Why Is The Category Of Weil-étale Sheaves Abelian With Enough Injectives?
In the fascinating realm of arithmetic geometry, understanding the cohomological properties of schemes over finite fields is paramount. Stephen Lichtenbaum's groundbreaking work on the Weil-étale topology has provided a powerful framework for studying these properties, particularly through the lens of Weil-étale cohomology. This article delves into a crucial aspect of this theory: why the category of Weil-étale sheaves is abelian with enough injectives. This property is fundamental for the well-definedness and utility of Weil-étale cohomology, enabling us to construct derived functors and extract valuable arithmetic information.
Exploring the Category of Weil-étale Sheaves
To understand why Weil-étale sheaves form an abelian category with sufficient injectives, it is crucial to first define and explore the category itself. Let's consider a scheme X of finite type over a finite field k. The Weil-étale topology, unlike the classical étale topology, incorporates information about the Frobenius endomorphism, a central player in the arithmetic of finite fields. This incorporation is achieved by considering Weil étale coverings, which are étale morphisms equipped with an action of the Weil group of k. Understanding the properties of Weil-étale sheaves requires us to understand the underlying topological structure and how sheaves behave within this structure.
A Weil-étale sheaf on X is a sheaf on the Weil-étale site of X. This site consists of Weil-étale coverings of X, and the sheaves are contravariant functors satisfying a gluing axiom analogous to the gluing axiom for sheaves in the usual étale topology. However, the presence of the Weil group action introduces a subtle difference. This Weil group action on Weil-étale sheaves encodes crucial arithmetic information about the scheme X and its relationship to the base field k. The structure of the category of Weil-étale sheaves is significantly influenced by this action, contributing to its abelian nature and the existence of enough injectives.
Weil-étale sheaves serve as a bridge connecting the geometric structure of the scheme X with the arithmetic properties of the finite field k. The category of Weil-étale sheaves, denoted by Sh(XW), is the category whose objects are Weil-étale sheaves on X and whose morphisms are morphisms of sheaves. Showing that this category is abelian involves demonstrating the existence of kernels, cokernels, and that monomorphisms and epimorphisms behave as expected in an abelian category. The existence of enough injectives, on the other hand, means that every object in Sh(XW) can be embedded in an injective object. This property is essential for doing homological algebra in the category, allowing us to define derived functors like the Weil-étale cohomology functors.
The Abelian Nature: A Symphony of Exactness
Proving that the category of Weil-étale sheaves is abelian is a cornerstone result that underpins the entire theory of Weil-étale cohomology. An abelian category, by definition, is a category that satisfies several crucial properties. These include the existence of zero objects, the existence of kernels and cokernels for every morphism, and the requirement that monomorphisms are kernels and epimorphisms are cokernels. Essentially, an abelian category provides a setting where homological algebra can be effectively carried out, which is vital for studying the intricate structures encoded within Weil-étale sheaves.
To demonstrate the abelian nature of Sh(XW), we must carefully construct kernels and cokernels for morphisms of Weil-étale sheaves. The presence of the Weil group action complicates these constructions, requiring us to ensure that the kernels and cokernels we define also inherit a compatible Weil group action. This careful handling of the Weil group action is critical for maintaining the structure of Weil-étale sheaves throughout our constructions. The process involves showing that the kernel and cokernel, constructed pointwise, satisfy the sheaf condition and possess a natural Weil group action that makes them Weil-étale sheaves themselves.
The abelian property of Weil-étale sheaves guarantees that short exact sequences behave predictably. A sequence of sheaves is exact if the image of each morphism coincides with the kernel of the next. This exactness property is crucial for cohomological calculations because it allows us to relate the cohomology of different sheaves within the sequence. In the context of Weil-étale cohomology, exact sequences of Weil-étale sheaves give rise to long exact sequences in cohomology, which are powerful tools for computing and understanding cohomology groups. The abelian nature thus provides a robust algebraic framework for studying Weil-étale sheaves and their cohomological properties.
Understanding the kernels and cokernels of morphisms between Weil-étale sheaves is essential for proving that the category is abelian. This involves constructing these objects explicitly and verifying that they satisfy the universal properties that define kernels and cokernels. The construction process often involves taking kernels and cokernels at the level of presheaves and then sheafifying the result. The Weil group action must be carefully considered at each step to ensure that the resulting kernel and cokernel are indeed Weil-étale sheaves. This rigorous verification demonstrates the abelian structure of the category, paving the way for the application of homological algebra techniques.
Enough Injectives: A Reservoir for Cohomology
Beyond its abelian nature, the category of Weil-étale sheaves possesses another vital property: it has enough injectives. This means that for any Weil-étale sheaf F, there exists an injective Weil-étale sheaf I and a monomorphism F → I. The existence of enough injectives is paramount because it allows us to compute derived functors, most notably the Weil-étale cohomology functors. Injective objects serve as building blocks for resolutions, which are essential for defining derived functors in a way that is independent of the specific resolution chosen. The presence of enough injectives in the category of Weil-étale sheaves is thus a cornerstone for the well-definedness and computational tractability of Weil-étale cohomology.
To demonstrate the existence of enough injectives, one often leverages Grothendieck's theorem, which provides sufficient conditions for a category of sheaves to have enough injectives. These conditions typically involve checking that the underlying site has enough points and that certain limits and colimits exist in the category of sheaves. The proof that Weil-étale sheaves have enough injectives is technically involved, requiring a careful analysis of the Weil-étale site and the properties of sheaves on this site. It often involves constructing injective objects by embedding sheaves into larger sheaves that are known to be injective, such as the sheaf of sections of a flasque sheaf. The crucial step is to ensure that the constructed injective objects possess a compatible Weil group action, making them bona fide injective objects in the category of Weil-étale sheaves.
Injective Weil-étale sheaves are crucial for defining the Weil-étale cohomology functors. These functors, denoted by HiW(X, F), measure the higher cohomological properties of the Weil-étale sheaf F on the scheme X. They are defined as the derived functors of the global sections functor, which takes a Weil-étale sheaf to the group of its global sections. The computation of these cohomology groups often involves taking an injective resolution of F, applying the global sections functor to the resolution, and then taking the cohomology of the resulting complex. The existence of enough injectives guarantees that such resolutions exist and that the resulting cohomology groups are well-defined.
The injectivity of Weil-étale sheaves plays a vital role in various aspects of Weil-étale cohomology theory. For instance, injective resolutions are used to establish long exact sequences in cohomology, which are fundamental tools for relating the cohomology of different sheaves. Furthermore, the injectivity property is essential for proving various duality theorems and other structural results in the theory. The ability to work with injective objects allows mathematicians to effectively manipulate and compute Weil-étale cohomology, making it a powerful tool for studying arithmetic properties of schemes over finite fields.
Implications for Weil-étale Cohomology
The fact that the category of Weil-étale sheaves is abelian with enough injectives has profound implications for the study of Weil-étale cohomology. It provides the necessary foundation for defining and manipulating cohomology groups, establishing long exact sequences, and proving various duality theorems. Weil-étale cohomology, in turn, serves as a powerful tool for investigating arithmetic properties of schemes over finite fields, such as the distribution of rational points and the behavior of zeta functions. The abelian nature and the existence of enough injectives are not just technical conditions; they are the bedrock upon which the entire theory of Weil-étale cohomology is built.
One of the key applications of Weil-étale cohomology is its connection to the zeta function of a scheme over a finite field. The zeta function, a central object in arithmetic geometry, encodes information about the number of points on the scheme over finite field extensions. Weil-étale cohomology provides a cohomological interpretation of the zeta function, relating it to the alternating sum of the traces of the Frobenius endomorphism acting on the cohomology groups. This connection allows us to use cohomological techniques to study the zeta function and to derive important results about its properties, such as the Weil conjectures.
The abelian category structure of Weil-étale sheaves enables us to use techniques from homological algebra to study the cohomology groups. For example, long exact sequences in cohomology, which arise from short exact sequences of sheaves, are powerful tools for computing and relating cohomology groups. Furthermore, the abelian nature allows us to define and study various operations on cohomology, such as cup products, which provide additional algebraic structure. These algebraic tools, made possible by the abelian category structure, are essential for unraveling the intricate arithmetic information encoded within Weil-étale cohomology.
The existence of enough injectives is essential for defining the Weil-étale cohomology functors as derived functors. This ensures that the cohomology groups are well-defined and independent of the choice of injective resolution. Moreover, the injectivity property allows us to prove various functorial properties of cohomology, such as the existence of long exact sequences and the compatibility of cohomology with certain operations on schemes. These functorial properties are crucial for applying Weil-étale cohomology in a variety of contexts and for relating it to other cohomological theories.
Conclusion: A Foundation for Arithmetic Exploration
In conclusion, the abelian nature of the category of Weil-étale sheaves, coupled with the existence of enough injectives, is not merely a technical detail but a fundamental property that underpins the entire edifice of Weil-étale cohomology. These properties provide the necessary framework for defining and manipulating cohomology groups, establishing long exact sequences, and proving duality theorems. Weil-étale cohomology, in turn, serves as a powerful tool for unraveling the intricate arithmetic properties of schemes over finite fields. The abelian nature and the abundance of injectives are thus essential for the ongoing exploration of the arithmetic landscape, providing mathematicians with a robust and versatile tool for studying the deep connections between geometry and number theory. Understanding why the category of Weil-étale sheaves is abelian with enough injectives is paramount for anyone venturing into the captivating world of arithmetic geometry and Weil-étale cohomology. The implications of these properties are far-reaching, impacting our ability to study zeta functions, rational points, and other crucial arithmetic invariants of schemes over finite fields. As research in this area continues to evolve, the fundamental nature of these properties will undoubtedly remain central to our understanding of arithmetic phenomena.