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 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...

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...

We study combinatorial aspects of the representation theory of the 0-Hecke algebra Hn(0), a deformation of the group algebra of the symmetric group Sn. We study the action of Hn(0) on the polynomial ring in n variables. We show that the coinvariant algebra of this action naturally carries the regular representation of Hn(0), giving an analogue of the well-known result for the symmetric group by...

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...

For t in N, E.Miller has defined a category of positively t-determined modules over the polynomial ring S in n variables. We consider the AuslanderReiten translate, Nt, on the (derived) category of such modules. A monomial ideal I is t-determined if each generator x has a ≤ t. We compute the multigraded cohomology and betti spaces of N k t (S/I) for every iterate k and also the S-module structu...

Generalizing the concepts of Stanley–Reisner and affine monoid algebras, one can associate to a rational pointed fan Σ in Rd the Zd-graded toric face ring K[Σ]. Assuming that K[Σ] is Cohen–Macaulay, the main result of this paper is to characterize the situation when its canonical module is isomorphic to a Zd-graded ideal of K[Σ]. From this result several algebraic and combinatorial consequences...

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...