Type sequences of one-dimensional local analytically irreducible rings (Q2879613)

Let \((R,\mathfrak{m})\) be a 1-dimensional local analytically irreducible Cohen-Macaulay ring, and let \(\overline{R}\) denote the integral closure of \(R\) in its field of quotients and \(C=R:\overline{R}\) denote the conductor If \(R\) is residually rational, i.e., \(R/\mathfrak{m}\) is isomorphic to the residue field of \(\overline{R}\), Matsuoka considered a sequence of \(r\) natural numbers \((t_1,\ldots,t_r)\), called the \textit{type sequence} of \(R\), that is naturally associated to \(R\) (here \(r=\lambda_R(R/C)\)). This sequence encodes pieces of information about \(R\). For instance, \(t_1\) is the Cohen-Macaulay type of \(R\), \(\sum\limits_{i=1}^{r} t_i=\lambda_R(\overline{R}/R)\), and the properties of \(R\) being Gorenstein or almost Gorenstein, or \(\lambda_R(\overline{R}/R)\) being maximal (with respect to a natural bound obtained from the type sequence) can be characterized using type sequences.NEWLINENEWLINEIn the present paper, the authors extend all the above mentioned results to the case of not necessarily residually rational 1-dimensional local analytically irreducible Cohen-Macaulay rings. In particular, the main result states that almost Gorenstein rings are characterized by a type sequence of the form \((t,n_1,\ldots,n_{r+l})\), whereas rings of maximal length by a type sequence of the form \((t,tn_1,\ldots,tn_{r+l})\), where the \(n_i\) are the dimensions of certain \(R/\mathfrak{m}\)-vector spaces. As a special case, the authors obtain an analogous characterization of Gorenstein and Kunz rings in terms of their type sequences. Several examples are provided to illustrate the main result.