Almost Gorenstein Hibi rings (Q1675069)

Let \(R = \bigoplus_{i \geq 0}R_i\) denote a standard graded Cohen-Macaulay ring. Let \(M\) denote a finitely generated graded \(R\)-module. Then \(\mu(M) \leq e(M)\) for the minimal number of generators \(\mu(M)\) and the multiplicity \(e(M)\). The ring \(R\) is called almost Gorenstein if there is a short exact sequence \(0 \to R \to K_R(a) \to C \to 0\) of graded \(R\)-modules such that \(C = 0\) or \(\mu(C) = e(C)\). Here \(K_R\) denotes the canonical module of \(R\) and \(a\) is the \(a\)-invariant of \(R\). Not that \(R\) is a Gorenstein ring if \(C = 0\). For the notion of almost Gorenstein rings see [\textit{S. Goto} et al., J. Pure Appl. Algebra 219, No. 7, 2666--2712 (2015; Zbl 1319.13017)] and the references there. Let \(k\) denote a field, \(H\) a distributive lattice, \(P\) the set of join-irreducible elements of \(H\) and \(\mathcal{R}_k(H)\) the Hibi ring of \(k\) on \(H\) (see also [\textit{T. Hibi}, Adv. Stud. Pure Math. 11, 93--109 (1987; Zbl 0654.13015)]). The author characterizes Hibi rings to be (1) level, non-Gorenstein and almost Gorenstein, (2) non-level and almost Gorenstein in terms of the partially ordered set defining the Hibi ring. Moreover he proves a criterion of a ladder determinantal ideal of \(2 \times 2\)-minors to be Gorenstein and almost Gorenstein in terms of the shape of the ladder. Recall that a ladder determinantal ring defined by \(2 \times 2\)-minors is a Hibi ring.