Minimal resolutions of monomial ideals

Abstract

An explicit, closed-form combinatorial minimal free resolution of an arbitrary monomial ideal $I$ in a polynomial ring in $n$ variables over a field of characteristic 0 (and almost all positive characteristics) is defined canonically, without any choices, using higher-dimensional generalizations of combined spanning trees for cycles and cocycles (hedges) in the upper Koszul simplicial complexes of $I$ at lattice points in ${\mathbb{Z}}^n$. The differentials in these sylvan resolutions are expressed as matrices whose entries are sums over lattice paths of weights determined combinatorially by sequences of hedges (hedgerows) along each lattice path. This combinatorics enters via an explicit matroidal expression for Moore–Penrose pseudoinverses as weighted averages of splittings defined by hedges. The translation from Moore–Penrose combinatorics to free resolutions relies on Wall complexes, which construct minimal free resolutions of graded ideals from vertical splittings of Koszul bicomplexes. The algebra of Wall complexes applied to individual hedgerows yields explicit but noncanonical combinatorial minimal free resolutions of arbitrary monomial ideals in any characteristic.