Ascent and descent for finite sequences of commuting endomorphisms

Homological techniques involving the Koszul complex are used to define and explore two invariants, ascent and descent, for a finite sequence of commut ing endomorphism of a module. It is shown in particular that, as in the cas e of a single endomorphism, if ascent and descent are both finite then they are equal, and that this finiteness condition is equivalent to a certain s trong Fitting type property.