Buchsbaum Stanley--Reisner rings with minimal multiplicity

In this paper, we study non-Cohen–Macaulay Buchsbaum Stanley–Reisner rings with linear free resolution. In particular, for given integers $c$, $d$, $q$ with $c\ge 1$, $2\le q\le d$, we give an upper bound $h_{c,d,q}$ on the dimension of the unique non-vanishing homology ${\tilde{H}}_{q-2}(\mathrm{\Delta };k)$ of a $d$-dimensional Buchsbaum ring $k[\mathrm{\Delta }]$ with $q$-linear resolution and codimension $c$. Also, we discuss about existence for such Buchsbaum rings with ${\dim }_k{\tilde{H}}_{q-2}(\mathrm{\Delta };k)=h$ for any $h$ with $0 \le h \le h_{c,d,q}$,andproveanexistencetheoreminthecaseof$q=d=3$ using the notion of Cohen–Macaulay linear cover. On the other hand, we introduce the notion of Buchsbaum Stanley–Reisner rings with minimal multiplicity of type $q$, which extends the notion of Buchsbaum rings with minimal multiplicity defined by Goto. As an application, we give many examples of Buchsbaum Stanley–Reisner rings with $q$-linear resolution.