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

Abstract

In this work we study some topological aspects of commutative algebra. More precisely, we study characterizations and properties of linearly compact modules on semilocal rings and prove, for example, that these modules satisfy a topological duality. In addition, there is a duality between homology and local cohomology modules for these modules. Another important result of this work is to extend Grothendieck's vanishing and non-vanishing theorems to linearly compact and semi-discrete modules on semilocal rings, as well as to prove the artinianess of local cohomology modules and the noetherianess of local homology modules.