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Chapter 2: Problem 2

A morphism \(f: X \rightarrow Y\) of schemes is quasi-compact if there is a cover of \(Y\) by open affines \(V_{i}\) such that \(f^{-1}\left(V_{i}\right)\) is quasi-compact for each \(i .\) Show that \(f\) is quasicompact if and only if for every open affine subset \(V \subseteq Y, f^{-1}(V)\) is quasi-compact.

Short Answer

Expert verified
The statement is true. A morphism \(f: X \rightarrow Y\) of schemes is quasi-compact if and only if for every open affine subset \(V \subseteq Y\), \(f^{-1}(V)\) is quasi-compact.

Step by step solution

01

Direct Part

We begin with the first part of the proof, which is to show that if \(f : X \rightarrow Y\) is quasi-compact, then for any open affine \(V \subseteq Y\), \(f^{-1}(V)\) is quasi-compact. To do this, we note from the definition of a quasi-compact morphism that there exists an open cover \(\{V_i\}_{i \in I}\) of \(Y\) by open affines such that for each \(i\), \(f^{-1}(V_i)\) is quasi-compact. For a fixed affine \(V\), we have \(V = V \cap \bigcup_{i \in I}V_i\). Since the \(V_i\)s are an open cover of \(Y\), we can write \(V\) as a union of the sets \(V \cap V_i\) for \(i\) in some index set. Now, since both \(V\) and each \(V_i\) are affine, the sets \(V \cap V_i\) are also affine (because the intersection of affine sets is affine), and their preimages under \(f\) are a cover of \(f^{-1}(V)\). Since \(f^{-1}(V_i)\) is quasi-compact for all \(i\), \(f^{-1}(V \cap V_i)\) must be quasi-compact as well. The union of quasi-compact sets is quasi-compact, so \(f^{-1}(V)\) is quasi-compact. This concludes the first part of the proof.
02

Converse Part

Next, we need to show the converse direction: if for every open affine subset \(V \subseteq Y\), \(f^{-1}(V)\) is quasi-compact, then \(f\) is quasi-compact. To do this, let \(\{V_i\}_{i \in I}\) be an open affine cover of \(Y\). Given our assumption, \(f^{-1}(V_i)\) is quasi-compact for each index \(i\). But then by the definition of a quasi-compact morphism, \(f\) must be quasi-compact. This completes the second part of the proof, and hence the result.

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Key Concepts

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

Schemes in Algebraic Geometry
In algebraic geometry, schemes are a fundamental mathematical structure that generalize the concept of algebraic varieties. Essentially, they are geometric objects that encode both the algebraic and topological properties of solutions to systems of polynomial equations. Schemes are built using the language of commutative algebra and category theory, which allows them to handle more complicated situations, like singularities, that classical varieties cannot.

At the heart of a scheme is the concept of a 'locally ringed space', which means that each point on a scheme has a 'local ring' of functions defined near it. These local rings help in understanding the local properties of the scheme. Schemes are constructed by gluing together pieces called 'affine schemes', which relate directly to algebraic things called 'rings'.

  • Affine schemes are determined by commutative rings, and open subsets of schemes are almost always related to 'localization' of these rings.
  • It鈥檚 important for students to grasp that schemes combine both algebraic and topological data.
  • Understanding schemes is crucial for modern algebraic geometry and is used to study the properties of algebraic objects.
Open Affine Subsets
Open affine subsets are key building blocks of schemes in algebraic geometry. Think of them as analogous to open intervals in real analysis, but on the level of algebraic structures. They are 'open' in the sense that they respect the topological structure, and 'affine' refers to their algebraic properties, essentially behaving like the algebraic geometry equivalent of Euclidean space.

An 'open affine subset' can be thought of as the spectrum of a ring, which is a set of prime ideals, that has a natural topological structure called Zariski topology. Open affine subsets have a pivotal role:

  • They allow for local analysis within the global context of a scheme.
  • Each open affine subset corresponds to a ring, providing a link between geometry (the shape of the set) and algebra (equations defining the set).
  • They are the setting for studying local properties of schemes, such as regularity or singularity at certain points.
Morphism Preimages
In the context of schemes in algebraic geometry, a morphism is a function between schemes, and the preimage describes the inverse mapping of subsets through the morphism. Just like in set theory, where the preimage of a function refers to all the points that map to a particular set, morphism preimages in algebraic geometry refer to the pull-back of subsets via morphisms between schemes.

The concept of preimages becomes particularly important when discussing properties such as 'quasi-compactness'. A morphism is quasi-compact if the preimage of any open affine subset remains a 'tame' or manageable set, specifically a quasi-compact set. This concept proves useful for several reasons:

  • It provides insights into how spaces 'interact' with each other under a morphism.
  • Quasi-compact morphisms preserve certain desirable properties, and their preimages help analyze the continuity and finiteness aspects of geometric mappings.
  • Understanding preimages is necessary to grasp the behavior of functions between intricate geometric structures like schemes.

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Most popular questions from this chapter

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.]

Show that a morphism of sheaves is an isomorphism if and only if it is both injective and surjective.

The Grothendieck Group of a Nonsingular Curce. Let \(X\) be a nonsingular curve over an algebraically closed field \(k .\) We will show that \(K(X) \cong \operatorname{Pic} X \oplus \mathbf{Z},\) in several steps. (a) For any divisor \(D=\sum n_{i} P_{i}\) on \(X\), let \(\psi(D)=\sum n_{i \hat{i}}\) ' \(\left(k\left(P_{i}\right)\right) \in K(X),\) where \(k\left(P_{i}\right)\) is the skyscraper sheaf \(k\) at \(P_{t}\) and 0 elsewhere. If \(D\) is an effective divisor, let \(\mathrm{C}_{D}\) be the structure sheaf of the associated subscheme of codimension \(1,\) and show that \(\psi(D)=\dot{\gamma}\left(C_{D}\right) .\) Then use (6.18) to show that for any \(D, \psi(D)\) depends only on the linear equivalence class of \(D,\) so \(\psi\) defines a homomorphism \(\psi: \mathrm{Cl} X \rightarrow K(X)\) (b) For any coherent sheaf \(\mathscr{F}\) on \(X\), show that there exist locally free sheaves \(\delta_{0}\) and \(\mathscr{E}_{1}\) and an exact sequence \(0 \rightarrow \mathscr{E}_{1} \rightarrow \mathscr{B}_{0} \rightarrow \mathscr{H} \rightarrow 0 .\) Let \(r_{0}=\operatorname{rank} \delta_{0}\) \(r_{1}=\operatorname{rank} \delta_{1},\) and define det \(\tilde{\mathscr{H}}=\left(\bigwedge^{r_{0}} \mathscr{E}_{0}\right) \otimes\left(\bigwedge^{r_{1}} \delta_{1}\right)^{-1} \in \operatorname{Pic} X .\) Here \(\wedge \mathrm{de}\) notes the exterior power (Ex. 5.16). Show that det \(\mathscr{H}\) is independent of the resolution chosen, and that it gives a homomorphism det: \(K(X) \rightarrow\) Pic \(X\) Finally show that if \(D\) is a divisor, then \(\operatorname{det}(\psi(D))=\mathscr{L}(D)\) (c) If \(\mathscr{F}\) is any coherent sheaf of rank \(r,\) show that there is a divisor \(D\) on \(X\) and an exact sequence \(0 \rightarrow \mathscr{P}(D)^{\oplus r} \rightarrow \mathscr{H} \rightarrow \mathscr{I} \rightarrow 0\), where \(\mathscr{J}\) is a torsion sheaf. Con- clude that if \(\mathscr{F}\) is a sheaf of rank \(r,\) then \(\gamma(\mathscr{F})-r \gamma\left(\mathcal{O}_{X}\right) \in \operatorname{Im} \psi\) (d) Using the maps \(\psi,\) det, rank,and \(1 \mapsto \gamma\left(C_{X}\right)\) from \(\mathbf{Z} \rightarrow K(X),\) show that \(K(X) \cong\) Pic \(X \oplus \mathbf{Z}\)

Let \(\mathscr{P}\) be a property of morphisms of schemes such that: (a) a closed immersion has \(\mathscr{P}\) (b) a composition of two morphisms having \(\mathscr{P}\) has \(\mathscr{P}\) (c) \(\mathscr{P}\) is stable under base extension. Then show that: (d) a product of morphisms having \(\mathscr{P}\) has \(\mathscr{P}\) (e) if \(f: X \rightarrow Y\) and \(g: Y \rightarrow Z\) are two morphisms, and if \(g\) fhas \(\mathscr{P}\) and \(g\) is separated, then \(f\) has \(\mathscr{P}\) (f) If \(f: X \rightarrow Y\) has \(\mathscr{P},\) then \(f_{\text {icd }}: X_{\text {idd }} \rightarrow Y_{\text {tad }}\) has \(\mathscr{P}\) \([\text {Hint}:\) For (e) consider the graph morphism \(\Gamma_{f}: X \rightarrow X \times_{z} Y\) and note that it is obtained by base extension from the diagonal morphism \(\left.\Delta: Y \rightarrow Y \times_{Z} Y .\right]\)

Show that a morphism \(f: X \rightarrow Y\) is finite if and only if for erery ' open affine subset \(V=\operatorname{Spec} B\) of \(Y, f^{-1}(V)\) is affine, equal to Spec \(A,\) where \(A\) is a finite \(B\) -module.
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