Neighbors, Generic Sets and Scarf-Buchberger Hypersurfaces

TL;DR

This paper generalizes the construction of the Scarf complex to broader modules using a new simplicial complex derived from subsets of al, relating to PL hypersurfaces and providing tools for resolutions in algebraic geometry.

Contribution

It introduces a new simplicial complex al(N(A)) for generic sets A in al^n, generalizing staircase surfaces and Buchberger graphs to arbitrary dimensions, with topological and resolution applications.

Findings

al(N(A)) is a locally finite simplicial complex for generic A.

The barycentric subdivision of al(N(A)) triangulates a PL hypersurface.

al(N(A)) can be used to construct locally finite free resolutions.

Abstract

The present paper is motivated by the need to generalize the construction of the Scarf complex in order to give combinatorial resolutions of a much broader class of modules than just the monomial ideals. For any subset $A\subseteq \mathbb{R}^n$, let $\mathfrak{N}(A)$ denote the collection of all subsets $B\subseteq A$ such that there is no $a\in A$ that is strictly less than the supremum of $B$ in all coordinates. We show that if $A\subseteq \mathbb{Z}^n$ is generic (in a sense appropriate for this context), then $\mathfrak{N}(A)$ is a locally finite simplicial complex. Moreover, if $A$ is generic, then the barycentric subdivision of $\mathfrak{N}(A)$ is equivalent to a triangulation of a PL hypersurface in $\mathbb{R}^n$. This gives us natural generalizations of the notions of ``staircase surface'' and ``Buchberger graph,'' described by Miller and Sturmfels, to arbitrary dimension. (This seems to be a new result, even in the well-studied case that $A$ is a finite subset of $\mathbb{N}^n$.) We give examples that show that when $A$ is infinite, $\mathfrak{N}(A)$ may have complicated topology, but if there are at most finitely many elements of $A$ below any given $b\in \mathbb{R}^n$, then $\mathfrak{N}(A)$ is locally contractible. $\mathfrak{N}(A)$ can therefore be used to construct locally finite free resolutions of sub-$k[\mathbb{N}^n]$-modules of the group algebra $k[\mathbb{R}^n]$ ($k$ is a field). We prove various additional facts about the structure of $\mathfrak{N}(A)$