  We give an explicit quadratic Grobner basis for generalized Chow rings of
supersolvable built lattices, with the help of the operadic structure on
geometric lattices introduced in a previous article. This shows that the
generalized Chow rings associated to minimal building sets of supersolvable
lattices are Koszul. As another consequence, we get that the cohomology
algebras of the components of the extended modular operad in genus 0 are
Koszul.
