The Buchberger resolution

TL;DR

The paper introduces the Buchberger resolution, a new graded free resolution for monomial ideals that generalizes the Buchberger graph and connects to combinatorial and algebraic properties of these ideals.

Contribution

It defines the Buchberger resolution, linking it to the Buchberger graph and the Scarf resolution, and explores its combinatorial structure and conjectures.

Findings

The Buchberger resolution coincides with the Scarf resolution for generic monomial ideals.

It encodes the combinatorics of the Buchberger algorithm.

The underlying simplicial complex's structure is not fully understood and is conjectured to relate to clique complexes.

Abstract

We define the Buchberger resolution, which is a graded free resolution of a monomial ideal in a polynomial ring. Its construction uses a generalization of the Buchberger graph and encodes much of the combinatorics of the Buchberger algorithm. The Buchberger resolution is a cellular resolution that coincides with the Scarf resolution for generic monomial ideals, which is the case when it is minimal. The simplicial complex underlying the Buchberger resolution is of interest for its own sake and its combinatorics is not fully understood. We close with a conjecture on the clique complex of the Buchberger graph.