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.

‎let $i$ be an ideal in a regular local ring $(r,n)$‎, ‎we will find‎ ‎bounds on the first and the last betti numbers of‎ ‎$(a,m)=(r/i,n/i)$‎. ‎if $a$ is an artinian ring of the embedding‎ ‎codimension $h$‎, ‎$i$ has the initial degree $t$ and $mu(m^t)=1$‎, ‎we call $a$ a {it $t-$extended stretched local ring}‎. ‎this class of‎ ‎local rings is a natural generalization of the class of stretched ...

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.

We show that monomial ideals generated in degree two satisfy a conjecture by Eisenbud, Green and Harris. In particular we give a partial answer to a conjecture of Kalai by proving that h-vectors of flag Cohen-Macaulay simplicial complexes are h-vectors of Cohen-Macaulay balanced simplicial complexes.

‎In this paper‎, ‎we give some necessary conditions for an $r$-partite graph such that the edge ring of the graph is Cohen-Macaulay‎. ‎It is proved that if there exists a cover of an $r$-partite Cohen-Macaulay graph by disjoint cliques of size $r$‎, ‎then such a cover is unique‎.

Let G be a simple (i.e., no loops and no multiple edges) graph. We investigate the question of how to modify G combinatorially to obtain a sequentially CohenMacaulay graph. We focus on modifications given by adding configurations of whiskers to G, where to add a whisker one adds a new vertex and an edge connecting this vertex to an existing vertex in G. We give various sufficient conditions and...

The theorem of Hochster and Roberts says that for any module V of a linearly reductive group G over a eld K the invariant ring KV ] G is Cohen-Macaulay. We prove the following converse: if G is a reductive group and KV ] G is Cohen-Macaulay for any module V , then G is linearly reductive.

Let I be a monomial ideal of the polynomial ring S = K[x1, . . . , x4] over a field K. Then S/I is sequentially Cohen-Macaulay if and only if S/I is pretty clean. In particular, if S/I is sequentially Cohen-Macaulay then I is a Stanley ideal.