ladic cohomology

General Introduction
See also Sheaf cohomology, Weil cohomology, Etale cohomology
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Online References
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Paper References
Katz: Review of adic cohomology (in the Motives volumes)
Katz: Independence of and Weak Lefschetz
Tate: Conjectures on Algebraic Cycles in adic cohomology (in the Motives volumes)
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Definition
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Properties
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Standard theorems
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Open Problems
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Connections to Number Theory
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Computations and Examples
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History and Applications
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Some Research Articles
Deligne: Decompositions... (Motives vol) has a section.
Ekedahl: On the adic formalism
Kahn: We prove some finiteness theorems for the \'étale cohomology, BorelMoore homology and cohomology with proper supports of schemes of finite type over a finite or padic field. This yields vanishing results for their ladic cohomology, proving part of a conjecture of Jannsen.
Jannsen: On the $l$adic cohomology of varieties over number fields and its Galois cohomology. Galois groups over $Q$ (Berkeley, CA, 1987), 315360.
Jannsen: On the Galois cohomology of $l$adic representations attached to varieties over local or global fields. Séminaire de Théorie des Nombres, Paris 198687, 165182. Let $K$ be a local or global field, $\overline K$ its separable closure, and $X$ an algebraic variety over $K$. The author presents three conjectures on Galois cohomology of the étale cohomology groups of . Statements of conjectures: see Mathscinet  they failed to save.
He states with or without proofs that all conjectures except (1b) are true for $\roman{spec}\,K$ or a form of the standard cellular varieties, that (3) is true for $i=1$, and that (1) holds for $i=1$ if $X$ is an elliptic curve defined over, and having CM by, $K$ and $l$ is a regular prime for $X$.
Analogues of (1) and (3) in the function field case are proved and interrelations between the three conjectures and those of \n A. Be\u\i linson\en [in Current problems in mathematics, Vol. 24 (Russian), 181238, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984; MR0760999 (86h:11103)] and \n J.M. Fontaine\en [Ann. of Math. (2) 115 (1982), no. 3, 529577; MR0657238 (84d:14010)] are discussed.
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