Macdonald dual in the semilocal case and applications to local homology and local cohomology

This paper extends Macdonald’s duality theory and Cuong-Nam’s local homology theory – which is in some sense dual to the local cohomology theory of A. Grothendieck [Gro1] – to semilocal noetherian rings equipped with the $J$-adic topology. By overcoming the key structural obstructions of the semilocal setting via Zöschinger’s structure theory and semilocal Matlis duality, this work establishes topological limit duality isomorphisms and links local homology with local cohomology. Ultimately, this framework provides the foundational machinery required to extend the classical results of Macdonald [Mac] and Cuong-Nam [C-N1] to the semilocal setting.