Conditions For Cokernel Of A Morphism To Be A Vector Bundle

In algebraic geometry, understanding the properties of vector bundles and their morphisms is crucial for studying the geometry of schemes. One fundamental question arises when we consider a morphism defined by a section of a locally free sheaf. Specifically, when is the cokernel of such a morphism a vector bundle? This question delves into the interplay between sheaf theory, vector bundles, and the underlying scheme structure. This article aims to provide a comprehensive exploration of this topic, offering insights and conditions under which the cokernel of a morphism induced by a section becomes a vector bundle. Throughout this discussion, we will explore essential concepts such as locally free sheaves, sections, morphisms, and cokernels, and their roles in determining the vector bundle structure of the cokernel.

To address the central question, it's essential to establish a solid foundation by defining the key concepts involved. Let's begin by elucidating the definitions of schemes, locally free sheaves, sections, and cokernels. A scheme in algebraic geometry is a topological space equipped with a sheaf of rings, generalizing the notion of an algebraic variety. Schemes provide a flexible framework for studying algebraic structures and their geometric properties. A locally free sheaf on a scheme $X$ is a sheaf of $\mathcal{O}_{X}$-modules that is locally isomorphic to a free module of finite rank. In simpler terms, this means that for every point on the scheme, there exists a neighborhood where the sheaf behaves like a vector space. Locally free sheaves are the sheaf-theoretic analogs of vector bundles, playing a crucial role in many geometric constructions. A section of a sheaf $\mathcal{E}$ on a scheme $X$ is a morphism from the structure sheaf $\mathcal{O}_{X}$ to $\mathcal{E}$, denoted as $s: \mathcal{O}_{X} \rightarrow \mathcal{E}$. Sections provide global information about the sheaf, akin to global functions on a manifold. Lastly, the cokernel of a morphism $f: \mathcal{A} \rightarrow \mathcal{B}$ between sheaves is defined as the quotient sheaf $\operatorname{coker}(f) = \mathcal{B} / \operatorname{im}(f)$, where $\operatorname{im}(f)$ is the image of $f$. The cokernel captures the part of the target sheaf that is not reached by the morphism, offering valuable insights into the structure of the morphism and the sheaves involved. Understanding these fundamental concepts is pivotal for analyzing the conditions under which the cokernel of a morphism defined by a section becomes a vector bundle.

Consider a scheme $X$ and a locally free sheaf $\mathcal{E}$ of finite rank on $X$. Let $s \in \Gamma(X, \mathcal{E})$ be a section of $\mathcal{E}$. This section $s$ corresponds to a morphism $s: \mathcal{O}_{X} \rightarrow \mathcal{E}$. Our primary objective is to determine when the cokernel of this morphism, denoted as $\operatorname{coker}(s) = \mathcal{E} / \operatorname{im}(s)$, is also a vector bundle. This question bridges the gap between abstract sheaf theory and concrete geometric properties. To dissect this problem, we need to examine the local behavior of the section $s$ and how it interacts with the structure of the sheaf $\mathcal{E}$. The cokernel, being a quotient sheaf, inherits properties from both $\mathcal{E}$ and the image of $s$. If the image of $s$ behaves predictably, it increases the likelihood that the cokernel will have the desired locally free structure. However, if the section $s$ has singularities or behaves irregularly, the cokernel may exhibit more complex behavior, potentially losing its locally free nature. This nuanced interplay between the section and the sheaf makes the question of when the cokernel is a vector bundle both interesting and challenging. The subsequent sections will delve into specific conditions and criteria that ensure the cokernel's locally free property, providing a comprehensive understanding of this fundamental concept in algebraic geometry.

To ascertain when the cokernel of a morphism defined by a section is a vector bundle, we need to establish specific conditions that govern this behavior. The central criterion revolves around the notion of the section being nowhere vanishing or, more generally, having a constant rank. Let's delve into these conditions in detail. A section $s \in \Gamma(X, \mathcal{E})$ is said to be nowhere vanishing if, for every point $x \in X$, the image $s(x)$ in the fiber $\mathcal{E}_{x}$ is not zero. In other words, the section does not vanish at any point on the scheme. This condition ensures that the morphism $s: \mathcal{O}_{X} \rightarrow \mathcal{E}$ is injective at every stalk, which is a crucial step in proving that the cokernel is locally free. However, the nowhere vanishing condition is often too restrictive. A more general condition involves the rank of the section. The rank of a section $s$ at a point $x$ is defined as the rank of the induced map $s_{x}: \mathcal{O}_{X,x} \rightarrow \mathcal{E}_{x}$ between the stalks. If the rank of $s$ is constant across the scheme, it provides a uniform behavior that is conducive to the cokernel being locally free. Specifically, if the rank of $s$ is constantly 1 and $s$ is locally a non-zero divisor, then the cokernel $\operatorname{coker}(s)$ is locally free. This condition is pivotal because it allows for situations where the section may vanish at certain points, but its behavior is still controlled and predictable. Moreover, the concept of being a non-zero divisor is crucial. A section $s$ is a non-zero divisor if, for every open set $U$ in $X$, the multiplication map $\mathcal{O}_{X}(U) \rightarrow \mathcal{E}(U)$ is injective. This condition ensures that the section does not introduce any artificial relations in the sheaf $\mathcal{E}$, which could disrupt the locally free structure of the cokernel. In summary, the key conditions for the cokernel to be a vector bundle are the constant rank of the section and the non-zero divisor property. These conditions collectively ensure that the quotient sheaf $\operatorname{coker}(s)$ behaves well locally, thus guaranteeing its locally free nature.

To fully appreciate the conditions under which the cokernel of a morphism defined by a section is a vector bundle, let's dissect the concepts of constant rank and non-zero divisors in greater detail. The constant rank condition is paramount in ensuring the uniformity of the morphism $s: \mathcal{O}_{X} \rightarrow \mathcal{E}$. If the rank of $s$ varies across the scheme $X$, it introduces irregularities that can prevent the cokernel from being locally free. The rank of $s$ at a point $x$ is the dimension of the image of the map $s_{x}: \mathcal{O}_{X,x} \rightarrow \mathcal{E}_{x}$, where $\mathcal{O}_{X,x}$ and $\mathcal{E}_{x}$ are the stalks of the sheaves at $x$. When the rank is constant, say $r$, it implies that the image of $s$ forms a subsheaf of $\mathcal{E}$ that behaves consistently across $X$. This consistency is crucial for the quotient sheaf $\operatorname{coker}(s)$ to inherit a well-defined locally free structure. For instance, if $\mathcal{E}$ has rank $n$ and the section $s$ has constant rank 1, the cokernel will have rank $n-1$, which is locally free if $\mathcal{E}$ is locally free. The non-zero divisor condition is equally significant. A section $s$ is a non-zero divisor if the multiplication map $s: \mathcal{O}_{X}(U) \rightarrow \mathcal{E}(U)$ is injective for every open set $U \subset X$. This condition prevents the introduction of torsion elements in the cokernel. Torsion elements are sections that are annihilated by some non-zero element in the structure sheaf, and their presence can disrupt the locally free nature of a sheaf. By ensuring that $s$ is a non-zero divisor, we guarantee that the morphism induced by $s$ is injective, which is a necessary condition for the cokernel to have a clean quotient structure. Moreover, the non-zero divisor condition ensures that the image of $s$ is a well-behaved subsheaf of $\mathcal{E}$, making the quotient $\mathcal{E} / \operatorname{im}(s)$ more likely to be locally free. In practice, checking the non-zero divisor condition often involves verifying that $s$ does not vanish identically on any open subset of $X$. This can be done by examining the local equations defining $s$ and ensuring that they do not share common factors. In conclusion, the interplay between constant rank and the non-zero divisor property is vital for ensuring that the cokernel of a morphism defined by a section is a vector bundle. These conditions provide the necessary uniformity and injectivity to guarantee the locally free structure of the cokernel.

To solidify our understanding of when the cokernel of a morphism defined by a section is a vector bundle, let's explore some illustrative examples. These examples will showcase how the conditions of constant rank and the non-zero divisor property come into play in different scenarios. Example 1: Consider the affine line $\mathbb{A}^{1} = \operatorname{Spec}(k[t])$, where $k$ is a field. Let $\mathcal{E} = \mathcal{O}_{\mathbb{A}^{1}}^{2}$ be a free sheaf of rank 2. Define a section $s: \mathcal{O}_{\mathbb{A}^{1}} \rightarrow \mathcal{E}$ by $s(1) = (t, 0)$. This section corresponds to the morphism $s: \mathcal{O}_{\mathbb{A}^{1}} \rightarrow \mathcal{O}_{\mathbb{A}^{1}}^{2}$ given by $f \mapsto (tf, 0)$. The rank of $s$ at a point $x \in \mathbb{A}^{1}$ corresponding to the maximal ideal $(t - a)$ is 1 if $a \neq 0$, and 0 if $a = 0$. Thus, the rank of $s$ is not constant. The cokernel of $s$ is $\operatorname{coker}(s) = \mathcal{O}_{\mathbb{A}^{1}}^{2} / \operatorname{im}(s)$. At the point $t = 0$, the cokernel is not locally free because the rank of the cokernel jumps. This example illustrates the importance of the constant rank condition. Example 2: Let $X = \mathbb{P}^{1}$, the projective line, and let $\mathcal{E} = \mathcal{O}_{\mathbb{P}^{1}}(1)$. Consider a section $s \in \Gamma(X, \mathcal{E})$ that vanishes at a single point. This section corresponds to a morphism $s: \mathcal{O}_{\mathbb{P}^{1}} \rightarrow \mathcal{O}_{\mathbb{P}^{1}}(1)$. The rank of $s$ is 1 everywhere except at the point where $s$ vanishes, where it is 0. Again, the rank is not constant, and the cokernel will not be locally free. Example 3: Let $X = \mathbb{P}^{1}$ and $\mathcal{E} = \mathcal{O}_{\mathbb{P}^{1}}^{2}$. Consider a section $s: \mathcal{O}_{\mathbb{P}^{1}} \rightarrow \mathcal{E}$ given by $s(1) = (x, y)$, where $x$ and $y$ are homogeneous coordinates on $\mathbb{P}^{1}$. The section $s$ is nowhere vanishing, and its rank is constantly 1. The cokernel $\operatorname{coker}(s)$ is locally free of rank 1, and it is isomorphic to $\mathcal{O}_{\mathbb{P}^{1}}(1)$. This example demonstrates a case where the conditions are met, and the cokernel is indeed a vector bundle. Example 4: Let $X = \operatorname{Spec}(k[x, y])$, and let $\mathcal{E} = \mathcal{O}_{X}^{2}$. Consider the section $s: \mathcal{O}_{X} \rightarrow \mathcal{E}$ given by $s(1) = (x, y)$. The section $s$ is a non-zero divisor, and its rank is constantly 1 away from the origin. However, at the origin, the rank drops to 0. The cokernel is not locally free at the origin. These examples highlight the significance of constant rank and the non-zero divisor property in determining whether the cokernel of a morphism defined by a section is a vector bundle. They also illustrate how deviations from these conditions can lead to cokernels that are not locally free.

While the conditions of constant rank and the non-zero divisor property are crucial for ensuring that the cokernel of a morphism defined by a section is a vector bundle, it is equally important to be aware of counterexamples and potential pitfalls. These counterexamples help illustrate the necessity of these conditions and the subtle ways in which they can be violated. Counterexample 1: Consider the affine plane $\mathbb{A}^{2} = \operatorname{Spec}(k[x, y])$ and the free sheaf $\mathcal{E} = \mathcal{O}_{\mathbb{A}^{2}}^{2}$. Let $s: \mathcal{O}_{\mathbb{A}^{2}} \rightarrow \mathcal{E}$ be the section defined by $s(1) = (x, y)$. Although the components $x$ and $y$ do not have a common factor, the section $s$ vanishes at the origin $(0, 0)$. The rank of $s$ is 1 away from the origin but drops to 0 at the origin. Consequently, the cokernel $\operatorname{coker}(s) = \mathcal{O}_{\mathbb{A}^{2}}^{2} / \operatorname{im}(s)$ is not locally free at the origin. This counterexample underscores the importance of the constant rank condition. If the rank of the section varies, even at a single point, the cokernel may fail to be locally free. Counterexample 2: Let $X = \operatorname{Spec}(k[t])$ be the affine line, and let $\mathcal{E} = \mathcal{O}_{X}$. Consider the section $s: \mathcal{O}_{X} \rightarrow \mathcal{E}$ defined by multiplication by $t$. This section corresponds to the morphism $f \mapsto tf$. The rank of $s$ is 1 everywhere, but $s$ is a zero divisor in the sense that the image of $s$ is the ideal $(t)$, and the cokernel is $k[t]/(t) \cong k$, which is a skyscraper sheaf supported at the origin. This skyscraper sheaf is not locally free, highlighting that even when the rank is constant, the non-zero divisor condition is crucial. Counterexample 3: Consider the scheme $X = \operatorname{Spec}(k[x, y]/(xy))$ and the sheaf $\mathcal{E} = \mathcal{O}_{X}$. Let $s: \mathcal{O}_{X} \rightarrow \mathcal{E}$ be the section defined by $s(1) = x$. The rank of $s$ is 1 away from the axes $x = 0$ and $y = 0$, but it is 0 along the axis $x = 0$. The cokernel is not locally free along the axis $x = 0$, demonstrating that the geometry of the scheme itself can influence the locally free nature of the cokernel. These counterexamples serve as valuable lessons, emphasizing that both the constant rank and the non-zero divisor conditions are essential, and the failure of either can lead to cokernels that are not vector bundles. They also highlight the importance of considering the global geometry of the scheme and the specific properties of the section when analyzing the cokernel.

The question of when the cokernel of a morphism defined by a section is a vector bundle has significant implications and applications in algebraic geometry. Understanding these conditions allows us to construct and analyze vector bundles more effectively, which are fundamental objects in the study of geometric spaces. Applications in Intersection Theory: In intersection theory, vector bundles and their sections play a crucial role in defining intersection products. When a section of a vector bundle vanishes, the vanishing locus provides geometric information about the intersection of subvarieties. The cokernel of the morphism defined by such a section helps in understanding the structure of the quotient sheaf, which can be related to the normal bundle of the intersection. For example, if we have two subvarieties $V$ and $W$ of a smooth variety $X$, their intersection $V \cap W$ can be studied by considering a section of a vector bundle on $X$ whose vanishing locus is $V \cap W$. The cokernel then provides information about the normal bundle of the intersection, which is essential for computing intersection multiplicities and other intersection-theoretic invariants. Applications in Moduli Spaces: Vector bundles also appear prominently in the construction of moduli spaces, which parameterize geometric objects such as curves, surfaces, and vector bundles themselves. The conditions under which a cokernel is a vector bundle are vital in ensuring that certain constructions in moduli theory are well-behaved. For instance, in the moduli space of stable vector bundles on a curve, understanding the cokernels of morphisms between vector bundles is essential for determining the stability conditions and the geometry of the moduli space. Further Directions: There are several avenues for further research and exploration related to this topic. One direction is to consider more general conditions beyond constant rank and the non-zero divisor property. For example, one could investigate cases where the rank of the section is allowed to vary in a controlled manner, such as along a specific subvariety. Another direction is to study the cokernels of morphisms defined by multiple sections or by sections of complexes of sheaves. This leads to more intricate questions about the derived category and the behavior of cokernels in derived categories. Additionally, the study of cokernels in the context of specific geometric settings, such as toric varieties or flag varieties, can yield interesting results and connections to other areas of mathematics. The question of when a cokernel is a vector bundle is a gateway to deeper explorations in algebraic geometry and offers fertile ground for future research.

In conclusion, determining when the cokernel of a morphism defined by a section is a vector bundle is a fundamental question in algebraic geometry with broad implications. The key conditions for the cokernel to be locally free are that the section has constant rank and is a non-zero divisor. These conditions ensure the uniformity and injectivity necessary for the quotient sheaf to inherit a well-defined locally free structure. Through illustrative examples and counterexamples, we have demonstrated the necessity of these conditions and the subtle ways in which they can be violated. The applications of this understanding extend to various areas, including intersection theory and the construction of moduli spaces, highlighting the practical significance of this theoretical question. Further research can explore more general conditions, consider morphisms defined by multiple sections, and delve into specific geometric settings to uncover deeper connections and applications. The exploration of cokernels and their properties remains a vibrant area of research in algebraic geometry, offering rich insights into the structure and behavior of sheaves and vector bundles on schemes.