## 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

### Keywords:

$$p$$-adic cohomology theory; continuous cohomology

Zbl 0581.14012
Full Text:

### References:

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