Constructing monomial ideals with a given minimal resolution

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 [11]. Subsequently, we explain how, in some of these cases [5, 6] where questions still remain, this point of view can be applied. We also prove an equivalence for trees between the notion of maximal defined in [6] and the notion of being maximal in a Betti stratum.