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Chapter 2: Problem 6

Let \(X\) be an integral scheme. Show that the local ring \(\mathscr{O}_{\xi}\) of the generic point of \(X\) is a field. It is called the function field of \(X,\) and is denoted by \(K(X) .\) Show also that if \(U=\operatorname{Spec} A\) is any open affine subset of \(X,\) then \(K(X)\) is isomorphic to the quotient field of \(A\)

Short Answer

Expert verified
In an integral scheme \(X\), the local ring of the generic point \(\xi\), denoted as \(K(X)\), is a field. Further, if \(U = operatorname{Spec} A\) is any open affine subset of \(X\), then the function field of the scheme, \(K(X)\), is isomorphic to the quotient field of the ring \(A\). This is because the generic point of the scheme is also a generic point of its subset, hence the local ring at these points is isomorphic to their respective quotient fields.

Step by step solution

01

Understand the local ring and the generic point

Recall that the generic point of a scheme \(X\) is a point \(\xi\) such that \(\overline{{\{\xi\}}}\) = \(X\). The local ring of a scheme at a point is a ring \(\mathscr{O}_{\xi}\) that determines the local behavior of the scheme at that point. In this case, we need to show that the local ring of the generic point \(\xi\) of \(X\) is a field, denoted \(K(X)\).
02

Proof that \(\mathscr{O}_{\xi}\) is a field

Let's consider a nonzero element \(f\) in \(\mathscr{O}_{\xi}\), a section over some neighborhood \(U\) of the generic point \(\xi\) such that \(f(\xi) 鈮� 0\). Now, \(f(\xi)\) is a regular function that is not zero at \(\xi\) and thus invertible in the local ring. So every nonzero element of \(\mathscr{O}_{\xi}\) is invertible, which means \(\mathscr{O}_{\xi}\) is a field.
03

Understand the affine scheme \(U\) and the quotient field of \(A\)

Open affine subset \(U\) of \(X\) can be expressed as \(U = operatorname{Spec} A\), where \(A\) is the coordinate ring of \(U\). The quotient field of \(A\) is the field of fraction of another integral domain \(A\). Now, our task is to show that \(K(X)\) is isomorphic to the quotient field of \(A\).
04

Proof of isomorphism between \(K(X)\) and the quotient field of \(A\)

We need to show the isomorphism between the function field \(K(X)\) and the quotient field \(Q(A)\) of \(A\). For this, we note that the generic point \(\xi\) of \(X\) is also a generic point of the open set \(U\). Hence, the local ring of 鈥peratorname{Spec} A at \(\xi\) is \(A\) localized at \(\xi\), which is isomorphic to the quotient field \(Q(A)\) as \(A\) is an integral domain. Therefore, we obtain \(K(X)=Q(A)\), and they are isomorphic.

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Key Concepts

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

Integral Schemes
Integral schemes are a key concept in algebraic geometry. They are irreducible and reduced schemes, meaning that they consist of a single "piece" without any "holes" or reducible components, and they have no nilpotent elements.
These schemes arise when we look at varieties or spaces that are connected and have no extra layers or overlapping segments.
  • **Irreducible**: Cannot be broken into simpler components.
  • **Reduced**: No nilpotent elements, that is, elements that would become zero when raised to some power.
Integral schemes are significant because they mirror real-world geometric objects like curves and surfaces without singularities. In practice, they help us focus on more manageable, core parts of geometric structures, making them easier to study and understand.
Local Rings
Local rings play a crucial role in understanding schemes because they provide insight into the behavior of schemes at specific points.
A local ring can be thought of as a ring with a single maximal ideal, giving it a "localized" property that helps us examine how a scheme behaves very close to a particular point.
Here are key features:
  • **Maximal ideal**: Contains one unique maximal ideal.
  • **Localized behavior**: Focuses analysis on a small neighborhood around a point.
In the context of the exercise, we see how the local ring at a generic point becomes a field. This signifies that at that prime location, every non-zero local function has an inverse, simplifying the analysis of the scheme's structure.
Generic Points
The concept of generic points is intriguing as it centres around the idea of capturing the essence of an entire scheme.
A generic point is a special kind of point whose closure is the entire topological space of the scheme.
Why are generic points important?
  • **Represents all points**: The closure of a generic point is the full space, reflecting the entire scheme.
  • **Simplifies analysis**: Studying a generic point provides insights into the whole space without having to look at individual points separately.
Understanding generic points allows mathematicians to focus on the "big picture" properties of a scheme, such as how structures behave across the entire space rather than getting bogged down by local intricacies.
Affine Schemes
Affine schemes form one of the most fundamental building blocks in algebraic geometry.
They are constructed from rings, particularly by considering \Spec A\, where \(A\) is a commutative ring. This process gives us a topological space that closely mimics the algebraic properties of the ring.
Key points about affine schemes:
  • **Construction**: Formed using the spectrum of a ring, \Spec A\.
  • **Intersection of algebra and geometry**: Connects algebraic concepts with geometric ones.
  • **Open subsets**: Allow for the oversight of scheme properties over smaller regions.
These schemes provide a bridge between the abstractness of algebra and the visualization possible with geometry, proving to be invaluable in simplifying problems by translating them into the language of ring theory.

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Most popular questions from this chapter

Fibres of a Morphism. (a) If \(f: X \rightarrow Y\) is a morphism, and \(y \in Y\) a point, show that \(\operatorname{sp}\left(X_{y}\right)\) is homeomorphic to \(f^{-1}(y)\) with the induced topology. (b) Let \(X=\operatorname{Spec} k[s, t] /\left(s-t^{2}\right),\) let \(Y=\operatorname{Spec} k[s],\) and let \(f: X \rightarrow Y\) be the morphism defined by sending \(s \rightarrow s .\) If \(y \in Y\) is the point \(a \in k\) with \(a \neq 0,\) show that the fibre \(X_{y}\) consists of two points, with residue field \(k .\) If \(y \in Y\) corresponds to \(0 \in k,\) show that the fibre \(X_{y}\) is a non-reduced one-point scheme. If \(\eta\) is the generic point of \(Y\), show that \(X_{\eta}\) is a one-point scheme, whose residue field is an extension of degree two of the residue field of \(\eta .\) (Assume \(k\) algebraically closed.)

A Complete Nonprojective Variety. Let \(k\) be an algebraically closed field of char \(\neq 2 .\) Let \(C \subseteq \mathbf{P}_{k}^{2}\) be the nodal cubic curve \(y^{2} z=x^{3}+x^{2} z .\) If \(P_{0}=(0,0,1)\) is the singular point, then \(C-P_{0}\) is isomorphic to the multiplicative group \(\mathbf{G}_{m}=\operatorname{Spec} k\left[t, t^{-1}\right](\mathrm{E} \mathrm{x} .6 .7) .\) For each \(a \in k, a \neq 0,\) consider the translation of \(\mathbf{G}_{m}\) given by \(t \mapsto a t .\) This induces an automorphism of \(C\) which we denote by \(\varphi_{a}\) Now consider \(C \times\left(\mathbf{P}^{1}-\\{0\\}\right)\) and \(C \times\left(\mathbf{P}^{1}-\\{\infty\\}\right) .\) We glue their open subsets \(C \times\left(\mathbf{P}^{1}-\\{0, x\\}\right)\) by the isomorphism \(\varphi:\langle P, u\rangle \mapsto\left\langle\varphi_{u}(P), u\right\rangle\) for \(P \in C, u \in \mathbf{G}_{m}=\mathbf{P}^{1}-\\{0, x\\} .\) Thus we obtain a scheme \(X,\) which is our example. The projections to the second factor are compatible with \(\varphi,\) so there is a natural morphism \(\pi: X \rightarrow \mathbf{P}^{1}\) (a) Show that \(\pi\) is a proper morphism, and hence that \(X\) is a complete variety over \(k\) (b) Use the method of \((\text { Ex. } 6.9)\) to show that \(\operatorname{Pic}\left(C \times \mathbf{A}^{1}\right) \cong \mathbf{G}_{m} \times \mathbf{Z}\) and \(\operatorname{Pic}\left(C \times\left(\mathbf{A}^{1}-\\{0\\}\right)\right) \cong \mathbf{G}_{m} \times \mathbf{Z} \times \mathbf{Z}\) [Hint: If \(A\) is a domain and if denotes the group of units, then \(\left.(A[u])^{*} \cong A^{*} \text { and }\left(A\left[u, u^{-1}\right]\right)^{*} \cong A^{*} \times \mathbf{Z} .\right]\) (c) Now show that the restriction map \(\operatorname{Pic}\left(C \times \mathbf{A}^{1}\right) \rightarrow \operatorname{Pic}\left(C \times\left(\mathbf{A}^{1}-\\{0\\}\right)\right)\) is of the form \(\langle t, n\rangle \mapsto\langle t, 0 . n\rangle,\) and that the automorphism \(\varphi\) of \(C \times\left(\mathbf{A}^{1}-\\{0\\}\right)\) induces a map of the form \(\langle t, d, n\rangle \mapsto\langle t, d+n, n\rangle\) on its Picard group. (d) Conclude that the image of the restriction map Pic \(X \rightarrow \operatorname{Pic}(C \times\\{0\\})\) consists entirely of divisors of degree 0 on \(C .\) Hence \(X\) is not projective over \(k\) and \(\pi\) is not a projective morphism.

Complete Intersections in \(\mathbf{P}^{n}\). A closed subscheme \(Y\) of \(\mathbf{P}_{k}^{n}\) is called a (strict, global) complete intersection if the homogeneous ideal \(I\) of \(Y\) in \(S=k\left[x_{0}, \ldots, x_{n}\right]\) can be generated by \(r=\operatorname{codim}\left(Y, \mathbf{P}^{n}\right)\) elements (I, Ex. 2.17). (a) Let \(Y\) be a closed subscheme of codimension \(r\) in \(\mathbf{P}^{n}\). Then \(Y\) is a complete intersection if and only if there are hypersurfaces (i.e., locally principal subschemes of codimension 1) \(H_{1}, \ldots, H_{r},\) such that \(Y=H_{1} \cap \ldots \cap H_{r}\) as schemes, i.e., \(\mathscr{I}_{Y}=\mathscr{I}_{H_{1}}+\ldots+\mathscr{I}_{H_{r}} .[\text { Hint }:\) Use the fact that the unmixedness theorem \(\text { holds in }S \text { (Matsumura }[2, \mathrm{p} .107]) .]\) (b) If \(Y\) is a complete intersection of dimension \(\geqslant 1\) in \(P^{n},\) and if \(Y\) is normal, then \(Y\) is projectively normal (Ex. 5.14). \([\text {Hint}: \text { Apply }(8.23)\) to the affine cone over \(Y .]\) (c) With the same hypotheses as (b), conclude that for all \(l \geqslant 0\), the natural map \(\Gamma\left(\mathbf{P}^{n}, \mathcal{O}_{\mathbf{p}^{n}}(l)\right) \rightarrow \Gamma\left(Y, \mathcal{O}_{\mathbf{Y}}(l)\right)\) is surjective. In particular, taking \(l=0,\) show that \(Y\) is connected. (d) Now suppose given integers \(d_{1}, \ldots, d_{r} \geqslant 1,\) with \(r

The real importance of the notion of constructible subsets derives from the following theorem of Chevalley-see Cartan and Chevalley [1, expos茅 7] and see also Matsumura \([2, \mathrm{Ch} .2, \$ 6]:\) let \(f: X \rightarrow Y\) be a morphism of finite type of noetherian schemes. Then the image of any constructible subset of \(X\) is a constructible subset of \(Y\). In particular, \(f(X),\) which need not be either open or closed, is a constructible subset of \(Y\). Prove this theorem in the following steps. (a) Reduce to showing that \(f(X)\) itself is constructible, in the case where \(X\) and \(Y\) are affine, integral noetherian schemes, and \(f\) is a dominant morphism. (b) In that case, show that \(f(X)\) contains a nonempty open subset of \(Y\) by using the following result from commutative algebra: let \(A \subseteq B\) be an inclusion of noetherian integral domains, such that \(B\) is a finitely generated \(A\) -algebra. Then given a nonzero element \(b \in B,\) there is a nonzero element \(a \in A\) with the following property: if \(\varphi: A \rightarrow K\) is any homomorphism of \(A\) to an algebraically closed field \(K,\) such that \(\varphi(a) \neq 0,\) then \(\varphi\) extends to a homomorphism \(\varphi^{\prime}\) of \(B\) into \(K,\) such that \(\varphi^{\prime}(b) \neq 0 .[\) Hint: Prove this algebraic result by induction on the number of generators of \(B\) over \(A\). For the case of one generator, prove the result directly. In the application, take \(b=1 .]\) (c) Now use noetherian induction on \(Y\) to complete the proof. (d) Give some examples of morphisms \(f: X \rightarrow Y\) of varieties over an algebraically closed field \(k,\) to show that \(f(X)\) need not be either open or closed.

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.]
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