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Chapter 2: Problem 14

(a) Let \(S\) be a graded ring. Show that Proj \(S=\varnothing\) if and only if every element of \(S_{+}\) is nilpotent. (b) Let \(\varphi: S \rightarrow T\) be a graded homomorphism of graded rings (preserving degrees) Let \(U=\left\\{p \in \operatorname{Proj} T | p \geqslant \varphi\left(S_{+}\right)\right\\} .\) Show that \(U\) is an open subset of Proj \(T\) and show that \(\varphi\) determines a natural morphism \(f: U \rightarrow\) Proj \(S\). (c) The morphism \(f\) can be an isomorphism even when \(\varphi\) is not. For example, suppose that \(\varphi_{d}: S_{d} \rightarrow T_{d}\) is an isomorphism for all \(d \geqslant d_{0},\) where \(d_{0}\) is an integer. Then show that \(U=\operatorname{Proj} T\) and the morphism \(f: \operatorname{Proj} T \rightarrow\) Proj \(S\) is an isomorphism. (d) Let \(V\) be a projective variety with homogeneous coordinate ring \(S(\mathrm{I}, \$ 2)\). Show that \(t(V) \cong \operatorname{Proj} S\).

Short Answer

Expert verified
The solution involves proving that the empty set of a graded ring Proj \(S\) is equivalent to every element in \(S_{+}\) being nilpotent. \(U=\left\{p \in \operatorname{Proj} T | p \geqslant \varphi\left(S_{+}\right)\right\}\) is proven to be an open subset of Proj \(T\). Subsequently, the condition \(\varphi_{d}: S_{d} \rightarrow T_{d}\) for all \(d \geqslant d_{0}\) being an isomorphism, ensures \(U=\operatorname{Proj} T\). Lastly, the concept of projective variety is used to show that \(t(V) \cong \operatorname{Proj} S\) by aligning the definitions.

Step by step solution

01

- Prove Proj \(S=\varnothing\) is equivalent to every element of \(S_{+}\) being nilpotent

The first part of the problem requires proving an equivalence, so two proof directions are required. If Proj \(S=\varnothing\), then for every homogeneous element \(f \in S_{+}\), the locus D(f) is empty. As D(f) is defined as the complement in Proj\(S\) of the set of homogeneous prime ideals containing \(f\), we can infer that every homogeneous prime ideal contains \(f\). Recall that an element in a ring is nilpotent if some power of it is 0. Since \(f\) is in every homogeneous prime ideal, it implies that \(f\) and all its powers are nilpotent. Conversely, if every element of \(S_{+}\) is nilpotent, then every homogeneous prime ideal of \(S\) must be the entire \(S_{+}\) and hence, Proj \(S=\varnothing\).
02

- Show that \(U\) is an open subset of Proj \(T\)

We have a graded homomorphism \(\varphi: S \rightarrow T\), a set \(U=\left\{p \in \operatorname{Proj} T | p \geqslant \varphi\left(S_{+}\right)\right\}\). Our goal is to prove that \(U\) is an open set of Proj \(T\). Let \(f \in T_{+}\) such that D(f) is included in \(U\). Getting that \(D(f)\) is in \(U\) means that for any \(p \in D(f)\), \(p \geqslant \varphi(S_{+})\). Recalling the definition of basic opens sets, we can state that any principal open set D(f) is a subset of \(U\) as every \(p \in D(f)\) satisfies \(p \geqslant \varphi(S_{+})\), hence making \(U\) an open subset of Proj \(T\).
03

- Prove that \(U=\operatorname{Proj} T\)

We are given that \(\varphi_{d}: S_{d} \rightarrow T_{d}\) is an isomorphism for all \(d \geqslant d_{0}\). Since \(\varphi_d\) is an isomorphism, it implies that if \(p \in\) Proj \(T\), then \(p \geqslant \varphi(S_{+})\) and \(p\) is part of the set \(U\). Hence, \(U=\operatorname{Proj} T\).
04

鈥� Show that \(t(V) \cong \operatorname{Proj} S\)

The key here is to notice that \(t(V)\) in the context of projective algebraic variety is given by the image of \(V\) under \(t\) and that \(\operatorname{Proj} S\) is a set of all prime ideals of the graded ring \(S\). From the definitions, if these sets are identified via \(t: V \rightarrow \operatorname{Proj} S\), a natural isomorphism is formed.

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Key Concepts

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

Graded Ring
A graded ring is a mathematical structure that is composed of a direct sum of abelian groups indexed by non-negative integers: \( R = \bigoplus_{n=0}^{\infty} R_n \). The key feature of a graded ring is its grading, which provides a way to separate elements by their 'degrees'.
Each piece \( R_n \) is called the 'degree \( n \)' component of \( R \). For a ring to be graded, the product of elements in different components must sum to an element in another component: \( R_i \cdot R_j \subseteq R_{i+j} \).

Understanding graded rings is important as they naturally arise in the study of polynomial rings and across algebraic geometry. They allow for a systematic way to work with the components by degree, which facilitates the classification and analysis of algebraic varieties. Many geometric constructions, such as Proj, are inherently connected to graded rings.
Proj Construction
The Proj construction is a fundamental tool in algebraic geometry used to build projective schemes from graded rings. Given a graded ring \( S \), the construction closely mirrors the process of creating an affine scheme but adapts it to accommodate the projective setting.
The set \( \text{Proj} \,S \) is the set of all homogeneous prime ideals in \( S \) that do not contain the irrelevant ideal \( S_+ \), where \( S_+ = \bigoplus_{n>0} S_n \). This set is topologized using the Zariski topology and equipped with a structure sheaf.

One of the significant properties of Proj is that it creates a projective variety that is invariant under scaling of coordinates, embodying the first principles of projective geometry. It's a versatile construction because it naturally integrates the concept of homogeneity from graded rings into the wider landscape of projective algebraic geometry.
Graded Homomorphism
A graded homomorphism is a kind of ring homomorphism between two graded rings that respects their grading structure. Specifically, if \( S = \bigoplus_{n=0}^\infty S_n \) and \( T = \bigoplus_{n=0}^\infty T_n \) are two graded rings, a homomorphism \( \varphi: S \to T \) is graded if, whenever \( s \in S_n \), then \( \varphi(s) \in T_n \) for all \( n \).
Preserving the degree in homomorphisms is crucial because it ensures that the algebraic operations between elements of specific degrees are consistent with the grading structure. This property allows for richer interaction paradigms between graded structures, leading to natural morphisms between their corresponding Proj constructions.

The concept of graded homomorphisms is instrumental when transitioning between different spaces in algebraic geometry. They enable the projection and transformation of structural information from one graded ring to another while maintaining the inherent projective relationships defined by their gradings.
Projective Variety
A projective variety is a type of algebraic variety that can be embedded into projective space, the space of equivalent classes of points \((x_0:x_1:...:x_n)\), where not all \(x_i\) are zero, and scalars define equivalence. Projective varieties arise naturally when considering solutions to homogeneous polynomial equations.
They are important as they generalize the notion of curves and surfaces into the projective framework, capturing geometric phenomena like intersections at infinity and symmetry under linear transformations.

Projective varieties are central to many advanced topics in algebraic geometry since they can be associated with a homogeneous coordinate ring. Through Proj construction of this ring, one can retrieve the projective variety or scheme. This linkage allows algebraic geometers to use powerful algebraic methods to explore projective varieties, leading to deeper insights into their geometric properties.
  • Affine varieties given by solutions to polynomial equations extend naturally to projective varieties.
  • The behavior of projective varieties is typically more regular, particularly with respect to intersections.

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Most popular questions from this chapter

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 \(n\)

Closed Subschemes. (a) Closed immersions are stable under base extension: if \(f: Y \rightarrow X\) is a closed immersion, and if \(X^{\prime} \rightarrow X\) is any morphism, then \(f^{\prime}: Y \times_{X} X^{\prime} \rightarrow X^{\prime}\) is also a closed immersion. (b) If \(Y\) is a closed subscheme of an affine scheme \(X=\operatorname{Spec} A\), then \(Y\) is also affine, and in fact \(Y\) is the closed subscheme determined by a suitable ideal \(\mathfrak{a} \subseteq A\) as the image of the closed immersion \(\operatorname{Spec} A / \mathfrak{a} \rightarrow \operatorname{Spec} A\). [Hints: First show that \(Y\) can be covered by a finite number of open affine subsets of the form \(D\left(f_{i}\right) \cap Y,\) with \(f_{i} \in A .\) By adding some more \(f_{i}\) with \(D\left(f_{i}\right) \cap Y=\varnothing\) if necessary, show that we may assume that the \(D\left(f_{i}\right)\) cover \(X .\) Next show that \(f_{1}, \ldots, f_{r}\) generate the unit ideal of \(A .\) Then use (Ex. 2.17 b) to show that \(Y\) is affine, and (Ex. \(2.18 \mathrm{d}\) ) to show that \(Y\) comes from an ideal \(\mathfrak{a} \subseteq\) A. .] Note: We will give another proof of this result using sheaves of ideals later (5.10). (c) Let \(Y\) be a closed subset of a scheme \(X\), and give \(Y\) the reduced induced subscheme structure. If \(Y^{\prime}\) is any other closed subscheme of \(X\) with the same underlying topological space, show that the closed immersion \(Y \rightarrow X\) factors through \(Y^{\prime} .\) We express this property by saying that the reduced induced structure is the smallest subscheme structure on a closed subset. (d) Let \(f: Z \rightarrow X\) be a morphism. Then there is a unique closed subscheme \(Y\) of \(X\) with the following property: the morphism \(f\) factors through \(Y\), and if \(Y^{\prime}\) is any other closed subscheme of \(X\) through which \(f\) factors, then \(Y \rightarrow X\) factors through \(Y^{\prime}\) also. We call \(Y\) the scheme-theoretic image of \(f\). If \(Z\) is a reduced scheme, then \(Y\) is just the reduced induced structure on the closure of the image \(f(Z)\)

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.

Product Schemes. (a) Let \(X\) and \(Y\) be schemes over another scheme \(S\). Use (8.10) and (8.11) to show that \(\Omega_{X \times_{s} Y / S} \cong p_{1}^{*} \Omega_{X / S} \oplus p_{2}^{*} \Omega_{Y / S}\) (b) If \(X\) and \(Y\) are nonsingular varieties over a field \(k\), show that \(\omega_{X \times Y} \cong p_{1}^{*} \omega_{X} \otimes\) \(p_{2}^{*} \omega_{Y}\) (c) Let \(Y\) be a nonsingular plane cubic curve, and let \(X\) be the surface \(Y \times Y\) Show that \(p_{g}(X)=1\) but \(p_{a}(X)=-1\) (I, Ex. 7.2). This shows that the arithmetic genus and the geometric genus of a nonsingular projective variety may be different.

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
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