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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
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 ...
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]
The five lemma is often applied to long exact sequences: when computing homology or cohomology of a given object, one typically employs a simpler subobject whose homology/cohomology is known, and arrives at a long exact sequence which involves the unknown homology groups of the original object. This alone is often not sufficient to determine ...
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:
Generalizing the previous example, one can compute Ext groups when the first module is the quotient of a commutative ring by any regular sequence, using the Koszul complex. [5] For example, if R is the polynomial ring k[x 1,...,x n] over a field k, then Ext * R (k,k) is the exterior algebra S over k on n generators in Ext 1. Moreover, Ext *