Grothendieck Duality and Transitivity I: Formal Schemes

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.