Grothendieck duality under Spec Z
Abstract
We define the derived category of a concrete category in a way which extends the usual definition of the derived category of a ring, and we prove that the bounded-below derived category of M0 (an approximation, used by e.g. Connes and Consani, to " of the field with one element") is the stable homotopy category of connective spectra. We also describe some basic features of Grothendieck duality for the map from Z to M0, or, what comes to the same thing, the map from Z to of the sphere spectrum; these basic features include a computation of the homology of the dualizing complex f!(S) of abelian groups associated to the sphere spectrum.
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