In this paper, we use a characterization of R-modules N such that fdRN = pdRN to characterize Cohen-Macaulay rings in terms of various dimensions. This is done by setting N to be the dth local cohomology functor of R with respect to the maximal ideal where d is the Krull dimension of R.

Let [Formula: see text] be a simple undirected graph. The family of all matchings forms simplicial complex called the matching text]. Here, we give classification graphs with Gorenstein complex. Also study when is Cohen–Macaulay and, in certain classes graphs, fully characterize those which have In particular, graph girth at least five or complete Cohen–Macaulay.

We study Stanley decompositions and show that Stanley’s conjecture on Stanley decompositions implies his conjecture on partitionable Cohen-Macaulay simplicial complexes. We also prove these conjectures for all Cohen-Macaulay monomial ideals of codimension 2 and all Gorenstein monomial ideals of codimension 3.

Let D be a weighted oriented graph, whose underlying graph is G , and let I (D) its edge ideal. If has no 3-, 5-, or 7-cycles, K?nig, we characterize when unmixed. 3- 5-cycles, Cohen–Macaulay. We prove that unmixed if only Cohen–Macaulay girth greater than 7 K?nig 4-cycles.

A tetrahedral curve is a (usually nonreduced) curve in P defined by an unmixed, height two ideal generated by monomials. We characterize when these curves are arithmetically Cohen-Macaulay by associating a graph to each curve and, using results from combinatorial commutative algebra and Alexander duality, relating the structure of the complementary graph to the Cohen-Macaulay property.

We study Stanley decompositions and show that Stanley’s conjecture on Stanley decompositions implies his conjecture on partitionable Cohen-Macaulay simplicial complexes. We also prove these conjectures for all Cohen-Macaulay monomial ideals of codimension 2 and all Gorenstein monomial ideals of codimension 3.