Abstract:
A resolution of an ideal is an exact sequence decomposing the ideal. Exact sequences appear naturally when calculating the cohomology of a topological space. A cellular resolution of an ideal is a topological space providing the exact sequence resolving the ideal. It is easy to find huge polyhedral complexes supporting a cellular resolutions, but we want to capture the structure of the minimal resolutions, since the graded betti numbers of the ideals are then easily calculated.

The resolutions are also topological objects in a more abstract sense: They are objects in a category with a good homotopy theory. This raises the question that one should understand the maps not only between ideals, and between their resolutions - but also construct cellular resolutions with a rich structure of maps between them.

In a joint work with Anton Dochtermann we have studied this for monomial ideals. Using cellular resolutions we describe explicit minimal resolutions for almost all monomial ideals with a linear resolutions. Special cases of this resolution have been studied by Corso, Nagel, and Reiner. The cell complexes involved are closely related to those used by Lovász for obstructions of graph colorings. Joins of these complexes are then used to resolve any monomial ideal; by this we provide a new explicit and general construction for resolutions of monomial ideals.