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


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


Zbl 0581.14012
Full Text: DOI EuDML


[1] Bousfield, A.K., Kan, D.M.: Homotopy limits, completions and localizations. Lecture Notes in Mathematics 304. Berlin, Heidelberg, New York: Springer 1972 · Zbl 0259.55004
[2] Bredon, G.: Sheaf theory. New York: McGraw-Hill 1967 · Zbl 0158.20505
[3] Dwyer, W.G., Friedlander, E.M.: Algebraic and étaleK-theory. Trans. Am. Math. Soc.292, 247-280 (1985) · Zbl 0581.14012
[4] Eilenberg, S., MacLane, S.: Cohomology theory of abstract groups. I. Ann. Math.48, 51-78 (1947) · Zbl 0029.34001
[5] Gray, B.I.: Spaces of the samen-type, for alln. Topology5, 241-243 (1966) · Zbl 0149.20102
[6] Grothendieck, A.: Sur quelques points d’algèbre homologique. Tôhoku Math. J.9, 119-221 (1957) · Zbl 0118.26104
[7] Grothendieck, A.: La théorie des classes de Chern: Bull Soc. Math. Fr.86, 137-154 (1958) · Zbl 0091.33201
[8] Harrison, D.K.: Infinite abelian groups and homological methods. Ann. Math.69, 366-391 (1956) · Zbl 0100.02901
[9] Milne, J.S.: Étale cohomology, Princeton Mathematical Series 33, Princeton, 1980 · Zbl 0433.14012
[10] Roos, J.-E.: Sur les foncteurs dérivés de \(\underleftarrow {\lim }\) . Applications. C.R. Acad. Sci. Ser. I252, 3702-3704 (1961) · Zbl 0102.02501
[11] Roos, J.-E.: Sur les foncteurs dérivés des produits infinis dans les catégories de Grothendieck. Exemples et contre-exemples. C.R. Acad. Sci. Ser. I263, 895-898 (1966) · Zbl 0163.26805
[12] Serre, J.-P.: Cohomologie Galoisienne. Lecture Notes in Mathematics 5. Berlin, Göttingen, Heidelberg, New York: Springer 1964 · Zbl 0143.05901
[13] Soulé, Ch.: Operations on étaleK-theory. Applications. Lecture Notes in Mathematics 966, 271-303. Berlin, Heidelberg, New York: Springer 1982
[14] Tate, J.: Relations betweenK 2 and Galois cohomology. Invent. Math.36, 257-274 (1976) · Zbl 0359.12011
[15] Tate, J.: Algebraic cycles and poles of zeta functions. Arithmetical algebraic geometry (Purdue Lafayette 1963). Schilling, O.F.G. (ed.). New York: Harper & Row 1965
[16] Grothendieck, A., et al.: Théorie des topos et cohomologie étale des schémas. Tome 3. Lecture Notes in Mathematics 305. Berlin, Heidelberg, New York: Springer 1973
[17] Deligne, P., et al.: Cohomologie étale. Lecture Notes in Mathematics 569. Berlin, Heidelberg, New York: Springer 1977 · Zbl 0349.14008
[18] Grothendieck A., et al.: Cohomologiel-adique et fonctionsL. Lecture Notes in Mathematics 589. Berlin, Heidelberg, New York: Springer 1982
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