41 research outputs found
Surprising occurrences of order structures in mathematics
Order and symmetry are main structural principles in mathematics. We give
five examples where on the face of it order is not apparent, but deeper
investigations reveal that they are governed by order structures. These
examples are finite topologies, associative algebras, subgroups of matrix
groups, ideals in polynomial rings, and classes of bipartite graphs.Comment: 23 page
The cone of Betti diagrams of bigraded artinian modules of codimension two
We describe the positive cone generated by bigraded Betti diagrams of
artinian modules of codimension two, whose resolutions become pure of a given
type when taking total degrees. If the differences of these total degrees, p
and q, are relatively prime, the extremal rays are parametrised by order ideals
in N^2 contained in the region px + qy < (p-1)(q-1). We also consider some
examples concerning artinian modules of codimension three.Comment: 15 page
Triangulations of polygons and stacked simplicial complexes: separating their StanleyâReisner ideals
A triangulation of a polygon has an associated StanleyâReisner ideal. We obtain a full algebraic and combinatorial understanding of these ideals and describe their separated models. More generally, we do this for stacked simplicial complexes, in particular for stacked polytopes.publishedVersio
Borel Degenerations of Arithmetically Cohen-Macaulay curves in P^3
We investigate Borel ideals on the Hilbert scheme components of
arithmetically Cohen-Macaulay (ACM) codimension two schemes in P^n. We give a
basic necessary criterion for a Borel ideal to be on such a component. Then
considering ACM curves in P^3 on a quadric we compute in several examples all
the Borel ideals on their Hilbert scheme component. Based on this we conjecture
which Borel ideals are on such a component, and for a range of Borel ideals we
prove that they are on the component.Comment: 20 pages, shorter and more effective versio
PBW-deformations of N-Koszul algebras
For a quotient algebra of the tensor algebra we give explicit conditions
on its relations for being a PBW-deformation of an -Koszul algebra .
We show there is a one-one correspondence between such deformations and a class
of -structures on the Yoneda algebra of . We
compute the PBW-deformations of the algebra whose relations are the
anti-symmetrizers of degree and also of cubic Artin-Schelter algebras.Comment: 35 pages. Some minor correction