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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
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 ...
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 *
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
The Gysin sequence is a long exact sequence not only for the de Rham cohomology of differential forms, but also for cohomology with integral coefficients. In the integral case one needs to replace the wedge product with the Euler class with the cup product, and the pushforward map no longer corresponds to integration.