Let F be a field and let G be a finite graph with a total ordering on its edge set. Richard Stanley noted that the Stanley-Reisner ring F (G) of the broken circuit complex of G is CohenMacaulay. Jason Brown gave an explicit description of a homogeneous system of parameters for F (G) in terms of fundamental cocircuits in G. So F (G) modulo this hsop is a finite dimensional vector space. We conje...

In this thesis, we undertake a combinatorial study of certain aspects of triangulations of cartesian products of simplices, particularly in relation to their relevance in toric algebra and to their underlying product structure. The first chapter reports joint work with Samu Potka. The object of study is a class of homogeneous toric ideals called cut ideals of graphs, that were introduced by Stu...

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...

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 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...

We discuss simple partitioning phenomena for intersection rings of certain toric varieties, and apply this to prove the following results. ̋ We generalize a result of Gräbe, proving that Stanley–Reisner rings of homology spheres are Gorenstein. ̋ We give a new and direct proof of a theorem of the authors due to which the cohomology rings of matroids are Poincare duality algebras. ̋ We prove a conj...