\mathcal{F}\) the adjunction mapping \(Q(\mathcal{F}) \to \mathcal{F}\) is an isomorphism.

Proof

Part (1) means \(\QCoh(\mathcal{O}_X)\) (a) has all colimits, (b) filtered colimits are exact, and (c) has a generator, see Injectives, Section 079A. By Schemes, Section 01LA colimits in \(\QCoh(\mathcal{O}_X)\) exist and agree with colimits in \(\textit{Mod}(\mathcal{O}_X)\). By Modules, Lemma 01AH filtered colimits are exact. Hence (a) and (b) hold. To construct a generator \(U\), pick a cardinal \(\kappa\) as in Lemma 077N. Pick a collection \((\mathcal{F}_t)_{t \in T}\) of \(\kappa\)-generated quasi-coherent sheaves as in Lemma 077M. Set \(U = \bigoplus_{t \in T} \mathcal{F}_t\). Since every object of \(\QCoh(\mathcal{O}_X)\) is a filtered colimit of \(\kappa\)-generated quasi-coherent modules, i.e., of objects isomorphic to \(\mathcal{F}_t\), it is clear that \(U\) is a generator. The assertions on limits and injectives hold in any Grothendieck abelian category, see Injectives, Theorem 079H and Lemma 07D8.

Proof of (2). To construct \(Q\) we use the following general procedure. Given an object \(\mathcal{F}\) of \(\textit{Mod}(\mathcal{O}_X)\) we consider the functor \[\QCoh(\mathcal{O}_X)^{opp} \longrightarrow \textit{Sets},\quad \mathcal{G} \longmapsto \Hom_X(\mathcal{G}, \mathcal{F})\] This functor transforms colimits into limits, hence is representable, see Injectives, Lemma 07D7. Thus there exists a quasi-coherent sheaf \(Q(\mathcal{F})\) and a functorial isomorphism \(\Hom_X(\mathcal{G}, \mathcal{F}) = \Hom_X(\mathcal{G}, Q(\mathcal{F}))\) for \(\mathcal{G}\) in \(\QCoh(\mathcal{O}_X)\). By the Yoneda lemma (Categories, Lemma 001P) the construction \(\mathcal{F} \leadsto Q(\mathcal{F})\) is functorial in \(\mathcal{F}\). By construction \(Q\) is a right adjoint to the inclusion functor. The fact that \(Q(\mathcal{F}) \to \mathcal{F}\) is an isomorphism when \(\mathcal{F}\) is quasi-coherent is a formal consequence of the fact that the inclusion functor \(\QCoh(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X)\) is fully faithful.

Sections with support in a closed subset

Given any topological space \(X\), a closed subset \(Z \subset X\), and an abelian sheaf \(\mathcal{F}\) you can take the subsheaf of sections whose support is contained in \(Z\). If \(X\) is a scheme, \(Z\) a closed subscheme, and \(\mathcal{F}\) a quasi-coherent module there is a variant where you take sections which are scheme theoretically supported on \(Z\). However, in the scheme setting you have to be careful because the resulting \(\mathcal{O}_X\)-module may not be quasi-coherent.

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. Let \(U \subset X\) be an open subscheme. The following are equivalent:

\(U\) is retrocompact in \(X\),

\(U\) is quasi-compact,

\(U\) is a finite union of affine opens, and

there exists a finite type quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_X\) such that \(X \setminus U = V(\mathcal{I})\) (set theoretically).

Proof

The equivalence of (1), (2), and (3) follows from Lemma 054D. Assume (1), (2), (3). Let \(T = X \setminus U\). By Schemes, Lemma 01J3 there exists a unique quasi-coherent sheaf of ideals \(\mathcal{J}\) cutting out the reduced induced closed subscheme structure on \(T\). Note that \(\mathcal{J}|_U = \mathcal{O}_U\) which is an \(\mathcal{O}_U\)-modules of finite type. By Lemma 01PF there exists a quasi-coherent subsheaf \(\mathcal{I} \subset \mathcal{J}\) which is of finite type and has the property that \(\mathcal{I}|_U = \mathcal{J}|_U\). Then \(X \setminus U = V(\mathcal{I})\) and we obtain (4). Conversely, if \(\mathcal{I}\) is as in (4) and \(W = \Spec(R) \subset X\) is an affine open, then \(\mathcal{I}|_W = \widetilde{I}\) for some finitely generated ideal \(I \subset R\), see Lemma 01PB. It follows that \(U \cap W = \Spec(R) \setminus V(I)\) is quasi-compact, see Algebra, Lemma 00F6. Hence \(U \subset X\) is retrocompact by Lemma 07ZL.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Consider the sheaf of \(\mathcal{O}_X\)-modules \(\mathcal{F}'\) which associates to every open \(U \subset X\) \[\mathcal{F}'(U) = \{s \in \mathcal{F}(U) \mid \mathcal{I}s = 0\}\] Assume \(\mathcal{I}\) is of finite type. Then

\(\mathcal{F}'\) is a quasi-coherent sheaf of \(\mathcal{O}_X\)-modules,

on any affine open \(U \subset X\) we have \(\mathcal{F}'(U) = \{s \in \mathcal{F}(U) \mid \mathcal{I}(U)s = 0\}\), and

\(\mathcal{F}'_x = \{s \in \mathcal{F}_x \mid \mathcal{I}_x s = 0\}\).

Proof

It is clear that the rule defining \(\mathcal{F}'\) gives a subsheaf of \(\mathcal{F}\) (the sheaf condition is easy to verify). Hence we may work locally on \(X\) to verify the other statements. In other words we may assume that \(X = \Spec(A)\), \(\mathcal{F} = \widetilde{M}\) and \(\mathcal{I} = \widetilde{I}\). It is clear that in this case \(\mathcal{F}'(U) = \{x \in M \mid Ix = 0\} =: M'\) because \(\widetilde{I}\) is generated by its global sections \(I\) which proves (2). To show \(\mathcal{F}'\) is quasi-coherent it suffices to show that for every \(f \in A\) we have \(\{x \in M_f \mid I_f x = 0\} = (M')_f\). Write \(I = (g_1, \ldots, g_t)\), which is possible because \(\mathcal{I}\) is of finite type, see Lemma 01PB. If \(x = y/f^n\) and \(I_fx = 0\), then that means that for every \(i\) there exists an \(m \geq 0\) such that \(f^mg_ix = 0\). We may choose one \(m\) which works for all \(i\) (and this is where we use that \(I\) is finitely generated). Then we see that \(f^mx \in M'\) and \(x/f^n = f^mx/f^{n + m}\) in \((M')_f\) as desired. The proof of (3) is similar and omitted.

Definition

Let \(X\) be a scheme. Let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals of finite type. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. The subsheaf \(\mathcal{F}' \subset \mathcal{F}\) defined in Lemma 01PO above is called the subsheaf of sections annihilated by \(\mathcal{I}\).

Lemma

Let \(f : X \to Y\) be a quasi-compact and quasi-separated morphism of schemes. Let \(\mathcal{I} \subset \mathcal{O}_Y\) be a quasi-coherent sheaf of ideals of finite type. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(\mathcal{F}' \subset \mathcal{F}\) be the subsheaf of sections annihilated by \(f^{-1}\mathcal{I}\mathcal{O}_X\). Then \(f_*\mathcal{F}' \subset f_*\mathcal{F}\) is the subsheaf of sections annihilated by \(\mathcal{I}\).

Proof

Omitted. (Hint: The assumption that \(f\) is quasi-compact and quasi-separated implies that \(f_*\mathcal{F}\) is quasi-coherent so that Lemma 01PO applies to \(\mathcal{I}\) and \(f_*\mathcal{F}\).)

For an abelian sheaf on a topological space we have discussed the subsheaf of sections with support in a closed subset in Modules, Remark 01AY. For quasi-coherent modules this submodule isn’t always a quasi-coherent module, but if the closed subset has a retrocompact complement, then it is.

Lemma

Let \(X\) be a scheme. Let \(Z \subset X\) be a closed subset. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Consider the sheaf of \(\mathcal{O}_X\)-modules \(\mathcal{F}'\) which associates to every open \(U \subset X\) \[\mathcal{F}'(U) = \{s \in \mathcal{F}(U) \mid \text{the support of }s\text{ is contained in }Z \cap U\}\] If \(X \setminus Z\) is a retrocompact open of \(X\), then

for an affine open \(U \subset X\) there exist a finitely generated ideal \(I \subset \mathcal{O}_X(U)\) such that \(Z \cap U = V(I)\),

for \(U\) and \(I\) as in (1) we have \(\mathcal{F}'(U) = \{x \in \mathcal{F}(U) \mid I^nx = 0 \text{ for some } n\}\),

\(\mathcal{F}'\) is a quasi-coherent sheaf of \(\mathcal{O}_X\)-modules.

Proof

Part (1) is Algebra, Lemma 00F6. Let \(U = \Spec(A)\) and \(I\) be as in (1). Then \(\mathcal{F}|_U\) is the quasi-coherent sheaf associated to some \(A\)-module \(M\). We have \[\mathcal{F}'(U) = \{x \in M \mid x = 0\text{ in }M_\mathfrak p \text{ for all }\mathfrak p \not \in Z\}.\] by Modules, Definition 01AT. Thus \(x \in \mathcal{F}'(U)\) if and only if \(V(\text{Ann}(x)) \subset V(I)\), see Algebra, Lemma 07Z5. Since \(I\) is finitely generated this is equivalent to \(I^n x = 0\) for some \(n\). This proves (2).

Proof of (3). Observe that given \(U \subset X\) open there is an exact sequence \[0 \to \mathcal{F}'(U) \to \mathcal{F}(U) \to \mathcal{F}(U \setminus Z)\] If we denote \(j : X \setminus Z \to X\) the inclusion morphism, then we observe that \(\mathcal{F}(U \setminus Z)\) is the sections of the module \(j_*(\mathcal{F}|_{X \setminus Z})\) over \(U\). Thus we have an exact sequence \[0 \to \mathcal{F}' \to \mathcal{F} \to j_*(\mathcal{F}|_{X \setminus Z})\] The restriction \(\mathcal{F}|_{X \setminus Z}\) is quasi-coherent. Hence \(j_*(\mathcal{F}|_{X \setminus Z})\) is quasi-coherent by Schemes, Lemma 01LC and our assumption that \(j\) is quasi-compact (any open immersion is separated). Hence \(\mathcal{F}'\) is quasi-coherent as a kernel of a map of quasi-coherent modules, see Schemes, Section 01LA.

Definition

Let \(X\) be a scheme. Let \(T \subset X\) be a closed subset whose complement is retrocompact in \(X\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. The quasi-coherent subsheaf \(\mathcal{F}' \subset \mathcal{F}\) defined in Lemma 07ZP is called the subsheaf of sections supported on \(T\).

Lemma

Let \(f : X \to Y\) be a quasi-compact and quasi-separated morphism of schemes. Let \(Z \subset Y\) be a closed subset such that \(Y \setminus Z\) is retrocompact in \(Y\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(\mathcal{F}' \subset \mathcal{F}\) be the subsheaf of sections supported in \(f^{-1}Z\). Then \(f_*\mathcal{F}' \subset f_*\mathcal{F}\) is the subsheaf of sections supported in \(Z\).

Proof

Omitted. (Hint: First show that \(X \setminus f^{-1}Z\) is retrocompact in \(X\) as \(Y \setminus Z\) is retrocompact in \(Y\). Hence Lemma 07ZP applies to \(f^{-1}Z\) and \(\mathcal{F}\). As \(f\) is quasi-compact and quasi-separated we see that \(f_*\mathcal{F}\) is quasi-coherent. Hence Lemma 07ZP applies to \(Z\) and \(f_*\mathcal{F}\). Finally, match the sheaves directly.)

Sections of quasi-coherent sheaves

Here is a computation of sections of a quasi-coherent sheaf on a quasi-compact open of an affine spectrum.

Lemma

Let \(A\) be a ring. Let \(I \subset A\) be a finitely generated ideal. Let \(M\) be an \(A\)-module. Then there is a canonical map \[\colim_n \Hom_A(I^n, M) \longrightarrow \Gamma(\Spec(A) \setminus V(I), \widetilde{M}).\] This map is always injective. If for all \(x \in M\) we have \(Ix = 0 \Rightarrow x = 0\) then this map is an isomorphism. In general, set \(M_n = \{x \in M \mid I^nx = 0\}\), then there is an isomorphism \[\colim_n \Hom_A(I^n, M/M_n) \longrightarrow \Gamma(\Spec(A) \setminus V(I), \widetilde{M}).\]

Proof

Since \(I^{n + 1} \subset I^n\) and \(M_n \subset M_{n + 1}\) we can use composition via these maps to get canonical maps of \(A\)-modules \[\Hom_A(I^n, M) \longrightarrow \Hom_A(I^{n + 1}, M)\] and \[\Hom_A(I^n, M/M_n) \longrightarrow \Hom_A(I^{n + 1}, M/M_{n + 1})\] which we will use as the transition maps in the systems. Given an \(A\)-module map \(\varphi : I^n \to M\), then we get a map of sheaves \(\widetilde{\varphi} : \widetilde{I^n} \to \widetilde{M}\) which we can restrict to the open \(\Spec(A) \setminus V(I)\). Since \(\widetilde{I^n}\) restricted to this open gives the structure sheaf we get an element of \(\Gamma(\Spec(A) \setminus V(I), \widetilde{M})\). We omit the verification that this is compatible with the transition maps in the system \(\Hom_A(I^n, M)\). This gives the first arrow. To get the second arrow we note that \(\widetilde{M}\) and \(\widetilde{M/M_n}\) agree over the open \(\Spec(A) \setminus V(I)\) since the sheaf \(\widetilde{M_n}\) is clearly supported on \(V(I)\). Hence we can use the same mechanism as before.

Next, we work out how to define this arrow in terms of algebra. Say \(I = (f_1, \ldots, f_t)\). Then \(\Spec(A) \setminus V(I) = \bigcup_{i = 1, \ldots, t} D(f_i)\). Hence \[0 \to \Gamma(\Spec(A) \setminus V(I), \widetilde{M}) \to \bigoplus\nolimits_i M_{f_i} \to \bigoplus\nolimits_{i, j} M_{f_if_j}\] is exact. Suppose that \(\varphi : I^n \to M\) is an \(A\)-module map. Consider the vector of elements \(\varphi(f_i^n)/f_i^n \in M_{f_i}\). It is easy to see that this vector maps to zero in the second direct sum of the exact sequence above. Whence an element of \(\Gamma(\Spec(A) \setminus V(I), \widetilde{M})\). We omit the verification that this description agrees with the one given above.

Let us show that the first arrow is injective using this description. Namely, if \(\varphi\) maps to zero, then for each \(i\) the element \(\varphi(f_i^n)/f_i^n\) is zero in \(M_{f_i}\). In other words we see that for each \(i\) we have \(f_i^m\varphi(f_i^n) = 0\) for some \(m \geq 0\). We may choose a single \(m\) which works for all \(i\). Then we see that \(\varphi(f_i^{n + m}) = 0\) for all \(i\). It is easy to see that this means that \(\varphi|_{I^{t(n + m - 1) + 1}} = 0\) in other words that \(\varphi\) maps to zero in the \(t(n + m - 1) + 1\)st term of the colimit. Hence injectivity follows.

Note that each \(M_n = 0\) in case we have \(Ix = 0 \Rightarrow x = 0\) for \(x \in M\). Thus to finish the proof of the lemma it suffices to show that the second arrow is an isomorphism.

Let us attempt to construct an inverse of the second map of the lemma. Let \(s \in \Gamma(\Spec(A) \setminus V(I), \widetilde{M})\). This corresponds to a vector \(x_i/f_i^n\) with \(x_i \in M\) of the first direct sum of the exact sequence above. Hence for each \(i, j\) there exists \(m \geq 0\) such that \(f_i^m f_j^m (f_j^n x_i - f_i^n x_j) = 0\) in \(M\). We may choose a single \(m\) which works for all pairs \(i, j\). After replacing \(x_i\) by \(f_i^mx_i\) and \(n\) by \(n + m\) we see that we get \(f_j^nx_i = f_i^nx_j\) in \(M\) for all \(i, j\). Let us introduce \[K_n = \{x \in M \mid f_1^nx = \ldots = f_t^nx = 0\}\] We claim there is an \(A\)-module map \[\varphi : I^{t(n - 1) + 1} \longrightarrow M/K_n\] which maps the monomial \(f_1^{e_1} \ldots f_t^{e_t}\) with \(\sum e_i = t(n - 1) + 1\) to the class modulo \(K_n\) of the expression \(f_1^{e_1} \ldots f_i^{e_i - n} \ldots f_t^{e_t}x_i\) where \(i\) is chosen such that \(e_i \geq n\) (note that there is at least one such \(i\)). To see that this is indeed the case suppose that \[\sum\nolimits_{E = (e_1, \ldots, e_t), |E| = t(n - 1) + 1} a_E f_1^{e_1} \ldots f_t^{e_t} = 0\] is a relation between the monomials with coefficients \(a_E\) in \(A\). Then we would map this to \[z = \sum\nolimits_{E = (e_1, \ldots, e_t), |E| = t(n - 1) + 1} a_E f_1^{e_1} \ldots f_{i(E)}^{e_{i(E)} - n} \ldots f_t^{e_t}x_{i(E)}\] where for each multiindex \(E\) we have chosen a particular \(i(E)\) such that \(e_{i(E)} \geq n\). Note that if we multiply this by \(f_j^n\) for any \(j\), then we get zero, since by the relations \(f_j^nx_i = f_i^nx_j\) above we get \[\begin{align*} f_j^nz & = \sum\nolimits_{E = (e_1, \ldots, e_t), |E| = t(n - 1) + 1} a_E f_1^{e_1} \ldots f_j^{e_j + n} \ldots f_{i(E)}^{e_{i(E)} - n} \ldots f_t^{e_t}x_{i(E)} \\ & = \sum\nolimits_{E = (e_1, \ldots, e_t), |E| = t(n - 1) + 1} a_E f_1^{e_1} \ldots f_t^{e_t}x_j = 0. \end{align*}\] Hence \(z \in K_n\) and we see that every relation gets mapped to zero in \(M/K_n\). This proves the claim.

Note that \(K_n \subset M_{t(n - 1) + 1}\). Hence the map \(\varphi\) in particular gives rise to an \(A\)-module map \(I^{t(n - 1) + 1} \to M/M_{t(n - 1) + 1}\). This proves the second arrow of the lemma is surjective. We omit the proof of injectivity.

Example

We will give two examples showing that the first displayed map of Lemma 01PM is not an isomorphism.

Let \(k\) be a field. Consider the ring \[A = k[x, y, z_1, z_2, \ldots]/(x^nz_n).\] Set \(I = (x)\) and let \(M = A\). Then the element \(y/x\) defines a section of the structure sheaf of \(\Spec(A)\) over \(D(x) = \Spec(A) \setminus V(I)\). We claim that \(y/x\) is not in the image of the canonical map \(\colim \Hom_A(I^n, A) \to A_x = \mathcal{O}(D(x))\). Namely, if so it would come from a homomorphism \(\varphi : I^n \to A\) for some \(n\). Set \(a = \varphi(x^n)\). Then we would have \(x^m(xa - x^ny) = 0\) for some \(m > 0\). This would mean that \(x^{m + 1}a = x^{m + n}y\). This would mean that \(\varphi(x^{n + m + 1}) = x^{m + n}y\). This leads to a contradiction because it would imply that \[0 = \varphi(0) = \varphi(z_{n + m + 1} x^{n + m + 1}) = x^{m + n}y z_{n + m + 1}\] which is not true in the ring \(A\).

Let \(k\) be a field. Consider the ring \[A = k[f, g, x, y, \{a_n, b_n\}_{n \geq 1}]/ (fy - gx, \{a_nf^n + b_ng^n\}_{n \geq 1}).\] Set \(I = (f, g)\) and let \(M = A\). Then \(x/f \in A_f\) and \(y/g \in A_g\) map to the same element of \(A_{fg}\). Hence these define a section \(s\) of the structure sheaf of \(\Spec(A)\) over \(D(f) \cup D(g) = \Spec(A) \setminus V(I)\). However, there is no \(n \geq 0\) such that \(s\) comes from an \(A\)-module map \(\varphi : I^n \to A\) as in the source of the first displayed arrow of Lemma 01PM. Namely, given such a module map set \(x_n = \varphi(f^n)\) and \(y_n = \varphi(g^n)\). Then \(f^mx_n = f^{n + m - 1}x\) and \(g^my_n = g^{n + m - 1}y\) for some \(m \geq 0\) (see proof of the lemma). But then we would have \(0 = \varphi(0) = \varphi(a_{n + m}f^{n + m} + b_{n + m}g^{n + m}) = a_{n + m}f^{n + m - 1}x + b_{n + m}g^{n + m - 1}y\) which is not the case in the ring \(A\).

We will improve on the following lemma in the Noetherian case, see Cohomology of Schemes, Lemma 01YB.

Lemma

Let \(X\) be a quasi-compact scheme. Let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals of finite type. Let \(Z \subset X\) be the closed subscheme defined by \(\mathcal{I}\) and set \(U = X \setminus Z\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. The canonical map \[\colim_n \Hom_{\mathcal{O}_X}(\mathcal{I}^n, \mathcal{F}) \longrightarrow \Gamma(U, \mathcal{F})\] is injective. Assume further that \(X\) is quasi-separated. Let \(\mathcal{F}_n \subset \mathcal{F}\) be subsheaf of sections annihilated by \(\mathcal{I}^n\). The canonical map \[\colim_n \Hom_{\mathcal{O}_X}(\mathcal{I}^n, \mathcal{F}/\mathcal{F}_n) \longrightarrow \Gamma(U, \mathcal{F})\] is an isomorphism.

Proof

Let \(\Spec(A) = W \subset X\) be an affine open. Write \(\mathcal{F}|_W = \widetilde{M}\) for some \(A\)-module \(M\) and \(\mathcal{I}|_W = \widetilde{I}\) for some finite type ideal \(I \subset A\). Restricting the first displayed map of the lemma to \(W\) we obtain the first displayed map of Lemma 01PM. Since we can cover \(X\) by a finite number of affine opens this proves the first displayed map of the lemma is injective.

We have \(\mathcal{F}_n|_W = \widetilde{M_n}\) where \(M_n \subset M\) is defined as in Lemma 01PM (details omitted). The lemma guarantees that we have a bijection \[\colim_n \Hom_{\mathcal{O}_W}( \mathcal{I}^n|_W, (\mathcal{F}/\mathcal{F}_n)|_W) \longrightarrow \Gamma(U \cap W, \mathcal{F})\] for any such affine open \(W\).

To see the second displayed arrow of the lemma is bijective, we choose a finite affine open covering \(X = \bigcup_{j = 1, \ldots, m} W_j\). The injectivity follows immediately from the above and the finiteness of the covering. If \(X\) is quasi-separated, then for each pair \(j, j'\) we choose a finite affine open covering \[W_j \cap W_{j'} = \bigcup\nolimits_{k = 1, \ldots, m_{jj'}} W_{jj'k}.\] Let \(s \in \Gamma(U, \mathcal{F})\). As seen above for each \(j\) there exists an \(n_j\) and a map \(\varphi_j : \mathcal{I}^{n_j}|_{W_j} \to (\mathcal{F}/\mathcal{F}_{n_j})|_{W_j}\) which corresponds to \(s|_{U \cap W_j}\). By the same token for each triple \((j, j', k)\) there exists an integer \(n_{jj'k}\) such that the restriction of \(\varphi_j\) and \(\varphi_{j'}\) as maps \(\mathcal{I}^{n_{jj'k}} \to \mathcal{F}/\mathcal{F}_{n_{jj'k}}\) agree over \(W_{jj'k}\). Let \(n = \max\{n_j, n_{jj'k}\}\) and we see that the \(\varphi_j\) glue as maps \(\mathcal{I}^n \to \mathcal{F}/\mathcal{F}_n\) over \(X\). This proves surjectivity of the map.

Ample invertible sheaves

Recall from Modules, Lemma 01CY that given an invertible sheaf \(\mathcal{L}\) on a locally ringed space \(X\), and given a global section \(s\) of \(\mathcal{L}\) the set \(X_s = \{x \in X \mid s \not \in \mathfrak m_x\mathcal{L}_x\}\) is open. A general remark is that \(X_s \cap X_{s'} = X_{ss'}\), where \(ss'\) denote the section \(s \otimes s' \in \Gamma(X, \mathcal{L} \otimes \mathcal{L}')\).

Definition

Let \(X\) be a scheme. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. We say \(\mathcal{L}\) is ample if

\(X\) is quasi-compact, and

for every \(x \in X\) there exists an \(n \geq 1\) and \(s \in \Gamma(X, \mathcal{L}^{\otimes n})\) such that \(x \in X_s\) and \(X_s\) is affine.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Let \(n \geq 1\). Then \(\mathcal{L}\) is ample if and only if \(\mathcal{L}^{\otimes n}\) is ample.

Proof

This follows from the fact that \(X_{s^n} = X_s\).

Lemma

Let \(X\) be a scheme. Let \(\mathcal{L}\) be an ample invertible \(\mathcal{O}_X\)-module. For any closed subscheme \(Z \subset X\) the restriction of \(\mathcal{L}\) to \(Z\) is ample.

Proof

This is clear since a closed subset of a quasi-compact space is quasi-compact and a closed subscheme of an affine scheme is affine (see Schemes, Lemma 01IH).

Lemma

Let \(X\) be a scheme. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Let \(s \in \Gamma(X, \mathcal{L})\). For any affine \(U \subset X\) the intersection \(U \cap X_s\) is affine.

Proof

This translates into the following algebra problem. Let \(R\) be a ring. Let \(N\) be an invertible \(R\)-module (i.e., locally free of rank 1). Let \(s \in N\) be an element. Then \(V = \{\mathfrak p \mid s \not \in \mathfrak p N\}\) is an affine open subset of \(\Spec(R)\).

Let \(A = \bigoplus_{n \geq 0} A_n\) be the symmetric algebra of \(N\) (which is commutative) and view \(s\) as an element of \(A_1\). Set \(B = A/(s - 1)A\). This is an \(R\)-algebra whose construction commutes with any base change \(R \to R'\). Thus \(B' = B \otimes_R R'\) is the zero ring if \(s\) maps to zero in \(N' = N \otimes_R R'\). It follows that if \(x \in \Spec(R) \setminus V\), then \(B \otimes_R \kappa(x) = 0\). We conclude that \(\Spec(B) \to \Spec(R)\) factors through \(V\) as the fibres over \(x \not \in V\) are empty. On the other hand, if \(\Spec(R') \subset V\) is an affine open, then \(s\) maps to a basis element of \(N'\) and we see that \(B' = R'[s]/(s - 1) \cong R'\). It follows that \(\Spec(B) \to V\) is an isomorphism and \(V\) is indeed affine.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{L}\) and \(\mathcal{M}\) be invertible \(\mathcal{O}_X\)-modules. If

\(\mathcal{L}\) is ample, and

the open sets \(X_t\) where \(t \in \Gamma(X, \mathcal{M}^{\otimes m})\) for \(m > 0\) cover \(X\),

then \(\mathcal{L} \otimes \mathcal{M}\) is ample.

Proof

We check the conditions of Definition 01PS. As \(\mathcal{L}\) is ample we see that \(X\) is quasi-compact. Let \(x \in X\). Choose \(n \geq 1\), \(m \geq 1\), \(s \in \Gamma(X, \mathcal{L}^{\otimes n})\), and \(t \in \Gamma(X, \mathcal{M}^{\otimes m})\) such that \(x \in X_s\), \(x \in X_t\) and \(X_s\) is affine. Then \(s^mt^n \in \Gamma(X, (\mathcal{L} \otimes \mathcal{M})^{\otimes nm})\), \(x \in X_{s^mt^n}\), and \(X_{s^mt^n}\) is affine by Lemma 01PV.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Assume the open sets \(X_s\), where \(s \in \Gamma(X, \mathcal{L}^{\otimes n})\) and \(n \geq 1\), form a basis for the topology on \(X\). Then among those opens, the open sets \(X_s\) which are affine form a basis for the topology on \(X\).

Proof

Let \(x \in X\). Choose an affine open neighbourhood \(\Spec(R) = U \subset X\) of \(x\). By assumption, there exists a \(n \geq 1\) and a \(s \in \Gamma(X, \mathcal{L}^{\otimes n})\) such that \(X_s \subset U\). By Lemma 01PV above the intersection \(X_s = U \cap X_s\) is affine. Since \(U\) can be chosen arbitrarily small we win.

Lemma

Let \(X\) be a scheme and \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Assume for every point \(x\) of \(X\) there exists \(n \geq 1\) and \(s \in \Gamma(X, \mathcal{L}^{\otimes n})\) such that \(x \in X_s\) and \(X_s\) is affine. Then \(X\) is separated.

Proof

We show first that \(X\) is quasi-separated. By assumption we can find a covering of \(X\) by affine opens of the form \(X_s\). By Lemma 01PV, the intersection of any two such sets is affine, so Schemes, Lemma 01KO implies that \(X\) is quasi-separated.

To show that \(X\) is separated, we can use the valuative criterion, Schemes, Lemma 01L0. Thus, let \(A\) be a valuation ring with fraction field \(K\) and consider two morphisms \(f, g : \Spec(A) \to X\) such that the two compositions \(\Spec(K) \to \Spec(A) \to X\) agree. As \(A\) is local, there exists \(p, q \ge 1\), \(s \in \Gamma(X, \mathcal{L}^{\otimes p})\), and \(t \in \Gamma(X, \mathcal{L}^{\otimes q})\) such that \(X_s\) and \(X_t\) are affine, \(f(\Spec A) \subseteq X_s\), and \(g(\Spec A) \subseteq X_t\). We now replace \(s\) by \(s^q\), \(t\) by \(t^p\), and \(\mathcal{L}\) by \(\mathcal{L}^{\otimes pq}\). This is harmless as \(X_s = X_{s^q}\) and \(X_t = X_{t^p}\), and now \(s\) and \(t\) are both sections of the same sheaf \(\mathcal{L}\).

The quasi-coherent module \(f^*\mathcal{L}\) corresponds to an \(A\)-module \(M\) and \(g^*\mathcal{L}\) corresponds to an \(A\)-module \(N\) by our classification of quasi-coherent modules over affine schemes (Schemes, Lemma 01IA). The \(A\)-modules \(M\) and \(N\) are locally free of rank \(1\) (Lemma 05JM) and as \(A\) is local they are free (Algebra, Lemma 00JJ). Therefore we may identify \(M\) and \(N\) with \(A\)-submodules of \(M \otimes_A K\) and \(N \otimes_A K\). The equality \(f|_{\Spec(K)} = g|_{\Spec(K)}\) determines an isomorphism \(\phi \colon M \otimes_A K \to N \otimes_A K\).

Let \(x \in M\) and \(y \in N\) be the elements corresponding to the pullback of \(s\) along \(f\) and \(g\), respectively. These satisfy \(\phi(x \otimes 1) = y \otimes 1\). The image of \(f\) is contained in \(X_s\), so \(x \not\in \mathfrak{m}_A M\), that is, \(x\) generates \(M\). Hence \(\phi\) determines an isomorphism of \(M\) with the submodule of \(N\) generated by \(y\). Arguing symmetrically using \(t\), \(\phi^{-1}\) determines an isomorphism of \(N\) with a submodule of \(M\). Consequently \(\phi\) restricts to an isomorphism of \(M\) and \(N\). Since \(x\) generates \(M\), its image \(y\) generates \(N\), implying \(y \not\in \mathfrak{m}_A N\). Therefore \(g(\Spec(A)) \subseteq X_s\). Because \(X_s\) is affine, it is separated by Schemes, Lemma 01KN, and we conclude \(f = g\).

Lemma

Let \(X\) be a scheme. If there exists an ample invertible sheaf on \(X\) then \(X\) is separated.

Proof

Follows immediately from Lemma 01PY and Definition 01PS.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Set \(S = \Gamma_*(X, \mathcal{L})\) as a graded ring. If every point of \(X\) is contained in one of the open subschemes \(X_s\), for some \(s \in S_{+}\) homogeneous, then there is a canonical morphism of schemes \[f : X \longrightarrow Y = \text{Proj}(S),\] to the homogeneous spectrum of \(S\) (see Constructions, Section 01M3). This morphism has the following properties

\(f^{-1}(D_{+}(s)) = X_s\) for any \(s \in S_{+}\) homogeneous,

there are \(\mathcal{O}_X\)-module maps \(f^*\mathcal{O}_Y(n) \to \mathcal{L}^{\otimes n}\) compatible with multiplication maps, see Constructions, Equation (01MO),

the composition \(S_n \to \Gamma(Y, \mathcal{O}_Y(n)) \to \Gamma(X, \mathcal{L}^{\otimes n})\) is the identity map, and

for every \(x \in X\) there is an integer \(d \geq 1\) and an open neighbourhood \(U \subset X\) of \(x\) such that \(f^*\mathcal{O}_Y(dn)|_U \to \mathcal{L}^{\otimes dn}|_U\) is an isomorphism for all \(n \in \mathbf{Z}\).

Proof

Denote \(\psi : S \to \Gamma_*(X, \mathcal{L})\) the identity map. We are going to use the triple \((U(\psi), r_{\mathcal{L}, \psi}, \theta)\) of Constructions, Lemma 01NK. By assumption the open subscheme \(U(\psi)\) of equals \(X\). Hence \(r_{\mathcal{L}, \psi} : U(\psi) \to Y\) is defined on all of \(X\). We set \(f = r_{\mathcal{L}, \psi}\). The maps in part (2) are the components of \(\theta\). Part (3) follows from condition (2) in the lemma cited above. Part (1) follows from (3) combined with condition (1) in the lemma cited above. Part (4) follows from the last statement in Constructions, Lemma 01NK since the map \(\alpha\) mentioned there is an isomorphism.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Set \(S = \Gamma_*(X, \mathcal{L})\). Assume (a) every point of \(X\) is contained in one of the open subschemes \(X_s\), for some \(s \in S_{+}\) homogeneous, and (b) \(X\) is quasi-compact. Then the canonical morphism of schemes \(f : X \longrightarrow \text{Proj}(S)\) of Lemma 01PZ above is quasi-compact with dense image.

Proof

To prove \(f\) is quasi-compact it suffices to show that \(f^{-1}(D_{+}(s))\) is quasi-compact for any \(s \in S_{+}\) homogeneous. Write \(X = \bigcup_{i = 1, \ldots, n} X_i\) as a finite union of affine opens. By Lemma 01PV each intersection \(X_s \cap X_i\) is affine. Hence \(X_s = \bigcup_{i = 1, \ldots, n} X_s \cap X_i\) is quasi-compact. Assume that the image of \(f\) is not dense to get a contradiction. Then, since the opens \(D_+(s)\) with \(s \in S_+\) homogeneous form a basis for the topology on \(\text{Proj}(S)\), we can find such an \(s\) with \(D_+(s) \not = \emptyset\) and \(f(X) \cap D_+(s) = \emptyset\). By Lemma 01PZ this means \(X_s = \emptyset\). By Lemma 01PW this means that a power \(s^n\) is the zero section of \(\mathcal{L}^{\otimes n\deg(s)}\). This in turn means that \(D_+(s) = \emptyset\) which is the desired contradiction.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Set \(S = \Gamma_*(X, \mathcal{L})\). Assume \(\mathcal{L}\) is ample. Then the canonical morphism of schemes \(f : X \longrightarrow \text{Proj}(S)\) of Lemma 01PZ is an open immersion with dense image.

Proof

By Lemma 01PY we see that \(X\) is quasi-separated. Choose finitely many \(s_1, \ldots, s_n \in S_{+}\) homogeneous such that \(X_{s_i}\) are affine, and \(X = \bigcup X_{s_i}\). Say \(s_i\) has degree \(d_i\). The inverse image of \(D_{+}(s_i)\) under \(f\) is \(X_{s_i}\), see Lemma 01PZ. By Lemma 01PW the ring map \[(S^{(d_i)})_{(s_i)} = \Gamma(D_{+}(s_i), \mathcal{O}_{\text{Proj}(S)}) \longrightarrow \Gamma(X_{s_i}, \mathcal{O}_X)\] is an isomorphism. Hence \(f\) induces an isomorphism \(X_{s_i} \to D_{+}(s_i)\). Thus \(f\) is an isomorphism of \(X\) onto the open subscheme \(\bigcup_{i = 1, \ldots, n} D_{+}(s_i)\) of \(\text{Proj}(S)\). The image is dense by Lemma 01Q0.

Lemma

Let \(X\) be a scheme. Let \(S\) be a graded ring. Assume \(X\) is quasi-compact, and assume there exists an open immersion \[j : X \longrightarrow Y = \text{Proj}(S).\] Then \(j^*\mathcal{O}_Y(d)\) is an invertible ample sheaf for some \(d > 0\).

Proof

This is Constructions, Lemma 01MW.

Proposition

Let \(X\) be a quasi-compact scheme. Let \(\mathcal{L}\) be an invertible sheaf on \(X\). Set \(S = \Gamma_*(X, \mathcal{L})\). The following are equivalent:

\(\mathcal{L}\) is ample,

the open sets \(X_s\), with \(s \in S_{+}\) homogeneous, cover \(X\) and the associated morphism \(X \to \text{Proj}(S)\) is an open immersion,

the open sets \(X_s\), with \(s \in S_{+}\) homogeneous, form a basis for the topology of \(X\),

the open sets \(X_s\), with \(s \in S_{+}\) homogeneous, which are affine form a basis for the topology of \(X\),

for every quasi-coherent sheaf \(\mathcal{F}\) on \(X\) the sum of the images of the canonical maps \[\Gamma(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) \otimes_{\mathbf{Z}} \mathcal{L}^{\otimes -n} \longrightarrow \mathcal{F}\] with \(n \geq 1\) equals \(\mathcal{F}\),

same property as (01Q8) with \(\mathcal{F}\) ranging over all quasi-coherent sheaves of ideals,

\(X\) is quasi-separated and for every quasi-coherent sheaf \(\mathcal{F}\) of finite type on \(X\) there exists an integer \(n_0\) such that \(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}\) is globally generated for all \(n \geq n_0\),

\(X\) is quasi-separated and for every quasi-coherent sheaf \(\mathcal{F}\) of finite type on \(X\) there exist integers \(n > 0\), \(k \geq 0\) such that \(\mathcal{F}\) is a quotient of a direct sum of \(k\) copies of \(\mathcal{L}^{\otimes - n}\), and

same as in (01QB) with \(\mathcal{F}\) ranging over all sheaves of ideals of finite type on \(X\).

Proof

Lemma 01Q1 is (01Q4) \(\Rightarrow\) (01Q5). Lemmas 01PT and 01Q2 provide the implication (01Q4) \(\Leftarrow\) (01Q5). The implications (01Q5) \(\Rightarrow\) (01Q7) \(\Rightarrow\) (01Q6) are clear from Constructions, Section 01M3. Lemma 01PX is (01Q6) \(\Rightarrow\) (01Q4). Thus we see that the first 4 conditions are all equivalent.

Assume the equivalent conditions (1) – (4). Note that in particular \(X\) is separated (as an open subscheme of the separated scheme \(\text{Proj}(S)\)). Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(X\). Choose \(s \in S_{+}\) homogeneous such that \(X_s\) is affine. We claim that any section \(m \in \Gamma(X_s, \mathcal{F})\) is in the image of one of the maps displayed in (01Q8) above. This will imply (01Q8) since these affines \(X_s\) cover \(X\). Namely, by Lemma 01PW we may write \(m\) as the image of \(m' \otimes s^{-n}\) for some \(n \geq 1\), some \(m' \in \Gamma(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n})\). This proves the claim.

Clearly (01Q8) \(\Rightarrow\) (01Q9). Let us assume (01Q9) and prove \(\mathcal{L}\) is ample. Pick \(x \in X\). Let \(U \subset X\) be an affine open which contains \(x\). Set \(Z = X \setminus U\). We may think of \(Z\) as a reduced closed subscheme, see Schemes, Section 01IZ. Let \(\mathcal{I} \subset \mathcal{O}_X\) be the quasi-coherent sheaf of ideals corresponding to the closed subscheme \(Z\). By assumption (01Q9), there exists an \(n \geq 1\) and a section \(s \in \Gamma(X, \mathcal{I} \otimes \mathcal{L}^{\otimes n})\) such that \(s\) does not vanish at \(x\) (more precisely such that \(s \not \in \mathfrak m_x \mathcal{I}_x \otimes \mathcal{L}_x^{\otimes n}\)). We may think of \(s\) as a section of \(\mathcal{L}^{\otimes n}\). Since it clearly vanishes along \(Z\) we see that \(X_s \subset U\). Hence \(X_s\) is affine, see Lemma 01PV. This proves that \(\mathcal{L}\) is ample. At this point we have proved that (1) – (6) are equivalent.

Assume the equivalent conditions (1) – (6). In the following we will use the fact that the tensor product of two sheaves of modules which are globally generated is globally generated without further mention (see Modules, Lemma 01AO). By (1) we can find elements \(s_i \in S_{d_i}\) with \(d_i \geq 1\) such that \(X = \bigcup_{i = 1, \ldots, n} X_{s_i}\). Set \(d = d_1\ldots d_n\). It follows that \(\mathcal{L}^{\otimes d}\) is globally generated by \[s_1^{d/d_1}, \ldots, s_n^{d/d_n}.\] This means that if \(\mathcal{L}^{\otimes j}\) is globally generated then so is \(\mathcal{L}^{\otimes j + dn}\) for all \(n \geq 0\). Fix a \(j \in \{0, \ldots, d - 1\}\). For any point \(x \in X\) there exists an \(n \geq 1\) and a global section \(s\) of \(\mathcal{L}^{j + dn}\) which does not vanish at \(x\), as follows from (01Q8) applied to \(\mathcal{F} = \mathcal{L}^{\otimes j}\) and ample invertible sheaf \(\mathcal{L}^{\otimes d}\). Since \(X\) is quasi-compact there we may find a finite list of integers \(n_i\) and global sections \(s_i\) of \(\mathcal{L}^{\otimes j + dn_i}\) which do not vanish at any point of \(X\). Since \(\mathcal{L}^{\otimes d}\) is globally generated this means that \(\mathcal{L}^{\otimes j + dn}\) is globally generated where \(n = \max\{n_i\}\). Since we proved this for every congruence class mod \(d\) we conclude that there exists an \(n_0 = n_0(\mathcal{L})\) such that \(\mathcal{L}^{\otimes n}\) is globally generated for all \(n \geq n_0\). At this point we see that if \(\mathcal{F}\) is globally generated then so is \(\mathcal{F} \otimes \mathcal{L}^{\otimes n}\) for all \(n \geq n_0\).

We continue to assume the equivalent conditions (1) – (6). Let \(\mathcal{F}\) be a quasi-coherent sheaf of \(\mathcal{O}_X\)-modules of finite type. Denote \(\mathcal{F}_n \subset \mathcal{F}\) the image of the canonical map of (01Q8). By construction \(\mathcal{F}_n \otimes \mathcal{L}^{\otimes n}\) is globally generated. By (01Q8) we see \(\mathcal{F}\) is the sum of the subsheaves \(\mathcal{F}_n\), \(n \geq 1\). By Modules, Lemma 01BB we see that \(\mathcal{F} = \sum_{n = 1, \ldots, N} \mathcal{F}_n\) for some \(N \geq 1\). It follows that \(\mathcal{F} \otimes \mathcal{L}^{\otimes n}\) is globally generated whenever \(n \geq N + n_0(\mathcal{L})\) with \(n_0(\mathcal{L})\) as above. We conclude that (1) – (6) implies (01QA).

Assume (01QA). Let \(\mathcal{F}\) be a quasi-coherent sheaf of \(\mathcal{O}_X\)-modules of finite type. By (01QA) there exists an integer \(n \geq 1\) such that the canonical map \[\Gamma(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) \otimes_{\mathbf{Z}} \mathcal{L}^{\otimes -n} \longrightarrow \mathcal{F}\] is surjective. Let \(I\) be the set of finite subsets of \(\Gamma(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n})\) partially ordered by inclusion. Then \(I\) is a directed partially ordered set. For \(i = \{s_1, \ldots, s_{r(i)}\}\) let \(\mathcal{F}_i \subset \mathcal{F}\) be the image of the map \[\bigoplus\nolimits_{j = 1, \ldots, r(i)} \mathcal{L}^{\otimes -n} \longrightarrow \mathcal{F}\] which is multiplication by \(s_j\) on the \(j\)th factor. The surjectivity above implies that \(\mathcal{F} = \colim_{i \in I} \mathcal{F}_i\). Hence Modules, Lemma 01BB applies and we conclude that \(\mathcal{F} = \mathcal{F}_i\) for some \(i\). Hence we have proved (01QB). In other words, (01QA) \(\Rightarrow\) (01QB).

The implication (01QB) \(\Rightarrow\) (01QC) is trivial.

Finally, assume (01QC). Let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. By Lemma 01PG (this is where we use the condition that \(X\) be quasi-separated) we see that \(\mathcal{I} = \colim_\alpha I_\alpha\) with each \(I_\alpha\) quasi-coherent of finite type. Since by assumption each of the \(I_\alpha\) is a quotient of negative tensor powers of \(\mathcal{L}\) we conclude the same for \(\mathcal{I}\) (but of course without the finiteness or boundedness of the powers). Hence we conclude that (01QC) implies (01Q9). This ends the proof of the proposition.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{L}\) be an ample invertible \(\mathcal{O}_X\)-module. Let \(i : X' \to X\) be a morphism of schemes. Assume at least one of the following conditions holds

\(i\) is a quasi-compact immersion,

\(X'\) is quasi-compact and \(i\) is an immersion,

\(i\) is quasi-compact and induces a homeomorphism between \(X'\) and \(i(X')\),

\(X'\) is quasi-compact and \(i\) induces a homeomorphism between \(X'\) and \(i(X')\).

Then \(i^*\mathcal{L}\) is ample on \(X'\).

Proof

Observe that in cases (1) and (3) the scheme \(X'\) is quasi-compact as \(X\) is quasi-compact by Definition 01PS. Thus it suffices to prove (2) and (4). Since (2) is a special case of (4) it suffices to prove (4).

Assume condition (4) holds. For \(s \in \Gamma(X, \mathcal{L}^{\otimes d})\) denote \(s' = i^*s\) the pullback of \(s\) to \(X'\). Note that \(s'\) is a section of \((i^*\mathcal{L})^{\otimes d}\). By Proposition 01Q3 the opens \(X_s\), for \(s \in \Gamma(X, \mathcal{L}^{\otimes d})\), form a basis for the topology on \(X\). Since \(X'_{s'} = i^{-1}(X_s)\) by Modules, Remark 0H78 and since \(X' \to i(X')\) is a homeomorphism, we conclude the opens \(X'_{s'}\) form a basis for the topology of \(X'\). Hence \(i^*\mathcal{L}\) is ample by Proposition 01Q3.

Lemma

Let \(S\) be a quasi-separated scheme. Let \(X\), \(Y\) be schemes over \(S\). Let \(\mathcal{L}\) be an ample invertible \(\mathcal{O}_X\)-module and let \(\mathcal{N}\) be an ample invertible \(\mathcal{O}_Y\)-module. Then \(\mathcal{M} = \text{pr}_1^*\mathcal{L} \otimes_{\mathcal{O}_{X \times_S Y}} \text{pr}_2^*\mathcal{N}\) is an ample invertible sheaf on \(X \times_S Y\).

Proof

The morphism \(i : X \times_S Y \to X \times Y\) is a quasi-compact immersion, see Schemes, Lemma 01KR. On the other hand, \(\mathcal{M}\) is the pullback by \(i\) of the corresponding invertible module on \(X \times Y\). By Lemma 0B3E it suffices to prove the lemma for \(X \times Y\). We check (1) and (2) of Definition 01PS for \(\mathcal{M}\) on \(X \times Y\).

Since \(X\) and \(Y\) are quasi-compact, so is \(X \times Y\). Let \(z \in X \times Y\) be a point. Let \(x \in X\) and \(y \in Y\) be the projections. Choose \(n > 0\) and \(s \in \Gamma(X, \mathcal{L}^{\otimes n})\) such that \(X_s\) is an affine open neighbourhood of \(x\). Choose \(m > 0\) and \(t \in \Gamma(Y, \mathcal{N}^{\otimes m})\) such that \(Y_t\) is an affine open neighbourhood of \(y\). Then \(r = \text{pr}_1^*s \otimes \text{pr}_2^*t\) is a section of \(\mathcal{M}\) with \((X \times Y)_r = X_s \times Y_t\). This is an affine open neighbourhood of \(z\) and the proof is complete.

Affine and quasi-affine schemes

Lemma

Let \(X\) be a scheme. Then \(X\) is quasi-affine if and only if \(\mathcal{O}_X\) is ample.

Proof

Suppose that \(X\) is quasi-affine. Set \(A = \Gamma(X, \mathcal{O}_X)\). Consider the open immersion \[j : X \longrightarrow \Spec(A)\] from Lemma 01P9. Note that \(\Spec(A) = \text{Proj}(A[T])\), see Constructions, Example 01MI. Hence we can apply Lemma 01Q2 to deduce that \(\mathcal{O}_X\) is ample.

Suppose that \(\mathcal{O}_X\) is ample. Note that \(\Gamma_*(X, \mathcal{O}_X) \cong A[T]\) as graded rings. Hence the result follows from Lemmas 01Q1 and 01P9 taking into account that \(\Spec(A) = \text{Proj}(A[T])\) for any ring \(A\) as seen above.

Lemma

Let \(X\) be a quasi-affine scheme. For any quasi-compact immersion \(i : X' \to X\) the scheme \(X'\) is quasi-affine.

Proof

This can be proved directly without making use of the material on ample invertible sheaves; we urge the reader to do this on a napkin. Since \(X\) is quasi-affine, we have that \(\mathcal{O}_X\) is ample by Lemma 01QE. Then \(\mathcal{O}_{X'}\) is ample by Lemma 0B3E. Then \(X'\) is quasi-affine by Lemma 01QE.

Lemma

Let \(X\) be a scheme. Suppose that there exist finitely many elements \(f_1, \ldots, f_n \in \Gamma(X, \mathcal{O}_X)\) such that

each \(X_{f_i}\) is an affine open of \(X\), and

the ideal generated by \(f_1, \ldots, f_n\) in \(\Gamma(X, \mathcal{O}_X)\) is equal to the unit ideal.

Then \(X\) is affine.

Proof

Assume we have \(f_1, \ldots, f_n\) as in the lemma. We may write \(1 = \sum g_i f_i\) for some \(g_j \in \Gamma(X, \mathcal{O}_X)\) and hence it is clear that \(X = \bigcup X_{f_i}\). (The \(f_i\)’s cannot all vanish at a point.) Since each \(X_{f_i}\) is quasi-compact (being affine) it follows that \(X\) is quasi-compact. Hence we see that \(X\) is quasi-affine by Lemma 01QE above. Consider the open immersion \[j : X \to \Spec(\Gamma(X, \mathcal{O}_X)),\] see Lemma 01P9. The inverse image of the standard open \(D(f_i)\) on the right hand side is equal to \(X_{f_i}\) on the left hand side and the morphism \(j\) induces an isomorphism \(X_{f_i} \cong D(f_i)\), see Lemma 01P8. Since the \(f_i\) generate the unit ideal we see that \(\Spec(\Gamma(X, \mathcal{O}_X)) = \bigcup_{i = 1, \ldots, n} D(f_i)\). Thus \(j\) is an isomorphism.

Quasi-coherent sheaves and ample invertible sheaves

Theme of this section: in the presence of an ample invertible sheaf every quasi-coherent sheaf comes from a graded module.

Situation

Let \(X\) be a scheme. Let \(\mathcal{L}\) be an ample invertible sheaf on \(X\). Set \(S = \Gamma_*(X, \mathcal{L})\) as a graded ring. Set \(Y = \text{Proj}(S)\). Let \(f : X \to Y\) be the canonical morphism of Lemma 01PZ. It comes equipped with a \(\mathbf{Z}\)-graded \(\mathcal{O}_X\)-algebra map \(\bigoplus f^*\mathcal{O}_Y(n) \to \bigoplus \mathcal{L}^{\otimes n}\).

The following lemma is really a special case of the next lemma but it seems like a good idea to point out its validity first.

Lemma

In Situation 01QH. The canonical morphism \(f : X \to Y\) maps \(X\) into the open subscheme \(W = W_1 \subset Y\) where \(\mathcal{O}_Y(1)\) is invertible and where all multiplication maps \(\mathcal{O}_Y(n) \otimes_{\mathcal{O}_Y} \mathcal{O}_Y(m) \to \mathcal{O}_Y(n + m)\) are isomorphisms (see Constructions, Lemma 01MU). Moreover, the maps \(f^*\mathcal{O}_Y(n) \to \mathcal{L}^{\otimes n}\) are all isomorphisms.

Proof

By Proposition 01Q3 there exists an integer \(n_0\) such that \(\mathcal{L}^{\otimes n}\) is globally generated for all \(n \geq n_0\). Let \(x \in X\) be a point. By the above we can find \(a \in S_{n_0}\) and \(b \in S_{n_0 + 1}\) such that \(a\) and \(b\) do not vanish at \(x\). Hence \(f(x) \in D_{+}(a) \cap D_{+}(b) = D_{+}(ab)\). By Constructions, Lemma 01MU we see that \(f(x) \in W_1\) as desired. By Constructions, Lemma 01NK which was used in the construction of the map \(f\) the maps \(f^*\mathcal{O}_Y(n_0) \to \mathcal{L}^{\otimes n_0}\) and \(f^*\mathcal{O}_Y(n_0 + 1) \to \mathcal{L}^{\otimes n_0 + 1}\) are isomorphisms in a neighbourhood of \(x\). By compatibility with the algebra structure and the fact that \(f\) maps into \(W\) we conclude all the maps \(f^*\mathcal{O}_Y(n) \to \mathcal{L}^{\otimes n}\) are isomorphisms in a neighbourhood of \(x\). Hence we win.

Recall from Modules, Definition 01CV that given a locally ringed space \(X\), an invertible sheaf \(\mathcal{L}\), and a \(\mathcal{O}_X\)-module \(\mathcal{F}\) we have the graded \(\Gamma_*(X, \mathcal{L})\)-module \[\Gamma_*(X, \mathcal{L}, \mathcal{F}) = \bigoplus\nolimits_{n \in \mathbf{Z}} \Gamma(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}).\] The following lemma says that, in Situation 01QH, we can recover a quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) from this graded module. Take a look also at Constructions, Lemma 03GM where we prove this lemma in the special case \(X = \mathbf{P}^n_R\).

Lemma

In Situation 01QH. Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(X\). Set \(M = \Gamma_*(X, \mathcal{L}, \mathcal{F})\) as a graded \(S\)-module. There are isomorphisms \[f^*\widetilde{M} \longrightarrow \mathcal{F}\] functorial in \(\mathcal{F}\) such that \(M_0 \to \Gamma(\text{Proj}(S), \widetilde{M}) \to \Gamma(X, \mathcal{F})\) is the identity map.

Proof

Let \(s \in S_{+}\) be homogeneous such that \(X_s\) is affine open in \(X\). Recall that \(\widetilde{M}|_{D_{+}(s)}\) corresponds to the \(S_{(s)}\)-module \(M_{(s)}\), see Constructions, Lemma 01M7. Recall that \(f^{-1}(D_{+}(s)) = X_s\). As \(X\) carries an ample invertible sheaf it is quasi-compact and quasi-separated, see Section 01PR. By Lemma 01PW there is a canonical isomorphism \(M_{(s)} = \Gamma_*(X, \mathcal{L}, \mathcal{F})_{(s)} \to \Gamma(X_s, \mathcal{F})\). Since \(\mathcal{F}\) is quasi-coherent this leads to a canonical isomorphism \[f^*\widetilde{M}|_{X_s} \to \mathcal{F}|_{X_s}\] Since \(\mathcal{L}\) is ample on \(X\) we know that \(X\) is covered by the affine opens of the form \(X_s\). Hence it suffices to prove that the displayed maps glue on overlaps. Proof of this is omitted.

Remark

With assumptions and notation of Lemma 01QJ. Denote the displayed map of the lemma by \(\theta_\mathcal{F}\). Note that the isomorphism \(f^*\mathcal{O}_Y(n) \to \mathcal{L}^{\otimes n}\) of Lemma 01QI is just \(\theta_{\mathcal{L}^{\otimes n}}\). Consider the multiplication maps \[\widetilde{M} \otimes_{\mathcal{O}_Y} \mathcal{O}_Y(n) \longrightarrow \widetilde{M(n)}\] see Constructions, Equation (01MQ). Pull this back to \(X\) and consider \[\xymatrix{ f^*\widetilde{M} \otimes_{\mathcal{O}_X} f^*\mathcal{O}_Y(n) \ar[r] \ar[d]_{\theta_\mathcal{F} \otimes \theta_{\mathcal{L}^{\otimes n}}} & f^*\widetilde{M(n)} \ar[d]^{\theta_{\mathcal{F} \otimes \mathcal{L}^{\otimes n}}} \\ \mathcal{F} \otimes \mathcal{L}^{\otimes n} \ar[r]^{\text{id}} & \mathcal{F} \otimes \mathcal{L}^{\otimes n} }\] Here we have used the obvious identification \(M(n) = \Gamma_*(X, \mathcal{L}, \mathcal{F} \otimes \mathcal{L}^{\otimes n})\). This diagram commutes. Proof omitted.

It should be possible to deduce the following lemma from Lemma 01QJ (or conversely) but it seems simpler to just repeat the proof.

Lemma

Let \(S\) be a graded ring such that \(X = \text{Proj}(S)\) is quasi-compact. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Set \(M = \bigoplus_{n \in \mathbf{Z}} \Gamma(X, \mathcal{F}(n))\) as a graded \(S\)-module, see Constructions, Section 01MM. The map \[\widetilde{M} \longrightarrow \mathcal{F}\] of Constructions, Lemma 0B5I is an isomorphism. If \(X\) is covered by standard opens \(D_+(f)\) where \(f\) has degree \(1\), then the induced maps \(M_n \to \Gamma(X, \mathcal{F}(n))\) are the identity maps.

Proof

Since \(X\) is quasi-compact we can find homogeneous elements \(f_1, \ldots, f_n \in S\) of positive degrees such that \(X = D_+(f_1) \cup \ldots \cup D_+(f_n)\). Let \(d\) be the least common multiple of the degrees of \(f_1, \ldots, f_n\). After replacing \(f_i\) by a power we may assume that each \(f_i\) has degree \(d\). Then we see that \(\mathcal{L} = \mathcal{O}_X(d)\) is invertible, the multiplication maps \(\mathcal{O}_X(ad) \otimes \mathcal{O}_X(bd) \to \mathcal{O}_X((a + b)d)\) are isomorphisms, and each \(f_i\) determines a global section \(s_i\) of \(\mathcal{L}\) such that \(X_{s_i} = D_+(f_i)\), see Constructions, Lemmas 01MU and 01MV. Thus \(\Gamma(X, \mathcal{F}(ad)) = \Gamma(X, \mathcal{F} \otimes \mathcal{L}^{\otimes a})\). Recall that \(\widetilde{M}|_{D_{+}(f_i)}\) corresponds to the \(S_{(f_i)}\)-module \(M_{(f_i)}\), see Constructions, Lemma 01M7. Since the degree of \(f_i\) is \(d\), the isomorphism class of \(M_{(f_i)}\) depends only on the homogeneous summands of \(M\) of degree divisible by \(d\). More precisely, the isomorphism class of \(M_{(f_i)}\) depends only on the graded \(\Gamma_*(X, \mathcal{L})\)-module \(\Gamma_*(X, \mathcal{L}, \mathcal{F})\) and the image \(s_i\) of \(f_i\) in \(\Gamma_*(X, \mathcal{L})\). The scheme \(X\) is quasi-compact by assumption and separated by Constructions, Lemma 01MC. By Lemma 01PW there is a canonical isomorphism \[M_{(f_i)} = \Gamma_*(X, \mathcal{L}, \mathcal{F})_{(s_i)} \to \Gamma(X_{s_i}, \mathcal{F}).\] The construction of the map in Constructions, Lemma 0B5I then shows that it is an isomorphism over \(D_+(f_i)\) hence an isomorphism as \(X\) is covered by these opens. We omit the proof of the final statement.

Finding suitable affine opens

In this section we collect some results on the existence of affine opens in more and less general situations.

Lemma

Let \(X\) be a quasi-separated scheme. Let \(Z_1, \ldots, Z_n\) be pairwise distinct irreducible components of \(X\), see Topology, Section 004U. Let \(\eta_i \in Z_i\) be their generic points, see Schemes, Lemma 01IS. There exist affine open neighbourhoods \(\eta_i \in U_i\) such that \(U_i \cap U_j = \emptyset\) for all \(i \not = j\). In particular, \(U = U_1 \cup \ldots \cup U_n\) is an affine open containing all of the points \(\eta_1, \ldots, \eta_n\).

Proof

Let \(V_i\) be any affine open containing \(\eta_i\) and disjoint from the closed set \(Z_1 \cup \ldots \hat Z_i \ldots \cup Z_n\). Since \(X\) is quasi-separated for each \(i\) the union \(W_i = \bigcup_{j, j \not = i} V_i \cap V_j\) is a quasi-compact open of \(V_i\) not containing \(\eta_i\). We can find open neighbourhoods \(U_i \subset V_i\) containing \(\eta_i\) and disjoint from \(W_i\) by Algebra, Lemma 00EV. Finally, \(U\) is affine since it is the spectrum of the ring \(R_1 \times \ldots \times R_n\) where \(R_i = \mathcal{O}_X(U_i)\), see Schemes, Lemma 01I5.

Remark

Lemma 01ZV above is false if \(X\) is not quasi-separated. Here is an example. Take \(R = \mathbf{Q}[x, y_1, y_2, \ldots]/((x-i)y_i)\). Consider the minimal prime ideal \(\mathfrak p = (y_1, y_2, \ldots)\) of \(R\). Glue two copies of \(\Spec(R)\) along the (not quasi-compact) open \(\Spec(R) \setminus V(\mathfrak p)\) to get a scheme \(X\) (glueing as in Schemes, Example 01JD). Then the two maximal points of \(X\) corresponding to \(\mathfrak p\) are not contained in a common affine open. The reason is that any open of \(\Spec(R)\) containing \(\mathfrak p\) contains infinitely many of the “lines” \(x = i\), \(y_j = 0\), \(j \not = i\) with parameter \(y_i\). Details omitted.

Notwithstanding the example above, for “most” finite sets of irreducible closed subsets one can apply Lemma 01ZV above, at least if \(X\) is quasi-compact. This is true because \(X\) contains a dense open which is separated.

Lemma

Let \(X\) be a quasi-compact scheme. There exists a dense open \(V \subset X\) which is separated.

Proof

Say \(X = \bigcup_{i = 1, \ldots, n} U_i\) is a union of \(n\) affine open subschemes. We will prove the lemma by induction on \(n\). It is trivial for \(n = 1\). Let \(V' \subset \bigcup_{i = 1, \ldots, n - 1} U_i\) be a separated dense open subscheme, which exists by induction hypothesis. Consider \[V = V' \amalg (U_n \setminus \overline{V'}).\] It is clear that \(V\) is separated and a dense open subscheme of \(X\).

It turns out that, even if \(X\) is quasi-separated as well as quasi-compact, there does not exist a separated, quasi-compact dense open, see Examples, Lemma 086I. Here is a slight refinement of Lemma 01ZV above.

Lemma

Let \(X\) be a quasi-separated scheme. Let \(Z_1, \ldots, Z_n\) be pairwise distinct irreducible components of \(X\). Let \(\eta_i \in Z_i\) be their generic points. Let \(x \in X\) be arbitrary. There exists an affine open \(U \subset X\) containing \(x\) and all the \(\eta_i\).

Proof

Suppose that \(x \in Z_1 \cap \ldots \cap Z_r\) and \(x \not \in Z_{r + 1}, \ldots, Z_n\). Then we may choose an affine open \(W \subset X\) such that \(x \in W\) and \(W \cap Z_i = \emptyset\) for \(i = r + 1, \ldots, n\). Note that clearly \(\eta_i \in W\) for \(i = 1, \ldots, r\). By Lemma 01ZV we may choose affine opens \(U_i \subset X\) which are pairwise disjoint such that \(\eta_i \in U_i\) for \(i = r + 1, \ldots, n\). Since \(X\) is quasi-separated the opens \(W \cap U_i\) are quasi-compact and do not contain \(\eta_i\) for \(i = r + 1, \ldots, n\). Hence by Algebra, Lemma 00EV we may shrink \(U_i\) such that \(W \cap U_i = \emptyset\) for \(i = r + 1, \ldots, n\). Then the union \(U = W \cup \bigcup_{i = r + 1, \ldots, n} U_i\) is disjoint and hence (by Schemes, Lemma 01I5) a suitable affine open.

Lemma

Let \(X\) be a scheme. Assume either

The scheme \(X\) is quasi-affine.

The scheme \(X\) is isomorphic to a locally closed subscheme of an affine scheme.

There exists an ample invertible sheaf on \(X\).

The scheme \(X\) is isomorphic to a locally closed subscheme of \(\text{Proj}(S)\) for some graded ring \(S\).

Then for any finite subset \(E \subset X\) there exists an affine open \(U \subset X\) with \(E \subset U\).

Proof

By Properties, Definition 01P6 a quasi-affine scheme is a quasi-compact open subscheme of an affine scheme. Any affine scheme \(\Spec(R)\) is isomorphic to \(\text{Proj}(R[X])\) where \(R[X]\) is graded by setting \(\deg(X) = 1\). By Proposition 01Q3 if \(X\) has an ample invertible sheaf then \(X\) is isomorphic to an open subscheme of \(\text{Proj}(S)\) for some graded ring \(S\). Hence, it suffices to prove the lemma in case (4). (We urge the reader to prove case (2) directly for themselves.)

Thus assume \(X \subset \text{Proj}(S)\) is a locally closed subscheme where \(S\) is some graded ring. Let \(T = \overline{X} \setminus X\). Recall that the standard opens \(D_{+}(f)\) form a basis of the topology on \(\text{Proj}(S)\). Since \(E\) is finite we may choose finitely many homogeneous elements \(f_i \in S_{+}\) such that \[E \subset D_{+}(f_1) \cup \ldots \cup D_{+}(f_n) \subset \text{Proj}(S) \setminus T\] Suppose that \(E = \{\mathfrak p_1, \ldots, \mathfrak p_m\}\) as a subset of \(\text{Proj}(S)\). Consider the ideal \(I = (f_1, \ldots, f_n) \subset S\). Since \(I \not \subset \mathfrak p_j\) for all \(j = 1, \ldots, m\) we see from Algebra, Lemma 00JS that there exists a homogeneous element \(f \in I\), \(f \not \in \mathfrak p_j\) for all \(j = 1, \ldots, m\). Then \(E \subset D_{+}(f) \subset D_{+}(f_1) \cup \ldots \cup D_{+}(f_n)\). Since \(D_{+}(f)\) does not meet \(T\) we see that \(X \cap D_{+}(f)\) is a closed subscheme of the affine scheme \(D_{+}(f)\), hence is an affine open of \(X\) as desired.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{L}\) be an ample invertible sheaf on \(X\). Let \[E \subset W \subset X\] with \(E\) finite and \(W\) open in \(X\). Then there exists an \(n > 0\) and a section \(s \in \Gamma(X, \mathcal{L}^{\otimes n})\) such that \(X_s\) is affine and \(E \subset X_s \subset W\).

Proof

The reader can modify the proof of Lemma 01ZY to prove this lemma; we will instead deduce the lemma from it. By Lemma 01ZY we can choose an affine open \(U \subset W\) such that \(E \subset U\). Consider the graded ring \(S = \Gamma_*(X, \mathcal{L}) = \bigoplus_{n \geq 0} \Gamma(X, \mathcal{L}^{\otimes n})\). For each \(x \in E\) let \(\mathfrak p_x \subset S\) be the graded ideal of sections vanishing at \(x\). It is clear that \(\mathfrak p_x\) is a prime ideal and since some power of \(\mathcal{L}\) is globally generated, it is clear that \(S_{+} \not \subset \mathfrak p_x\). Let \(I \subset S\) be the graded ideal of sections vanishing on all points of \(X \setminus U\). Since the sets \(X_s\) form a basis for the topology we see that \(I \not \subset \mathfrak p_x\) for all \(x \in E\). By (graded) prime avoidance (Algebra, Lemma 00JS) we can find \(s \in I\) homogeneous with \(s \not \in \mathfrak p_x\) for all \(x \in E\). Then \(E \subset X_s \subset U\) and \(X_s\) is affine by Lemma 01PV.

Lemma

Let \(X\) be a quasi-affine scheme. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Let \(E \subset W \subset X\) with \(E\) finite and \(W\) open. Then there exists an \(s \in \Gamma(X, \mathcal{L})\) such that \(X_s\) is affine and \(E \subset X_s \subset W\).

Proof

The proof of this lemma has a lot in common with the proof of Algebra, Lemma 00DS. Say \(E = \{x_1, \ldots, x_n\}\). If \(E = W = \emptyset\), then \(s = 0\) works. If \(W \not = \emptyset\), then we may assume \(E \not = \emptyset\) by adding a point if necessary. Thus we may assume \(n \geq 1\). We will prove the lemma by induction on \(n\).

Base case: \(n = 1\). After replacing \(W\) by an affine open neighbourhood of \(x_1\) in \(W\), we may assume \(W\) is affine. Combining Lemmas 01QE and Proposition 01Q3 we see that every quasi-coherent \(\mathcal{O}_X\)-module is globally generated. Hence there exists a global section \(s\) of \(\mathcal{L}\) which does not vanish at \(x_1\). On the other hand, let \(Z \subset X\) be the reduced induced closed subscheme on \(X \setminus W\). Applying global generation to the quasi-coherent ideal sheaf \(\mathcal{I}\) of \(Z\) we find a global section \(f\) of \(\mathcal{I}\) which does not vanish at \(x_1\). Then \(s' = fs\) is a global section of \(\mathcal{L}\) which does not vanish at \(x_1\) such that \(X_{s'} \subset W\). Then \(X_{s'}\) is affine by Lemma 01PV.

Induction step for \(n > 1\). If there is a specialization \(x_i \leadsto x_j\) for \(i \not = j\), then it suffices to prove the lemma for \(\{x_1, \ldots, x_n\} \setminus \{x_i\}\) and we are done by induction. Thus we may assume there are no specializations among the \(x_i\). By either Lemma 01ZY or Lemma 09NV we may assume \(W\) is affine. By induction we can find a global section \(s\) of \(\mathcal{L}\) such that \(X_s \subset W\) is affine and contains \(x_1, \ldots, x_{n - 1}\). If \(x_n \in X_s\) then we are done. Assume \(s\) is zero at \(x_n\). By the case \(n = 1\) we can find a global section \(s'\) of \(\mathcal{L}\) with \(\{x_n\} \subset X_{s'} \subset W \setminus \overline{\{x_1, \ldots, x_{n - 1}\}}\). Here we use that \(x_n\) is not a specialization of \(x_1, \ldots, x_{n - 1}\). Then \(s + s'\) is a global section of \(\mathcal{L}\) which is nonvanishing at \(x_1, \ldots, x_n\) with \(X_{s + s'} \subset W\) and we conclude as before.

Lemma

Let \(X\) be a scheme and \(x \in X\) a point. There exists an affine open neighbourhood \(U \subset X\) of \(x\) such that the canonical map \(\mathcal{O}_X(U) \to \mathcal{O}_{X, x}\) is injective in each of the following cases:

\(X\) is integral,

\(X\) is locally Noetherian,

\(X\) is reduced and has a finite number of irreducible components.

Proof

After translation into algebra, this follows from Algebra, Lemma 0BX1.

Lemma

Let \(U\), \(V\) be affine schemes and let \(W \to U\) and \(W \to V\) be open immersions. For any \(w \in W\) there exists an affine open neighbourhood \(W' \subset W\) of \(w\) such that \(W'\) maps to a standard open of both \(U\) and \(V\).

Proof

Let \(X\) be the glueing of \(U\) and \(V\) along \(W\) and apply Schemes, Lemma 01IW.

But we only list those properties here which we have not already dealt with separately somewhere else.↩︎
This may be nonstandard terminology. If \(G\) is a finite group or group scheme then sometimes a \(G\)-scheme denotes a scheme equipped with an action of \(G\).↩︎
Nicely explained in a blog post by Akhil Mathew.↩︎
This functor is sometimes called the coherator.↩︎