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

TL;DR

This paper explores the connection between concrete and abstract approaches to Grothendieck Duality, demonstrating their equivalence and extending key theorems to broader classes of scheme maps.

Contribution

It provides a semi-expository account linking explicit and categorical duality theories, proving the Ideal Theorem for a wider class of scheme maps.

Findings

Equivalence of concrete and abstract Grothendieck Duality theories

Extension of the Ideal Theorem to arbitrary essentially-finite-type maps

Clarification of relations among differential forms, residues, and duality

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.