Étale duality for constructible sheaves on arithmetic schemes

Abstract.

In this note we relate the following three topics for arithmetic schemes: a general duality for étale constructible torsion sheaves, a theory of étale homology, and the arithmetic complexes of Gersten–Bloch–Ogus type defined by K. Kato (1986).

In brief, there is an absolute duality using certain dualizing sheaves on these schemes, we describe and characterize the dualizing sheaves to some extent, relate them to symbol maps, define étale homology via the dualizing sheaves, and show that the niveau spectral sequence for the latter, constructed by the method of Bloch and Ogus (1974), leads to the complexes defined by Kato. Some of these relations may have been expected by experts, and some have been used implicitly in the literature, although we do not know any explicit reference for statements or proofs. Moreover, the main results are used in a crucial way in a paper by Jannsen and Saito (2003). So a major aim is to fill a gap in the literature, and a special emphasis is on precise formulations, including the determination of signs. But the general picture developed here may be of interest itself.