The Multiplicity Conjecture for Barycentric Subdivisions

Abstract

Abstract. For a simplicial complex ∆ we study the effect of barycentric subdivision on ring theoretic invariants of its Stanley-Reisner ring. In particular, for Stanley-Reisner rings of barycentric subdivisions we verify a conjecture by Huneke and Herzog & Srinivasan, that relates the multiplicity of a standard graded k-algebra to the product of the maximal and minimal shifts in its minimal free resolution up to the height. On the way to proving the conjecture we develop new and list well known results on behavior of dimension, Hilbert series, multiplicity, local cohomology, depth and regularity when passing from the Stanley-Reisner ring of ∆ to the one of its barycentric subdivision.

Face Ring Multiplicity via CM-Connectivity Sequences

“…For instance, (reduced) order complexes of all of the following posets satisfy this condition: face posets of 2-CM cell complexes with the intersection property (that is, intersection of any two faces is a face; this class includes face posets of all polytopes and face posets of all 2-CM simplicial complexes), geometric lattices, supersolvable lattices with nonzero Möbius function on every interval, rank selected subposets of any of these. We remark that the multiplicity upper bound conjecture for the order complexes of face posets of all simplicial complexes was very recently verified by Kubitzke and Welker [18].…”

Graded Betti numbers of Cohen-Macaulay modules and the multiplicity conjecture

“…The conjecture gives lower and upper bounds for the multiplicity of a graded algebra in terms of the lowest degrees and highest degrees where the Betti numbers are non-zero. The conjecture has been verified in a number of interesting cases, including the ones mentioned above where the structure of the minimal free resolution is known (see Herzog and Srinivasan [6], Migliore-Nagel [13], Migliore, Nagel and Römer [14], [15], Kubitzke and Welker [9]). However, the meaning of the conjecture itself has been a bit mysterious, since the multiplicity of an algebra is a very coarse invariant, while the graded Betti numbers are much finer.…”

Algebraic Properties of Clique Complexes of Line Graphs

“…. , we can get a square-free monomial ideal J generated in degree 2, such that S/I is Cohen-Macaulay (CM for short) if and only if S/J is so (see [3] and also [7]). But then J is the edge ideal of a graph and this shows why it is important to study algebraic properties of edge ideals of graphs.…”

“…The family of cliques of a graph G forms a simplicial complex which is called the clique complex of G and is denoted by ∆(G). Algebraic properties of simplicial complexes in general also has got a wide attention recently, see for example [3,7,9,12,15] and the references therein. If we denote the Stanley-Reisner ideal of ∆ by I ∆ , then we have I ∆(G) = I(G), where G denotes the complement of the graph G. Thus studying clique complexes of graphs algebraically, is another way to study algebraic properties of graphs.…”