We give a purely algebraic proof of the hypersurface case of the Toric Residue Mirror Conjecture recently proposed by Batyrev and Materov.

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 ...

It is a well known fact that a supersolvable lattice is ELoshellable. Hence a supersolvable lattice (resp., its Stanley-Reisner ring) is Cohen-Macaulay. We prove that if L is a supersolvable lattice such that all intervals have non-vanishing Mt~bius number, then for an arbitrary element x e L the poser L {x} is also Cohen-Macaulay. Posets with this property are called 2-Cohen-Macaulay posets. I...

We study diagrams associated with a finite simplicial complex K, in various algebraic and topological categories. We relate their colimits to familiar structures in algebra, combinatorics, geometry and topology. These include: right-angled Artin and Coxeter groups (and their complex analogues, which we call circulation groups); Stanley-Reisner algebras and coalgebras; Davis and Januszkiewicz’s ...

We study local cohomology of rings of global sections of sheafs on the Alexandrov space of a partially ordered set. We give a criterion for a splitting of the local cohomology groups into summands determined by the cohomology of the poset and the local cohomology of the stalks. The face ring of a rational pointed fan can be considered as the ring of global sections of a flasque sheaf on the fac...

In 1988 Kalai construct a large class of simplicial spheres, called squeezed spheres, and in 1991 presented a conjecture about generic initial ideals of Stanley–Reisner ideals of squeezed spheres. In the present paper this conjecture will be proved. In order to prove Kalai’s conjecture, based on the fact that every squeezed (d−1)-sphere is the boundary of a certain d-ball, called a squeezed d-b...

We consider stratifications of schemes by subvarieties, and introduce the notion of a stratification generated by a collection of (reduced) subschemes. We recall some basics of Gröbner degenerations (just for lex order) and Stanley-Reisner theory. We then state the main theorem, that the lex init of a Kazhdan-Lusztig variety in BottSamelson coördinates is the Stanley-Reisner scheme of a subword...