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 . 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
Top images from around the web for Sheaves and their properties
Category:Group homomorphisms - Wikimedia Commons View original
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 , 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 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 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 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 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 Er page of a spectral sequence is a bigraded complex with differentials dr of bidegree (r,1−r). The cohomology of dr is isomorphic to the Er+1 page
A spectral sequence is said to converge to a graded object H∗ if there exists an r0 such that Erp,q≅Er0p,q for all r≥r0 and Hn≅⨁p+q=nEr0p,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 E2 page is given by E2p,q=Hp(Y,Rqf∗F), and it converges to the sheaf cohomology Hp+q(X,F)
The Grothendieck spectral sequence is associated with a composition of functors F ∘ G. Its E2 page is given by E2p,q=(RpF)((RqG)(A)), and it converges to Rp+q(F∘G)(A)
The Serre spectral sequence is associated with a fibration F→E→B and a sheaf F on E. Its E2 page is given by E2p,q=Hp(B,Rqf∗F), where f:E→B is the projection, and it converges to the sheaf cohomology Hp+q(E,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 square of spaces and relates the cohomology of the total space to the cohomology of the base spaces and the fiber product