p-adic étale Tate twists and arithmetic duality☆

Abstract

In this paper, we define, for arithmetic schemes with semistable reduction, p-adic objects playing the roles of Tate twists inétale topology, and establish their fundamental properties.Résumé: Dans ce papier, nous definissions, pour les schémas arithmétiquesà réduction semistable, des objets p-adiques jouant les rôles de twistsà la Tate en topologieétale, et nousétablissons leurs propriétés fondamentales.

Higher-dimensional arithmetic using <i>p</i>-adic étale Tate twists

“…Although T4 is not among Beilinson-Lichtenbaum axioms, it is a natural property to be satisfied. In the following theorem, 'unique' means 'unique up to a unique isomorphism' (see [Sat3] In case X/B is smooth, this object is already considered by Schneider [Sch]. We have T r (0) Z/p r Z and T r (n) Rj * µ ⊗n p r for n > dim(X).…”

“…[Mi2], II.2). In this section, we give the statements and an outline of the duality results that the author proved in [Sat3], §6 (see also [JSS] for other duality results).…”

“…[Sat3], Theorem 2.7): ). By (1.2.2), it turns out that the sheaf H n (T r (n) X ) is generated by ι * (U 1 M n r ) and the image of (G m , X ) ⊗n , so that the image of the natural cup-product map [Sat2].…”

“…By works of Levine ([Le1], [Le2]), these two objects are regarded as strong candidates for the motivic complexes Γ (n)é t X and Γ (n) Zar X , respectively, so that we can expect the following conjecture (cf. [Sat3], Conjecture 0.4):…”

“…In this paper, we like to survey recent results of the author [Sat2], [Sat3] and his joint work with Shuji Saito [SaSa] (cf. [JSS]).…”