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Continuous étale cohomology. (English) Zbl 0649.14011
The author shows how to construct a very well-behaved \(p\)-adic cohomology theory, called continuous cohomology by deriving the left exact functor \[ \{\text{inverse system }(F_n)\text{ of étale sheaves on }X\} \to \text{ abelian groups } (F_n) \mapsto \lim_{\overset \leftarrow n}H_0(X,F_n). \] The construction, when applied to locally constant sheaves, \(F_n\), gives the continuous étale cohomology theory of W. G. Dwyer and E. M. Friedlander [Trans. Am. Math. Soc. 292, 247–280 (1985; Zbl 0581.14012)]. However, the author’s construction applies to arbitrary sheaves while enjoying all the desirable properties of a cohomology theory (e.g. Hochschild-Serre spectral sequences, Chern classes, a Milnor \(\lim_{\leftarrow}^ 1\) sequence to relate it to \(\ell\)-adic cohomology). All in all, continuous cohomology looks to be one way around a number of technical difficulties in \(\ell\)-adic cohomology.
Reviewer: V.P.Snaith

MSC:
14F20 Étale and other Grothendieck topologies and (co)homologies
14F30 \(p\)-adic cohomology, crystalline cohomology
18G10 Resolutions; derived functors (category-theoretic aspects)
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
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