Glossary

9.1 Sheaf cohomology

3 min read•august 7, 2024

is a powerful tool in algebraic topology that measures how local data on a space fits together globally. It extends the idea of sheaves, which track local information, to capture more complex relationships between local and global properties.

This topic introduces key concepts like derived functors and , which are essential for computing cohomology. We'll explore how these methods reveal deep insights about topological spaces and their associated sheaves.

Sheaf Theory Fundamentals

Sheaf and Presheaf Concepts

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  • Sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space
  • Consists of a topological space XX and a sheaf FF on XX associating data to each open set U⊆XU \subseteq X in a way that is compatible with restriction
  • Sheaf axioms ensure local properties can be uniquely glued together
  • Presheaf is a similar concept to a sheaf but without the gluing axiom
    • Associates data to open sets of a topological space and is compatible with restriction
    • May not allow for unique gluing of local sections

Étalé Space and Sheaf of Sections

  • Étalé space is a topological space EE along with a local homeomorphism p:E→Xp: E \to X
    • Provides a useful perspective for studying sheaves (p−1(U)p^{-1}(U) corresponds to F(U)F(U))
    • Allows for visualizing how the data varies over the base space XX
  • Sheaf of sections is a fundamental example of a sheaf
    • Given a continuous map f:Y→Xf: Y \to X, the sheaf of sections associates to each open set U⊆XU \subseteq X the set of continuous sections of ff over UU
    • Encodes the idea of continuously varying data over a space

Cohomology Constructions

Derived Functor and Čech Cohomology

  • cohomology is a general method for constructing cohomology theories
    • Starts with a left exact functor FF (such as the global section functor Γ\Gamma) and derives right derived functors RiFR^iF
    • The ii-th derived functor RiFR^iF applied to a sheaf FF gives the ii-th Hi(X,F)H^i(X, F)
  • Čech cohomology is a concrete realization of sheaf cohomology using open covers
    • Given an open cover U={Ui}\mathcal{U} = \{U_i\} of XX, defines cochains as functions on intersections of open sets
    • The Čech complex gives a flasque resolution of the sheaf, allowing for computation of cohomology

Injective Resolutions and Cohomology Groups

  • is a tool for computing derived functor cohomology
    • An injective resolution of a sheaf FF is an exact sequence 0→F→I0→I1→⋯0 \to F \to I^0 \to I^1 \to \cdots where each IiI^i is an injective sheaf
    • Existence of injective resolutions allows for the computation of RiFR^iF as the cohomology of the complex F(I∙)F(I^\bullet)
  • Cohomology groups Hi(X,F)H^i(X, F) are the central objects of study in sheaf cohomology
    • Measure the global obstruction to solving a geometric problem related to the sheaf FF (such as finding global sections)
    • Encode topological and geometric information about the space XX and the sheaf FF

Special Sheaves

Flasque and Acyclic Sheaves

  • is a special type of sheaf with vanishing higher cohomology
    • Defined by the property that any section over an open set can be extended to a global section
    • Flasque sheaves are used in constructing resolutions to compute cohomology
    • Examples include the sheaf of locally constant functions and the sheaf of sections of a vector bundle
  • is a sheaf that is "invisible" to the cohomology functor
    • A sheaf FF is acyclic if Hi(X,F)=0H^i(X, F) = 0 for all i>0i > 0
    • Acyclic sheaves play a role in the machinery of spectral sequences and hypercohomology
    • Examples include flasque sheaves and soft sheaves (sheaves where sections extend over closed subsets)

Key Terms to Review (20)

Acyclic Sheaf: An acyclic sheaf is a type of sheaf on a topological space that has vanishing cohomology groups for all open covers. This means that the higher cohomology groups of the sheaf are zero, which indicates that it behaves nicely with respect to cohomological computations. Acyclic sheaves are important because they simplify the process of computing sheaf cohomology, allowing one to use local data effectively.
Artin-Rees Lemma: The Artin-Rees Lemma is a crucial result in commutative algebra that provides conditions under which an ideal's power intersects with a submodule of a module. It essentially states that for an ideal and a finitely generated module, there exists some power of the ideal such that its intersection with any submodule can be controlled by a power of the ideal, linking it to properties of local cohomology and modules. This lemma is particularly useful when dealing with the Tor functor and sheaf cohomology, as it helps manage the behavior of ideals in these contexts.
Čech Cohomology: Čech cohomology is a mathematical tool used to study the properties of topological spaces through the lens of sheaves. It provides a way to compute cohomological invariants that are useful in various branches of mathematics, including algebraic geometry and topology. The method relies on covering a space with open sets and analyzing the sections of sheaves over these covers to extract important topological information.
Cohomological Dimension: Cohomological dimension is a measure of the 'size' of the cohomology groups of a module or a sheaf, indicating the highest degree in which non-trivial cohomology occurs. This concept helps in understanding how different algebraic structures behave with respect to cohomological techniques, serving as a critical tool in both algebraic geometry and group theory, as well as in analyzing the behavior of modules over rings.
Cohomology Group: A cohomology group is an algebraic structure associated with a topological space or more generally with a sheaf that encodes information about the space's shape and its features. Cohomology groups allow for the classification of cohomology classes, which are equivalence classes of cochains that capture the notion of how various functions behave over open covers of the space. They play a vital role in understanding the properties of sheaves and their global sections through cohomological methods.
Derived functor: A derived functor is a concept in homological algebra that extends the idea of a functor by associating it with a sequence of objects (called derived objects) which capture the 'homological information' of the original functor. This allows one to study properties of functors that are not preserved under direct application, particularly in the context of resolving modules and understanding exact sequences.
Eilenberg–Moore Spectral Sequence: The Eilenberg–Moore spectral sequence is a powerful tool in homological algebra that arises when computing the homology of a fibration or a more general type of 'fibrational' structure. This spectral sequence connects the derived functors of sheaf cohomology with the properties of the underlying topological space and its associated sheaves, making it particularly useful in understanding sheaf cohomology in a deeper way.
Five Lemma: The Five Lemma is a key result in homological algebra that provides a method for proving the isomorphism of homology groups in the context of a commutative diagram of chain complexes. It connects the properties of morphisms in exact sequences, enabling one to deduce information about the mapping of objects based on the behavior of their images and kernels. This lemma is essential for understanding how exact sequences function and is closely linked to various other concepts, including exactness and the structure of derived functors.
Flasque Sheaf: A flasque sheaf is a type of sheaf where the restriction maps between open sets are surjective. This property ensures that any section over an open set can be extended to any larger open set containing it, making flasque sheaves particularly useful in sheaf cohomology. Flasque sheaves are often employed to analyze the behavior of cohomological functors and to compute cohomology groups efficiently.
Henri Cartan: Henri Cartan was a prominent French mathematician known for his significant contributions to algebraic topology and homological algebra, particularly in developing the theory of sheaves and derived functors. His work laid essential groundwork for later developments in category theory and cohomology, impacting various mathematical areas including group cohomology and spectral sequences.
Injective resolution: An injective resolution is a type of exact sequence of injective modules that allows one to represent a module as an extension by injective modules. This concept is crucial for understanding how injective modules can be used to study other modules and their homological properties. The construction of injective resolutions provides a way to compute derived functors, including Ext, and plays an important role in various homological contexts, such as sheaf cohomology and the determination of homological dimensions.
Jean-Pierre Serre: Jean-Pierre Serre is a prominent French mathematician known for his work in algebraic geometry, topology, and number theory. His contributions have had a significant impact on various areas of mathematics, particularly through the development of cohomological methods and the theory of derived categories.
Long exact sequence: A long exact sequence is a sequence of abelian groups or modules connected by homomorphisms, which satisfies exactness at every point, indicating that the image of each homomorphism equals the kernel of the next. This concept is crucial in understanding the behavior of homology and cohomology theories, allowing one to relate different algebraic structures through their exact sequences and facilitating computations in various contexts.
Projective Dimension: Projective dimension is a homological invariant that measures the complexity of a module by determining the length of the shortest projective resolution of that module. It connects deeply with various concepts in homological algebra, such as resolutions, derived functors, and cohomology theories, showcasing how modules relate to projective modules and their role in understanding other homological properties.
Quasi-coherent sheaves: Quasi-coherent sheaves are a type of sheaf on a topological space that locally look like sheaves of modules over a ring. They provide a way to relate geometric structures to algebraic properties, making them crucial in the study of schemes and algebraic geometry. By being able to connect local and global data, quasi-coherent sheaves play an essential role in understanding the cohomological properties of spaces.
Riemann-Roch Theorem: The Riemann-Roch Theorem is a fundamental result in algebraic geometry that connects the geometry of a smooth projective curve to its function theory, specifically providing a way to calculate dimensions of certain spaces of meromorphic functions and differentials. It establishes a relationship between the genus of a curve, the degree of a divisor, and the dimensions of the associated cohomology groups, offering deep insights into sheaf cohomology.
Sheaf: A sheaf is a mathematical structure that associates data to the open sets of a topological space in a way that locally resembles a function but satisfies certain gluing conditions. It captures the idea of local data and how it can be pieced together to form global information, making it essential in areas such as algebraic geometry and homological algebra.
Sheaf Cohomology: Sheaf cohomology is a mathematical tool that studies the global properties of sheaves on topological spaces by analyzing the derived functors of sheaf homomorphisms. This concept bridges algebraic geometry and topology, providing insights into how local data can combine to reveal global phenomena. It plays a critical role in understanding various structures, such as complex varieties, differential forms, and the relationships between algebraic and topological invariants.
Spectral sequence: A spectral sequence is a mathematical tool that allows one to compute homology or cohomology groups by systematically breaking down complex objects into simpler pieces. It is built from a sequence of approximations that converge to a desired object, providing a way to handle filtered complexes and understand their properties through successive stages of computation. This method finds significant applications in various areas, including homological algebra and sheaf cohomology.
Vanishing theorems: Vanishing theorems are results in algebraic geometry and sheaf theory that provide conditions under which certain cohomology groups, particularly sheaf cohomology groups, vanish. These theorems are essential in understanding the relationship between geometric properties of spaces and their cohomological invariants, and they play a crucial role in simplifying complex calculations in sheaf cohomology by showing that certain groups are zero.
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