11.2 The Čech-to-derived functor spectral sequence

The Čech-to-derived functor spectral sequence bridges Čech cohomology and derived functor cohomology for sheaves. It uses a double complex to compute sheaf cohomology, connecting local and global perspectives on cohomological information.

This powerful tool allows us to understand sheaf cohomology through different lenses. By relating Čech and derived functor approaches, it provides insights into the structure of sheaves and their cohomology on topological spaces.

Čech-to-Derived Functor Spectral Sequence

Construction of the Spectral Sequence

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• The Čech-to-derived functor spectral sequence computes the sheaf cohomology of a sheaf $F$ on a topological space $X$
• Involves a double complex $C^{p,q} = [C^p(U, I^qF)](https://www.fiveableKeyTerm:c^p(u,_i^qf))$
  • $U$ is an open cover of $X$
  • $I^qF$ is the $q$-th injective resolution of $F$
  • $C^p(U, -)$ denotes the Čech complex
• Spectral sequence arises from two filtrations on the double complex
  • "Horizontal" filtration by $p$
  • "Vertical" filtration by $q$
• $E_1$ page of the spectral sequence: $E_1^{p,q} = H^q(C^p(U, I^*F))$
  • Cohomology groups of the Čech complex with coefficients in the injective resolution
• $E_2$ page: $E_2^{p,q} = H^p(X, [H^q(I^*F)](https://www.fiveableKeyTerm:h^q(i^*f)))$
  • $H^q(I^*F)$ is the $q$-th derived functor of $F$
• Spectral sequence converges to the sheaf cohomology $[H^*(X, F)](https://www.fiveableKeyTerm:h^*(x,_f))$

Convergence and Edge Homomorphisms

• The Čech-to-derived functor spectral sequence converges to the sheaf cohomology $H^*(X, F)$
• Convergence is often denoted by $E_∞^{p,q}$, representing the associated graded object of $H^*(X, F)$ with respect to a suitable filtration
• Edge homomorphisms of the spectral sequence relate the sheaf cohomology to the Čech cohomology and derived functor cohomology
  • Provide a way to extract information about $H^*(X, F)$ from the $E_2$ page
• In favorable situations, the spectral sequence may degenerate at the $E_2$ page
  • All differentials on later pages are zero
  • Sheaf cohomology can be read off directly from the $E_2$ page

Computing Sheaf Cohomology

Using the Spectral Sequence

• The Čech-to-derived functor spectral sequence computes the sheaf cohomology groups $H^n(X, F)$ for a sheaf $F$ on a topological space $X$
• $E_2$ page provides a way to compute sheaf cohomology in terms of Čech cohomology and derived functor cohomology
  • $E_2^{p,q} = H^p(X, H^q(I^*F))$ relates Čech cohomology ($p$-index) to derived functor cohomology ($q$-index)
• Differentials on the $E_r$ pages for $r ≥ 2$ give relations between the cohomology groups
  • Used to determine the sheaf cohomology
• Example: If the spectral sequence degenerates at the $E_2$ page, sheaf cohomology can be read off directly
  • Degeneration means all differentials on later pages are zero

Favorable Situations and Degeneration

• In some cases, the Čech-to-derived functor spectral sequence may degenerate at the $E_2$ page
  • Happens when all differentials on later pages ($E_r$ for $r ≥ 3$) are zero
  • Implies that the $E_2$ page directly gives the sheaf cohomology
• Degeneration at the $E_2$ page is a favorable situation for computing sheaf cohomology
  • Allows for direct computation without the need to consider higher differentials
• Examples of situations where degeneration may occur:
  • When the space $X$ has suitable acyclicity properties (Stein spaces)
  • When the sheaf $F$ is flasque or soft

Čech vs Derived Functor Cohomology

Relationship between Čech and Derived Functor Cohomology

• The Čech-to-derived functor spectral sequence connects Čech cohomology and derived functor cohomology
• Čech cohomology $H^p(U, F)$:
  • Computed using open covers
  • Measures the global behavior of the sheaf $F$
• Derived functor cohomology $H^q(X, F)$:
  • Defined using injective resolutions
  • Captures the sheaf-theoretic cohomology
• $E_2$ page of the spectral sequence: $E_2^{p,q} = H^p(X, H^q(I^*F))$
  • Relates Čech cohomology ($p$-index) to derived functor cohomology ($q$-index)

Coincidence and Comparison

• When the spectral sequence degenerates at the $E_2$ page, Čech cohomology and derived functor cohomology coincide
  • Degeneration implies that the differentials on later pages are zero
  • $E_2$ page directly gives the sheaf cohomology
• In general, the spectral sequence provides a way to compute derived functor cohomology using Čech cohomology and additional data from the differentials
  • Differentials give relations between the cohomology groups
  • Can be used to extract information about the derived functor cohomology
• Comparison between Čech and derived functor cohomology:
  • Čech cohomology is more computable and intuitive
  • Derived functor cohomology is more abstract and sheaf-theoretic
  • Spectral sequence bridges the gap between the two approaches

Spectral Sequence Terms and Differentials

Meaning of the Terms

• The terms in the Čech-to-derived functor spectral sequence have specific meanings related to the cohomology of the sheaf $F$
• $E_1$ page terms $E_1^{p,q}$:
  • Represent the cohomology of the Čech complex with coefficients in the injective resolution of $F$
  • Computed as $E_1^{p,q} = H^q(C^p(U, I^*F))$
• $E_2$ page terms $E_2^{p,q}$:
  • Represent the Čech cohomology of the derived functors of $F$
  • Given by $E_2^{p,q} = H^p(X, H^q(I^*F))$
• Higher page terms $E_r^{p,q}$ for $r ≥ 3$:
  • Obtained by taking cohomology with respect to the differentials
  • Represent successive approximations to the sheaf cohomology

Role of the Differentials

• The differentials in the Čech-to-derived functor spectral sequence provide additional relations between the cohomology groups
• Differentials on the $E_r$ pages: $d_r^{p,q}: E_r^{p,q} → E_r^{p+r,q-r+1}$
  • Maps between the terms on the $E_r$ page
  • Satisfy the condition $d_r^{p+r,q-r+1} ∘ d_r^{p,q} = 0$ (differentials square to zero)
• Differentials on the $E_2$ page: $d_2^{p,q}: H^p(X, H^q(I^*F)) → H^{p+2}(X, H^{q-1}(I^*F))$
  • Relate the Čech cohomology and derived functor cohomology
  • Provide a way to compute the derived functor cohomology using Čech cohomology
• Kernel and image of the differentials:
  • Determine the successive pages of the spectral sequence
  • Used to compute the higher page terms $E_r^{p,q}$ for $r ≥ 3$
  • Ultimately determine the sheaf cohomology $H^*(X, F)$
• Convergence of the spectral sequence:
  • Represented by $E_∞^{p,q}$
  • Obtained by taking the limit of the higher page terms
  • Represents the associated graded object of the sheaf cohomology $H^*(X, F)$ with respect to a suitable filtration