Georgia Institute of Technology College of Sciences

One way to analyze a module is to consider its minimal free resolution and take a look its Betti numbers. In general, computing minimal free resolution is not so easy, but in case of some certain modules, computing the Betti numbers become relatively easy by using a Hochster's formula (with the associated simplicial complex. Besides, Mumford introduced Castelnuovo-Mumford regularity. The regularity controls when the Hilbert function of the variety becomes a polynomial. (In other words, the regularity represents how much the module is irregular). We can define the regularity in terms of Betti numbers and we may see some properties for some certain ideals using its associated simplicial complex and homology. In this talk, I will review the Stanley-Reisner ideals, the (graded) betti-numbers, and Hochster's formula. Also, I am going to introduce the Castelnuovo-Mumford regularity in terms of Betti numbers and then talk about a useful technics to analyze the Betti-table (using the Hochster's formula and Mayer-Vietories sequence).