The Buchberger resolution

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.