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Chapter 3: Problem 5

If \(x\) is a point of a scheme \(X,\) we define an 茅tale neighborhood of \(x\) to be an 茅tale morphism \(f: U \rightarrow X,\) together with a point \(x^{\prime} \in U\) such that \(f\left(x^{\prime}\right)=x .\) As an example of the use of 茅tale neighborhoods, prove the following: if \(\overline{\mathscr{H}}\) is a coherent sheaf on \(X,\) and if every point of \(X\) has an 茅tale neighborhood \(f: U \rightarrow X\) for which \(f * \mathscr{F}\) is a free \(\mathscr{C}_{U}\) -module, then \(\mathscr{F}\) is locally free on \(X.\)

Short Answer

Expert verified
Therefore, the sheaf \(\mathscr{F}\) is locally free on the scheme \(X\) under the given condition that every point of \(X\) has an 茅tale neighborhood such that the pullback of \(\mathscr{F}\) is a free module.

Step by step solution

01

Understand the Definitions

Firstly, become familiar with the definition of an 茅tale neighborhood and a free sheaf. An 茅tale morphism is a concept in algebraic geometry that is a type of morphism of schemes which is formally smooth of relative dimension one. A sheaf is free if it is locally isomorphic to a direct sum of copies of the structure sheaf.
02

Understand the Condition

Remember that the condition presented states that for every point of \(X\), there exists an 茅tale neighborhood, \(f: U \rightarrow X\) such that \(f^*F\) is a free \(O_U\)-module.
03

Prove that the Sheaf is Locally Free

By taking the condition from Step 2, it is clear that for every \(x \in X\), there exists an open neighborhood \(U\) (in the 茅tale topology) of \(x\) within \(X\) where \(\mathscr{F}\) is free. This implies that \(\mathscr{F}\) is locally free.

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Key Concepts

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

茅tale neighborhoods
脡tale neighborhoods are essential concepts in algebraic geometry. They involve an 茅tale morphism, which is a special type of morphism that is "nice" in a certain way鈥攕imilar to how open maps function in topology. An 茅tale morphism, denoted as \( f: U \rightarrow X \), is characterized by two main properties:
  • It is flat, meaning that it preserves the dimension in a specific context, akin to a function with full rank in linear algebra.
  • It is unramified, implying that it behaves like an open map but in the context of algebraic geometry.

For a point \( x \in X \), an 茅tale neighborhood is provided by \( f \) together with a corresponding point \( x' \in U \) such that \( f(x') = x \). The existence of such neighborhoods around every point allows for more generalized local analysis of the structure of schemes. It gives a way to transfer local properties from a simpler context (like \( U \)) onto the more complicated structure of \( X \).
locally free sheaves
In algebraic geometry, sheaves are fundamental objects used to systematically capture local data about algebraic varieties or schemes. When we say a sheaf is locally free, it's somewhat analogous to saying a vector space is finite-dimensional. A sheaf \( \mathscr{F} \) on a scheme \(X\) is locally free if, for every point \(x\) in \(X\), there exists a neighborhood where \( \mathscr{F} \) resembles a free module. This means that locally, our sheaf looks like a collection of functions.
  • "Free" implies that within a specified neighborhood, the sheaf resembles a direct sum of copies of the structure sheaf \( \mathscr{O}_X \).
  • The concept of local freeness allows us to work with complicated spaces by considering simpler, more understandable pieces.

So, transitioning from coherent to locally free鈥攍ike in our initial exercise鈥攊s about finding these convenient local fields where the abstract turns tangible.
coherent sheaves
Coherent sheaves are another central concept in algebraic geometry. They generalize the notion of vector bundles, which have fixed dimensions.Unlike general sheaves, coherent sheaves have controlled and predictable behaviors. They have finitely generated sections locally and thus are more structured.
  • They help in executing geometric operations like intersections because of their manageable properties.
  • Mathematically, a coherent sheaf is one where, for any open set, the sections can be generated by a finite number of elements.

In our context, the exercise explores conditions under which a coherent sheaf \( \mathscr{H} \) can become a locally free sheaf by utilizing 茅tale neighborhoods.
algebraic geometry
Algebraic geometry stands as a fascinating domain where geometry and algebra unite. It studies solutions of systems of algebraic equations using abstract concepts like schemes, sheaves, and morphisms. One of its main interests is to explore and resolve the properties of "spaces" defined by these equations.
  • The language of schemes allows mathematicians to work with geometric objects that might have singularities or other complexities.
  • Using coherent sheaves and 茅tale morphisms facilitates a deeper understanding of these geometric objects.

This magic mix of geometry and algebra allows for the tackling of intricate problems in both mathematics and related fields, such as number theory and topology.

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Most popular questions from this chapter

Let \(X=\mathbf{P}_{k}^{1},\) with \(k\) an infinite field. (a) Show that there does not exist a projective object \(\mathscr{P} \in \mathfrak{M}\) od \((X)\), together with a surjective map \(\mathscr{P} \rightarrow \mathscr{O}_{X} \rightarrow 0 .\) [Hint: Consider surjections of the form \(\mathscr{O}_{V} \rightarrow\) \(k(x) \rightarrow 0,\) where \(x \in X\) is a closed point, \(V\) is an open neighborhood of \(x\) and \(\mathscr{O}_{V}=j_{!}\left(\left.\mathcal{O}_{X}\right|_{V}\right),\) where \(j: V \rightarrow X\) is the inclusion. (b) Show that there does not exist a projective object \(\mathscr{P}\) in either \(\mathbb{Z} \operatorname{co}(X)\) or \(\operatorname{Cob}(X)\) together with a surjection \(\mathscr{P} \rightarrow \mathscr{O}_{X} \rightarrow 0 .\) [Hint: Consider surjections of the form \(\mathscr{L} \rightarrow \mathscr{L} \otimes k(x) \rightarrow 0,\) where \(x \in X\) is a closed point, and \(\mathscr{L}\) is an invertible sheaf on \(X .]\)

Let \(Y\) be a hypersurface in \(X=P_{k}^{N}\) with \(N \geqslant 4 .\) Let \(\hat{X}\) be the formal completion of \(X\) along \(Y(\mathrm{II}, \S 9) .\) Prove that the natural map Pic \(\hat{X} \rightarrow\) Pic \(Y\) is an isomorphism. \(\left[\text {Hint}: \text { Use }(\mathrm{II}, \mathrm{Ex} .9 .6), \text { and then study the maps Pic } X_{n+1} \rightarrow \text { Pic } X_{n} \text { for each } n\right.\) using (Ex. 4.6) and (Ex. 5.5).]

Cohomological Interpretation of Depth. If \(A\) is a ring, a an ideal, and \(M\) an \(A\) module, then depth\(_{\mathrm{a}} M\) is the maximum length of an \(M\) -regular sequence \(x_{1}, \ldots, x_{r}\) with all \(x_{i} \in a .\) This generalizes the notion of depth introduced in (II, \(\S 8\) ). (a) Assume that \(A\) is noetherian. Show that if \(\operatorname{depth}_{\mathfrak{a}} M \geqslant 1,\) then \(\Gamma_{a}(M)=0\) and the converse is true if \(M\) is finitely generated. [Hint: When \(M\) is finitely generated, both conditions are equivalent to saying that \(a\) is not contained in any associated prime of \(M .]\) (b) Show inductively, for \(M\) finitely generated, that for any \(n \geqslant 0,\) the following conditions are equivalent: (i) \(\operatorname{depth}_{a} M \geqslant n\) (ii) \(H_{n}^{i}(M)=0\) for all \(i

Let \(X\) be a projective scheme over a field \(k,\) and let \(\overline{\mathscr{I}}\) be a coherent sheaf on \(X\) We define the Euler characteristic of \(\mathscr{F}\) by $$\chi(\mathscr{F})=\sum(-1)^{i} \operatorname{dim}_{k} H^{i}(X, \overline{\mathscr{F}})$$ If$$0 \rightarrow \mathscr{F}^{\prime} \rightarrow \mathscr{F} \rightarrow \mathscr{F}^{\prime \prime} \rightarrow 0$$ is a short exact sequence of coherent sheaves on \(X,\) show that \(\chi(\mathscr{F})=Y\left(\mathscr{F}^{\prime}\right)+\) \(x\left(\mathscr{F}^{\prime \prime}\right)\)

Let \(\left(X, \mathcal{O}_{X}\right)\) be a ringed space, and let \(\mathscr{F}^{\prime}, \mathscr{F}^{\prime \prime} \in \mathbb{M}_{\mathrm{O}} \mathfrak{d}(X) .\) An extension of \(\mathscr{F}^{\prime \prime} \mathrm{by}\) \(\mathscr{F}^{\prime}\) is a short exact sequence $$0 \rightarrow \mathscr{F}^{\prime} \rightarrow \mathscr{F} \rightarrow \mathscr{F}^{\prime \prime} \rightarrow 0$$ in \(\operatorname{Mod}(X) .\) Two extensions are isomorphic if there is an isomorphism of the short exact sequences, inducing the identity maps on \(\mathscr{F}^{\prime}\) and \(\mathscr{F}^{\prime \prime} .\) Given an extension as above consider the long exact sequence arising from Hom \(\left(\mathscr{F}^{\prime \prime}, \cdot\right),\) in particular the map $$\delta: \operatorname{Hom}\left(\mathscr{F}^{\prime \prime}, \mathscr{F}^{\prime \prime}\right) \rightarrow \operatorname{Ext}^{1}\left(\mathscr{F}^{\prime \prime}, \mathscr{F}^{\prime}\right)$$ and let \(\xi \in \operatorname{Ext}^{1}\left(\mathscr{F}^{\prime \prime}, \mathscr{F}^{\prime}\right)\) be \(\delta\left(1_{\mathscr{F}^{\prime \prime}}\right) .\) Show that this process gives a one-to-one correspondence between isomorphism classes of extensions of \(\mathscr{F}^{\prime \prime}\) by \(\mathscr{F}^{\prime},\) and elements of the group \(\operatorname{Ext}^{1}\left(\mathscr{F}^{\prime \prime}, \mathscr{F}^{\prime}\right) .\) For more details, see, e.g., Hilton and Stammbach [1, Ch. III].
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