In mathematics, rigid cohomology is a p-adic cohomology theory introduced by Berthelot (1986). It extends crystalline cohomology to schemes that need not be proper or smooth, and extends Monsky–Washnitzer cohomology to non-affine varieties. For a scheme X of finite type over a perfect field k, there are rigid cohomology groups Hi rig(X/K) which are finite dimensional vector spaces over the field K of fractions of the ring of Witt vectors of k. More generally one can define rigid cohomology with compact supports, or with support on a closed subscheme, or with coefficients in an overconvergent isocrystal.
If X is smooth and proper over k the rigid cohomology groups are the same as the crystalline cohomology groups.
The name "rigid cohomology" comes from its relation to rigid analytic spaces.
Kedlaya (2006) used rigid cohomology to give a new proof of the Weil conjectures.
In mathematics, rigidcohomology is a p-adic cohomology theory introduced by Berthelot (1986). It extends crystalline cohomology to schemes that need not...
version of algebraic de Rham cohomology. Crystalline cohomology only works well for smooth proper schemes. Rigidcohomology extends it to more general schemes...
extensions of Qp, when the buildings are given a suitably coarse topology. Rigidcohomology Non-Archimedean analysis by S. Bosch, U. Güntzer, R. Remmert ISBN 3-540-12546-9...
mathematician at the University of Rennes. He developed crystalline cohomology and rigidcohomology. Berthelot died on 7 December 2023. Berthelot, Pierre Cohomologie...
another proof using the Fourier transform, replacing etale cohomology with rigidcohomology. Deligne (1980) was able to prove the hard Lefschetz theorem...
most often used to define cohomology theories, they have found other applications as well, such as to John Tate's theory of rigid analytic geometry. There...
quasi-coherent cohomology and G ∙ {\displaystyle G^{\bullet }} a bounded below complex of O Y {\displaystyle O_{Y}} -modules with coherent cohomology. Here the...
Noetherian schemes. Čech cohomology and sheaf cohomology agree on an affine open cover. This makes it possible to compute the sheaf cohomology of P S n {\displaystyle...
(see also Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, vol. 323...
dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of the pattern that leave it unchanged. In three dimensions...
functions on the group by evaluating the function on the summed elements. The cohomology algebra (over a field K {\displaystyle K} ) of a Lie group G {\displaystyle...
properties of smooth manifolds, including notions of size, distance, and rigid shape. By comparison differential topology is concerned with coarser properties...
understanding of the deformation theory of schemes, and results about the cohomology of complex manifolds inspired the formulation of the Weil conjectures...
arithmetic algebraic geometry, particularly on the development of p-adic cohomology theories." Aleksandr Logunov – "For novel techniques to study solutions...
ordinary cohomology mod p with a generalized cohomology theory, such as complex cobordism or, more usually, a piece of it called Brown–Peterson cohomology. The...
=0.} These operators and their properties form the basis for Dolbeault cohomology and many aspects of Hodge theory. On a star-shaped domain of a complex...
computer: some discouraging experiences". Arnold's Mathematical Seminar. Milne, James S. (1980). Étale cohomology. Princeton University Press. pp. xiii, 66....
geometry as a closed form. As Euclidean space is contractible, its de Rham cohomology vanishes, so every closed form is also an exact form, and can be expressed...
non-Archimedean field (e.g. p-adic field), refining Tate's notion of a rigid analytic space. In the complex case, algebraic geometry begins by defining...
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