Neighbors, Generic Sets and Scarf-Buchberger Hypersurfaces

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 $\mathfrak{N}(A)$