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Chapter 2: Problem 5

Describe Spec \(\mathbf{Z}\), and show that it is a final object for the category of schemes. i.e., each scheme \(X\) admits a unique morphism to Spec \(\mathbf{Z}\).

Short Answer

Expert verified
Spec \( \mathbf{Z} \) can be understood as the set \(\{ (0), (2), (3), (5), (7), \ldots \}\), the set of all prime numbers and 0, considering that \(\mathbf{Z}\) only has two prime ideals. It is a final object in the category of schemes because for any scheme X there is a unique morphism \( X \rightarrow \) Spec \( \mathbf{Z} \).

Step by step solution

01

Describe Spec \( \mathbf{Z} \)

Since \(\mathbf{Z}\) only has two prime ideals, the ideal (0) and (p) for any prime p. The Spec \( \mathbf{Z} \) is the set of these ideals, that can be seen as \(\{ (0), (2), (3), (5), (7), \ldots \}\), i.e., the set of all prime numbers and 0.
02

Proving Spec \( \mathbf{Z} \) as a Final Object

To show Spec \( \mathbf{Z} \) is a final object in the category of schemes, we need to prove that for every scheme \( X \), there is a unique morphism \( X \rightarrow \) Spec \( \mathbf{Z} \). By definition, over any scheme \( X \), the structure sheaf of Spec \( \mathbf{Z} \) is constant, it's the sheaf associated to the constant presheaf with value \( \mathbf{Z} \). Hence, there is a unique morphism of schemes \( X \rightarrow \) Spec \( \mathbf{Z} \), given on an open set \( U \subseteq X \) by the unique ring homomorphism \( \mathbf{Z} \rightarrow \mathcal{O}(U) \). This shows that Spec \( \mathbf{Z} \) is a final object in the category of schemes.

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Key Concepts

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

Scheme Theory
Scheme theory forms the backbone of modern algebraic geometry, building upon the concept of a variety by allowing 'singularities' and incorporating prime ideals as points. Think of schemes as geometric spaces that are defined not only by their points but also by the 'functions' that live on them. In essence, a scheme is a topological space teamed up with a structure sheaf, which assigns rings of functions to every open set.
The specific example of Spec \(\mathbb{Z}\) illustrates these ideas vividly. It consists of points corresponding to prime ideals, giving us an elegant geometric picture of arithmetic properties of integers.
Prime Ideals
Prime ideals are a cornerstone of scheme theory and algebraic geometry. A prime ideal in a ring like \(\mathbb{Z}\) is a subset of elements that is not the whole ring, and it observes a key property: if the product of two elements is in the ideal, then at least one of these elements is also in the ideal. The prime ideals of \(\mathbb{Z}\) correspond to the zero ideal and the ideals generated by prime numbers, representing the 'points' of Spec \(\mathbb{Z}\).
Understanding the role of prime ideals in schemes helps in visualizing algebraic structures as geometric entities and forms the foundation for discussing morphisms and mappings between different schemes.
Category of Schemes
In the category of schemes, each object is a scheme and morphisms between these objects follow specific algebraic rules. It's an abstract framework that helps mathematicians to deal with schemes in a structured way.
Within this setting, Spec \(\mathbb{Z}\) holds a special place as a final object, meaning that every scheme maps to it in a unique way鈥攋ust like all roads lead to Rome. This uniqueness is crucial for understanding how schemes relate to each other and how they can be transformed through morphisms.
Structure Sheaf
A structure sheaf is an essential part of a scheme that links the geometrical with the algebraic. For each open subset of a scheme, the structure sheaf assigns a ring of 'functions,' allowing us to 'do algebra' locally on the scheme.
The structure sheaf of Spec \(\mathbb{Z}\) is particularly simple: it assigns the integer ring \(\mathbb{Z}\) to any open set. This simplicity is what enables the unique morphisms we find in Spec \(\mathbb{Z}\) as a final object in the category of schemes, neatly tying together the algebraic structure with the geometrical intuition.
Morphism of Schemes
A morphism of schemes is a way of transforming one scheme into another, involving both a continuous function between the underlying topological spaces and a compatible family of ring homomorphisms for the structure sheaves.
Every scheme 'wants' to map to Spec \(\mathbb{Z}\) in its own unique way through such a morphism. It is akin to saying there is a specific and unique perspective from which to view all other schemes from the vantage point of Spec \(\mathbb{Z}\). This unique mapping perspective is not only foundational for understanding how different algebraic structures interact with one another but also emphasizes the universality of \(\mathbb{Z}\) within the world of schemes.

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Most popular questions from this chapter

Let \(R\) be a discrete valuation ring with quotient field \(K\), and let \(X=\operatorname{Spec} R\). (a) To give an \(\mathscr{O}_{x}\) -module is equivalent to giving an \(R\) -module \(M,\) a \(K\) -vector space \(L,\) and a homomorphism \(\rho: M \otimes_{R} K \rightarrow L\) (b) That \(\mathscr{O}_{x}\) -module is quasi-coherent if and only if \(\rho\) is an isomorphism.

Subshect with Supports. Let \(Z\) be a closed subset of \(X\), and let \(\mathscr{F}\) be a sheaf on \(X\) We define \(\Gamma_{X}(X, \overline{\mathscr{F}})\) to be the subgroup of \(\Gamma(X, \overline{\mathscr{H}})\) consisting of all sections whose support (Ex. 1.14 ) is contained in \(Z\). (a) Show that the presheaf \(V \mapsto \Gamma_{z \cap v}\left(V,\left.\bar{y}\right|_{V}\right)\) is a sheaf. It is called the subsheaf of \(\overline{\mathscr{F}}\) with supports in \(Z,\) and is denoted by \(\mathscr{H}_{Z}^{0} \cdot \overline{\mathscr{F}}\) ). (b) Let \(U=X-Z,\) and let \(j: U \rightarrow X\) be the inclusion. Show there is an exact sequence of sheaves on \(X\) $$0 \rightarrow \mathscr{H}_{Z}^{0}(\mathscr{F}) \rightarrow \mathscr{H} \rightarrow i_{*}\left(\left.\mathscr{F}\right|_{C}\right)$$ Furthermorc, if \(\mathscr{F}\) is flasque, the \(\operatorname{map} \mathscr{F} \rightarrow i_{*}\left(\left.\mathscr{F}\right|_{l}\right)\) is surjective

Let \(A\) be a ring, let \(X=\operatorname{Spec} A\). let \(f \in A\) and let \(D(f) \subseteq X\) be the open complement of \(V\) ( \((f)\) ). Show that the locally ringed space \(\left(D(f),\left.C_{X}\right|_{D u_{1}}\right)\) is isomorphic to Spec \(A_{f}\).

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.

Support. Recall the notions of support of a section of a sheaf, support of a sheaf, and subsheaf with supports from (Ex. 1.14 ) and (Ex. 1.20 ). (a) Let \(A\) be a ring, let \(M\) be an \(A\) -module, let \(X=\operatorname{Spec} A,\) and let \(\mathscr{F}=\tilde{M}\) For any \(m \in M=\Gamma(X, \overline{\mathscr{F}}),\) show that Supp \(m=V(\text { Ann } m),\) where Ann \(m\) is the annihilator of \(m=\\{a \in A | a m=0\\}\) (b) Now suppose that \(A\) is noetherian, and \(M\) finitely generated. Show that \(\operatorname{Supp} \mathscr{F}=V(\operatorname{Ann} M)\) (c) The support of a coherent sheaf on a noetherian scheme is closed. (d) For any ideal a \(\subseteq A,\) we define a submodule \(\Gamma_{\mathrm{a}}(M)\) of \(M\) by \(\Gamma_{\mathrm{a}}(M)=\) \(\left\\{m \in M | a^{n} m=0 \text { for some } n>0\right\\} .\) Assume that \(A\) is noetherian, and \(M\) any \(A\) -module. Show that \(\Gamma_{\mathrm{a}}(M)^{\sim} \cong \mathscr{H}_{Z}^{0}(\mathscr{F}),\) where \(Z=V(\mathrm{a})\) and \(\mathscr{F}=\tilde{M}\) \([\text {Hint}: \text { Use (Ex. } 1.20)\) and (5.8) to show a priori that \(\mathscr{H}_{Z}^{0}(\mathscr{F})\) is quasi-coherent. Then show that \(\left.\Gamma_{\mathrm{a}}(M) \cong \Gamma_{\mathrm{z}}(\mathscr{F}) .\right]\) (e) Let \(X\) be a noetherian scheme, and let \(Z\) be a closed subset. If \(\mathscr{F}\) is a quasicoherent (respectively, coherent) \(O_{X}\) -module, then \(\mathscr{H}_{Z}^{0}(\mathscr{F})\) is also quasicoherent (respectively, coherent).
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