Initial Ideals of Truncated Homogeneous Ideals
نویسنده
چکیده
Denote by R the power series ring in countably many variables over a eld K; then R 0 is the smallest sub-algebra of R that contains all homogeneous elements. It is a fact that a homogeneous, nitely generated ideal J in R 0 have an initial ideal gr(J), with respect to an arbitrary admissible order, that is locally nitely generated in the sense that dimK gr(J) d P d?1 j=1 R 0 j gr(J) d?j < 1 for all total degrees d. Furthermore, gr(J) is locally nitely generated even under the weaker hypothesis that J is homogeneous and locally nitely generated. In this paper, we investigate the relation between gr(J) and the sequence of initial ideals of the \truncated" ideals n (J) Kx1; : : : ; xn]. It is shown that gr(J) is reconstructible from said sequence. More precisely, it is shown that for all g there exists an N(g) such that T g gr(J) = T g gr(n (J)) e whenever n > N(g); here T denotes the total-degree ltration.
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