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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 ...
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
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.
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 ...
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]
An exact sequence (or exact complex) is a chain complex whose homology groups are all zero. This means all closed elements in the complex are exact. A short exact sequence is a bounded exact sequence in which only the groups A k, A k+1, A k+2 may be nonzero. For example, the following chain complex is a short exact sequence.
The above short exact sequences specifying the relative chain groups give rise to a chain complex of short exact sequences. An application of the snake lemma then yields a long exact sequence
This follows from the naturality of the sequence produced by the snake lemma. If is a commutative diagram with exact rows, then the snake lemma can be applied twice, to the "front" and to the "back", yielding two long exact sequences; these are related by a commutative diagram of the form