We show that a finite regular cell complex with the intersection property is a Cohen-Macaulay space iff the top enriched cohomology module is the only nonvanishing one. We prove a comprehensive generalization of Balinski’s theorem on convex polytopes. Also we show that for any Cohen-Macaulay cell complex as above, although there is now generalization of the Stanley-Reisner ring of simplicial co...

Generalizing the notion of a Koszul algebra, a graded kalgebra A is K2 if its Yoneda algebra ExtA(k, k) is generated as an algebra in cohomology degrees 1 and 2. We prove a strong theorem about K2 factor algebras of Koszul algebras and use that theorem to show the Stanley-Reisner face ring of a simplicial complex ∆ is K2 whenever the Alexander dual simplicial complex ∆∗ is (sequentially) Cohen-...

Consider the general n-gon with vertices at the points 1,2, . . . ,n. Then its suspension involves two more vertices, say at n+1 and n+2. Let R be the polynomial ring k[x1,x2, . . . ,xn], where k is any field. Then we can associate an ideal I to our suspension in the Stanley-Reisner sense. In this paper, we find a free minimal resolution and the Betti numbers of the R-module R/I.

Unprojection theory analyzes and constructs complicated commutative rings in terms of simpler ones. Our main result is that, on the algebraic level of Stanley-Reisner rings, stellar subdivisions of Gorenstein* simplicial complexes correspond to unprojections of type Kustin-Miller. As an application of our methods we study the minimal resolution of Stanley– Reisner rings associated to stacked po...

Normal affine algebraic varieties in characteristic 0 are uniquely determined (up to isomorphism) by the Lie algebra of derivations of their coordinate ring. This was shown by Siebert [Si] and, independently, by Hauser and the third author [HM]. In both papers the assumption of normality is essential. There are nonisomorphic non-normal varieties with isomorphic Lie algebras. The third author [M...

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...

Given a matroid M represented by a linear subspace L ⊂ C (equivalently by an arrangement of n hyperplanes in L), we define a graded ring R(L) which degenerates to the Stanley-Reisner ring of the broken circuit complex for any choice of ordering of the ground set. In particular, R(L) is Cohen-Macaulay, and may be used to compute the h-vector of the broken circuit complex of M . We give a geometr...

We combine work of Cox on the homogeneous coordinate ring of a toric variety and results of Eisenbud-Mustaţǎ-Stillman and Mustaţǎ on cohomology of toric and monomial ideals to obtain a formula for computing χ(OX (D)) for a divisor D on a complete simplicial toric variety XΣ. The main point is to use Alexander duality to pass from the toric irrelevant ideal, which appears in the computation of χ...

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...