arXiv subjects
Publications and source records attributed to Shouta Tounai.
In this paper, we give a simple criterion for a locally distributive semilattice to have CAT(0) orthoscheme complex. Namely, the orthoscheme complex of a locally distributive semilattice S is CAT(0) if and only if S is a flag semilattice, that is, any pairwise bounded triple of S is bounded.
An affine monoid is an additive monoid which is cancellative, pointed and finitely generated. An affine monoid $Λ$ has the partial order defined by $λ\le λ+ μ$. The Frobenius complex is the order complex of an open interval of $Λ$ with respect to this partial order. The reduced homology of the Frobenius complex is related to the torsion group of the monoid algebra $K[Λ]$. In this paper, we pay attention to homotopy types of Frobenius complexes, and we express the homotopy types of the Frobenius complexes of $Λ$ in terms of those of $Λ_1$ and $Λ_2$ when $Λ$ is an affine monoid obtained by gluing two affine monoids $Λ_1$ and $Λ_2$ with one relation. We also state an application to the Poincaré series of the torsion group of the monoid algebra.
A finitely generated additive submonoid $Λ$ of ${\mathbb N}^d$ has the partial order defined by $λ\le λ+ μ$ for $λ, μ\in Λ$. The Frobenius complex is the order complex of an open interval of $Λ$. In this paper, we express the homotopy type of the Frobenius complex of $Λ[ρ/ r]$ in terms of those of $Λ$, where $Λ[ρ/ r]$ is the additive monoid which is added the $r$-th part of $ρ$ to $Λ$. As applications, we determine the homotopy type of the Frobenius complex of some submonoids of $\mathbb N$, for example, the submonoid generated by a finite geometric sequence. We also state an application to the multi-graded Poincaré series.
A submonoid A of N^d has a natural order defined by a <= a + b for elements a and b of A. The Frobenius complex is the order complex of an open interval of A with respect to this order. In this paper, the homotopy type of the Frobenius complex of A is determined when A is the submonoid of N generated by two relatively prime integers, or the submonoid of N^2 generated by three elements of which any two are linearly independent. As an application, the multigraded Poincare series of the quotient algebra K[x, y, z] / (x^p y^q - z^r) over a field K is determined and proved to be rational.