In mathematics, crystalline cohomology is a Weil cohomology theory for schemes X over a base field k. Its values Hn(X/W) are modules over the ring W of Witt vectors over k. It was introduced by Alexander Grothendieck (1966, 1968) and developed by Pierre Berthelot (1974).
Crystalline cohomology is partly inspired by the p-adic proof in Dwork (1960) of part of the Weil conjectures and is closely related to the algebraic version of de Rham cohomology that was introduced by Grothendieck (1963). Roughly speaking, crystalline cohomology of a variety X in characteristic p is the de Rham cohomology of a smooth lift of X to characteristic 0, while de Rham cohomology of X is the crystalline cohomology reduced mod p (after taking into account higher Tors).
The idea of crystalline cohomology, roughly, is to replace the Zariski open sets of a scheme by infinitesimal thickenings of Zariski open sets with divided power structures. The motivation for this is that it can then be calculated by taking a local lifting of a scheme from characteristic p to characteristic 0 and employing an appropriate version of algebraic de Rham cohomology.
Crystalline cohomology only works well for smooth proper schemes. Rigid cohomology extends it to more general schemes.
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In mathematics, crystallinecohomology is a Weil cohomology theory for schemes X over a base field k. Its values Hn(X/W) are modules over the ring W of...
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the notion of étale cohomology, which led to the proof of the Weil conjectures. Crystallinecohomology and many other cohomology theories in algebraic...
In mathematics, rigid cohomology is a p-adic cohomology theory introduced by Berthelot (1986). It extends crystallinecohomology to schemes that need not...
been used to define other cohomology theories since then, such as ℓ-adic cohomology, flat cohomology, and crystallinecohomology. While Grothendieck topologies...
essentially the same as crystallinecohomology. In nonzero characteristic p Ogus (1975) showed that it is closely related to etale cohomology with mod p coefficients...
Weil cohomology or Weil cohomology theory is a cohomology satisfying certain axioms concerning the interplay of algebraic cycles and cohomology groups...
geometry and related branches of mathematics, cyclic homology and cyclic cohomology are certain (co)homology theories for associative algebras which generalize...
mathematician at the University of Rennes. He developed crystallinecohomology and rigid cohomology. Berthelot died on 7 December 2023. Berthelot, Pierre...
singular cohomology, de Rham cohomology, ℓ-adic cohomology, and crystallinecohomology. Scholze and Dustin Clausen proposed a program for condensed mathematics...
such that for all n the slopes of the Newton polygon of the nth crystallinecohomology are all n/2 (de Jong 2014). For special classes of varieties such...
fundamental tool in the theory of PD differential operators and crystallinecohomology, where it is used to overcome technical difficulties which arise...
scientist Cynthia Dwork, historian Deborah Dwork, and Andrew Dwork. Crystallinecohomology Gross–Koblitz formula Langlands–Deligne local constant p-adic gamma...
the Birch and Swinnerton-Dyer conjecture. CrystallinecohomologyCrystallinecohomology is a p-adic cohomology theory in characteristic p, introduced by...
of the sought-after étale cohomology (as well as other refined theories such as flat cohomology and crystallinecohomology). At this point—about 1964—the...
an Algebraic Surface". Worse pathologies related to p-torsion in crystallinecohomology were explored by Luc Illusie (Ann. Sci. Ec. Norm. Sup. (4) 12 (1979)...
concerns the theory of the cotangent complex and deformations, crystallinecohomology and the De Rham–Witt complex, and logarithmic geometry. In 2012...
Alexander Grothendieck Crystallinecohomology: A p-adic cohomology theory in characteristic p invented to fill the gap left by étale cohomology which is deficient...
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