11.4 Sheaf operations and derived functors

Sheaf operations and derived functors are crucial tools in algebraic topology. They allow us to manipulate and analyze sheaves, which track local data on topological spaces. These operations help us understand how information flows between different parts of a space.

Derived functors measure how well sheaf operations preserve exactness. They give us a way to extend sheaf cohomology beyond just global sections. This deeper understanding of sheaves connects local and global properties, bridging different areas of mathematics.

Basic sheaf operations

Sheaves and their properties

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• A sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space
• Sheaves satisfy gluing conditions that allow local data to be uniquely determined by their restrictions to smaller open sets
• The stalk of a sheaf F at a point x, denoted F_x, captures the local behavior of the sheaf near x
• Sheaf morphisms are maps between sheaves that preserve the sheaf structure and can be defined locally
• Sheaves form an abelian category, which allows for the use of homological algebra techniques

Operations on sheaves

• For a continuous map f: X → Y and a sheaf F on X, the direct image sheaf fF on Y is defined by (fF)(V) = F(f^(-1)(V)) for each open set V in Y
• The inverse image sheaf f^(-1)F is defined as the sheafification of the presheaf U ↦ lim F(V) where the limit is taken over all open sets V containing f(U)
• The restriction of a sheaf F on X to an open subset U, denoted F|U, is defined by (F|U)(V) = F(V) for each open set V in U
• The extension by zero of a sheaf F on an open set U to X is denoted by j!F, where j is the inclusion map. It is the sheaf on X that agrees with F on U and is zero outside of U
• Sheaf operations such as direct sums, tensor products, and sheaf Hom can be defined using the corresponding operations on the stalks or local sections

Derived functors of sheaf operations

Derived functors and exactness

• Derived functors are a way to measure the failure of a functor to be exact. They provide additional information about the sheaves involved
• The right derived functors of a left exact functor F are denoted by R^iF and can be computed using injective resolutions
• The left derived functors of a right exact functor G are denoted by L_iG and can be computed using projective resolutions
• Derived functors preserve exactness in the appropriate sense: R^iF preserves exactness of injective resolutions, while L_iG preserves exactness of projective resolutions
• The long exact sequence of derived functors is a powerful tool for computing and comparing the cohomology of sheaves

Important derived functors

• The derived functors of the global section functor are the sheaf cohomology functors H^i(X, F)
• The derived functors of the direct image functor f* are the higher direct image functors R^if*, which can be used to compute sheaf cohomology on the target space Y
• The derived functors of the inverse image functor f^(-1) are the higher inverse image functors L_if^(-1), which play a role in the base change theorem
• The derived functors of the tensor product and sheaf Hom functors are the Tor and Ext functors, respectively
• The local-to-global spectral sequence relates the cohomology of a sheaf to the cohomology of its stalks

Computing derived functors

Simple cases and examples

• For a constant sheaf A on a contractible space X, the sheaf cohomology H^i(X, A) is isomorphic to A for i=0 and is zero for i>0
• For a locally constant sheaf F on a circle S^1, the sheaf cohomology H^0(S^1, F) is isomorphic to the global sections of F, H^1(S^1, F) is isomorphic to the coinvariants of the monodromy action on a stalk, and H^i(S^1, F) = 0 for i>1
• For a sheaf F on a CW complex X, the sheaf cohomology H^i(X, F) can be computed using a cellular resolution of F, which is a complex of sheaves constructed from the cellular structure of X
• The Čech cohomology of a sheaf F with respect to an open cover U of X is isomorphic to the sheaf cohomology H^i(X, F) when the cover is sufficiently fine (Leray theorem)
• The cohomology of a sheaf on a projective variety can often be computed using the Čech complex associated to a suitable affine open cover

Interpreting sheaf cohomology

• The interpretation of sheaf cohomology often depends on the specific context, such as the geometry or topology of the underlying space, or the algebraic properties of the sheaf
• In some cases, sheaf cohomology has a geometric interpretation, such as the classification of certain vector bundles or the obstruction to the existence of global sections
• Sheaf cohomology can also be related to other cohomology theories, such as singular cohomology or de Rham cohomology, via comparison theorems
• The vanishing or non-vanishing of certain sheaf cohomology groups can provide information about the structure of the underlying space or the properties of the sheaf
• In the context of algebraic geometry, sheaf cohomology is a fundamental tool for studying the geometry of varieties and their subvarieties

Spectral sequences for sheaf cohomology

Spectral sequences and their structure

• A spectral sequence is a tool for computing homology or cohomology by successive approximations. It consists of a sequence of pages, each containing a bigraded complex, with differentials between the pages
• The $E_r$ page of a spectral sequence is a bigraded complex with differentials $d_r$ of bidegree $(r, 1-r)$. The cohomology of $d_r$ is isomorphic to the $E_{r+1}$ page
• A spectral sequence is said to converge to a graded object $H^*$ if there exists an $r_0$ such that $E_r^{p,q} \cong E_{r_0}^{p,q}$ for all $r \geq r_0$ and $H^n \cong \bigoplus_{p+q=n} E_{r_0}^{p,q}$
• The differentials and convergence of a spectral sequence often provide additional information about the relationship between the objects involved, such as the topology of the spaces or the properties of the sheaves
• Spectral sequences can be constructed from various sources, such as double complexes, filtrations, or exact couples

Important spectral sequences

• The Leray spectral sequence is associated with a continuous map f: X → Y and a sheaf F on X. Its $E_2$ page is given by $E_2^{p,q} = H^p(Y, R^qf*F)$, and it converges to the sheaf cohomology $H^{p+q}(X, F)$
• The Grothendieck spectral sequence is associated with a composition of functors F ∘ G. Its $E_2$ page is given by $E_2^{p,q} = (R^pF)((R^qG)(A))$, and it converges to $R^{p+q}(F ∘ G)(A)$
• The Serre spectral sequence is associated with a fibration $F \to E \to B$ and a sheaf $\mathcal{F}$ on $E$. Its $E_2$ page is given by $E_2^{p,q} = H^p(B, R^qf_*\mathcal{F})$, where $f: E \to B$ is the projection, and it converges to the sheaf cohomology $H^{p+q}(E, \mathcal{F})$
• The Hodge-de Rham spectral sequence relates the Hodge cohomology and de Rham cohomology of a complex manifold or algebraic variety
• The Eilenberg-Moore spectral sequence is associated with a pullback square of spaces and relates the cohomology of the total space to the cohomology of the base spaces and the fiber product