Narrow Results

We classify Cohen–Macaulay graphs of girth at least 5 and planar Gorenstein graphs of girth at least 4. Moreover, such graphs are also vertex decomposable.

We graph-theoretically characterize the class of graphs $G$ such that $I{(G)}^2$ are Buchsbaum.

Let $R$ be a Cohen–Macaulay local ring. We prove that the $n$th syzygy module of a maximal Cohen–Macaulay $R$-module cannot have a semidualizing direct summand for every $n\ge 1$. In particular, it follows that $R$ is Gorenstein if and only if some syzygy of a canonical module of $R$ has a nonzero free direct summand. We also give a number of necessary and sufficient conditions for a Cohen–Macaulay local ring of minimal multiplicity to be regular or Gorenstein. These criteria are based on vanishing of certain Exts or Tors involving syzygy modules of the residue field.

Let $G$ be a connected simple graph, with $n$ vertices whose homogeneous monomial subring is $S$. We prove that if $S$ is normal and Gorenstein, then $G$ is unmixed with cover number $\lceil \frac{n}{2}\rceil$ and $G$ has a strong $\lceil \frac{n}{2}\rceil$-$\tau$-reduction. Also, if $S$ is normal and $G$ is unmixed whose cover number is $\lceil \frac{n}{2}\rceil$, we give sufficient conditions for $S$ to be Gorenstein.

Let R=k[x₁,…,x_n], where k is a field. The path ideal (of length t ≥ 2) of a directed graph G is the monomial ideal, denoted by I_t(G), whose generators correspond to the directed paths of length t in G. Let Γ be a directed rooted tree. We characterize all such trees whose path ideals are unmixed and Cohen-Macaulay. Moreover, we show that R/I_t(Γ) is Gorenstein if and only if the Stanley-Reisner simplicial complex of I_t(Γ) is a matroid.

Let $T$ be a tilting module. In this paper, Gorenstein $\sigma [T]$-injective modules are introduced and some of their basic properties are studied. Moreover, some characterizations of rings over which all modules are Gorenstein $\sigma [T]$-injective are given. For instance, we show that every $R$-module is Gorenstein $\sigma [T]$-injective if and only if every flat $R$-module is Gorenstein $\sigma [T]$-injective if and only if $R$ is a $\sigma [T]$-injective module on itself.