Exact zero divisors with applications to free resolutions over Artinian rings

This dissertation considers local rings (R, [special characters omitted], k) containing an exact zero divisor a, that is, an element a in R satisfying (0 :R a) ≅ R/aR. We establish the descent and ascent of homological and structural properties of the change of rings given by factoring out an exact zero divisor. Important results on this change of rings include the persistence of the Krull dimension, depth, type, several homological dimensions (projective dimension, injective dimension, etc...) and the comparative study of the free resolutions of modules over R and R/aR. In chapter 2, we establish that exact zero divisors preserve invariants whose extreme cases characterize the regularity properties in the chain of implications: Complete Intersection ⇒ Gorenstein ⇒ Cohen-Macaulay. In Chapter 3, we give an application to the study of finitely generated modules through their free resolution over local artinian Gorenstein rings. We point out that generic Gorenstein standard graded k-algebra of socle degree 3 admit an exact zero divisor and we establish the rationality of Poincaré series for finite modules. In addition, we prove that such rings are Koszul and give results on the existence of Koszul modules over such rings