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Chapter 2: Problem 5

Let \(f: X \rightarrow Y\) be a morphism of schemes. (a) Show by example that if \(\mathscr{F}\) is coherent on \(X\), then \(f_{*} \mathscr{F}\) need not be coherent on \(Y\), even if \(X\) and \(Y\) are varieties over a field \(k\) (b) Show that a closed immersion is a finite morphism \((\S 3)\) (c) If \(f\) is a finite morphism of noetherian schemes, and if \(\mathscr{F}\) is coherent on \(X\) then \(f_{*} \mathscr{F}\) is coherent on \(Y\).

Short Answer

Expert verified
(a) Counterexample: if \(X = \mathbb{P}^1_k\), \(Y = Spec(k)\), and \(\mathscr{F}\) is the structure sheaf of \(X\), then \(f_{*}\mathscr{F}\) is not coherent on \(Y\), though \(\mathscr{F}\) is coherent on \(X\). \n(b) A closed immersion is a finite morphism because it is proper, quasi-finite, quasi-compact, and separated, all properties of finite morphisms. \n(c) If \(f\) is a finite morphism of noetherian schemes, then a coherent sheaf \(\mathscr{F}\) on \(X\) will also be coherent on \(Y\). This is because the properties of the sheaf are forwarded by the morphism, and the noetherian schemes ensure the finiteness conditions required for coherency.

Step by step solution

01

Showing Coherency of a Specific Example

To show coherency of \(\mathscr{F}\) on \(X\), you need a specific example. Let's choose a case where \(X = \mathbb{P}^1_k\) and \(Y = Spec(k)\), and let \(\mathscr{F}\) be the structure sheaf of \(X\). The pushforward \(f_{*}\mathscr{F}\) isn't coherent (as a \(k\)-module) for this particular case, even though \(\mathscr{F}\) is coherent on X. This is because the global sections of \(f_{*} \mathscr{F}\) can be infinite dimensional over \(k\), which means \(f_{*} \mathscr{F}\) is not a finite type module on \(Y\).
02

Showing a Closed Immersion is a Finite Morphism

Next, it's needed to provide a demonstration that a closed immersion is a finite morphism. A closed immersion by definition is proper and quasi-finite. Furthermore, a finite morphism is also proper and quasi-finite, and since every closed immersion is indeed quasi-compact and separated, it can be concluded that a closed immersion is a finite morphism.
03

Coherency with a Finite Morphism of Noetherian Schemes

A coherent sheaf \(\mathscr{F}\) on \(X\) will also be coherent on \(Y\) when \(f\) is a finite morphism of noetherian schemes. Basically, if \(f\) is finite, then the condition that \(\mathscr{F}\) is coherent on \(X\) helps ensure it remains coherent after being pushed forward by \(f\) onto \(Y\). This is due to the way the properties of finiteness and noetherian schemes interact: properties of the sheaf transfer through a finite morphism, and noetherian schemes guarantee that the necessary finiteness conditions are met.

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Key Concepts

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

Coherent Sheaf
A coherent sheaf can be thought of as a structured set of data that glorifies neatness and regularity over a space, like a scheme. To bring clarity, let's break this down:
  • A sheaf is essentially a collection of data, such as functions, defined in a consistent way on open sets of a topological space.
  • "Coherent" refers to the idea that this collection is not just any random data, but rather fits together in a smooth, consistent manner.
  • In algebraic geometry, a coherent sheaf often arises as a consistent system of algebraic equations over a space.
Coherent sheaves are useful because they generalize and carry many important properties, such as being finite type, which means the data on small open sets can be generated by finitely many sections. This makes them a beautiful algebraic object, encapsulating structure while being operationally handy.

However, as in the exercise example, even if a sheaf \( \mathscr{F} \) is coherent on one scheme \( X \), its pushforward, the transformation of data from one scheme to another scheme \( Y \) via a function \( f \), may not always reflect the same level of neatness. This is due to the intrinsic nature of how spaces expand or contract when connected through functions or morphisms.
Finite Morphism
A finite morphism is an elegant concept intertwining algebraic structure between two schemes. Visualize it as a very controlled, almost predictable, transformation between spaces. Here's what it's all about:
  • In essence, a morphism \( f: X \to Y \) of schemes is called finite when the pre-image of any affine open subset of \( Y \) is covered by finitely many affine open subsets of \( X \), each of which corresponds to a finite algebra.
  • It ensures that the mapping doesn't 'blow up' in unexpected ways - everything happening on \( X \) is reflected in a simple, controlled way through \( f \).

Finite morphisms play a pivotal role, especially when discussing the relationship between coherent sheaves on different schemes, as they help in preserving the "finiteness" conditions that coherent sheaves need.

In the discussion of the exercise, a finite morphism allows us to deduce that if a sheaf \( \mathscr{F} \) is coherent on \( X \), it continues to maintain this structure when mapped to \( Y \), provided \( f \) itself is finely controlled, like through finite morphisms.
Closed Immersion
A closed immersion is an explicit, often straightforward, way one space (or scheme) sits inside another. It鈥檚 as if one space is wearing the other like a snug glove.
  • Formally, a closed immersion \( f: X \hookrightarrow Y \) is a morphism of schemes that identifies \( X \) as a closed subset of \( Y \), complete with the induced scheme structure.
  • In simpler terms, think of \( X \) being tightly "glued" into \( Y \), so there's no room for misrepresentation.
  • The importance of closed immersions is profound in many algebraic contexts, not least because they are both proper and quasi-finite, making them quintessentially finite morphisms.

So, when you think about closed immersions, think about one mathematical space perfectly inhabiting another, much like a jigsaw piece perfectly fitting into its spot in the puzzle.

This intrinsic tidy embedding ensures that the properties of \( X \) translate neatly into \( Y \), conserving many essential mathematical principles and relationships like finiteness and coherence when dealing with sheaves.

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Most popular questions from this chapter

A morphism \(f: X \rightarrow Y\), with \(Y\) irreducible, is generically finite if \(f^{-1}(\eta)\) is a finite set, where \(\eta\) is the generic point of \(Y .\) A morphism \(f: X \rightarrow Y\) is dominant if \(f(X)\) is dense in \(Y\). Now let \(f: X \rightarrow Y\) be a dominant, generically finite morphism of finite type of integral schemes. Show that there is an open dense subset \(U \subseteq Y\) such that the induced morphism \(f^{-1}(U) \rightarrow U\) is finite. \([\text {Hint}:\) First show that the function field of \(X \text { is a finite field extension of the function field of } Y .]\)

Let \(X\) be a separated scheme over an affine scheme \(S\). Let \(U\) and \(V\) be open affine subsets of \(X\). Then \(U \cap V\) is also affine. Give an example to show that this fails if \(X\) is not separated.

Let \(X\) be a variety of dimension \(n\) over \(k .\) Let \(\mathscr{E}\) be a locally free sheaf of \(\operatorname{rank}>n\) on \(X,\) and let \(V \subseteq \Gamma(X, \mathscr{E})\) be a vector space of global sections which generate \(\mathscr{E} .\) Then show that there is an element \(s \in V\), such that for each \(x \in X,\) we have \(s_{x} \notin \mathrm{m}_{x} \mathscr{E}_{x} .\) Conclude that there is a morphism \(\mathscr{O}_{x} \rightarrow \mathscr{E}\) giving rise to an exact sequence \\[ 0 \rightarrow \mathscr{O}_{X} \rightarrow \mathscr{E} \rightarrow \mathscr{E}^{\prime} \rightarrow 0 \\] where \(\mathscr{E}^{\prime}\) is also locally free. \([\text {Hint}:\) Use a method similar to the proof of Bertini's theorem \((8.18) .]\)

Constructible Sets. Let \(X\) be a Zariski topological space. A constructible subset of \(X\) is a subset which belongs to the smallest family \(\mathfrak{Y}\) of subsets such that (1) every open subset is in \(\mathfrak{F},(2)\) a finite intersection of elements of \(\mathfrak{F}\) is in \(\mathfrak{F},\) and (3) the complement of an element of \(\mathfrak{F}\) is in \(\mathfrak{F}\) (a) A subset of \(X\) is locally closed if it is the intersection of an open subset with a closed subset. Show that a subset of \(X\) is constructible if and only if it can be written as a finite disjoint union of locally closed subsets. (b) Show that a constructible subset of an irreducible Zariski space \(X\) is dense if and only if it contains the generic point. Furthermore, in that case it contains a nonempty open subset. (c) A subset \(S\) of \(X\) is closed if and only if it is constructible and stable under specialization. Similarly, a subset \(T\) of \(X\) is open if and only if it is constructible and stable under generization. (d) If \(f: X \rightarrow Y\) is a continuous map of Zariski spaces, then the inverse image of any constructible subset of \(Y\) is a constructible subset of \(X\)

Let \(A\) be a ring. Show that the following conditions are equivalent: (i) Spec \(A\) is disconnected : (ii) there exist nonzero elements \(e_{1}, e_{2} \in A\) such that \(e_{1} e_{2}=0, e_{1}^{2}=e_{1}, e_{2}^{2}=e_{2}\) \(e_{1}+e_{2}=1\) (these elements are called orthogonal idempotents): (iii) \(A\) is isomorphic to a direct product \(A_{1} \times A_{2}\) of two nonzero rings.
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