Overconvergent de Rham-Witt cohomology

Abstract

Ann. Scient. Éc. Norm. Sup. 4 e série, t. 44, 2011, p. 197

Overconvergent Witt vectors

“…Therefore we speak of overconvergent Witt vectors. In this paper we study the rings of overconvergent Witt vectors systematically and prove in particular all facts used in [2]. Similar growth conditions of Witt vectors were used by de Jong [5] in his work on homomorphisms of p-divisible groups and by Kedlaya [7] in his work on the Crew Conjecture.…”

“…This is done for overconvergent Witt di¤erentials over a perfect field in [2]. For Witt vectors the argument given here is more elementary and works over an integral domain R. The basic idea due to Meredith is the same as for Witt di¤erentials.…”

“…A much deeper fact that we use in [2] Further, we show (see Corollary 2.46): Let A be a finitely generated algebra over K. Let B ¼ A½T= À f ðTÞ Á be a finite étale A-algebra, where f ðTÞ A A½T is a monic polynomial of degree n such that f 0 ðTÞ is a unit in B. We denote by t the residue class of T in B.…”

“…The de Rham-Witt complex of Deligne-Illusie is a complex of W ðO X Þ-modules whose hypercohomology is the crystalline cohomology of X . In [2] we constructed a subcomplex of the de Rham-Witt complex whose hypercohomology is the rigid cohomology of X defined by Berthelot [1]. For this we put a growth condition on Witt vectors which is inspired by the work of Monsky and Washnitzer.…”

Comparison between overconvergent de Rham-Witt and crystalline cohomology for projective and smooth varieties

“…Let X be a smooth variety over a perfect field k of characteristic . In we constructed an overconvergent de Rham‐Witt complex as a suitable sub‐complex of the completed de Rham‐Witt complex of Deligne‐Illusie. It is a Zariski sheaf of differential graded algebras.…”