Poset Resolutions of Monomial Ideals

We introduce the class of lattice-linear monomial ideals and use the LCM-lattice to give an explicit construction for their minimal free resolution. The class of lattice-linear ideals includes (among others) the class of monomial ideals with linear free resolution and the class of Scarf monomial ideals. Our main tool is a new construction by Tchernev that produces from a map of posets $\eta :P\text{lra}\text{mbb}{N}^n$ a sequence of multigraded modules and maps.