On rings with linear resolutions

Abstract

Let R denote a positively graded noetherian ring over a field k, R = @EoRi with R, = k and m the irrelevant maximal ideal generated by forms of degree 1, or let R be the m-adic completion of such a ring. If we consider X as a homomorphic image of a polynomial ring, resp. of a power series ring over k with a homogeneous defining ideal, one may ask whether R has a (minimal) linear resolution as an S-module. Cohen-Macaulay rings R of maximal embedding dimension, e.g., have a linear resolution as Sally showed in [9]. She also proved formulas for the Hilbert series HR and the Poincari: series Pi, Pz. Schenzel [lo] later characterized CM-rings R with a linear resolution and called these rings “extremal.” He also gave formulas for HR and Pi. S. Goto (See (3.1) below) proved, besides other facts, the linearity of the resolution for Buchsbaum rings of maximal embedding dimension (i.e., edim R = dim R + e(R) + I(R) 1; I(R) denotes the invariant of the Buchsbaum ring R) and a formula for HH. In Section 1 we generalize these results and characterize rings with a linear resolution, which have a parameter system that forms a d-sequence. In the Buchsbaum case we then show that the H,, Pi and Pf of such rings are determined by the embedding dimension, the Krull dimension, the dimensions of the local cohomology modules as k-vector spaces and what we call the type of R. In Section 2 we determine all Buchsbaum rings R = k[ [R,]] of maximal embedding dimension and multiplicity 2. Sally proved in [8] that, for a local CM-ring (R, m) of maximal embedding dimension, the associated graded ring is also CM of maximal embedding dimension. Goto showed a similar result for Buchsbaum rings of