## Trace and relative duality for arithmetic $${\mathcal D}$$-modules. (Trace et dualité relative pour les $${\mathcal D}$$-modules arithmétiques.)(French)Zbl 1083.14017

Adolphson, Alan (ed.) et al., Geometric aspects of Dwork theory. Vol. I, II. Berlin: Walter de Gruyter (ISBN 3-11-017478-2/hbk). 1039-1112 (2004).
The main purpose of this highly technical paper is to provide a complete detailed proof of the relative duality theorem for proper morphisms on smooth schemes of unequal characteristics. Some announcement of this result was given by the author in [C. R. Acad. Sci., Paris, Sér. I 319, No. 12, 1283–1286 (1994; Zbl 0829.14010) and C. R. Acad. Sci., Paris, Sér. I 321, No. 6, 751–754 (1995; Zbl 0876.14011)].
The key point of the proof is the construction of a trace-morphism for the residual complexes of $${\mathcal D}$$-modules, similar to the one established by Grothendieck and Hartshorne for $$O$$-modules, compatible with the usual trace-morphism. Also, a corresponding adjunction formula for proper morphisms is deduced.
For the entire collection see [Zbl 1047.14001].
Reviewer: Tan VoVan (Boston)

### MSC:

 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 32C38 Sheaves of differential operators and their modules, $$D$$-modules 14G20 Local ground fields in algebraic geometry 14F30 $$p$$-adic cohomology, crystalline cohomology

### Citations:

Zbl 0829.14010; Zbl 0876.14011