Free resolutions of Artinian Gorenstein Rings

In this paper, we will introduce and build a set of tools to help us manipulate free resolutions of Artinian Gorenstein graded rings. We introduce the Hilbert series of a graded ring to examine its layers and access information to be compiled in Betti tables, as well as some relevant tools, such as the Macaulay and Gotzmann theorems, that condition the different Hilbert functions associated with a ring. After introducing the minimal free resolution of an Artinian Gorenstein ring, we will build tools to connect them to the Betti table of the said ring. In that way, we get a set of useful tools to connect these different notions and tie them all up. By knowing only the Hilbert series of an Artinian Gorenstein, we are now able to narrow down the possible Betti tables and therefore the associated possible minimal resolutions.