Example: LCM-lattices

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.

i1 : R = QQ[a,b,c,d];

i2 : M = ideal(a^3*b^2*c, a^3*b^2*d, a^2*c*d, a*b*c^2*d, b^2*c^2*d);

o2 : Ideal of R

i3 : LM = lcmLattice M;

In particular, Gasharov, Peeva, and Welker provided a key connection between the lcm-lattice of a monomial ideal and its minimal free resolution. In particular, it is possible to use the lcm-lattice 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 $0$.

i4 : D1 = orderComplex(openInterval(LM, 1_R, a^2*b^2*c^2*d));

i5 : prune HH(D1)

o5 = -1 : 0

      0 : 0

      1 : 0

o5 : GradedModule

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

i6 : D2 = orderComplex(openInterval(LM, 1_R, a^3*b^2*c*d));

i7 : prune HH(D2)

o7 = -1 : 0  

            2
      0 : QQ

o7 : GradedModule

Example: LCM-lattices

See also