In this paper, we provide a simple proof for the fact that two simplicial complexes are isomorphic if and only if their associated Stanley-Reisner rings, or their associated facet rings are isomorphic as K-algebras. As a consequence, we show that two graphs are isomorphic if and only if their associated edge rings are isomorphic as K-algebras. Based on an explicit K-algebra isomorphism of two S...

A squarefree module over a polynomial ring S = k[x1, . . . , xn] is a generalization of a Stanley-Reisner ring, and allows us to apply homological methods to the study of monomial ideals more systematically. The category Sq of squarefree modules is equivalent to the category of finitely generated left Λ-modules, where Λ is the incidence algebra of the Boolean lattice 2. The derived category D(S...

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...

Combinatorial commutative algebra is a branch of combinatorics, discrete geometry, and commutative algebra. On the one hand, problems from combinatorics or discrete geometry are studied using techniques from commutative algebra; on the other hand, questions in combinatorics motivated various results in commutative algebra. Since the fundamental papers of Stanley (see [13] for the results) and H...

For a graph G=(V,E) the edge ring k[G] is k[x1,…,xn]/I(G), where n=|V| and I(G) generated by {xixj;{i,j}∈E}. The conjecture we treat following.Conjecture 1. If has 2-linear resolution, then projective dimension of K[G], pd (k[G]), equals maximal degree vertex in G.As far as know, this first mentioned paper Gitler Valencia [7, Conjecture 4.13], there it called Eliahou-Villarreal conjecture. trea...

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...

Given an infinite field k and a simplicial complex Δ, common theme in studying the f- h-vectors of Δ has been consideration Hilbert series Stanley–Reisner ring k[Δ] modulo generic linear system parameters Θ. Historically, these computations have restricted to special classes complexes (most typically triangulations spheres or manifolds). We provide compact topological expression hd−1a(Δ), dimen...