Algebraic properties of toric rings of graphs

Let $G = (V,E)$ be a simple graph. We investigate the Cohen-Macaulayness and algebraic invariants, such as the Castelnuovo-Mumford regularity and the projective dimension, of the toric ring $k[G]$ via those of toric rings associated to induced subgraphs of $G$.Comment: 18 pages; changed title and re-organized sections to better exhibit results; correct the last main theore

Pure simplicial complexes and well-covered graphs

A graph $G$ is called well-covered if all maximal independent sets of vertices have the same cardinality. A simplicial complex $\Delta$ is called pure if all of its facets have the same cardinality. Let $\mathcal G$ be the class of graphs with some disjoint maximal cliques covering all vertices. In this paper, we prove that for any simplicial complex or any graph, there is a corresponding graph in class $\mathcal G$ with the same well-coveredness property. Then some necessary and sufficient conditions are presented to recognize fast when a graph in the class $\cal G$ is well-covered or not. To do this characterization, we use an algebraic interpretation according to zero-divisor elements of the edge rings of graphs.Comment: 10 pages. arXiv admin note: substantial text overlap with arXiv:1009.524

On the Existence of Non-Free Totally Reflexive Modules

For a standard graded Cohen-Macaulay ring R, if the quotient R/(x) admits nonfree totally reflexive modules, where x is a system of parameters consisting of elements of degree one, then so does the ring R. A non-constructive proof of this statement was given in [16]. We give an explicit construction of the totally reflexive modules over R obtained from those over R/(x). We consider the question of which Stanley-Reisner rings of graphs admit nonfree totally reflexive modules and discuss some examples. For an Artinian local ring (R,m) with m3 = 0 and containing the complex numbers, we describe an explicit construction of uncountably many non-isomorphic indecomposable totally reflexive modules, under the assumption that at least one such non-free module exists. In addition, we generalize Rangel-Tracy rings. We prove that her results do not generalize. Specifically, the presentation of a totally reflexive module cannot be choosen generically in our generalizations

Regularity and multiplicity of toric rings of three-dimensional Ferrers diagrams

We investigate the Castelnuovo--Mumford regularity and the multiplicity of the toric ring associated to a three-dimensional Ferrers diagram. In particular, in the rectangular case, we are able to provide direct formulas for these two important invariants. Then, we compare these invariants for an accompanied pair of Ferrers diagrams under some mild conditions, and bound the Castelnuovo--Mumford regularity for more general cases.Comment: 22 pages, 2 figures and comments are welcom

A description of incidence rings of group automata

Group automata occur in the Krohn-Rhodes Decomposition Theorem and have been extensively investigated in the literature. The incidence rings of group automata were introduced by the first author in analogy with group rings and incidence rings of graphs. The main theorem of the present paper gives a complete description of the structure of incidence rings of group automata in terms of matrix rings over group rings and their natural modules. As a consequence, when the ground ring is a field, we can use known group algebra results to determine when the incidence algebra is prime, semiprime, Artinian or semisimple. We also offer sufficient conditions for the algebra to be semiprimitive