The LCM lattice of an Ideal is the set of all LCMs of subsets of the generators of the ideal with partial ordering given by divisibility. These are particularly useful in the study of resolutions of monomial ideals.



In particular, Gasharov, Peeva, and Welker provided a key connection between the lcmlattice of a monomial ideal and its minimal free resolution. In particular, it is possible to use the lcmlattice to compute the multigraded Betti numbers of the ideal.
In particular, in the first example we show the $i^{\rm th}$ Betti number associated to $a^2b^2c^2d$ is always $0$.


In the second example, we show that the $(1, a^3b^2cd)$ Betti number is $2$.

