Grothendieck Duality theories -- abstract and concrete, I: pseudo-coherent finite maps

Abstract

Grothendieck Duality -- the theory of the twisted inverse image pseudofunctor (-)^! over a suitable category of scheme-maps -- can be developed concretely, with emphasis on explicit constructions, or abstractly, with emphasis on category-theoretic considerations. It is not obvious that the two resulting theories are essentially the same. This is a semi-expository account of the connection between these approaches, a nontrivial matter involving some alluring relations, for instance among differential forms, residues and duality. In particular, it emerges that the culminating Ideal Theorem in Hartshorne's "Residues and Duality" holds for arbitrary essentially-finite-type maps of noetherian schemes and bounded-below complexes with quasi-coherent cohomology. What appears in this first part mostly concerns pseudo-coherent finite maps. The rest is being prepared.