Modules of Finite Projective Dimension and Singularities

In the first part of this thesis, we study liftings of modules of finite projective dimension. We introduce a notion of “Serre liftable” modules and deduce applications to multiplicity conjectures in local algebra. In the second part, we introduce and study a module of mixed K"ahler differentials for finite algebras over ramified discrete valuation rings of mixed characteristic. We state and prove a Jacobian criterion for computing the singular loci of such algebras.Advisors: Jack Jeffries and Mark Walker