Grothendieck Duality and Transitivity I: Formal Schemes

TL;DR

This paper extends Grothendieck duality and transitivity results to noetherian formal schemes, providing a natural transformation compatible with pseudofunctorial structures, with implications for residues and traces.

Contribution

It generalizes duality transformations to formal schemes and establishes their compatibility with composition, advancing the abstract theory of residues and traces.

Findings

Extended duality transformation to formal schemes.

Proved compatibility with pseudofunctorial structures.

Implications for residues and trace theory.

Abstract

For a proper map $f\colon X\to Y$ of noetherian ordinary schemes, one has a well-known natural transformation, ${\bf L}^*f^*(-)\overset{\bf L}{\otimes} f^!{\mathcal{O}}_Y\to f^!$, obtained via the projection formula, which extends, using Nagata's compactification, to the case where $f$ is separated and of finite type. In this paper we extend this transformation to the situation where $f$ is a pseudo-finite-type map of noetherian formal schemes which is a composite of compactifiable maps, and show it is compatible with the pseudofunctorial structures involved. This natural transformation has implications for the abstract theory of residues and traces, giving Fubini type results for iterated maps. These abstractions are rendered concrete in a sequel to this paper.