Partial quotients of moment-angle complexes are topological analogues smooth, not necessarily compact toric varieties. In 1998, Buchstaber and Panov proposed a formula for the cohomology ring such partial quotient in terms torsion product involving corresponding Stanley-Reisner ring. We show that their gives correct cup if 2 is invertible chosen coefficient ring, but general. rectify this by de...

The research summarized in this thesis consists essentially of two parts. In the first, we generalize a coloring theorem of Baxter about triangulations of the plane (originally used to prove combinatorially Brouwer's fixed point theorem in two dimensions) to arbitrary dimensions and to oriented simplicial and cubical pseudomanifolds. We show that in a certain sense no other generalizations may ...

for an $n$-gon with vertices at points $1,2,cdots,n$, the betti numbers of its suspension, the simplicial complex that involves two more vertices $n+1$ and $n+2$, is known. in this paper, with a constructive and simple proof, wegeneralize this result to find the minimal free resolution and betti numbers of the $s$-module $s/i$ where  $s=k[x_{1},cdots, x_{n}]$ and $i$ is the associated ideal to ...

The notion of toric face rings generalizes both Stanley-Reisner rings and affine semigroup rings, and has been studied by Bruns, Römer, et.al. Here, we will show that, for a toric face ring R, the “graded” Matlis dual of a Cěch complex gives a dualizing complex. In the most general setting, R is not a graded ring in the usual sense. Hence technical argument is required.

We continue the study [2] on sheaves of rings on finite posets. We present examples where the ring of global sections coincide with toric faces rings, quotients of a polynomial ring by a monomial ideal and algebras with straightening laws. We prove a rank-selection theorem which generalizes the well-known rank-selection theorem of Stanley–Reisner rings. Finally, we determine an explicit present...

For a graph G, we show a theorem that establishes a correspondence between the fine Hilbert series of the Stanley-Reisner ring of the clique complex for the complementary graph of G and the fine subgraph polynomial of G. We obtain from this theorem some corollaries regarding the independent set complex and the matching complex.

Let ∆ be a triangulated homology ball whose boundary complex is ∂∆. A result of Hochster asserts that the canonical module of the Stanley–Reisner ring of ∆, F[∆], is isomorphic to the Stanley–Reisner module of the pair (∆, ∂∆), F[∆, ∂∆]. This result implies that an Artinian reduction of F[∆, ∂∆] is (up to a shift in grading) isomorphic to the Matlis dual of the corresponding Artinian reduction ...

We prove that sequentially Cohen-Macaulay rings in positive characteristic, as well as sequentially Cohen-Macaulay Stanley-Reisner rings in any characteristic, have trivial Lyubeznik table. Some other configurations of Lyubeznik tables are also provided depending on the deficiency modules of the ring.

Let ∆ be a stable simplicial complex on n vertexes. Over an arbitrary base field K, the symmetric algebraic shifted complex ∆s of ∆ is defined. It is proved that the Betti numbers of the Stanley-Reisner ideals in the polynomial ring K[x1, x2, . . . , xn] of the symmetric algebraic shifted, exterior algebraic shifted and combinatorial shifted complexes of ∆ are equal.