4.4 Long exact sequence in cohomology

Long exact sequences in cohomology connect different sheaf cohomology groups, revealing relationships between global behaviors of sheaves on topological spaces. They arise from short exact sequences of sheaves and provide a powerful tool for computing and understanding cohomology groups.

These sequences use induced morphisms and connecting homomorphisms to link cohomology groups of different degrees. By studying these connections, we gain insights into the structure of spaces and sheaves, making long exact sequences essential for advanced sheaf theory analysis.

Connecting sheaf cohomology groups

• Sheaf cohomology groups $H^i(X, \mathcal{F})$ measure the global behavior of a sheaf $\mathcal{F}$ on a topological space $X$
• The long exact sequence connects different cohomology groups, providing a powerful tool for computing and understanding their relationships
• Studying the connections between cohomology groups reveals important properties of the underlying space and sheaves

Morphisms of sheaf cohomology

Induced morphisms

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• A morphism of sheaves $f: \mathcal{F} \to \mathcal{G}$ induces morphisms between their cohomology groups $f^*: H^i(X, \mathcal{F}) \to H^i(X, \mathcal{G})$ for each degree $i$
• Induced morphisms preserve the structure and properties of cohomology groups
• Enables the study of how sheaf morphisms affect the global behavior of sheaves

Functoriality of cohomology

• Sheaf cohomology is a contravariant functor from the category of sheaves to the category of abelian groups
• Composition of sheaf morphisms corresponds to the composition of induced morphisms in cohomology
• Identity morphisms induce identity maps on cohomology groups

Defining the long exact sequence

Exactness properties

• A sequence of abelian groups and homomorphisms is exact if the image of each homomorphism equals the kernel of the next
• Exactness captures the precise relationships between the groups in the sequence
• Short exact sequences, $0 \to A \to B \to C \to 0$, play a crucial role in constructing long exact sequences

Connecting homomorphisms

• In a long exact sequence, connecting homomorphisms link cohomology groups of different degrees
• Connecting homomorphisms arise from the snake lemma applied to short exact sequences of sheaves
• The connecting homomorphism $\delta^i: H^i(X, \mathcal{G}) \to H^{i+1}(X, \mathcal{F})$ measures the obstruction to lifting sections

Constructing the long exact sequence

Short exact sequences of sheaves

• Consider a short exact sequence of sheaves $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$ on a topological space $X$
• The morphisms in the short exact sequence induce morphisms between the corresponding cohomology groups
• The snake lemma applied to the induced morphisms yields a long exact sequence in cohomology

Associated long exact sequence

• The long exact sequence associated to a short exact sequence of sheaves has the form: $0 \to H^0(X, \mathcal{F}) \to H^0(X, \mathcal{G}) \to H^0(X, \mathcal{H}) \to H^1(X, \mathcal{F}) \to \cdots$
• The sequence continues with alternating cohomology groups of $\mathcal{F}$, $\mathcal{G}$, and $\mathcal{H}$, connected by induced morphisms and connecting homomorphisms
• The long exact sequence provides a powerful tool for computing cohomology groups and understanding their relationships

Applying the long exact sequence

Computing cohomology groups

• The long exact sequence can be used to compute unknown cohomology groups from known ones
• If two out of three consecutive terms in the sequence are known, the third can often be determined using exactness properties
• The sequence can be broken into smaller exact sequences for easier computation

Relating different cohomologies

• The long exact sequence relates the cohomology groups of different sheaves on the same space
• It allows for the comparison of cohomology groups arising from different constructions (Čech, derived functor)
• Studying the relationships between different cohomologies provides insights into the structure of the space and sheaves

Long exact sequence vs spectral sequences

Comparison of approaches

• Both long exact sequences and spectral sequences are tools for computing cohomology groups
• Long exact sequences arise from short exact sequences of sheaves, while spectral sequences come from filtered complexes or double complexes
• Spectral sequences provide a more general and powerful framework for computing cohomology

Advantages and limitations

• Long exact sequences are simpler and more straightforward to use when applicable
• Spectral sequences can handle more complex situations and provide more detailed information about the structure of cohomology groups
• However, spectral sequences can be more challenging to compute and interpret compared to long exact sequences

Naturality of the long exact sequence

Commutative diagrams

• The long exact sequence is natural with respect to morphisms of short exact sequences of sheaves
• Given a commutative diagram of short exact sequences, the induced morphisms on cohomology form a commutative diagram of long exact sequences
• Naturality allows for the comparison of long exact sequences arising from different short exact sequences

Functorial properties

• The construction of the long exact sequence is functorial, meaning it preserves composition and identity morphisms
• Functoriality ensures that the long exact sequence behaves well under sheaf morphisms and provides a consistent framework for studying cohomology

Examples and computations

Čech cohomology

• Čech cohomology is a concrete realization of sheaf cohomology based on open covers of a topological space
• The long exact sequence can be used to compute Čech cohomology groups by relating them to simpler cohomologies (constant sheaves, flasque sheaves)
• Čech cohomology is particularly useful for studying coherent sheaves on complex manifolds

Derived functor cohomology

• Derived functor cohomology is an abstract approach to sheaf cohomology using the language of derived categories and functors
• The long exact sequence in derived functor cohomology arises from the derived functors of the global sections functor
• Derived functor cohomology provides a more general and flexible framework for studying sheaf cohomology

Generalizations and variants

Relative cohomology

• Relative cohomology studies the cohomology of a sheaf on a pair $(X, A)$, where $A$ is a closed subspace of $X$
• The long exact sequence of a pair relates the relative cohomology to the absolute cohomology of $X$ and $A$
• Relative cohomology is useful for studying local properties of sheaves and for computing cohomology using excision

Hypercohomology

• Hypercohomology is a generalization of sheaf cohomology that allows for the study of complexes of sheaves
• The long exact sequence in hypercohomology relates the hypercohomology of a complex to the cohomology of its individual sheaves
• Hypercohomology is particularly useful in the study of derived categories and the derived functors of sheaves