On liftable and weakly liftable modules

Let $T$ be a Noetherian ring and $f$ a nonzerodivisor on $T$. We study concrete necessary and sufficient conditions for a module over $R=T/(f)$ to be weakly liftable to $T$, in the sense of Auslander, Ding and Solberg. We focus on cyclic modules and get various positive and negative results on the lifting and weak lifting problems. For a module over $T$ we define the loci for certain properties: liftable, weakly liftable, having finite projective dimension and study their relationships

Remarks on non-commutative crepant resolutions of complete intersections

We study obstructions to existence of non-commutative crepant resolutions, in the sense of Van den Bergh, over local complete intersections

Comparing complexities of pairs of modules

Let $R$ be a local ring and $M,N$ be finitely generated $R$-modules. The complexity of $(M,N)$, denoted by \cxx RMN, measures the polynomial growth rate of the number of generators of the modules \Ext nRMN. In this paper we study several basic equalities and inequalities involving complexities of different pairs of modules