Descent from the Form Ring and Buchsbaum Rings

One of the major problems in commutative algebra is to recover information about a commutative ring A from known properties of the form ring G := GA(q) = ⊕n≥0q /q with respect to some ideal q of A. There are Krull’s classical results saying that A is an integral domain resp. a normal domain if G is an integral domain resp. a normal domain. It follows from the work [1], [2], [8] that several other properties of a homological nature, like regularity, Cohen-Macaulayness, Gorensteinness etc., descend from G to A. In this note we want to pursue this point of view further. To this end let Q denote the homogeneous ideal of G generated by all the inital forms of element of q. For our purposes here we investigate the local cohomology modules H Q(G) and H • q (A) of G with respect to Q and of A with respect to q resp. For their definition and basic properties see [7]. The first result concerns the descent of the finiteness from H i Q(G) to H i q (A).