Search results
Results from the WOW.Com Content Network
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 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.
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by Jean Leray (1946a, 1946b), they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra.
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,
The cohomology of a compact Kähler manifold has a Hodge structure, and the nth cohomology group is pure of weight n. The cohomology of a complex variety (possibly singular or non-proper) has a mixed Hodge structure. This was shown for smooth varieties by Deligne (1971), Deligne (1971a) and in general by Deligne (1974).
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 ...
Motivic cohomology is an invariant of algebraic varieties and of more general schemes. It is a type of cohomology related to motives and includes the Chow ring of algebraic cycles as a special case. Some of the deepest problems in algebraic geometry and number theory are attempts to understand motivic cohomology.
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 .