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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

Extending a Sheaf by Zero. Let \(X\) be a topological space, let \(Z\) be a closed subset. let \(i: Z \rightarrow X\) be the inclusion, let \(U=X-Z\) be the complementary open subset and let \(j: U \rightarrow X\) be its inclusion. (a) Let \(\mathscr{J}\) be a sheaf on \(Z\). Show that the stalk \(\left(i_{*}, \overline{\mathscr{H}}\right)_{p}\) of the direct image sheaf on \(X\) is \(\mathscr{F}_{P}\) if \(P \in Z, 0\) if \(P \notin Z\). Hence we call \(i_{*}\). \(\bar{y}\) the sheaf obtained by extending of \(i_{*} \overline{\mathscr{H}},\) and say "consider \(\mathscr{F}\) as a sheaf on \(X\)," when we mean "consider \(i_{*}\). (b) Now let \(\overline{\mathscr{H}}\) be a sheaf on \(U\). Let \(j\), \((\overrightarrow{\mathscr{H}})\) be the sheaf on \(X\) associated to the presheaf \(V \mapsto \mathscr{F}(V)\) if \(V \subseteq U, V \mapsto 0\) otherwise. Show that the stalk \((j,(\mathscr{F}))_{P}\) is equal to \(\overline{\mathscr{I}}_{p}\) if \(P \in U, 0\) if \(P \notin U\), and show that \(j\), \(\overline{\mathscr{H}}\) is the only sheafon \(X\) which has this property, and whose restriction to \(U\) is \(\mathscr{F}\). We call \(j\). F. Fhe sheaf obtained by extending \(\mathscr{F}\) by zero outside \(U\) (c) Now let \(\mathscr{F}\) be a sheaf on \(X\). Show that there is an exact sequence of sheaves on \(X\) $$0 \rightarrow j \cdot\left(\left.\overline{\mathscr{H}}\right|_{c}\right) \rightarrow \overline{\mathscr{H}} \rightarrow i_{*}\left(\left.\mathscr{F}\right|_{Z}\right) \rightarrow 0$$

Extension of Coherent Sheaves. We will prove the following theorem in several steps: Let \(X\) be a noetherian scheme, let \(U\) be an open subset, and let \(\mathscr{F}\) be a coherent sheaf on \(U\). Then there is a coherent sheaf \(\mathscr{F}^{\prime}\) on \(X\) such that \(\left.\mathscr{F}^{\prime}\right|_{v} \cong \mathscr{F}\) (a) On a noetherian affine scheme, every quasi-coherent sheaf is the union of its coherent subsheaves. We say a sheaf \(\mathscr{F}\) is the union of its subsheaves \(\mathscr{F}\) if for every open set \(U\), the group \(\mathscr{F}(U)\) is the union of the subgroups ?\((U)\) (b) Let \(X\) be an affine noetherian scheme, \(U\) an open subset, and \(\mathscr{F}\) coherent on \(U .\) Then there exists a coherent sheaf \(\mathscr{F}^{\prime}\) on \(X\) with \(\left.\mathscr{F}^{\prime}\right|_{v} \cong \mathscr{F} .\) [Hint: Let \(\left.i: U \rightarrow X \text { be the inclusion map. Show that } i_{*} \mathscr{F} \text { is quasi-coherent, then use }(a) .\right]\) (c) With \(X, U, \mathscr{F}\) as in (b), suppose furthermore we are given a quasi-coherent sheaf \(\mathscr{G}\) on \(X\) such that \(\left.\mathscr{F} \subseteq \mathscr{G}\right|_{v} .\) Show that we can find \(\mathscr{F}^{\prime}\) a coherent subsheaf of \(\mathscr{G},\) with \(\left.\mathscr{F}^{\prime}\right|_{v} \cong \mathscr{F}\). [Hint: Use the same method, but replace \(i_{*} \mathscr{F}\) by \(\left.\rho^{-1}\left(i_{*} \mathscr{F}\right) \text { , where } \rho \text { is the natural } \operatorname{map} \mathscr{G} \rightarrow i_{*}\left(\left.\mathscr{G}\right|_{U}\right) .\right]\) (d) Now let \(X\) be any noetherian scheme, \(U\) an open subset, \(\mathscr{F}\) a coherent sheaf on \(U,\) and \(\mathscr{G}\) a quasi-coherent sheaf on \(X\) such that \(\left.\mathscr{F} \subseteq \mathscr{G}\right|_{V} .\) Show that there is a coherent subsheaf \(\mathscr{F}^{\prime} \subseteq \mathscr{G}\) on \(X\) with \(\left.\mathscr{F}^{\prime}\right|_{v} \cong \mathscr{F}\). Taking \(\mathscr{I}=i_{*} \mathscr{F}\) proves the result announced at the beginning. [Hint: Cover \(X\) with open affines, and extend over one of them at a time. (e) As an extra corollary, show that on a noetherian scheme, any quasi- coherent sheaf \(\mathscr{F}\) is the union of its coherent subsheaves. [Hint: If \(s\) is a section of \(\mathscr{F}\) over an open set \(U,\) apply (d) to the subsheaf of \(\left.\mathscr{F}\right|_{v}\) generated by s.]

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.

Support. Let \(\mathscr{F}\) be a sheaf on \(X\), and let \(s \in \mathscr{F}(U)\) be a section over an open set \(U\) The support of \(s\), denoted Supp s, is defined to be \(\left\\{P \in U | s_{P} \neq 0\right\\},\) where \(s_{P}\) denotes the germ of s in the stalk \(\overline{\mathscr{F}}_{p}\). Show that Supp s is a closed subset of \(U\). We define the support of \(\overline{\mathscr{F}}, \operatorname{Supp}, \overline{\mathscr{F}},\) to be \(\left\\{P \in X | \mathscr{F}_{P} \neq 0\right\\},\) It need not be a closed subset.

Tensor Operations on Sheaves. First we recall the definitions of various tensor operations on a module. Let \(A\) be a ring, and let \(M\) be an \(A\) -module. Let \(T^{\prime \prime}(M)\) be the tensor product \(M \otimes \ldots \otimes M\) of \(M\) with itself \(n\) times, for \(n \geqslant 1\). For \(n=0\) we put \(T^{0}(M)=A .\) Then \(T(M)=\bigoplus_{n \geqslant 0} T^{\prime \prime}(M)\) is a (noncommutative) \(A\) -algebra, which we call the tensor algebra of \(M .\) We define the symmetric algebra \(S(M)=\bigoplus_{n \geqslant 0} S^{\prime \prime}(M)\) of \(M\) to be the quotient of \(T(M)\) by the two-sided ideal generated by all expressions \(x \otimes y-y \otimes x,\) for all \(x, y \in M .\) Then \(S(M)\) is a commutative \(A\) -algebra. Its component \(S^{n}(M)\) in degree \(n\) is called the \(n\) th symmetric product of \(M .\) We denote the image of \(x \otimes y\) in \(S(M)\) by \(x y,\) for any \(x, y \in M .\) As an example, note that if \(M\) is a free \(A\) -module of rank \(r,\) then \(S(M) \cong\) \(A\left[x_{1}, \ldots, x_{r}\right]\). We define the exterior algebra \(\wedge(M)=\bigoplus_{n \geqslant 0} \wedge^{\prime \prime}(M)\) of \(M\) to be the quotient of \(T(M)\) by the two- sided ideal generated by all expressions \(x \otimes x\) for \(x \in M .\) Note that this ideal contains all expressions of the form \(x \otimes y+y \otimes x\) so that \(\wedge(M)\) is a skew commutative graded \(A\) -algebra. This means that if \(u \in\) \(\wedge^{r}(M)\) and \(v \in \Lambda^{s}(M),\) then \(u \wedge v=(-1)^{r s} v \wedge u\) (here we denote by \(\wedge\) the multiplication in this algebra; so the image of \(x \otimes y\) in \(\wedge^{2}(M)\) is denoted by \(x \wedge y\) ). The \(n\) th component \(\wedge^{\prime \prime}(M)\) is called the \(n\) th exterior power of \(M\). Now let \(\left(X, O_{X}\right)\) be a ringed space, and let \(\mathscr{F}\) be a sheaf of \(\mathcal{O}_{X}\) -modules. We define the tensor algebra, symmetric algebra, and exterior algebra of \(\mathscr{F}\) by taking the sheaves associated to the presheaf, which to each open 'set \(U\) assigns the corresponding tensor operation applied to \(\mathscr{F}(U)\) as an \(\mathscr{O}_{X}(U)\) -module. The results are \(\mathcal{O}_{X^{-}}\) algebras, and their components in each degree are \(\mathscr{C}_{X}\) -modules. (a) Suppose that \(\mathscr{F}\) is locally free of rank \(n\). Then \(T^{\prime}(\mathscr{F}), S^{\prime}(\mathscr{F})\), and \(\wedge^{\prime}(\mathscr{F})\) are also locally free, of ranks \(n^{\prime},\left(\begin{array}{c}m+r-1 \\ n-1\end{array}\right),\) and \(\left(\begin{array}{c}m \\ 2\end{array}\right)\) respectively. (b) Again let \(\mathscr{F}\) be locally free of rank \(n\). Then the multiplication \(\operatorname{map} \wedge \mathscr{F} \otimes\) \(\wedge^{n-r} \mathscr{F} \rightarrow \wedge^{n} \cdot \mathscr{F}\) is a perfect pairing for any \(r,\) i.c., it induces an isomorphism of \(\wedge^{\prime \prime} \mathscr{F}\) with \(\left(\wedge^{n-r} \mathscr{F}\right)^{\sim} \otimes \wedge^{\prime \prime} \mathscr{F}\). As a special case, note if \(\mathscr{F}\) has rank 2 then \(\mathscr{F} \cong \mathscr{F}^{\sim} \otimes \wedge^{2} \mathscr{F}\) (c) Let \(0 \rightarrow \mathscr{F}^{\prime} \rightarrow \mathscr{F} \rightarrow \mathscr{F}^{\prime \prime} \rightarrow 0\) be an exact sequence of locally free sheaves. Then for any \(r\) there is a finite filtration of \(S^{\prime}(\mathscr{F})\) \\[ S^{\prime}(\mathscr{F})=F^{0} \supseteq F^{1} \supseteq \ldots \supseteq F^{\prime} \supseteq F^{r+1}=0 \\] with quotients \\[ F^{p} / F^{p+1} \cong S^{p}\left(\mathscr{F}^{\prime}\right) \otimes S^{r-p}\left(\mathscr{F}^{\prime \prime}\right) \\] for each \(p\). (d) Same statement as (c), with exterior powers instead of symmetric powers. In particular, if \(\mathscr{F}^{\prime}, \mathscr{F}, \mathscr{F}^{\prime \prime}\) have ranks \(n^{\prime}, n, n^{\prime \prime}\) respectively, there is an isomorphism \(\wedge^{n} \mathscr{F} \cong \wedge^{n^{\prime} \mathscr{F}^{\prime}} \otimes \wedge^{n^{\prime \prime}} \mathscr{F}^{\prime \prime}\) (e) Let \(f: X \rightarrow Y\) be a morphism of ringed spaces, and let \(\mathscr{F}\) be an \(\mathscr{U}_{Y}\) -module. Then \(f^{*}\) commutes with all the tensor operations on \(\mathscr{F},\) i.e., \(f^{*}\left(S^{n}(\mathscr{F})\right)=\) \(S^{\prime \prime}\left(f^{*} \mathscr{F}\right)\) etc.
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