Hilbert function of local Artinian level rings in codimension two
The present paper deals with the problem of characterizing the possible Hilbert functions of local Artinian level rings of codimension two. The central theorem in the paper provides, given an ``admissible'' numerical function \(h\), an effective method to produce a zero-dimensional ideal of \(kx,y\) such that \(R/I\) is a local Artinian level ring having \(h\) as a Hilbert function. As a consequence, the author obtains the main result of the paper, which gives a characterization of the possible Hilbert functions of codimension two local Artinian level rings of given type and socle degree. The techniques used in the paper include the use of a similar notion to that of Gröbner bases in the formal power series ring, namely enhanced standard bases, for which the author gives a characterization in the case of ideals of height two. The author uses also the good properties of the generic initial ideal (Gin) of ideals in \(kx_1,\dots, x_n\). The results in the paper extend previous work by \textit{J. Briançon} [Invent. Math. 41, 45--89 (1977; Zbl 0353.14004)], \textit{A. Iarrobino} [Proc. Symp. Pure Math. 40, Part 1, 593--608 (1983; Zbl 0563.13010), Lect. Notes Math. 1124, 146--165 (1985; Zbl 0567.14001), J. Algebra 272, No. 2, 530--580 (2004; Zbl 1119.13015)] and \textit{J. V. Chipalkatti} and \textit{A. V. Geramita} [Mich. Math. J. 51, No. 1, 187--207 (2003; Zbl 1097.13514)].
scientific article; zbMATH DE number 5549367
scientific article