Cokernel Of A Morphism When Is It A Vector Bundle

Introduction

In algebraic geometry, understanding the properties of vector bundles and their relationships with morphisms is crucial. Specifically, this article delves into the conditions under which the cokernel of a morphism, defined by a section of a locally free sheaf on a scheme, forms a vector bundle. This exploration is fundamental in various contexts, including the study of singularities, moduli spaces, and the classification of sheaves. Let's delve deep into vector bundles and morphisms. A vector bundle can be thought of as a family of vector spaces parameterized by a topological space or, in the context of algebraic geometry, a scheme. These bundles play a significant role in representing geometric and topological information. A morphism, in the context of sheaf theory, is a map between sheaves that respects their underlying structure. When we consider a section of a locally free sheaf, it induces a natural morphism, and the properties of the cokernel of this morphism become highly relevant.

In this discussion, we consider $X$ as a scheme and $\mathcal{E}$ as a locally free sheaf of finite rank on $X$. Let $s \in \Gamma(X, \mathcal{E})$ be a section. This section corresponds to a morphism $s : \mathcal{O}_{X} \rightarrow \mathcal{E}$. The cokernel of this morphism, denoted as $\mathcal{E} / \text{Im}(s)$, is the quotient sheaf obtained by taking the quotient of $\mathcal{E}$ by the image of $s$. The central question here is: when is this cokernel, $\mathcal{E} / \text{Im}(s)$, also a vector bundle? Understanding the conditions under which this occurs is pivotal in various areas of algebraic geometry. This article aims to provide a comprehensive exploration of these conditions, offering insights and examples to clarify the underlying concepts. We will explore different scenarios and provide detailed explanations, making it accessible to both students and researchers in the field. We also aim to establish a clear understanding of the relationship between sections of locally free sheaves and the vector bundle structure of the resulting cokernel. This discussion will provide a foundation for further research and applications in more advanced topics in algebraic geometry.

Background and Definitions

To address the main question, it is essential to first establish a solid understanding of the foundational concepts. This involves defining key terms such as schemes, locally free sheaves, sections, morphisms, and cokernels. A scheme is a fundamental object in algebraic geometry, generalizing the notion of an algebraic variety by allowing for nilpotents in the structure sheaf. Schemes provide a powerful framework for studying geometric objects using algebraic tools. A locally free sheaf $\mathcal{E}$ on a scheme $X$ is a sheaf of $\mathcal{O}_{X}$-modules such that there exists an open cover ${U_{i}}$ of $X$ for which the restriction of $\mathcal{E}$ to each $U_{i}$ is isomorphic to a free $\mathcal{O}_{U_{i}}$-module of finite rank. Locally free sheaves are also known as vector bundles, as they behave analogously to vector bundles in differential geometry and topology. The rank of a locally free sheaf at a point $x \in X$ is the dimension of the vector space obtained by tensoring the stalk $\mathcal{E}_{x}$ with the residue field $k(x)$ over the stalk of the structure sheaf $\mathcal{O}_{X,x}$.

A section $s$ of a sheaf $\mathcal{E}$ on $X$ is an element of the global sections $\Gamma(X, \mathcal{E})$. In the case where $\mathcal{E}$ is a locally free sheaf, a section can be thought of as a generalization of a vector field. Given a section $s \in \Gamma(X, \mathcal{E})$, we can define a morphism $s : \mathcal{O}_{X} \rightarrow \mathcal{E}$ by mapping $1$ to $s$. This morphism is crucial in our context, as it allows us to study the cokernel of the induced map. The cokernel of a morphism $f : \mathcal{A} \rightarrow \mathcal{B}$ between sheaves is defined as the sheaf associated to the presheaf $U \mapsto \mathcal{B}(U) / \text{Im}(f(U))$. In simpler terms, the cokernel measures what is left of $\mathcal{B}$ after removing the image of $\mathcal{A}$ under $f$. Specifically, in our case, the cokernel of the morphism $s : \mathcal{O}_{X} \rightarrow \mathcal{E}$ is denoted as $\mathcal{E} / \text{Im}(s)$ and is defined locally. To illustrate this, consider an open set $U \subset X$. The sections of $\mathcal{E} / \text{Im}(s)$ over $U$ are given by the sections of $\mathcal{E}$ over $U$ modulo the sections that are in the image of the map $\mathcal{O}_{X}(U) \rightarrow \mathcal{E}(U)$ induced by $s$. This definition ensures that the cokernel is a sheaf, which is essential for further analysis.

Conditions for the Cokernel to be a Vector Bundle

The core question of when the cokernel $\mathcal{E} / \text{Im}(s)$ is a vector bundle hinges on the properties of the section $s$. A fundamental condition is that the section $s$ must be locally a non-zero divisor. This means that for every point $x \in X$, there exists an open neighborhood $U$ of $x$ such that $s$ does not vanish on $U$. More formally, a section $s \in \Gamma(X, \mathcal{E})$ is said to be locally a non-zero divisor if the induced map $s : \mathcal{O}_{X} \rightarrow \mathcal{E}$ is injective. This injectivity condition ensures that the image of $s$ is well-behaved, which is crucial for the cokernel to be locally free.

To delve deeper, let's consider the stalk of the cokernel at a point $x \in X$. The stalk of $\mathcal{E} / \text{Im}(s)$ at $x$ is given by $(\mathcal{E} / \text{Im}(s))_{x} \cong \mathcal{E}_{x} / (s(\mathcal{O}_{X,x}))$, where $\mathcal{E}_{x}$ is the stalk of $\mathcal{E}$ at $x$, and $s(\mathcal{O}_{X,x})$ is the image of the stalk of $\mathcal{O}_{X}$ at $x$ under the morphism induced by $s$. For $\mathcal{E} / \text{Im}(s)$ to be locally free, this quotient module must be a free $\mathcal{O}_{X,x}$-module for every $x \in X$. This condition is satisfied if and only if the map $s_{x} : \mathcal{O}_{X,x} \rightarrow \mathcal{E}_{x}$ is injective and the quotient module is locally free. Another way to state this condition is in terms of the rank of the cokernel. If the rank of $\mathcal{E}$ is $r$, then the cokernel $\mathcal{E} / \text{Im}(s)$ is a vector bundle of rank $r-1$ if and only if the section $s$ is locally a non-zero divisor and the rank of $\mathcal{E} / \text{Im}(s)$ is constant. This constant rank condition is essential to ensure that the cokernel behaves uniformly across the scheme $X$. In summary, the key conditions for the cokernel $\mathcal{E} / \text{Im}(s)$ to be a vector bundle are that the section $s$ must be locally a non-zero divisor, and the rank of the cokernel must be constant. These conditions ensure that the cokernel behaves well locally and globally, making it a vector bundle.

Examples and Counterexamples

To illustrate the concepts discussed, let's consider some examples and counterexamples. These examples will help to solidify the understanding of when the cokernel $\mathcal{E} / \text{Im}(s)$ is a vector bundle and when it is not.

Example 1: Let $X = \mathbb{A}^{1}_{k} = \text{Spec}(k[t])$, where $k$ is a field, and let $\mathcal{E} = \mathcal{O}_{X}^{2}$, which is a free sheaf of rank 2. Consider the section $s \in \Gamma(X, \mathcal{E})$ given by $s = (t, 0)$. This section corresponds to the morphism $s : \mathcal{O}_{X} \rightarrow \mathcal{O}_{X}^{2}$ defined by $1 \mapsto (t, 0)$. The image of this morphism is the subsheaf generated by $(t, 0)$. The cokernel $\mathcal{E} / \text{Im}(s)$ is then isomorphic to the sheaf associated to the module $k[t]^{2} / (t, 0) \cong k[t] / (t) \oplus k[t] \cong k \oplus k[t]$. This cokernel is not locally free at the point $(t) \in \text{Spec}(k[t])$, because the stalk at $(t)$ is $k \oplus k[t]_{(t)}$, which is not a free $k[t]_{(t)}$-module. Thus, in this case, the cokernel is not a vector bundle, because the section $s = (t, 0)$ vanishes at $t = 0$.

Example 2: Let $X = \mathbb{P}^{1}_{k}$, and let $\mathcal{E} = \mathcal{O}_{X}(1) \oplus \mathcal{O}_{X}(1)$. Consider the section $s \in \Gamma(X, \mathcal{E})$ given by the global section corresponding to the map $1 \mapsto (x, y)$, where $x$ and $y$ are homogeneous coordinates on $\mathbb{P}^{1}_{k}$. This corresponds to a morphism $s : \mathcal{O}_{X} \rightarrow \mathcal{O}_{X}(1) \oplus \mathcal{O}_{X}(1)$. The cokernel $\mathcal{E} / \text{Im}(s)$ is locally free because the section $(x, y)$ never vanishes simultaneously on $\mathbb{P}^{1}_{k}$. Indeed, the cokernel is isomorphic to $\mathcal{O}_{X}(2)$, which is a line bundle (a vector bundle of rank 1). This example illustrates a case where the cokernel is a vector bundle because the section does not vanish.

Counterexample: Let $X = \mathbb{A}^{2}_{k} = \text{Spec}(k[x, y])$, and let $\mathcal{E} = \mathcal{O}_{X}^{2}$. Consider the section $s \in \Gamma(X, \mathcal{E})$ given by $s = (x, y)$. This section corresponds to the morphism $s : \mathcal{O}_{X} \rightarrow \mathcal{O}_{X}^{2}$ defined by $1 \mapsto (x, y)$. The cokernel $\mathcal{E} / \text{Im}(s)$ is not locally free at the origin $(0, 0)$, because the ideal generated by $x$ and $y$ is not principal. At the origin, the stalk of the cokernel is not a free module, and thus, the cokernel is not a vector bundle. This counterexample highlights the importance of the section not vanishing for the cokernel to be a vector bundle.

These examples and counterexamples provide concrete illustrations of the conditions under which the cokernel of a morphism defined by a section is a vector bundle. They show that the properties of the section, particularly its vanishing behavior, play a crucial role in determining the structure of the cokernel. By carefully examining these cases, we can develop a deeper understanding of the relationship between sections and vector bundles in algebraic geometry.

Applications and Further Directions

The conditions under which the cokernel of a morphism is a vector bundle have significant applications in various areas of algebraic geometry. Understanding these conditions is crucial for studying the geometry of schemes and their sheaves. One important application is in the study of singularities. When a section $s$ vanishes, the cokernel $\mathcal{E} / \text{Im}(s)$ may fail to be locally free, indicating the presence of singularities. By analyzing the cokernel, one can gain insights into the nature of these singularities. Specifically, the non-locally free locus of the cokernel corresponds to the points where the section vanishes, providing valuable information about the geometry of the scheme.

Another application lies in the construction and study of moduli spaces. Moduli spaces parameterize geometric objects, such as vector bundles or sheaves, with certain properties. The conditions for the cokernel to be a vector bundle are essential in defining and studying these moduli spaces. For example, in the moduli space of stable vector bundles, the stability condition often involves conditions on the rank and degree of subsheaves, which can be related to the cokernel of morphisms. By ensuring that certain cokernels are vector bundles, one can guarantee that the moduli space has desirable properties, such as being smooth or having a well-defined dimension.

Furthermore, the study of cokernels is also relevant in the context of derived categories and homological algebra. The cokernel is a fundamental concept in the theory of chain complexes and derived functors. Understanding when the cokernel is locally free allows for the application of powerful tools from homological algebra to the study of sheaves and vector bundles. For instance, in the derived category of coherent sheaves, the cokernel plays a crucial role in defining distinguished triangles and studying extensions of sheaves. In terms of further research directions, one could explore the conditions under which higher-order cokernels or iterated cokernels are vector bundles. This involves considering more complex morphisms and sheaves and analyzing the resulting cokernels. Additionally, one could investigate the relationship between the cokernel and other sheaf operations, such as tensor products or exterior powers. These investigations could lead to new insights into the structure of sheaves and their applications in algebraic geometry.

Another direction for further research is to consider the case where the scheme $X$ has additional structure, such as being a smooth variety or a Calabi-Yau manifold. In these cases, the conditions for the cokernel to be a vector bundle may simplify or have additional implications. For example, on a smooth variety, one can use the cotangent bundle and its sections to study the geometry of the variety. The cokernel of the morphism induced by these sections can provide information about the tangent spaces and the singularities of the variety. In summary, the study of when the cokernel of a morphism is a vector bundle is a rich and multifaceted topic with applications in various areas of algebraic geometry. By continuing to explore these concepts and their implications, we can further advance our understanding of the geometry of schemes and their sheaves.

Conclusion

In conclusion, the question of when the cokernel of a morphism defined by a section is a vector bundle is a fundamental one in algebraic geometry, with far-reaching implications. This article has explored the key conditions for the cokernel $\mathcal{E} / \text{Im}(s)$ of a morphism $s : \mathcal{O}_{X} \rightarrow \mathcal{E}$ to be a vector bundle, where $\mathcal{E}$ is a locally free sheaf on a scheme $X$. The central condition is that the section $s$ must be locally a non-zero divisor, ensuring that the induced map is injective. Additionally, the rank of the cokernel must be constant, guaranteeing that the quotient sheaf behaves uniformly across the scheme. We've delved into the foundational definitions of schemes, locally free sheaves, sections, morphisms, and cokernels, providing a solid basis for understanding the main question. Through detailed examples and counterexamples, we've illustrated the practical implications of these conditions, highlighting cases where the cokernel is a vector bundle and those where it is not.

The examples demonstrated the importance of the section's vanishing behavior in determining the structure of the cokernel. For instance, a section that vanishes at certain points can lead to a cokernel that is not locally free at those points, thus not a vector bundle. Conversely, sections that do not vanish ensure that the cokernel remains locally free, resulting in a vector bundle. Furthermore, we have discussed various applications of these concepts in areas such as singularity theory, moduli spaces, and derived categories. The understanding of when cokernels are vector bundles is crucial for studying singularities, constructing moduli spaces with desirable properties, and applying tools from homological algebra to the study of sheaves. The exploration of further research directions, such as higher-order cokernels and the relationship between cokernels and other sheaf operations, opens avenues for new insights into the structure of sheaves and their applications.

In essence, the study of cokernels and their properties is a cornerstone of algebraic geometry, providing essential tools for analyzing the geometry of schemes and their sheaves. By thoroughly understanding the conditions under which cokernels are vector bundles, we can advance our knowledge of algebraic geometry and its applications in various mathematical domains. This article aims to serve as a comprehensive guide to this topic, offering clarity and depth for both students and researchers in the field. The exploration of these concepts not only enhances our theoretical understanding but also paves the way for practical applications and further research, solidifying the importance of this topic in the realm of algebraic geometry.