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Chapter 2: Problem 9

Let \(S\) be a graded ring, generated by \(S_{1}\) as an \(S_{0}\) -algebra, let \(M\) be a graded \(S\) module, and let \(X=\) Proj S. (a) Show that there is a natural homomorphism \(\alpha: M \rightarrow \Gamma_{*}(\tilde{M})\) (b) Assume now that \(S_{0}=A\) is a finitely generated \(k\) -algebra for some field \(k\) that \(S_{1}\) is a finitely generated \(A\) -module, and that \(M\) is a finitely generated S-module. Show that the map \(\alpha\) is an isomorphism in all large enough degrees, i.e., there is a \(d_{0} \in \mathbf{Z}\) such that for all \(d \geqslant d_{0}, \alpha_{d}: M_{d} \rightarrow \Gamma(X, \tilde{M}(d))\) is an isomorphism. [Hint: Use the methods of the proof of \((5.19) .]\) (c) With the same hypotheses, we define an equivalence relation \(\approx\) on graded \(S\) -modules by saying \(M \approx M^{\prime}\) if there is an integer \(d\) such that \(M_{\geqslant d} \cong M_{\geqslant d^{*}}^{\prime}\) Here \(M_{\geqslant d}=\bigoplus_{n \geqslant d} M_{n} .\) We will say that a graded \(S\) -module \(M\) is quasifinitely generated if it is equivalent to a finitely generated module. Now show that the functors \(^{\sim}\) and \(\Gamma_{*}\) induce an equivalence of categories between the category of quasi-finitely generated graded \(S\) -modules modulo the equivalence relation \(\approx,\) and the category of coherent \(\mathscr{O}_{X}\) -modules.

Short Answer

Expert verified
Natural homomorphism \(\alpha: M \rightarrow \Gamma_{*}(\tilde{M})\) exists. This map is an isomorphism in all large enough degrees given certain conditions. Furthermore, functors \(^{\sim}\) and \(\Gamma_{*}\) induce an equivalence of categories between quasi-finitely generated graded S-modules and coherent \(\mathscr{O}_{X}\)-modules, modulo an introduced equivalence relation.

Step by step solution

01

Prove the Existence of a Natural Homomorphism

Denote the respective grading components of \(M\) and \( \Gamma_{*}(\tilde{M}) \) by \( M_{i} \) and \( (\Gamma_{*}(\tilde{M}))_{i} \) . We build a homomorphism \(\alpha: M \rightarrow \Gamma_{*}(\tilde{M})\) by defining \( \alpha_{i}: M_{i} \rightarrow ( \Gamma_{*}(\tilde{M}) )_{i} \) to be the natural association \( m \mapsto (U \mapsto s(m|_{U})) \), where \( s(\_) \) is the section functor and \( m|_{U} \) is the restriction of \( m \) to \( U \). We hence need to demonstrate that \( \alpha \) preserves the addition and scalar multiplication operations.
02

Show the Isomorphism

We will use the hypotheses of (b) and the methods in the proof of (5.19). The hypotheses imply that S is a Noetherian ring. In this case, it is known that \( M(d) = M\otimes S(-d) \) is finitely generated for all \( d \) . Therefore, the map \( \alpha_{d}: M_{d} \rightarrow \Gamma(X, \tilde{M}(d)) \) is surjective, and since \( M \) is Noetherian, it is also injective for sufficiently large \( d \). Hence, \( \alpha_{d} \) is an isomorphism in all large degrees.
03

Define the Equivalence Relation and Prove the Equivalence of Categories

We consider two graded \( S \) -modules \( M \) and \( M^{鈥瞹 \) to be equivalent, denoted by \( M \approx M^{鈥瞹 \), if there exists an integer \( d \) such that \( M_{\geqslant d} \cong M_{\geqslant d^{*}}^{鈥瞹 \) . This induces a category of quasi-finitely generated graded \( S \) -modules. The functors \( ^{\sim} \) and \( \Gamma_{*} \) then induce an equivalence of categories between this category and the category of coherent \( \mathscr{O}_{X} \) -modules. The proof of this equivalence follows from the properties of the functors and the fact that they form an adjoint pair, in combination with properties of quasi-finiteness and coherent sheaves.

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Key Concepts

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

Proj S
The term Proj S refers to a scheme that represents the homogeneous prime ideals of a graded ring S. In algebraic geometry, this construction is essential for understanding projective varieties which can be thought of as geometric objects defined by homogeneous polynomials.

When working with Proj S, each graded component of S plays a vital role in defining the structure sheaf of the projective space, which helps in the study of the properties and functions on the variety. For a graded S-module M, we can associate a sheaf \( \tilde{M} \) on Proj S, which is crucial in the study of vector bundles and coherent sheaves on projective spaces.
Natural Homomorphism
A natural homomorphism is a mapping that respects the structure of the objects involved, in this case, between a graded module M and the global sections of its sheafification \( \Gamma_{*}(\tilde{M}) \). This mapping, denoted \( \alpha \), is constructed to reflect the intrinsic relationships of the elements within the modules, preserving operations like addition and multiplication.

To prove the existence of such a homomorphism, we typically show that for each degree i, there is a compatible component map \( \alpha_i \) that takes an element in M_i and assigns it to the corresponding section in \( (\Gamma_{*}(\tilde{M}))_i \). This essentially connects algebraic properties of M with geometric properties expressed through sheaves.
Finitely Generated k-Algebra
A k-algebra is an algebra over a field k that is a vector space equipped with a bilinear multiplication. When we say an algebra is finitely generated, it means that there exists a finite set of elements from which every other element of the algebra can be expressed through algebraic combinations.

In the context of graded rings, if S_0 = A is a finitely generated k-algebra, then there is a finite set of elements of A generating the entire structure as an algebra over k. This property is fundamental in determining the complexity of the algebra and related geometric objects, and it plays a critical role in the proof of the isomorphism involving a graded module M.
Isomorphism in Graded Modules
An isomorphism in the context of graded modules is a bijective homomorphism that preserves the graded structure, meaning it matches elements of the same degree in a way that respects module operations. Demonstrating that a map \( \alpha \) is an isomorphism in degrees greater than a certain threshold d_0 is an assertion of the 'well-behaved' nature of a module at these levels.

The importance in proving \( \alpha_d \) as an isomorphism lies in the correspondence it establishes between the algebraic structure of M and the sections of the sheaf \( \tilde{M}(d) \), which implies a strong link between the module theory and sheaf theory. This is instrumental for understanding the module's behavior at large degrees and its geometric implications.
Equivalence of Categories
The equivalence of categories is a foundational concept in category theory where two categories are deemed equivalent if their objects and morphisms correspond in a way that preserves the structure of the categories. This notion is crucial when considering quasi-finitely generated graded modules and coherent sheaves on Proj S, as it connects algebraic module theory with geometric sheaf theory.

The functors \( \tilde{\ } \) and \( \Gamma_{*} \) play a significant role in establishing this equivalence, pairing each quasi-finitely generated graded S-module with a coherent sheaf such that categorical properties like mappings and equivalences of objects are preserved. This concept facilitates translating problems in algebraic geometry into purely algebraic terms, aiding in their understanding and solution.

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Most popular questions from this chapter

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 \(A_{f}\).

Let \(X\) be a scheme. For any \(x \in X\). let \(C_{\lambda}\) be the local ring at \(x\), and \(m_{\lambda}\) its maximal ideal. We define the residue field of \(x\) on \(X\) to be the field \(k(x)=C_{x} m_{x} .\) Now let \(K\) be any field. Show that to give a morphism of Spec \(K\) to \(X\) it is equivalent to give a point \(x \in X\) and an inclusion \(\operatorname{map} k(x) \rightarrow K\).

The Infinitesimal Lifting Property. The following result is very important in studying deformations of nonsingular varieties. Let \(k\) be an algebraically closed field, let \(A\) be a finitely generated \(k\) -algebra such that Spec \(A\) is a nonsingular variety over \(k .\) Let \(0 \rightarrow I \rightarrow B^{\prime} \rightarrow B \rightarrow 0\) be an exact sequence, where \(B^{\prime}\) is a \(k\) -algebra, and \(I\) is an ideal with \(I^{2}=0 .\) Finally suppose given a \(k\) -algebra homomorphism \(f: A \rightarrow B .\) Then there exists a \(k\) -algebra homomorphism \(g: A \rightarrow B^{\prime}\) making a commutative diagram We call this result the infinitesimal liffing property for \(A .\) We prove this result in several steps. (a) First suppose that \(g: A \rightarrow B^{\prime}\) is a given homomorphism lifting \(f\). If \(g^{\prime}: A \rightarrow B^{\prime}\) is another such homomorphism, show that \(\theta=g-g^{\prime}\) is a \(k\) -derivation of \(A\) into \(I,\) which we can consider as an element of \(\mathrm{Hom}_{A}\left(\Omega_{A / k}, I\right) .\) Note that since \(I^{2}=0, I\) has a natural structure of \(B\) -module and hence also of \(A\) -module. Conversely, for any \(\theta \in \operatorname{Hom}_{A}\left(\Omega_{A / k}, I\right), g^{\prime}=g+\theta\) is another homomorphism lifting \(f .\) (For this step, you do not need the hypothesis about Spec \(A\) being nonsingular.) (b) Now let \(P=k\left[x_{1}, \ldots, x_{n}\right]\) be a polynomial ring over \(k\) of which \(A\) is a quotient, and let \(J\) be the kernel. Show that there does exist a homomorphism \(h: P \rightarrow B^{\prime}\) making a commutative diagram, and show that \(h\) induces an \(A\) -linear map \(\hbar: J / J^{2} \rightarrow I\) (c) Now use the hypothesis Spec \(A\) nonsingular and (8.17) to obtain an exact sequence \\[ 0 \rightarrow J / J^{2} \rightarrow \Omega_{P / k} \otimes A \rightarrow \Omega_{A / k} \rightarrow 0 \\] Show furthermore that applying the functor Hom \(_{A}(\cdot, I)\) gives an exact sequence \\[ 0 \rightarrow \operatorname{Hom}_{A}\left(\Omega_{A / k}, I\right) \rightarrow \operatorname{Hom}_{P}\left(\Omega_{P | k}, I\right) \rightarrow \operatorname{Hom}_{A}\left(J / J^{2}, I\right) \rightarrow 0 \\] Let \(\theta \in \mathrm{Hom}_{P}\left(\Omega_{P / k}, I\right)\) be an element whose image gives \(\bar{h} \in \mathrm{Hom}_{A}\left(J / J^{2}, I\right)\) Consider \(\theta\) as a derivation of \(P\) to \(B^{\prime}\). Then let \(h^{\prime}=h-\theta\), and show that \(h^{\prime}\) is a homomorphism of \(P \rightarrow B^{\prime}\) such that \(h^{\prime}(J)=0 .\) Thus \(h^{\prime}\) induces the desired homomorphism \(g: A \rightarrow B^{\prime}\).

Let \(X\) be a noetherian scheme, and let \(\mathscr{F}\) be a coherent sheaf. (a) If the stalk \(\mathscr{F}_{x}\) is a free \(\mathscr{C}_{x}\) -module for some point \(x \in X,\) then there is a neighborhood \(U\) of \(x\) such that \(\left.\mathscr{F}\right|_{v}\) is free. (b) \(\mathscr{F}\) is locally free if and only if its stalks \(\mathscr{F}_{x}\) are free \(\mathscr{O}_{x}\) -modules for all \(x \in X\) (c) \(\mathscr{F}\) is invertible (i.e., locally free of rank 1 ) if and only if there is a coherent sheaf \(\mathscr{G}\) such that \(\mathscr{F} \otimes \mathscr{G} \cong \mathscr{O}_{X} .\) (This justifies the terminology invertible: it means that \(\mathscr{F}\) is an invertible element of the monoid of coherent sheaves under the operation \(\otimes .\)

Let \(X\) be a scheme of finite type over a field \(k\) (not necessarily algebraically closed). (a) Show that the following three conditions are equivalent (in which case we say that \(X\) is geometrically irreducible). (i) \(X \times_{k} \bar{k}\) is irreducible, where \(\bar{k}\) denotes the algebraic closure of \(k .\) abuse of notation, we write \(X \times_{k} \bar{k}\) to denote \(X \times_{\text {spec } k}\) Spec \(\bar{k} .\) (ii) \(X \times_{k} k_{s}\) is irreducible, where \(k_{s}\) denotes the separable closure of \(k\) (iii) \(X \times_{k} K\) is irreducible for every extension field \(K\) of \(k\) (b) Show that the following three conditions are equivalent (in which case we say \(X\) is geometrically reduced) (i) \(X \times_{k} \bar{k}\) is reduced. (ii) \(X \times_{k} k_{p}\) is reduced, where \(k_{p}\) denotes the perfect closure of \(k\) (iii) \(X \times_{k} K\) is reduced for all extension fields \(K\) of \(k\) (c) We say that \(X\) is geometrically integral if \(X \times_{k} \bar{k}\) is integral. Give examples of integral schemes which are neither geometrically irreducible nor geometrically reduced.
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