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Chapter 2: Problem 10

Chow's Lemma. This result says that proper morphisms are fairly close to projective morphisms. Let \(X\) be proper over a noetherian scheme \(S\). Then there is a scheme \(X^{\prime}\) and a morphism \(g: X^{\prime} \rightarrow X\) such that \(X^{\prime}\) is projective over \(S,\) and there is an open dense subset \(L \subseteq X\) such that \(y\) induces an isomorphism of \(g^{-1}(U)\) to \(U\). Prove this result in the following steps. (a) Reduce to the case \(X\) irreducible. (b) Show that \(X\) can be covered by a finite number of open subsets \(U_{i}, i=1, \ldots, n\) each of which is quasi-projective over \(S\). Let \(U_{i} \rightarrow P_{i}\) be an open immersion of \(L_{i}\) into a scheme \(P_{i}\) which is projective over \(S\). (c) Let \(L=\bigcap U_{1},\) and consider the map \\[ f: U \rightarrow X \times_{s} P_{1} \times_{s} \cdots \times_{s} P_{n} \\] deduced from the given maps \(L \rightarrow X\) and \(U \rightarrow P_{1} .\) Let \(X^{\prime}\) be the closed image subscheme structure (Ex.3.11d) \(f(U)\). Let \(g: X^{\prime} \rightarrow X\) be the projection onto the first factor, and let \(h: X^{\prime} \rightarrow P=P_{1} \times_{s} \ldots \times_{s} P_{n}\) be the projection onto the product of the remaining factors. Show that \(h\) is a closed immersion. hence \(X\) ' is projective over \(S\). (d) Show that \(g^{-1}(L) \rightarrow l\) is an isomorphism. thus completing the proof.

Short Answer

Expert verified
The exercise aims at proving Chow's Lemma, which states that proper morphisms are close to being projective. Given the proposition that a scheme \(X\) is proper over a Noetherian scheme \(S\), the exercise is broken down into several steps which require reducing to the case \(X\) is irreducible, covering \(X\) by quasi-projective subsets, defining a map \(f\), showing \(h\) as a closed immersion and proving \(g^{-1}(L) \rightarrow L\) is an isomorphism. After successfully executing these steps, Chow's Lemma is proven.

Step by step solution

01

Reduction to the Case \(X\) is Irreducible

Start by considering \(X\) as a reduced scheme which is the finite union of its irreducible components \(X = X_1 \cup X_2 \cup \ldots \cup X_n\). If Chow's Lemma holds for each irreducible component of \(X\), then it holds for \(X\). Therefore, without loss of generality, it can be assumed that \(X\) is irreducible.
02

Cover \(X\) by Quasi-projective Subsets

Consider the subset \(X\) which can be covered by a finite number of open subsets \(U_i, i=1, \ldots, n\) each of which is quasi-projective over \(S\). Let \(U_i \rightarrow P_i\) be an open immersion of \(L_i\) into a scheme \(P_i\) which is projective over \(S\).
03

Defining Map \(f\)

Let \(L=\bigcap U_{1}\), and consider the map \(f: U \rightarrow X \times_{s} P_{1} \times_{s} \cdots \times_{s} P_{n}\) deduced from the given maps. Define \(X^{\prime}\) to be the closed image subscheme structure of \(f(U)\). Also, let \(g: X^{\prime} \rightarrow X\) and \(h: X^{\prime} \rightarrow P=P_{1} \times_{s} \ldots \times_{s} P_{n}\) be the projections onto the first factor and the remaining factors respectively.
04

Show Projective Morphisms

Here, it can be shown that \(h\) is a closed immersion. This can be proven by showing that \(X^{\prime}\) is projective over \(S\), using the property of projective morphism.
05

Isomorphism

Finally, provide a demonstration to prove that \(g^{-1}(L) \rightarrow L\) is an isomorphism. As a result, the proof for Chow's Lemma is complete.
06

Conclusion

In conclusion, this exercise essentially involves the use of schemes and morphisms to elucidate the theorem by Chow. It demands a comprehensive understanding of Algebraic Geometry, in particular, the notions of projective morphisms and Noetherian schemes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Proper Morphisms
A morphism between schemes is known as a "proper morphism" if it adheres to three specific conditions: it is of finite type, separated, and universally closed. Imagine you are dealing with continuous functions in calculus that maintain some 'closure' property, similarly proper morphisms hold certain accessibility and boundary characteristics when transitioning from one scheme to another.

  • Finite Type: It means the scheme is constructed from finitely generated algebra over the base.
  • Separated: A condition that can be compared to a 'function being injective', ensuring no overlap when mapped onto another scheme.
  • Universally Closed: This ensures that any base change retains 'closedness', akin to closed sets in topology.
Chow's Lemma basically assures that if you have a proper morphism, it can often be connected to a simpler projective morphism, allowing for easier manipulation and understanding of the schemes involved.
Projective Morphisms
Projective morphisms are akin to treating schemes as very structured geometric objects. When you hear 'projective', think of the projection of complex structures into something more digestible, much like projecting a 3D object onto a 2D plane.

  • They are essentially maps that can be embedded or realized within a projective space, which provides a robust framework for analysis in algebraic geometry.
  • This morphism not only projects one scheme into another but also ensures the enclosed geometric shape maintains its essence and complexity in relation to the base scheme.
This concept is pivotal in Chow's Lemma, as it uses projective morphisms to show that any proper morphism can maintain the required structural integrity when transitioned to projective forms.
Noetherian Schemes
In algebraic geometry, a Noetherian scheme is like a well-organized bookshelf: it is made up of Noetherian rings, which ensures a tidy and finite basis handle for computations. Noetherian schemes have several special properties, making them easier to work with in the realm of advanced mathematics.

  • Finite Ascending Chains: Any increasing sequence of ideal or closed subsets stabilizes.
  • Basis of Open Sets: Every open subset has a basis formed by open subsets associated with a finite number of generators.
Chow's Lemma relies on Noetherian schemes because they offer a controlled setting where the union of finite structures (like components of an exercise) can be handled efficiently, proving the lemma's claim.
Quasi-Projective Schemes
Quasi-projective schemes bring flexibility to algebraic geometry. Imagine it as a scheme that tiptoes between being affine and projective, offering certain benefits of both.

  • Open Subset of Projective Scheme: Quasi-projective schemes can be viewed as open subsets of projective schemes, thus inheriting some of their structured nature.
  • Balance of Constraints: They allow for embedding in projective space without being entirely confined to it, associated with paths through open immersions.
Within Chow's Lemma framework, quasi-projective schemes represent building blocks that can briskly adjust to the desired projective conditions, by leveraging their adaptability and innate properties.

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Most popular questions from this chapter

Let \(X\) be a scheme of finite type over a field \(k\) (not necessarily algebraically closed). (a) Show that the following three conditions are equivalent (in which case we say that \(X\) is geometrically irreducible). (i) \(X \times_{k} \bar{k}\) is irreducible, where \(\bar{k}\) denotes the algebraic closure of \(k .\) abuse of notation, we write \(X \times_{k} \bar{k}\) to denote \(X \times_{\text {spec } k}\) Spec \(\bar{k} .\) (ii) \(X \times_{k} k_{s}\) is irreducible, where \(k_{s}\) denotes the separable closure of \(k\) (iii) \(X \times_{k} K\) is irreducible for every extension field \(K\) of \(k\) (b) Show that the following three conditions are equivalent (in which case we say \(X\) is geometrically reduced) (i) \(X \times_{k} \bar{k}\) is reduced. (ii) \(X \times_{k} k_{p}\) is reduced, where \(k_{p}\) denotes the perfect closure of \(k\) (iii) \(X \times_{k} K\) is reduced for all extension fields \(K\) of \(k\) (c) We say that \(X\) is geometrically integral if \(X \times_{k} \bar{k}\) is integral. Give examples of integral schemes which are neither geometrically irreducible nor geometrically reduced.

(a) For any morphism of sheaves \(\varphi: \mathscr{I} \rightarrow \mathscr{G}\), show that for each point \(P,(\mathrm{ker} \varphi)_{p}=\) \(\operatorname{ker}\left(\varphi_{p}\right)\) and \((\operatorname{im} \varphi)_{p}=\operatorname{im}\left(\varphi_{p}\right)\) (b) Show that \(\varphi\) is injective (respectively, surjective) if and only if the induced map on the stalks \(\varphi_{p}\) is injective (respectively, surjective) for all \(P\). (c) Show that a sequence. \(\bar{y}^{\prime} \Rightarrow \bar{y}^{\prime+1} \rightarrow \ldots\) of sheaves and mor- phisms is exact if and only if for each \(P \in X\) the corresponding sequence of stalks is exact as a sequence of abelian groups.

Let \(\left(X, C_{X}\right)\) be a locally ringed space, and let \(f: \mathscr{L} \rightarrow . ll\) be a surjective map of invertible sheaves on \(X\). Show that \(f\) is an isomorphism. [Hint: Reduce to a question of modules over a local ring by looking at the stalks.]

Let \(S\) be a scheme, let \(X\) be a reduced scheme over \(S\), and let \(Y\) be a separated scheme over \(S\). Let \(f\) and \(g\) be two \(S\) -morphisms of \(X\) to \(Y\) which agree on an open dense subset of \(X .\) Show that \(f=g .\) Give examples to show that this result fails if either (a) \(X\) is nonreduced, or (b) \(Y\) is nonseparated. [Hint: Consider the \(\left.\operatorname{map} h: X \rightarrow Y \times_{S} Y \text { obtained from } f \text { and } g .\right].\)

Flasque Sheares. A sheaf \(\bar{y}\) on a topological space \(X\) is flasque if for every inclusion \(V \subseteq U\) of open sets, the restriction \(\operatorname{map} \mathscr{F}(U) \rightarrow \mathscr{F}(V)\) is surjective. (a) Show that a constant sheaf on an irreducible topological space is flasque. See (I, 81 ) for irreducible topological spaces. (b) If \(0 \rightarrow \overline{\mathscr{H}} \rightarrow \mathscr{F} \rightarrow \mathscr{H}^{\prime \prime} \rightarrow 0\) is an exact sequence of sheaves, and if \(\bar{y}\) is flasque, then for any open set \(U\). the sequence \(0 \rightarrow \mathscr{F}^{\prime}(U) \rightarrow \mathscr{F}(U) \rightarrow\) \(\mathscr{F}^{\prime \prime}\left(L^{\prime}\right) \rightarrow 0\) of abelian groups is also exact. (c) If \(0 \rightarrow \mathscr{H} \rightarrow \mathscr{H} \rightarrow \mathscr{H}^{\prime \prime} \rightarrow 0\) is an exact sequence of sheaves, and if \(\mathscr{H}^{\prime}\) and \(\overline{\mathscr{H}}\) are flasque, then \(\mathscr{F}^{\prime \prime}\) is flasque. (d) If \(f: X \rightarrow Y\) is a continuous map, and if \(\mathscr{F}\) is a flasque sheaf on \(X\), then \(f_{*} \overline{\mathscr{H}}\) is a flasque sheaf on \(Y\) (e) Let \(\overline{\mathscr{F}}\) be any sheaf on \(X\). We define a new sheaf \(\mathscr{G}\), called the sheaf of discontinuous sections of \(\mathscr{F}\) as follows. For each open set \(U \subseteq X, \mathscr{G}(U)\) is the set of
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