Free Complexes over the Exterior Algebra with Small Homology

Let M be a graded module over a standard graded polynomial ring S. The Total Rank Conjecture by Avramov-Buchweitz predicts the total Betti number of M should be at least the total Betti number of the residue field. Walker proved this is indeed true in a large number of cases. One could then try to push this result further by generalizing this conjecture to finite free complexes which is known as the Generalized Total Rank Conjecture. However, Iyengar and Walker constructed examples to show this generalized conjecture is not always true. In this thesis, we investigate other counterexamples of the Generalized Total Rank Conjecture and some of their properties. Under the BGG correspondence, a finite free graded complex over the exterior algebra with small homology corresponds to a free complex over the polynomial ring with a small total Betti number. Therefore, we focus on examples of finite free complexes over the exterior algebra with small homology. The main examples we consider are Koszul complexes of quadrics, and we show the Koszul complex of one general quadric and the Koszul complex of two general quadrics have the smallest possible homology among complexes over the exterior algebra with the same graded Poincaré series. Finally while analyzing these Koszul complexes, we notice the dimension of their total homology has a nice asymptotic behavior and investigate under what conditions other complexes have this same asymptotic behavior