Artinian level algebras of socle degree 4 (Q2636380)

Let \((A,\mathfrak m)\) be an Artinian local \(K\)-algebra, where \(K\) is a field. The \textit{socle} of \(A\) is \(\text{Soc}(A) = (0:\mathfrak m)\) and the \textit{socle degree} of \(A\) is \(\max \{ j \in \mathbb Z \;| \;\mathfrak m^j \neq 0\}\). The \textit{type} of \(A\) is the dimension of \(\text{Soc}(A)\), and \(A\) is \textit{level of type \(\tau\)} if \(\text{Soc}(A) = \mathfrak m^s\) and \(\dim_K \mathfrak m^s = \tau\). In particular, \(A\) is \textit{Gorenstein} if it is level of type 1. The \textit{Hilbert function} of \(A\) is \( h_i = h_i(A) := \dim_K \mathfrak m^i/\mathfrak m^{i+1}, \) namely the Hilbert function of the associated graded ring \(\mathrm{gr}_{\mathfrak m}(A) := \bigoplus_{i \geq 0} \mathfrak m^i / \mathfrak m^{i+1}\). The first main result is that a sequence \((1,3,h_2,h_3,h_4)\), with \(h_4 \geq 2\), is the Hilbert function of a local level \(K\)-algebra if and only if \(h_3 \leq 3h_4\). If \(h_4 = 1\) then \(h\) is a Gorenstein sequence if and only if \(h_3 \leq 3\) and \(h_2 \leq \binom{h_3+1}{2} + (3-h_3)\). Their second main result is a characterization more generally for Gorenstein sequences. When \(K\) is algebraically closed of characteristic zero, their third main result is that for \(h = (1,h_1,h_2,h_3,1)\) with \(h_2 \geq h_3\), \textit{every} local Gorenstein \(K\)-algebra with Hilbert function \(h\) is necessarily canonically graded if and only if \(h_1 = h_3\) and \(h_2 = \binom{h_1+1}{2}\).