Duality Foundations

Abstract

In this chapter, we discuss Grothendieck’s notion of a residual complex. This concept allows one to construct a duality theory in the proper Cohen-Macaulay case without projectiveness assumptions (although some proofs ultimately reduce via Chow’s Lemma to the analysis of projective space and finite maps, as treated in Chapter 2). The special role of CM maps are that these are exactly the morphisms for which one can define a relative dualizing sheaf (rather than a relative dualizing complex), generalizing the sheaf of top degree relative differential forms in the smooth case. The base change theory for dualizing sheaves is set up at the end of this chapter. This makes it possible to consider the base change compatibility of the trace map for proper CM morphisms, a problem we will address in Chapter 4.