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Sheaf cohomology gives a satisfactory general answer. Namely, let A be the kernel of the surjection B → C, giving a short exact sequence. of sheaves on X. Then there is a long exact sequence of abelian groups, called sheaf cohomology groups:
For example, one can define the cohomology of a topological space X with coefficients in any complex of sheaves, earlier called hypercohomology (but usually now just "cohomology"). From that point of view, sheaf cohomology becomes a sequence of functors from the derived category of sheaves on X to abelian groups.
If we take a series of short exact sequences linked by chain complexes (that is, a short exact sequence of chain complexes, or from another point of view, a chain complex of short exact sequences), then we can derive from this a long exact sequence (that is, an exact sequence indexed by the natural numbers) on homology by application of the zig ...
Let X be a topological space and A, B be two subspaces whose interiors cover X. (The interiors of A and B need not be disjoint.) The Mayer–Vietoris sequence in singular homology for the triad (X, A, B) is a long exact sequence relating the singular homology groups (with coefficient group the integers Z) of the spaces X, A, B, and the intersection A∩B. [8]
Another example comes from the holomorphic log complex on a complex manifold. [1] Let X be a complex algebraic manifold and j : X ↪ Y {\displaystyle j:X\hookrightarrow Y} a good compactification. This means that Y is a compact algebraic manifold and D = Y − X {\displaystyle D=Y-X} is a divisor on Y {\displaystyle Y} with simple normal ...
called the long exact sequence of cohomology with compact support. It has numerous applications, such as the Jordan curve theorem, which is obtained for X = R² and Z a simple closed curve in X. De Rham cohomology with compact support satisfies a covariant Mayer–Vietoris sequence: if U and V are open sets covering X, then
Tate cohomology enjoys similar features, such as long exact sequences, product structures. An important application is in class field theory , see class formation . Tate cohomology of finite cyclic groups , G = Z / n , {\displaystyle G=\mathbb {Z} /n,} is 2-periodic in the sense that there are isomorphisms
The exact triangles generalize the short exact sequences in an abelian category, as well as fiber sequences and cofiber sequences in topology. Much of homological algebra is clarified and extended by the language of triangulated categories, an important example being the theory of sheaf cohomology .