Article of Nekovar
Arithmetic cohomology over finite fields and special values of zeta-functions, by Thomas Geisser
arXiv: Experimental full text search
AAG (Arithmetic algebraic geometry)
Mixed (?), Charp
Arithmetic vs geometric CTs.
Ref: Nekovar, Beilinson’s height pairing article.
Absolute: VS. Geometric: VS with extra structure (“Tannakian”?).
Hochschild-Serre spectral sequence.
It seems geometric theories are related to the def of L-fn and zeta fn, but arithmetic theories capture orders of vanishing and maybe more info about the special values.
Absolute should be related to weight 0 part of geometric.
What is geometric/absolute cohom of Spec Z, its compactification, and the infinite prime?
Arithmetic cohomology can mean at least two things. The first is a notion of Geisser, who constructs a cohomology theory with compact supports for separated schemes of finite type over a finite field, which (assuming the Tate conjecture and rat=num) gives an integral model for l-adic cohomology with compact supports when l is different from p. These groups are expected to be finitely generated and related to special values of zeta functions for all schemes as above. This is a variant of Weil-etale cohomology, probably agreeing with it for smooth projective varieties, but being better behaved in general. For l=p his definition using the eh-topology gives a new theory which for smooth and proper schemes agrees with logarithmic de Rham-Witt cohomology. See also Arithmetic homology
Arithmetic cohomology is also sometimes used as a synonym for Absolute cohomology. See also Absolute cohomology, Geometric cohomology
nLab page on Arithmetic cohomology