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Chapter 2: Problem 6

Support. Recall the notions of support of a section of a sheaf, support of a sheaf, and subsheaf with supports from (Ex. 1.14 ) and (Ex. 1.20 ). (a) Let \(A\) be a ring, let \(M\) be an \(A\) -module, let \(X=\operatorname{Spec} A,\) and let \(\mathscr{F}=\tilde{M}\) For any \(m \in M=\Gamma(X, \overline{\mathscr{F}}),\) show that Supp \(m=V(\text { Ann } m),\) where Ann \(m\) is the annihilator of \(m=\\{a \in A | a m=0\\}\) (b) Now suppose that \(A\) is noetherian, and \(M\) finitely generated. Show that \(\operatorname{Supp} \mathscr{F}=V(\operatorname{Ann} M)\) (c) The support of a coherent sheaf on a noetherian scheme is closed. (d) For any ideal a \(\subseteq A,\) we define a submodule \(\Gamma_{\mathrm{a}}(M)\) of \(M\) by \(\Gamma_{\mathrm{a}}(M)=\) \(\left\\{m \in M | a^{n} m=0 \text { for some } n>0\right\\} .\) Assume that \(A\) is noetherian, and \(M\) any \(A\) -module. Show that \(\Gamma_{\mathrm{a}}(M)^{\sim} \cong \mathscr{H}_{Z}^{0}(\mathscr{F}),\) where \(Z=V(\mathrm{a})\) and \(\mathscr{F}=\tilde{M}\) \([\text {Hint}: \text { Use (Ex. } 1.20)\) and (5.8) to show a priori that \(\mathscr{H}_{Z}^{0}(\mathscr{F})\) is quasi-coherent. Then show that \(\left.\Gamma_{\mathrm{a}}(M) \cong \Gamma_{\mathrm{z}}(\mathscr{F}) .\right]\) (e) Let \(X\) be a noetherian scheme, and let \(Z\) be a closed subset. If \(\mathscr{F}\) is a quasicoherent (respectively, coherent) \(O_{X}\) -module, then \(\mathscr{H}_{Z}^{0}(\mathscr{F})\) is also quasicoherent (respectively, coherent).

Short Answer

Expert verified
The solutions to the different parts of the problem build on each other, starting from showing that the support of section of a sheaf and the support of a sheaf are respectively equal to \( V(\operatorname{Ann} m) \) and \( V(\operatorname{Ann} M) \). Maintaining coherence, it is further proved that the support of a coherent sheaf on a Noetherian scheme is a closed set. Additional concepts and properties, including the notion of a submodule and quasi-coherence, are used to point out their isomorphism and to prove that \( \mathscr{H}_{Z}^{0}(\mathscr{F}) \) retains the coherence property of \( \mathscr{F} \).

Step by step solution

01

Title

Start with part (a). Express \( \operatorname{Supp} m \) as the set of primes \( p \) of \( A \) such that the localization \( m_p \) is non-zero. Then use the definition of the annihilator \( \operatorname{Ann} m \) to obtain an equivalent description of \( V(\operatorname{Ann} m) \). This should lead to the conclusion that \( \operatorname{Supp} m = V(\operatorname{Ann} m) \).
02

Title

Proceed to part (b). First, recall the definition of the support of a sheaf. Then use the properties of the support of sections over affine schemes and properties of the spectrum of a Noetherian ring to show that \( \operatorname{Supp} \mathscr{F} = V(\operatorname{Ann} M) \).
03

Title

Take on part (c). Utilize the fact that a coherent sheaf on a Noetherian scheme is finitely generated, and apply the results of part (b) to prove that the support of the sheaf is a closed subset.
04

Title

Move on to part (d). Utilize the definition \( \Gamma_{\mathrm{a}}(M)^{\sim} \). Then use the hint to show that \( \Gamma_{\mathrm{a}}(M)^{\sim} \) is quasi-coherent. Then apply the definitions of \( \Gamma_{\mathrm{a}}(M) \) and \( \Gamma_{\mathrm{z}}(\mathscr{F}) \) to prove their isomorphism.
05

Title

Finally, tackle part (e). Use the definition and properties of \( \mathscr{H}_{Z}^{0}(\mathscr{F}) \) to assert that \( \mathscr{H}_{Z}^{0}(\mathscr{F}) \) is quasi-coherent. If \( \mathscr{F} \) is additionally coherent, show that \( \mathscr{H}_{Z}^{0}(\mathscr{F}) \) is also coherent.

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Key Concepts

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

Support of a Sheaf
Understanding the support of a sheaf is critical in algebraic geometry as it helps identify the 'locus' of activity or interest within a scheme. The support of a sheaf intuitively corresponds to the collection of points in a scheme where the sheaf does not vanish.

In the context of the exercise, for a given section of a sheaf, we think of the support as all the prime ideals (or geometric points if we're thinking geometrically) where the section is nonzero upon localization. The exercise guides you through showing that for a module, the support of a section is precisely the set of prime ideals containing its annihilator. This aligns with the intuition that the annihilator determines the 'zeros' of a section.

The exercise goes further to relate the support of a sheaf to the annihilator of the module it comes from, particularly in the setting of a Noetherian ring. This relationship is essential as it ties abstract sheaf-theoretic concepts to more concrete ring-theoretic properties which are often easier to handle.
Noetherian Scheme
Noetherian schemes are an important class of schemes characterized by their finiteness conditions. A Noetherian scheme comes from a Noetherian ring, which means every ascending chain of ideals eventually stabilizes - a condition translating to 'every subset has a maximal element'.

In the solutions, it鈥檚 shown how to utilize the Noetherian property to deduce properties about the support of a sheaf. Coherent sheaves on a Noetherian scheme, a central topic in algebraic geometry, are sheaves of modules that are locally finitely generated, a concept fundamental to the exercise at hand.

The Noetherian property ensures that the support of a coherent sheaf on a Noetherian scheme is always a closed subset. This result is important because closed sets in the Zariski topology are algebraic sets, and the concept of closure is tied to algebraic concepts like radical of ideals and irreducible components.
Quasi-Coherent Sheaf
A quasi-coherent sheaf is a type of sheaf that is often easier to work with due to its relationship with modules over a ring. Specifically, it generalizes the notion of a module over a ring to the setting of a scheme. In simple terms, a quasi-coherent sheaf on an affine scheme is analogous to a module over its coordinate ring.

The original exercise features the concept of quasi-coherent sheaves in several parts, illustrating the importance of this concept in deducing properties of related mathematical objects. For instance, when considering sheaves associated to modules affected by some ideal, understanding that these sheaves are quasi-coherent helps assert certain isomorphisms.

Furthermore, the exercise underscores that quasi-coherence is preserved under certain operations, such as taking sections with supports in a closed subset. This preservation property is incredibly significant in the larger scheme of structure-preserving operations, an overarching theme in algebraic geometry.

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Most popular questions from this chapter

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 \(k\) be a field of characteristic \(\neq 2 .\) Let \(f \in k\left[x_{1}, \ldots, x_{n}\right]\) be a square-free nonconstant polynomial, i.e., in the unique factorization of \(f\) into irreducible polynomials, there are no repeated factors. Let \(A=k\left[x_{1}, \ldots, x_{n}, z\right] /\left(z^{2}-f\right)\) Show that \(A\) is an integrally closed ring. [Hint: The quotient field \(K\) of .4 is just \(k\left(x_{1}, \ldots, x_{n}\right)[z]\left(z^{2}-f\right) .\) It is a Galois extension of \(k\left(x_{1}, \ldots, x_{n}\right)\) with Galois group \(\mathbf{Z}\) 2 \(\mathbf{Z}\) generated by \(\boldsymbol{z} \mapsto-z .\) If \(x=\underline{y}+h z \in K,\) where \(g, l_{1} \in h\left(x_{1}, \ldots, x_{n}\right)\) then the minimal polynomial of \(x\) is \(X^{2}-2 g X+\left(g^{2}-h^{2} f\right) .\) Now show that \(\alpha\) is integral over \(k\left[x_{1}, \ldots, x_{n}\right]\) if and only if \(g, l_{1} \in k\left[x_{1}, \ldots, x_{n}\right] .\) Conclude that \(\left.A \text { is the integral closure of } k\left[x_{1}, \ldots, x_{n}\right] \text { in } K .\right]\)

Flasque Sheares. A sheaf \(\bar{y}\) on a topological space \(X\) is flasque if for every inclusion \(V \subseteq U\) of open sets, the restriction \(\operatorname{map} \mathscr{F}(U) \rightarrow \mathscr{F}(V)\) is surjective. (a) Show that a constant sheaf on an irreducible topological space is flasque. See (I, 81 ) for irreducible topological spaces. (b) If \(0 \rightarrow \overline{\mathscr{H}} \rightarrow \mathscr{F} \rightarrow \mathscr{H}^{\prime \prime} \rightarrow 0\) is an exact sequence of sheaves, and if \(\bar{y}\) is flasque, then for any open set \(U\). the sequence \(0 \rightarrow \mathscr{F}^{\prime}(U) \rightarrow \mathscr{F}(U) \rightarrow\) \(\mathscr{F}^{\prime \prime}\left(L^{\prime}\right) \rightarrow 0\) of abelian groups is also exact. (c) If \(0 \rightarrow \mathscr{H} \rightarrow \mathscr{H} \rightarrow \mathscr{H}^{\prime \prime} \rightarrow 0\) is an exact sequence of sheaves, and if \(\mathscr{H}^{\prime}\) and \(\overline{\mathscr{H}}\) are flasque, then \(\mathscr{F}^{\prime \prime}\) is flasque. (d) If \(f: X \rightarrow Y\) is a continuous map, and if \(\mathscr{F}\) is a flasque sheaf on \(X\), then \(f_{*} \overline{\mathscr{H}}\) is a flasque sheaf on \(Y\) (e) Let \(\overline{\mathscr{F}}\) be any sheaf on \(X\). We define a new sheaf \(\mathscr{G}\), called the sheaf of discontinuous sections of \(\mathscr{F}\) as follows. For each open set \(U \subseteq X, \mathscr{G}(U)\) is the set of

Vector Bundles. Let \(Y\) be a scheme. \(A\) (geometric) vector bundle of rank \(n\) over \(Y\) is a scheme \(X\) and a morphism \(f: X \rightarrow Y\), together with additional data consisting of an open covering \(\left\\{U_{i}\right\\}\) of \(Y\), and isomorphisms \(\psi_{i}: f^{-1}\left(U_{i}\right) \rightarrow \mathbf{A}_{U_{i}}^{n}\) such that for any \(i, j,\) and for any open affine subset \(V=\operatorname{Spec} A \subseteq U_{i} \cap U_{j}\) the automorphism \(\psi=\psi_{j} \circ \psi_{i}^{-1}\) of \(\mathbf{A}_{V}^{n}=\operatorname{Spec} A\left[x_{1}, \ldots, x_{n}\right]\) is given by a linear automorphism \(\theta\) of \(A\left[x_{1}, \ldots, x_{n}\right],\) i.e., \(\theta(a)=a\) for any \(a \in A,\) and \(\theta\left(x_{i}\right)=\) \(\sum a_{i j} x_{j}\) for suitable \(a_{i j} \in A\) An isomorphism \(g:\left(X, f,\left\\{U_{i}\right\\},\left\\{\psi_{i}\right\\}\right) \rightarrow\left(X^{\prime}, f^{\prime},\left\\{U_{i}^{\prime}\right\\},\left\\{\psi_{i}^{\prime}\right\\}\right)\) of one vector bundle of rank \(n\) to another one is an isomorphism \(g: X \rightarrow X^{\prime}\) of the underlying schemes, such that \(f=f^{\prime} \circ g,\) and such that \(X, f,\) together with the covering of \(Y\) consisting of all the \(U_{i}\) and \(U_{i}^{\prime},\) and the isomorphisms \(\psi_{i}\) and \(\psi_{i}^{\prime} \circ g,\) is also a vector bundle structure on \(X\) (a) Let \(\mathscr{E}\) be a locally free sheaf of rank \(n\) on a scheme \(Y\). Let \(S(\mathscr{E})\) be the symmetric algebra on \(\mathscr{E},\) and let \(X=\operatorname{Spec} S(\mathscr{E}),\) with projection morphism \(f: X \rightarrow Y\) For each open affine subset \(U \subseteq Y\) for which \(\left.\mathscr{E}\right|_{U}\) is free, choose a basis of \(\mathscr{E}\) and let \(\psi: f^{-1}(U) \rightarrow \mathbf{A}_{U}^{n}\) be the isomorphism resulting from the identification of \(S(\mathscr{E}(U))\) with \(\mathscr{O}(U)\left[x_{1}, \ldots, x_{n}\right] .\) Then \((X, f,\\{U\\},\\{\psi\\})\) is a vector bundle of rank \(n\) over \(Y\), which (up to isomorphism) does not depend on the bases of \(\mathscr{E}_{U}\) chosen. We call it the geometric vector bundle associated to \(\delta,\) and denote it by \(\mathbf{V}(\mathscr{E})\). (b) For any morphism \(f: X \rightarrow Y\), a section of \(f\) over an open set \(U \subseteq Y\) is a morphism \(s: U \rightarrow X\) such that \(f \circ s=\) id \(_{U} .\) It is clear how to restrict sections to smaller open sets, or how to glue them together, so we see that the presheaf \(U \mapsto\\{\text { set of sections of } f \text { over } U\\}\) is a sheaf of sets on \(Y\), which we denote by \(\mathscr{S}(X / Y) .\) Show that if \(f: X \rightarrow Y\) is a vector bundle of \(\operatorname{rank} n,\) then the sheaf of sections \(\mathscr{S}(X / Y)\) has a natural structure of \(\mathscr{O}_{Y}\) -module, which makes it a locally free \(\mathscr{O}_{Y}\) -module of rank \(n\). [Hint: It is enough to define the module structure locally, so we can assume \(Y=\operatorname{Spec} A\) is affine, and \(X=\mathbf{A}_{Y}^{n} .\) Then a section \(s: Y \rightarrow X\) comes from an \(A\) -algebra homomorphism \(\theta: A\left[x_{1}, \ldots, x_{n}\right] \rightarrow\) \(A,\) which in turn determines an ordered \(n\) -tuple \(\left\langle\theta\left(x_{1}\right), \ldots, \theta\left(x_{n}\right)\right\rangle\) of elements of \(A .\) Use this correspondence between sections \(s\) and ordered \(n\) -tuples of elements of \(A \text { to define the module structure. }]\) (c) Again let \(\delta\) be a locally free sheaf of rank \(n\) on \(Y\), let \(X=\mathbf{V}(\delta)\), and let \(\mathscr{S}=\) \(\mathscr{S}(X / Y)\) be the sheaf of sections of \(X\) over \(Y\). Show that \(\mathscr{S} \cong \mathscr{E}^{\curlyvee},\) as follows. Given a section \(s \in \Gamma\left(V, \delta^{\curlyvee}\right)\) over any open set \(V\), we think of \(s\) as an element of \(\operatorname{Hom}\left(\left.\mathscr{E}\right|_{V}, \mathcal{O}_{V}\right) .\) So \(s\) determines an \(\mathscr{O}_{V^{-} \text {algebra homomorphism }} S\left(\left.\mathscr{E}\right|_{V}\right) \rightarrow \mathcal{O}_{V}\) This determines a morphism of spectra \(V=\operatorname{Spec} O_{V} \rightarrow \operatorname{Spec} S\left(\left.\mathscr{E}\right|_{V}\right)=\) \(f^{-1}(V),\) which is a section of \(X / Y .\) Show that this construction gives an isomorphism of \(\mathscr{E}^{\curlyvee}\) to \(\mathscr{S}\) (d) Summing up, show that we have established a one-to-one correspondence between isomorphism classes of locally free sheaves of rank \(n\) on \(Y\), and isomorphism classes of vector bundles of rank \(n\) over \(Y\). Because of this, we sometimes use the words "locally free sheaf" and "vector bundle" interchangeably, if no confusion seems likely to result.

Let \(A\) be a ring, let \(S=A\left[x_{0}, \ldots, x_{r}\right]\) and let \(X=\) Proj \(S\). We have seen that a homogeneous ideal \(I\) in \(S\) defines a closed subscheme of \(X\) (Ex. 3.12 ), and that conversely every closed subscheme of \(X\) arises in this way (5.16) (a) For any homogeneous ideal \(I \subseteq S\), we define the saturation \(I\) of \(I\) to be \(\left\\{s \in S | \text { for each } i=0, \ldots, r, \text { there is an } n \text { such that } x_{i}^{n} s \in I\right\\} .\) We say that \(I\) is saturated if \(I=I .\) Show that \(T\) is a homogeneous ideal of \(S\). (b) Two homogeneous ideals \(I_{1}\) and \(I_{2}\) of \(S\) define the same closed subscheme of \(X\) if and only if they have the same saturation. (c) If \(Y\) is any closed subscheme of \(X\), then the ideal \(\Gamma_{*}\left(\mathscr{I}_{Y}\right)\) is saturated. Hence it is the largest homogeneous ideal defining the subscheme \(Y\) (d) There is a \(1-1\) correspondence between saturated ideals of \(S\) and closed subschemes of \(X\).
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