Duality and flat base change on formal schemes

TL;DR

This paper develops several versions of Grothendieck Duality for unbounded complexes on noetherian formal schemes, providing new proofs and applications that unify various duality results in algebraic geometry.

Contribution

It introduces multiple formulations of duality theorems for formal schemes, adapting Deligne's method and Neeman's approach, with applications to classical duality results.

Findings

New proofs of duality theorems for formal schemes

A flat-base-change theorem for pseudo-proper maps

Unified framework for duality-related results

Abstract

We give several related versions of global Grothendieck Duality for unbounded complexes on noetherian formal schemes. The proofs, based on a non-trivial adaptation of Deligne's method for the special case of ordinary schemes, are reasonably self-contained, modulo the Special Adjoint Functor Theorem. An alternative approach, inspired by Neeman and based on recent results about "Brown Representability," is indicated as well. A section on applications and examples illustrates how these theorems synthesize a number of different duality-related results (local duality, formal duality, residue theorems, dualizing complexes...). A flat-base-change theorem for pseudo-proper maps leads in particular to sheafified versions of duality for bounded-below complexes with quasi-coherent homology. Thanks to Greenlees-May duality, the results take a specially nice form for proper maps and bounded-below complexes with coherent homology.