Dynamical systems on chain complexes and canonical minimal resolutions

TLDR: In this article, the authors introduce notions of vector field and its discrete time flow on a chain complex, and construct a minimal free resolution for every toric ring and every monomial ideal.

Abstract: We introduce notions of vector field and its (discrete time) flow on a chain complex. The resulting dynamical systems theory provides a set of tools with a broad range of applicability that allow, among others, to replace in a canonical way a chain complex with a "smaller" one of the same homotopy type. As applications we construct in an explicit, canonical, and symmetry-preserving fashion a minimal free resolution for every toric ring and every monomial ideal. Our constructions work in all characteristics and over any base field. A key subtle new point is that in certain finitely many positive characteristics (which depend on the object that is being resolved) a transcendental extension of the base field is produced before a resolution is obtained, while in all other characteristics the base field is kept unchanged. In the monomial case we show that such a transcendental base field extension cannot in general be avoided, and we conjecture that the same holds in the toric case.