Reductions of graded rings and pseudo-flat graded modules
The author develops a theory of reductions for graded rings, in generalizing Northcott-Rees' theory for ideals, e.g., the existence and the characterization of minimal reductions. To get these results, he defines a notion of analytic spread of graded modules and also introduces a class of pseudo-flat graded modules which satisfy milder conditions than those of normally flatness. The following structure theorem for pseudo-flat graded modules is one of the main theorems of this article: Let R be a local ring, \(A=\oplus_{n\geq 0}A_ n \) a graded noetherian ring generated over \(A_ 0=R\) by elements of degree one, and let \(M=\oplus_{n\in {\mathbb{Z}}}M_ n \) a finitely generated graded A-module. Then: Theorem 3.4. Assume that R is a reduced local ring with infinite residue field. Then M is pseudo-flat if and only if there is a polynomial \(sub-R-algebra\quad B\) of A such that M is finitely generated faithful B- module. The ``if'' part is also valid even if the residue field of R is a finite field. In particular, A is pseudo-flat if and only if A is a finite extension of a polynomial R-algebra.
scientific article; zbMATH DE number 4165026