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The Z-cohomology of RP 2a has an element y of degree 2 such that the whole cohomology is the direct sum of a copy of Z spanned by the element 1 in degree 0 together with copies of Z/2 spanned by the elements y i for i=1,...,a. The Z-cohomology of RP 2a+1 is the same together with an extra copy of Z in degree 2a+1. [10]
In algebraic geometry, a mixed Hodge structure is an algebraic structure containing information about the cohomology of general algebraic varieties.It is a generalization of a Hodge structure, which is used to study smooth projective varieties.
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold M using partial differential equations.The key observation is that, given a Riemannian metric on M, every cohomology class has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric.
Vector field corresponding to a differential form on the punctured plane that is closed but not exact, showing that the de Rham cohomology of this space is non-trivial.. In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form ...
In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors . Hochschild cohomology was introduced by Gerhard Hochschild ( 1945 ) for algebras over a field , and extended to algebras over more general rings by Henri Cartan and ...
In algebraic geometry, the theorem of absolute (cohomological) purity is an important theorem in the theory of étale cohomology. It states: [1] given a regular scheme X over some base scheme,: a closed immersion of a regular scheme of pure codimension r, an integer n that is invertible on the base scheme,
In algebraic geometry, a Weil cohomology or Weil cohomology theory is a cohomology satisfying certain axioms concerning the interplay of algebraic cycles and cohomology groups. The name is in honor of André Weil .
A commonly cited example is the Dwork construction of the Picard–Fuchs equation.Let (,,) be the elliptic curve + + =.Here, is a free parameter describing the curve; it is an element of the complex projective line (the family of hypersurfaces in dimensions of degree n, defined analogously, has been intensively studied in recent years, in connection with the modularity theorem and its ...