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Chapter 2: Problem 18

Constructible Sets. Let \(X\) be a Zariski topological space. A constructible subset of \(X\) is a subset which belongs to the smallest family \(\mathfrak{Y}\) of subsets such that (1) every open subset is in \(\mathfrak{F},(2)\) a finite intersection of elements of \(\mathfrak{F}\) is in \(\mathfrak{F},\) and (3) the complement of an element of \(\mathfrak{F}\) is in \(\mathfrak{F}\) (a) A subset of \(X\) is locally closed if it is the intersection of an open subset with a closed subset. Show that a subset of \(X\) is constructible if and only if it can be written as a finite disjoint union of locally closed subsets. (b) Show that a constructible subset of an irreducible Zariski space \(X\) is dense if and only if it contains the generic point. Furthermore, in that case it contains a nonempty open subset. (c) A subset \(S\) of \(X\) is closed if and only if it is constructible and stable under specialization. Similarly, a subset \(T\) of \(X\) is open if and only if it is constructible and stable under generization. (d) If \(f: X \rightarrow Y\) is a continuous map of Zariski spaces, then the inverse image of any constructible subset of \(Y\) is a constructible subset of \(X\)

Short Answer

Expert verified
In sum, understanding the concept of constructible subsets is crucial: with each part demonstrating their unique properties, their relationship with other elements in topology, and their behavior under transformations. These properties include the ability to be expressed as a disjoint union of locally closed sets, their behavior in dense sets, their relation to stable sets, and the impact of continuous maps. Addressing each of these aspects would provide a comprehensive understanding of constructible sets.

Step by step solution

01

Part (a) - Locally Closed Subsets

The idea here is to show that a subset of \(X\) is constructible if and only if it can be written as a finite disjoint union of locally closed subsets. A locally closed subset is an intersection of an open set and a closed set. Construct a proof that shows every constructible set can be expressed as a finite disjoint union of locally closed subsets and vice versa.
02

Part (b) - Dense Constructible Subsets

Now focus on the case of constructible subsets of an irreducible Zariski space \(X\). A set is called dense if the closure of the set is the whole space. Show that a constructible subset is dense in \(X\) if and only if it contains the generic point. Also demonstrate that in such case it contains a nonempty open subset.
03

Part (c) - Stable sets

Next let's take a look at stable sets. A set is stable under specialization (generization) if for any pair of points with one specializing (generalizing) to the other, both either belong to the set or do not. Show that a set is closed if and only if it is a constructible and stable under specialization. And similarly, show that a set is open if and only if it is a constructible and stable under generization.
04

Part (d) - Continuous Maps and Constructible Subsets

Lastly, suppose there is a continuous map \(f: X \rightarrow Y\) of Zariski spaces. With a continuous map, the inverse image of open (closed) sets are open (closed). The aim here is to prove that the inverse image of any constructible subset of \(Y\) is also a constructible subset of \(X\).

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Key Concepts

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

Constructible Sets
In Algebraic Geometry, a set is termed "constructible" if it belongs to a certain family of subsets within a Zariski topological space \(X\). The family, often denoted by \(\mathfrak{Y}\), is constructed with these key properties:
  • Includes all open subsets of \(X\)
  • Closed under finite intersections
  • Closed under complementation
This makes constructible sets quite flexible and broad. Understanding them requires recognizing that any constructible set can be outlined as a finite disjoint union of locally closed subsets. Such a decomposition proves the intuitive fact that these sets are built using open and closed sets' combinations. The journey to grasping constructible sets involves understanding this delicate interplay between set operations and topological properties, revealing their importance across various algebraic contexts.
Zariski Topology
The Zariski topology is a fundamental structure in Algebraic Geometry. It is distinctive for its definition of open and closed sets very differently compared to the standard topology. In a Zariski space, closed sets are basically algebraic sets, meaning they are zero sets of polynomials. Unlike the usual topology on \(\mathbb{R}^n\), here, the open sets are large, and the closed sets are small, providing a framework suited especially for algebraic varieties. This topological structure is crucial for studying the solutions to polynomial equations and serves as the foundation for defining constructible sets. Effective use of Zariski topology demands understanding how it interacts with other topological principles, influencing notions like compactness, connectedness, and separation. To put it simply, Zariski topology tailors itself naturally to the algebraic setting, making it indispensable for exploring the deeper geometric features of algebraic structures.
Locally Closed Subsets
Locally closed subsets blend open and closed topological properties in intriguing ways. Defined as intersections of open sets with closed sets, they maintain properties of both, making them robust for certain geometrical constructions. In simple terms, a subset \(A\) of a topological space \(X\) is locally closed if there exists an open set \(U\) and a closed set \(F\) in \(X\) such that \(A = U \cap F\). These subsets play a critical role in breaking down constructible sets into understandable parts. Their ability to describe regions of a space that are somehow restricted yet open enough for certain foliation patterns is why expressing a constructible set as a finite union of locally closed subsets helps in maneuvering the complex landscape of algebraic sets. This characteristic makes them pivotal in the manipulation and understanding of constructible sets.
Continuous Maps in Algebraic Geometry
Continuous maps between Zariski spaces present a fascinating aspect of algebraic geometry, emphasizing the invariant nature of constructibility. If you have a continuous map \(f: X \rightarrow Y\), key properties of the space are well-preserved. Specifically, the inverse image of a constructible subset of \(Y\) under \(f\) remains constructible in \(X\). This makes continuous maps incredibly useful and versatile in algebraic geometry, providing a pathway to transfer properties from one space to another while maintaining key structural aspects. Continuous maps facilitate work beyond just tracking subsets; they reveal how spaces relate to one another through intrinsic properties preservable by mapping. This quality speaks to the elegance and refinement of algebraic constructs that persist across different spaces within the Zariski topology framework.

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Most popular questions from this chapter

Let \(X\) be an integral scheme of finite type over a field \(k\), having function field \(K\) We say that a valuation of \(K / k\) (see \(I, \$ 6\) ) has center \(x\) on \(X\) if its valuation ring \(R\) dominates the local ring \(C_{x . X}\) (a) If \(X\) is separated over \(k\), then the center of any valuation of \(K / k\) on \(X\) (if it exists) is unique. (b) If \(X\) is proper over \(k\), then every valuation of \(K / k\) has a unique center on \(X\) \(*(\mathrm{c})\) Prove the converses of \((\mathrm{a})\) and \((\mathrm{b}) .[\text { Hint }: \text { While parts }(\mathrm{a}) \text { and }(\mathrm{b})\) follow quite easily from (4.3) and \((4.7),\) their converses will require some comparison of valuations in different fields. (d) If \(X\) is proper over \(k\), and if \(k\) is algebraically closed, show that \(\Gamma\left(X, C_{X}\right)=k\) This result generalizes (I, 3.4a). [Hint: Let \(a \in \Gamma\left(X, \mathscr{C}_{X}\right),\) with \(a \notin k .\) Show that there is a valuation ring \(R\) of \(K / k\) with \(a^{-1} \in \mathrm{m}_{R} .\) Then use (b) to get a contradiction. Note. If \(X\) is a variety over \(k,\) the criterion of (b) is sometimes taken as the definition of a complete variety.

Let \(X\) be a noetherian scheme, let \(\delta\) be a coherent locally free sheaf on \(X\), and let \(\pi: \mathbf{P}(\delta) \rightarrow X\) be the corresponding projective space bundle. Show that there is a natural 1-1 correspondence between sections of \(\pi\) (i.e., morphisms \(\sigma: X \rightarrow\) \(\mathbf{P}(\delta)\) such that \(\pi \quad \sigma=\) id \(_{\mathrm{x}}\) ) and quotient invertible sheaves \(\delta \rightarrow \mathscr{L} \rightarrow 0\) of \(\delta\)

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.

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.

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
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