On the adic formalism. (English) Zbl 0821.14010

The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. II, Prog. Math. 87, 197-218 (1990).
[For the entire collection see Zbl 0717.00009.]
This article is concerned with developing a formalism for complexes of \(\ell\)-adic sheaves instead of just for the sheaves themselves as was done by J. P. Jouanolou [Sémin. géométrie algébrique, 1965- 1966, SGA 5, Lect. Notes Ser. 589, Exposé V, 204-250 (1977; Zbl 0352.18019)]. The need for such a generalisation has become apparent from the theory of perverse sheaves, which by their definition are complexes of \(\ell\)-adic sheaves. When trying to carry through such an extension one is immediately faced with two problems. On the one hand it is clear already from the case of \(\ell\)-adic sheaves that – contrary to the case of torsion sheaves – one is not dealing with actual sheaves but rather inverse systems of sheaves. On the other hand one wants to pretend that one is dealing with sheaves and not some more abstract objects.
The task of the present paper is therefore to show that this is indeed possible – a formalism is developed which for all practical purposes behaves as if one were dealing with a derived category of complexes.


14F30 \(p\)-adic cohomology, crystalline cohomology
14F20 Étale and other Grothendieck topologies and (co)homologies
18E30 Derived categories, triangulated categories (MSC2010)
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)