Initial ideals of truncated homogeneous ideals

Abstract

Denote by R the power series ring in countably many variables over a field K; then Ŕ is the smallest sub-algebra of R that contains all homogeneous elements. It is a fact that a homogeneous, finitely generated ideal J in Ŕ have an initial ideal gr(J), with respect to an arbitrary admissible order, that is locally finitely generated in the sense that for all total degrees d. Furthermore, gr(J) is locally finitely generated even under the weaker hypothesis that J is homogeneous and locally finitely generated.

In this paper, we investigate the relation between gr(J) and the sequence of initial ideals of the“truncated”ideals . It is shown that gr(J) is reconstructive from said sequence. More precisely, it is shown that for all g there exists an N(g) such that here T denotes the total-degree filtration.