Artinian level modules of embedding dimension two (Q2504390)

Let \(R=k[x_{1},x_{2}, \ldots x_{n}]\), \(k\) a field, and consider \(R\) as a graded ring with unique graded maximal ideal \({\mathbf m}=(x_{1},x_{2}, \ldots x_{n})\), where each \(x_{i}\) has been given degree one. Assume all \(R\)-modules to be finitely generated and graded. An Artinian \(R\)-module \(M=M_{0} \oplus \ldots \oplus M_{c}\) is called a level module if it is generated by \(M_{0}\) and \(\text{Soc}(M)=M_{c}\). It is known that a sequence \((h_{0},h_{1}, \ldots h_{c})\) of positive integers with \(h_{0}=1\) is the Hilbert function of a graded Artinian level algebra of embedding dimension two if and only if \(h_{i-1}-2h_{i} +h_{i+1} \leq 0\) for all \(0 \leq i \leq c,\) where \(h_{-1}=h_{c+1}=0\) [\textit{A. Iarrobino}, J. Algebra 272, 530--580 (2004; Zbl 1119.13015) and \textit{J. V. Chipalkatti} and \textit{A. V. Geramita}, Mich. Math. J. 51, 187--207 (2003; Zbl 1097.13514)]. In the paper under review, this result is generalized to graded Artinian level modules. The proof of necessity parallels that for algebras and uses Betti numbers. Two proofs are provided for sufficiency. In the first proof, which works over infinite fields only, Macaulay's criterion for the characterization of Hilbert functions of a graded module, together with a deformation argument, is used. The second approach works over any field \(k\), is of a more combinatorial nature and uses a construction with monomial ideals in \(k[x,y]\). The final section shows that an Artinian quotient of monomial ideals may be lifted to an ideal in the homogeneous coordinate ring of a certain set of reduced points such that the lifted ideal and homogeneous coordinate ring coincide when the degrees are high enough. This can be applied to lift the level of quotients of monomial ideals used in the second proof of the main theorem mentioned above.