A way of constructing Gorenstein ideals from small Gorenstein ideals in local Gorenstein rings is shown by A. Kustin and M. Miller in 1983. In this talk, we show a variant of their construction for graded case and avoid the ring extension. We see that how the liaison theory helps us immensely to modify their construction.

Cellular resolutions have been an area of active interest in the study of monomial ideals in the past few years. A cellular resolution is a way of encoding the information of the free resolution of an ideal in a cell complex. We will study the Stanley-Reisner rings arising from the simplicial polytopes known as the cyclic polytopes. These polytopes have the interesting property that they maximize all entries in the f-vector of the set of polytopes with fixed dimension and vertices. I will explain a little of the background theory and then cover the main idea of the construction.