Constructing monomial ideals with a given minimal resolution

Abstract

This paper gives a description of various recent results, which construct monomial ideals with a given minimal free resolution. We show that these results are all instances of coordinatizing a finite atomic lattice, as found in~\cite {mapes}. Subsequently, we explain how, in some of these cases \cite {Faridi, Floystad1} where questions still remain, this point of view can be applied. We also prove an equivalence for trees between the notion of \textit {maximal} defined in~\cite {Floystad1} and the notion of being maximal in a Betti stratum.