Graded Complexes Over Power Series Rings

A common method in studying a commutative Noetherian local ring A is to find a regular subring R contained in A so that A becomes a finitely generated R-module, and in this way one can obtain some information about the original ring by applying what is known about regular local rings. By the structure theorems of Cohen, if A is complete and contains a field, there will always exist such a subring R, and R will be a power series ring k[[X1, …, Xn]] = k[[X]] over a field k. In this paper we show that if R is chosen properly, the ring A (or, more generally, an A-module M), will have a comparatively simple structure as an R-module. More precisely, A (or M) will have a free resolution which resembles the Koszul complex on the variables (X1, …, Xn) = (X); such a complex will be called an (X)-graded complex and will be given a precise definition below.