6 research outputs found
Cohen-Macaulayness of monomial ideals and symbolic powers of Stanley-Reisner ideals
We present criteria for the Cohen-Macaulayness of a monomial ideal in terms
of its primary decomposition. These criteria allow us to use tools of graph
theory and of linear programming to study the Cohen-Macaulayness of monomial
ideals which are intersections of prime ideal powers. We can characterize the
Cohen-Macaulayness of the second symbolic power or of all symbolic powers of a
Stanley-Reisner ideal in terms of the simplicial complex. These
characterizations show that the simplicial complex must be very compact if some
symbolic power is Cohen-Macaulay. In particular, all symbolic powers are
Cohen-Macaulay if and only if the simplicial complex is a matroid complex. We
also prove that the Cohen-Macaulayness can pass from a symbolic power to
another symbolic powers in different ways.Comment: The published version of this paper contains a gap in the proofs of
Theorem 2.5 and Theorem 3.5. This version corrects the proofs with almost the
same arguments. Moreover, we have to modify the definition of tight complexes
in Theorem 2.5. These changes don't affect other things in the published
version. A corrigendum has been sent to the journa
Local topology of the free complex of a two-dimensional generalized convex shelling
AbstractA generalized convex shelling was introduced by Kashiwabara et al. for their representation theorem of convex geometries. Motivated by the work by Edelman and Reiner, we study local topology of the free complex of a two-dimensional separable generalized convex shelling. As a result, we prove a deletion of an element from such a complex is homotopy equivalent to a single point or two distinct points, depending on the dependency of the element to be deleted. Our result resolves an open problem by Edelman and Reiner for this case, and it can be seen as a first step toward the complete resolution from the viewpoint of a representation theorem for convex geometries by Kashiwabara et al
Matroid Representation of Clique Complexes
In this paper, we approach the quality of a greedy algorithm for the maximum weighted clique problem from the viewpoint of matroid theory. More precisely, we consider the clique complex of a graph (the collection of all cliques of the graph), which is also called a flag complex, and investigate the minimum number k such that the clique complex of a given graph can be represented as the intersection of k matroids. This number k can be regarded as a measure of "how complex a graph is with respect to the maximum weighted clique problem" since a greedy algorithm is a k-approximation algorithm for this problem. For any k > 0 we characterize..