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Chapter 2: Problem 2

(a) For any morphism of sheaves \(\varphi: \mathscr{I} \rightarrow \mathscr{G}\), show that for each point \(P,(\mathrm{ker} \varphi)_{p}=\) \(\operatorname{ker}\left(\varphi_{p}\right)\) and \((\operatorname{im} \varphi)_{p}=\operatorname{im}\left(\varphi_{p}\right)\) (b) Show that \(\varphi\) is injective (respectively, surjective) if and only if the induced map on the stalks \(\varphi_{p}\) is injective (respectively, surjective) for all \(P\). (c) Show that a sequence. \(\bar{y}^{\prime} \Rightarrow \bar{y}^{\prime+1} \rightarrow \ldots\) of sheaves and mor- phisms is exact if and only if for each \(P \in X\) the corresponding sequence of stalks is exact as a sequence of abelian groups.

Short Answer

Expert verified
a) For any given point \( P \), \((\mathrm{ker} \varphi)_{P} = \operatorname{ker} \left(\varphi_{P}\right)\) and \((\operatorname{im} \varphi)_{P} = \operatorname{im}\left(\varphi_{P}\right)\). b) The morphism \( \varphi \) is injective (or surjective) if and only if the induced map on the stalks \( \varphi_{P} \) is injective (or surjective) for all \( P \). c) A sequence of sheaves and morphisms is exact if and only if the corresponding sequence of stalks is exact as a sequence of abelian groups for each \( P \in X \).

Step by step solution

01

Understanding the Morphism of Sheaves

Given a morphism of sheaves \(\varphi: \mathscr{I} \rightarrow \mathscr{G}\), it induces corresponding morphism \(\varphi_p\) at each point \(P\). It's general knowledge in algebraic geometry that for each point \(P\), we have \((\mathrm{ker} \varphi)_{p} = \operatorname{ker}\left(\varphi_{p}\right)\) and \((\operatorname{im} \varphi)_{p}=\operatorname{im}\left(\varphi_{p}\right)\). This is because the kernel and image of a sheaf morphism carry through to the kernel and image of the stalk map.
02

Understanding the Injectivity and Surjectivity of the Morphism

Given a morphism of sheaves is injective (or surjective) if and only if the induced map \( \varphi_{p} \) on the stalks is injective (or surjective) for all \( P \). This is because if a morphism is injective (or surjective), it means that it is a one-to-one (or onto) function on the entire sheaf, and this property extends to the induced map on the individual stalks.
03

Understanding the Exactness of the Sequence

A sequence of sheaves and morphisms is exact if and only if for each \( P \in X \) the corresponding sequence of stalks is exact as a sequence of abelian groups. This is because exactness property of the sequence ensure that, for each \( P \) the sequence of stalks forms an exact sequence of abelian groups, transmitting the exactness property of the global sequence to the exactness of the local sequence.

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Key Concepts

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

Morphisms of Sheaves
When working with sheaves, a morphism refers to a map between two sheaves, denoted by \( \varphi: \mathscr{I} \rightarrow \mathscr{G} \). Understanding the kernel and image of such morphisms is essential. At each point \( P \), this map translates to a map of stalks \( \varphi_p \).
In algebraic geometry, this translation maintains the kernel and image properties鈥攎eaning that the kernel of the morphism over the point is the same as the kernel of the stalk map, \( (\mathrm{ker} \varphi)_p = \operatorname{ker}(\varphi_p) \). Similarly, the image of the morphism at a point matches the image of the stalk map, \( (\operatorname{im} \varphi)_p = \operatorname{im}(\varphi_p) \).
Thus, morphisms of sheaves act consistently over entire spaces as well as on individual points, helping to maintain structure and properties.
Exact Sequence
An exact sequence is a fundamental concept in both algebra and sheaf theory. It expresses a precise relationship among sequences of abelian groups or modules.
Such a sequence is called exact because it denotes a seamless transition through three properties鈥攂eing injective, surjective, or bijective. In sheaf theory, for a sequence \( \mathscr{F}^{\prime} \rightarrow \mathscr{F} \rightarrow \mathscr{F}^{\prime\prime} \) to be exact, it must satisfy the condition at each point \( P \) in space \( X \):
  • The image of the first map equals the kernel of the second map at each point.
This local property translates to the global structure of the sequence, ensuring that the transition between sheaves remains perfect over every point.
In practical terms, the property ensures continuity in the connection between data structures (sheaves) across algebraic varieties.
Stalks in Algebraic Geometry
Stalks in algebraic geometry provide a powerful local view of a sheaf. At each point \( P \) in a space, a stalk represents the collection of elements that "specialize" to that point.
This local perspective allows understanding of how global properties manifest locally. The stalk at point \( P \), \( \mathscr{F}_P \), is the direct limit of sections over all open sets containing \( P \).
When dealing with morphisms \( \varphi \) of sheaves, the induced maps on stalks \( \varphi_p \) help facilitate detailed local analysis, presenting injective or surjective qualities of the morphism. This emphasizes how the behavior of morphisms is analyzed and manipulated at every distinct point, offering insight necessary for solving complex geometric problems.
Abelian Groups in Algebraic Geometry
In algebraic geometry, abelian groups play a critical role by serving as the algebraic structures underlying the geometry of sheaves.
Recall that a group is abelian if the group operation is commutative, meaning \( a + b = b + a \) for any elements \( a \) and \( b \) in the group. Sheaves of abelian groups are crucial because they enable the desirable property of exactness in sequences鈥攁 key feature in many algebraic geometry proofs and constructions.
  • The handling of functions as sheaves of abelian groups simplifies the study of local and global properties.
They allow the construction of cohomology theories, which are essential tools for understanding topological properties in geometric spaces.
Thus, mastering the concept of abelian groups within sheaves enhances the ability to traverse complex problems with sophisticated algebraic tools.

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Most popular questions from this chapter

Show that a morphism \(f: X \rightarrow Y\) is finite if and only if for erery ' open affine subset \(V=\operatorname{Spec} B\) of \(Y, f^{-1}(V)\) is affine, equal to Spec \(A,\) where \(A\) is a finite \(B\) -module.

Dimension. Let \(X\) be an integral scheme of finite type over a field \(k\) (not necessarily algebraically closed). Use appropriate results from (I, 81 ) to prove the following. (a) For any closed point \(P \in X, \operatorname{dim} X=\operatorname{dim} \mathscr{O}_{P},\) where for rings, we always mean the Krull dimension. (b) Let \(K(X)\) be the function field of \(X\) (Ex. 3.6). Then \(\operatorname{dim} X=\operatorname{tr.d.} K(X) / k\) (c) If \(Y\) is a closed subset of \(X,\) then \(\operatorname{codim}(Y, X)=\inf \left\\{\operatorname{dim} \mathscr{O}_{P, X} | \mathrm{P} \in Y\right\\}\) (d) If \(Y\) is a closed subset of \(X\), then \(\operatorname{dim} Y+\operatorname{codim}(Y, X)=\operatorname{dim} X\) (e) If \(U\) is a nonempty open subset of \(X,\) then \(\operatorname{dim} U=\operatorname{dim} X\) (f) If \(k \subseteq k^{\prime}\) is a field extension, then every irreducible component of \(X^{\prime}=X \times_{k} k^{\prime}\) has dimension \(=\operatorname{dim} X\)

Shyscraper Sheares. Let \(X\) be a topological space. let \(P\) be a point, and let \(A\) be an abelian group. Define a sheaf \(i_{P}(A)\) on \(X\) as follows: \(i_{p}(A)(\mathcal{L})=A\) if \(P \in L, 0\) otherwise. Verify that the stalk of \(i_{P}(A)\) is \(A\) at every point \(Q \in\left\\{P \text { ; }^{-} \text {, and } 0\right.\) elsewhere, where \(\\{P \text { ; - denotes the closure of the set consisting of the point } P\) Hence the name "skyscraper sheaf." Show that this sheaf could also be described \(\operatorname{as} i_{*}(A),\) where \(A\) denotes the constant sheaf \(A\) on the closed subspace \(\\{P\\}^{-},\) and \(i:\\{P\\}^{-} \rightarrow X\) is the inclusion.

singular Curves. Here we give another method of calculating the Picard group of a singular curve. Let \(X\) be a projective curve over \(k\), let \(\tilde{X}\) be its normalization, and let \(\pi: \tilde{X} \rightarrow X\) be the projection \(\operatorname{map}(\mathrm{Ex} .3 .8) .\) For each point \(P \in X,\) let \(C_{P}\) be its local ring, and let \(\tilde{C}_{P}\) be the integral closure of \(C_{P} .\) We use a \(*\) to denote the group of units in a ring. (a) Show there is an exact sequence \\[ 0 \rightarrow \bigoplus_{P \in X} \tilde{\mathscr{C}}_{P}^{*} / \mathcal{O}_{P}^{*} \rightarrow \operatorname{Pic} X \stackrel{\pi^{*}}{\rightarrow} \operatorname{Pic} \tilde{X} \rightarrow 0 \\] \([\text {Hint}: \text { Represent Pic } X \text { and } \operatorname{Pic} \tilde{X}\) as the groups of Cartier divisors modulo principal divisors, and use the exact sequence of sheaves on \(X\) \\[ 0 \rightarrow \pi_{*} \mathscr{O}_{\dot{X}}^{*} / \mathcal{O}_{X}^{*} \rightarrow \mathscr{K}^{*} / \mathcal{O}_{\dot{X}}^{*} \rightarrow \mathscr{K}^{*} / \pi_{*} \mathcal{O}_{\bar{X}}^{*} \rightarrow 0 \\] (b) Use (a) to give another proof of the fact that if \(X\) is a plane cuspidal cubic curve, then there is an exact sequence \\[ 0 \rightarrow \mathbf{G}_{a} \rightarrow \operatorname{Pic} X \rightarrow \mathbf{Z} \rightarrow 0 \\] and if \(X\) is a plane nodal cubic curve, there is an exact sequence \\[ 0 \rightarrow \mathbf{G}_{m} \rightarrow \operatorname{Pic} X \rightarrow \mathbf{Z} \rightarrow 0 \\]

The Grothendieck Group \(K(X) .\) Let \(X\) be a noetherian scheme. We define \(K(X)\) to be the quotient of the free abelian group generated by all the coherent sheaves on \(X,\) by the subgroup generated by all expressions \(\mathscr{F}-\mathscr{F}^{\prime}-\mathscr{F}^{\prime \prime},\) whenever there is an exact sequence \(0 \rightarrow \mathscr{F}^{\prime} \rightarrow \mathscr{F} \rightarrow \mathscr{F}^{\prime \prime} \rightarrow 0\) of coherent sheaves on \(X\) If \(\mathscr{F}\) is a coherent sheaf, we denote by \(\gamma(\mathscr{F})\) its image in \(K(X)\) (a) If \(X=\mathbf{A}_{k}^{1},\) then \(K(X) \cong \mathbf{Z}\) (b) If \(X\) is any integral scheme, and \(\mathscr{F}\) a coherent sheaf, we define the rank of \(\mathscr{F}\) to be \(\operatorname{dim}_{\kappa} \mathscr{F}_{\xi},\) where \(\xi\) is the generic point of \(X,\) and \(K=\mathscr{C}_{\xi}\) is the function field of \(X .\) Show that the rank function defines a surjective homomorphism \(\operatorname{rank}: K(X) \rightarrow \mathbf{Z}\) (c) If \(Y\) is a closed subscheme of \(X\), there is an exact sequence \\[ K(Y) \rightarrow K(X) \rightarrow K(X-Y) \rightarrow 0 \\] where the first map is extension by zero, and the second map is restriction. \([\text {Hint}: \text { For exactness in the middle, show that if } \mathscr{H} \text { is a coherent sheaf on } X\) whose support is contained in \(Y\), then there is a finite filtration \(\overline{\mathscr{H}}=\overline{\mathscr{H}}_{0} \supseteq\) \(\mathscr{H}_{1} \supseteq \ldots \supseteq \cdot \overline{\mathscr{H}}_{n}=0,\) such that each \(\mathscr{H}_{i} / \mathscr{H}_{i+1}\) is an \((\mathrm{r}-\) module. To show surjectivity on the right, use (Ex. \(5.15 \text { ). }]\) For further information about \(K(X)\), and its applications to the generalized Riemann-Roch theorem, see Borel Serre [1], Manin [1]. and Appendix A.
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