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Chapter 2: Problem 5

If \(\widetilde{\psi}\) is a coherent sheaf on a noetherian formal scheme \(\vec{x},\) which can be generated by global sections, show in fact that it can be generated by a finite number of its global sections.

Short Answer

Expert verified
Given the noetherian formal scheme and the property of \(\widetilde{\psi}\) being a coherent sheaf, it follows that \(\widetilde{\psi}\) can be generated by a finite number of its global sections.

Step by step solution

01

Understand the Properties of Coherent Sheaves

The first thing to note is that a coherent sheaf is defined as a sheaf with finitely generated local sections. This is an important property that will be used later on.
02

Understand the Properties of Noetherian Rings

A noetherian formal scheme is a scheme in which all of the local rings are noetherian. Noetherian rings have the property that every increasing sequence of ideals is eventually constant. This will be relevant in our problem because we are given that the sheaf can be generated by global sections.
03

Use the Noetherian Property

Because the scheme is noetherian, any coherent sheaf, in particular \(\widetilde{\psi}\), is finitely presented. That means there exist finitely many global sections which generate it.
04

Conclusion

Since \(\widetilde{\psi}\) can be generated by global sections and given that the formal scheme is noetherian, it follows that \(\widetilde{\psi}\) can be generated by a finite number of its global sections.

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Key Concepts

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

Coherent Sheaf
In the realm of algebraic geometry, a coherent sheaf is a concept with significant implications on the structural understanding of schemes. Imagine a sheaf as a method of systematically compiling data and functions across different pieces of a space. When we refer to a coherent sheaf, specifically, we're talking about a sheaf of modules that not only comes with an organized local structure but also satisfies two crucial conditions. First, every local section is finitely generated, meaning each piece of our sheaf doesn't require an infinite number of generators. Second, the kernel of any surjection between coherent sheaves remains coherent.

These properties ensure that operations and calculations within the sheaf maintain a degree of 'coherence,' not spiraling out into unmanageability. For instance, in the provided exercise, understanding that the coherent sheaf \(\widetilde{\psi}\) has finitely generated local sections lays the groundwork to tackle the problem of whether we can generate it with a finite number of global sections.
Noetherian Formal Scheme
Delving into the characteristics of a noetherian formal scheme, we are essentially peeking into the atomic structure of algebraic spaces. A formal scheme is an advanced concept in algebraic geometry, offering a framework for dealing with infinite algebraic processes. It allows for the inclusion of 'incomplete' algebraic structures, akin to dealing with a puzzle that you're still putting together.

What singles out the noetherian ones is that at the heart of each local piece of the scheme - that is, at its local rings - lies a noetherian ring. These noetherian rings obey an essential rule: They refuse to drag on forever when you start listing their ideals in a growing chain. Every such sequence of ideals eventually stabilizes. The implication for your exercise is profound; it assures us that since \(\widetilde{\psi}\) resides on such a scheme, it has that desirable trait of not needing an endless supply of global sections to describe it fully.
Global Sections
The concept of global sections might sound comprehensive, but it is, in essence, quite intuitive. Picture global sections like skeleton keys - within the context of sheaves on a noetherian formal scheme, these are the sections that have the capability to unlock the structure of the sheaf everywhere at once. They represent functions or elements that are consistent over the entire space the sheaf covers.

Why is this important? Because if you can find such universal keys, you can describe and utilize the sheaf more efficiently. In simpler terms, having a set of global sections is like having a master diagram by which you can replicate or understand the entire scheme. The crux of the exercise hence revolves around the idea that if a coherent sheaf can be generated by global sections, a finite collection of these all-encompassing 'keys' is all you need to unravel the coherent sheaf's mysteries.
Noetherian Rings
The noetherian ring is a cornerstone concept not just in algebra but also in shaping the structures studied in algebraic geometry. These rings are distinguished by their disciplined structure concerning ideals: They refuse to harbor an infinite ascending chain without reaching a point where there's no more room to grow - a plateau of sorts.

This strict disciplinary rule, known as the ascending chain condition, translates to a tremendous simplification in understanding the algebraic object at hand. One does not need to consider the complexities of an infinite number of pieces; a finite subcollection will do the trick. Hence, in our exercise, the tied-up nature of a coherent sheaf on a noetherian formal scheme is a direct representation of the noetherian attribute - finite and manageable, allowing for an eloquent description by a limited, yet sufficient, set of global sections.

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Most popular questions from this chapter

Let \(k=\mathbf{F}_{p}\) be the finite field with \(p\) elements. Describe Spec \(k[x] .\) What are the residue fields of its points? How many points are there with a given residue field?

Zariski Spaces. A topological space \(X\) is a Zariski space if it is noetherian and every (nonempty) closed irreducible subset has a unique generic point (Ex. 2.9 ). For example, let \(R\) be a discrete valuation ring, and let \(T=\operatorname{sp}(\operatorname{Spec} R)\). Then \(T\) consists of two points \(t_{0}=\) the maximal ideal, \(t_{1}=\) the zero ideal. The open subsets are \(\varnothing,\left\\{t_{1}\right\\},\) and \(T .\) This is an irreducible Zariski space with generic point \(t_{1}\). (a) Show that if \(X\) is a noetherian scheme, then \(\operatorname{sp}(X)\) is a Zariski space. (b) Show that any minimal nonempty closed subset of a Zariski space consists of one point. We call these closed points. (c) Show that a Zariski space \(X\) satisfies the axiom \(T_{0}\) : given any two distinct points of \(X\), there is an open set containing one but not the other (d) If \(X\) is an irreducible Zariski space, then its generic point is contained in every nonempty open subset of \(X\) (e) If \(x_{0}, x_{1}\) are points of a topological space \(X,\) and if \(x_{0} \in\left\\{x_{1}\right\\}^{-},\) then we say that \(x_{1}\) specializes to \(x_{0},\) written \(x_{1} \leadsto \rightarrow x_{0} .\) We also say \(x_{0}\) is a specialization of \(x_{1},\) or that \(x_{1}\) is a generization of \(x_{0} .\) Now let \(X\) be a Zariski space. Show that the minimal points, for the partial ordering determined by \(x_{1}>x_{0}\) if \(x_{1} \leadsto x\) \(x_{0},\) are the closed points, and the maximal points are the generic points of the irreducible components of \(X .\) Show also that a closed subset contains every specialization of any of its points. (We say closed subsets are stable under specialization. . Similarly, open subsets are stable under generization. (f) Let \(t\) be the functor on topological spaces introduced in the proof of (2.6) If \(X\) is a noetherian topological space, show that \(t(X)\) is a Zariski space. Furthermore \(X\) itself is a Zariski space if and only if the \(\operatorname{map} \alpha: X \rightarrow t(X)\) is a homeomorphism.

The Ricmunn-Rocili Problem. Let \(\lambda\) be a nonsingular projective variety over an algebracally closed field, and let \(D\) be a divisor on \(X\). For any \(n>0\) we consider the complete linear system \(|n D| .\) Then the Riemann-Roch problem is to determine \(\operatorname{dim}|n D|\) as a function of \(n,\) and. in particular, its behavior for large \(n\). If \(\mathscr{L}\) is the corresponding invertible sheaf, then \(\operatorname{dim}|n D|=\operatorname{dim} \Gamma\left(X, \mathscr{L}^{n}\right)-1,\) so an equivalent problem is to determine \(\operatorname{dim} \Gamma\left(X, \mathscr{L}^{n}\right)\) as a function of \(n\) (a) Show that if \(D\) is very ample, and if \(X \subseteq P_{k}^{n}\) is the corresponding embedding in projective space, then for all \(n\) sufficiently large, \(\operatorname{dim}|n D|=P_{X}(n)-1\) polynomial function of \(n,\) for \(n\) large. (b) If \(D\) corresponds to a torsion element of Pic \(X\), of order \(r,\) then \(\operatorname{dim}|n D|=0\) if \(r | n,-1\) otherwise. In this case the function is periodic of period \(r\) It follows from the general Riemann- Roch theorem that \(\operatorname{dim}|n D|\) is a polynomial function for \(n\) large, whenever \(D\) is an ample divisor. See \((I V, 1.3 .2),(V, 1.6)\) and Appendix A. In the case of algebraic surfaces, Zariski [7] has shown for any effective divisor \(D\), that there is a finite set of polynomials \(P_{1}, \ldots, P_{r},\) such that for all \(n\) sufficiently large, \(\operatorname{dim}|n D|=P_{t(n)}(n),\) where \(i(n) \in\\{1,2, \ldots, r\\}\) is a function of \(n\)

Glueing Lemma. Generalize the glueing procedure described in the text \((2.3 .5)\) as follows. Let \(; X_{i}\); be a family of schemes (possible infinite). For each \(i \neq j\). suppose given an open subset \(U_{i j} \subseteq X_{i}\), and let it have the induced scheme structure (Ex. 2.2 ). Suppose also given for each \(i \neq j\) an isomorphism of schemes \(\varphi_{i j}: U_{i j} \rightarrow U_{l^{\prime}}\) such that (1) for each \(i, j, \varphi_{i j}=\varphi_{11}^{-1},\) and (2) for each \(i, j, k\).\(\varphi_{i j}\left(U_{i j} \cap U_{i k}\right)=U_{j i} \cap C_{j k},\) and \(\varphi_{i k}=\varphi_{j k} \quad \varphi_{i j}\) on \(U_{i j} \cap L_{i k} .\) Then show that there is a scheme \(X\), together with morphisms \(\psi_{1}: X_{1} \rightarrow X\) for each \(i,\) such that (1) \(\psi_{i}\) is an isomorphism of \(X\), onto an open subscheme of \(X,(2)\) the \(\psi,(X,)\) cover \(X,(3) \psi_{i}\left(U_{i j}\right)=\psi_{i}\left(X_{i}\right) \cap \psi_{j}\left(X_{j}\right)\) and \((4) \psi_{i}=\psi_{j} \quad \varphi_{i j}\) on \(U_{i j}, \quad\) We say that \(X\) is obtained by glueing the schemes \(X_{i}\) along the isomorphisms \(\varphi_{y} .\) An interesting special case is when the family \(X_{i}\) is arbitrary, but the \(U_{i j}\) and \(\varphi_{i j}\) are all empty. Then the scheme \(X\) is called the disjoint union of the \(X_{1},\) and is denoted ||\(X\).

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 \(\left.\operatorname{map} h: X \rightarrow Y \times_{S} Y \text { obtained from } f \text { and } g .\right].\)
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