Degreewise n-projective and n-flat model structures on chain complexes

In the paper “Cotorsion Pairs in C(R-Mod)”, the authors construct an abelian model structure on the category of chain complexes Ch®, where the class of cofibrant objects is given by the class of degreewise projective chain complexes. Using a generalization of a well known theorem by I. Kaplansky, we generalize the method used in that paper in order to obtain, for each integer n>1, a new abelian model structure on Ch®, where the class of cofibrant objects is the class of chain complexes whose terms have projective dimension smaller or equal than n (dw-n-projective complexes), provided the ring R is noetherian. We also present another method, based on the paper “Covers and Envelopes in Grothendieck Categories: Flat Covers of Complexes with Applications”, to construct this model structure. This method also works to construct an abelian model structure whose cofibrant objects are the dw-n-flat complexes.