Projective dimensions and extensions of modules from tilted to cluster-tilted algebras

We study the module categories of a tilted algebra C and the corresponding cluster-tilted algebra B=C⋉E where E is the C-C-bimodule ExtC2(DC,C). We investigate how various properties of a C-module are affected when considered in the module category of B. We give a complete characterization of the projective dimension of a C-module inside modB. If a C-module M satisfies ExtC1(M,M)=0, we show two sufficient conditions for M to satisfy ExtB1(M,M)=0. In particular, if MC is indecomposable and ExtC1(M,M)=0, we prove MB always satisfies ExtB1(M,M)=0.