Some applications of a criterion for artinianness of a module Original Research Article

A criterion previously shown by the author for a module to be artinian is first applied to quickly deduce that certain local cohomology modules are artinian and thereafter to determine the first nonartinian local cohomology module using the concept of filter-depth. Next homomorphisms having finite index are considered. The logarithmic law is proved for such homomorphisms, and the index of an endomorphism on a noetherian (artinian) module is shown to be nonnegative (resp. nonpositive). Finally finiteness results for Koszul (co) homology modules are given, which are extensions of Matijevicʹs classical theorem on maximal ideal transforms.