  An explicit combinatorial minimal free resolution of an arbitrary monomial
ideal $I$ in a polynomial ring in $n$ variables over a field of characteristic
$0$ 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 the
Moore-Penrose pseudoinverses of the differentials in any CW complex as weighted
averages of splittings defined by hedges. This "Hedge Formula" also yields a
projection formula from CW chains to boundaries. 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.
