Basic Compatibilities

Abstract

After reviewing’some preliminary general nonsense, we discuss the basic formalism of the functor f#: D_qc+ (Y) → D_qc (X) (resp. f D_qc+ (Y) → D_qc+ (X)) for a smooth (resp. finite) map f : X → Y between locally noetherian schemes. A ‘projective trace’ map Trpf is defined in case X=Pⁿ _Y and f is the canonical projection, and a ‘fundamental local isomorphism’ nf is defined in case f is a closed immersion which is a local complete intersection morphism of pure codimension. Most of this chapter is concerned with verifying several important non-trivial properties of these constructions. At the end of the chapter, these compatibilities are used to ‘glue’ the definitions of (.)# and (-) in the case of more general maps which are neither smooth nor finite.