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Hypercohomology
In homological algebra, the hyperhomology or hypercohomology (\mathbb_*(-), \mathbb^*(-)) is a generalization of (co)homology functors which takes as input not objects in an abelian category \mathcal but instead chain complexes of objects, so objects in \text(\mathcal). It is a sort of cross between the derived functor cohomology of an object and the homology of a chain complex since hypercohomology corresponds to the derived global sections functor \mathbf^*\Gamma(-). Hyperhomology is no longer used much: since about 1970 it has been largely replaced by the roughly equivalent concept of a derived functor between derived categories. Motivation One of the motivations for hypercohomology comes from the fact that there isn't an obvious generalization of cohomological long exact sequences associated to short exact sequences0 \to M' \to M \to M'' \to 0i.e. there is an associated long exact sequence0 \to H^0(M') \to H^0(M) \to H^0(M'')\to H^1(M') \to \cdots It turns out hypercoho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deligne Cohomology
In mathematics, Deligne cohomology is the hypercohomology of the Deligne complex of a complex manifold. It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordinary cohomology and intermediate Jacobians. For introductory accounts of Deligne cohomology see , , and . Definition The analytic Deligne complex Z(''p'')D, an on a complex analytic manifold ''X'' is0\rightarrow \mathbf Z(p)\rightarrow \Omega^0_X\rightarrow \Omega^1_X\rightarrow\cdots\rightarrow \Omega_X^ \rightarrow 0 \rightarrow \dotswhere Z(''p'') = (2π i)''p''Z. Depending on the context, \Omega^*_X is either the complex of smooth (i.e., ''C''∞) differential forms or of holomorphic forms, respectively. The Deligne cohomology is the ''q''-th hypercohomology of the Deligne complex. An alternative definition of this complex is given as the homotopy limit of the diagram\begin & & \mathbb \\ & & \downarrow \\ \Omega_X^ & \to & \Omega_X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |