Sequentially Cohen-Macaulayness of bigraded modules (Q527050)

A module is called sequentially Cohen-Macaulay if there is a filtration with Cohen-Macaulay factors and that the dimension of the factors is an increasing sequence. One may define a relative situation with respect to an ideal. In this regard we say a module is Cohen-Macaulay with respect to an ideal if its grade with respect to that ideal is the same as of its cohomological dimension with respect to that ideal. Also one may define the relative dimension with respect to an ideal. The relative notion of dimension in the paper is the cohomological dimension with respect to that ideal. To see this is compatible it is enough to apply Grothendieck's non-vanishing theorem: the cohomological dimension with respect to the maximal ideal of a local ring is the dimension of that module. The paper under review investigates sequentially Cohen-Macaulayness of a bi-graded module with respect to a graded ideal over a bi-graded polynomial ring and a hyper-surface ring.