We recall numerical criteria for Cohen–Macaulayness related to system of parameters and introduce monomial ideals König type which include the edge graphs. show that a ideal is if only its corresponding residue class ring admits whose elements are form $$x_i-x_j$$ . This provides an algebraic characterization use this special parameter systems study graphs order complex certain family posets. F...

This paper produces a recursive formula of the Betti numbers of certain Stanley-Reisner ideals (graph ideals associated to forests). This gives a purely combinatorial definition of the projective dimension of these ideals, which turns out to be a new numerical invariant of forests. Finally, we propose a possible extension of this invariant to general graphs. 0. Introduction Throughout this pape...

A few years ago, I defined a squarefree module over a polynomial ring S = k[x1, . . . , xn] generalizing the Stanley-Reisner ring k[∆] = S/I∆ of a simplicial complex ∆ ⊂ 2. This notion is very useful in the StanleyReisner ring theory. In this paper, from a squarefree S-module M , we construct the k-sheaf M on an (n − 1) simplex B which is the geometric realization of 2. For example, k[∆] is (th...

Let M be a monomial ideal in the polynomial ring S = k[x1, . . . , xn] over a field k. We are interested in the problem, first posed by Kaplansky in the early 1960’s, of finding a minimal free resolution of S/M over S. The difficulty of this problem is reflected in the fact that the homology of arbitrary simplicial complexes can be encoded (via the Stanley-Reisner correspondence [BH,Ho,St]) int...

Let ∆ be simplicial complex and let k[∆] denote the Stanley– Reisner ring corresponding to ∆. Suppose that k[∆] has a pure free resolution. Then we describe the Betti numbers and the Hilbert– Samuel multiplicity of k[∆] in terms of the h–vector of ∆. As an application, we derive a linear equation system and some inequalities for the components of the h–vector of the clique complex of an arbitra...

A finite projective plane, or more generally a finite linear space, has an associated incidence complex that gives rise to two natural algebras: the Stanley-Reisner ring R/IΛ and the inverse system algebra R/I∆. We give a careful study of both of these algebras. Our main results are a full description of the graded Betti numbers of both algebras in the more general setting of linear spaces (giv...

In this paper, we give an algebra–combinatorics formula of the Möbius transform for an abstract simplicial complex K on [m] = {1, ...,m} in terms of the Betti numbers of the Stanley–Reisner face ring of K. Furthermore, we employ a way of compressing K to estimate the lower bound of the sum of those Betti numbers by using this formula. As an application, associating with the moment-angle complex...

We extend the notion of face rings simplicial complexes and posets to case finite-length (possibly infinite) with a group action. The action on complex induces an ring, we prove that ring invariants is isomorphic quotient poset under mild condition action.We also identify class actions preserve homotopical Cohen–Macaulay property quotients. When acted-upon independence semimatroid, h-polynomial...

In this paper, we give an algebra-combinatorics formula of the Möbius transform for an abstract simplicial complex K on [m] = {1, ...,m} in terms of the Betti numbers of the Stanley-Reisner face ring of K. Furthermore, we employ a way of compressing K to estimate the lower bound of the sum of those Betti numbers by using this formula. As an application, associating with the moment-angle complex...