Inverse limits of Macaulay's inverse systems (Q1730201)

Let \(\mathbb{K}\) be an infinite field and \(P\) either \(\mathbb{K}[x_1,\dots,x_n]\) or \(\mathbb{K}[[x_1,\dots,x_n]]\) (formal power series). Let \(\hat{P} = \Hom_{P_0}(P,E)\), where \(E\) is the injective hull of \(\mathbb{K}\) over \(P_0\); we get a correspondence between graded ideals \(I\) of \(P\) and \(P\)-submodules \(I^\perp\) of \(\hat{P}\), under which the ideals \(I\) for which \(P/I\) is Artinian correspond to finitely generated submodules \(I^\perp\); \(I^\perp\) is known as the inverse system of \(I\) (by Macaulay). This correspondence as been extended to the case of positive dimension and such generalization is studied here, by using the varius socles in the inverse system. The main result is an explicit description of inverse limits of Macaulay's inverse systems, obtained by dividing out powers of a linear regular sequence.